study and modelling of lightweight floor structure
TRANSCRIPT
LICENTIATE T H E S I S
Department of Human Work ScienceDivision of Sound and vibration Study and Modelling of Lightweight
Floor Structure Regarding its Acoustic Properties
Mohammad Sazzad Mosharrof
ISSN: 1402-1757 ISBN 978-91-7439-125-1
Luleå University of Technology 2010
ISSN: 1402-1757 ISBN 978-91-7439-XXX-X Se i listan och fyll i siffror där kryssen är
Moham
mad Sazzad M
osharrof Study and Modelling of Lightw
eight Floor Structure Regarding its A
coustic Properties
Study and modelling of lightweight floor structure regarding its acoustic properties
Mohammad Sazzad Mosharrof
Division of Sound and vibration Department of Human Work Science
Luleå University of Technology SE-97187 Luleå, Sweden
Printed by Universitetstryckeriet, Luleå 2010
ISSN: 1402-1757 ISBN 978-91-7439-125-1
Luleå 2010
www.ltu.se
Acknowledgement This project was carried out in the division of Sound and Vibration in Luleå Technical University (LTU) and was financed by LTU, Vinnova, SkeWood, thanks to all of them. I would also like to thank my supervisor Professor Anders Ågren and assistant supervisor Fredrik Ljunggren for their support and supervision, all my colleges here for there supports, especially Rikard Ökvist, we shared our office and had a nice time together. A special thank to Assistant Professor Jonas Brunskog of Technical University of Denmark (DTU). Indeed I understood quite many things discussing with him during my short educational visit to DTU. The tour was financed by European Cooperation of Science and Technology (COST) and I am also thankful to COST. Finally, special thanks to all my family members and Muslim brothers living in Luleå.
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List and summary of the attached paper Paper 1
Mosharrof, M. S. Ljunggren, F. Ågren, A. Brunskog, J. Prediction model for the impact sound pressure level of decoupled lightweight floors; Proceedings of internoise 2009, Ottawa, 2009.
The paper talks about developing a model for the double plate decoupled structure. The analytical approach is applied here. Plates are decoupled i.e. different sets of beams are used for each plates separately. Air inside the cavity, formed in between the plates, is the only thing coupling the plates together. If no air is assumed both plates would vibrate independently. Other than the fluid loading, force acting to the upper plate is the reaction from the beams (pf1). Each beam will generate a reaction force toward the plate. pf1 is the total contribution of all the beams. A moment will also generate at each joint which in this paper is neglected. However the joints are assumed to be rigid meaning the lateral displacement of plate and beam at each joint is exactly the same. Same is true for the lower plate, the reaction forces from the beams pf2 acts towards the lower plate with rigid boundary condition and no moment consideration. When fluid loading is consider, air inside the cavity as well as in the surroundings, will interact with both the plates. Pressure inside the cavity is the pc where pc(x,y,0) and pc(x,y,d) represent the cavity pressure interacting with upper and lower plate respectively. pr is the interaction force of the surrounding air with the upper plate while pt is that with the lower plate. The model solves for the two plate displacements w1, w2 corresponds to plate 1 and plate 2 respectively. Kirchhof’s plate equation is considered to describe both plates with the interacting forces placed in the right hand sides. Fourier transform is used and thus the displacements are calculated in the wave number domain. All the transformed force terms are expressed in terms of the transformed plate displacements. Fluid interacting terms are calculated using the Helmholtz equation and condition that the vibration velocity at the plate surfaces and the particle velocities close to the plates are exactly the same. Forces from each beam are calculated using the Euler beam equation and are summed for all beams (infinity in theory). Poisson’s summation formula is applied. All the force terms are then inserted to the governing equations. Summations of the beam reactions still remain in undetermined forms which were determined by making use of variable change and summing up for all variables. With all these force terms evaluated two plate equations are now solved in the wave number domain. From the displacement of the lower plate, pressure in the receiving room is calculated using far field approach described in Cremer [32]. Comparison has been made on the predicted results between coupled and decoupled structure. Coupled results were taken from paper [4] while same material data was used to predict for decoupled set up. Comparison shows a significant improvement in the higher frequency where in the lower frequency region a peak seam to generate due the mass air mass resonance.
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Paper 2 Mosharrof, M. S. Brunskog, J. Ljunggren, F. Ågren, A. Improved prediction model for the impact sound level of lightweight floors - introducing decoupled floor-ceiling and beam-plate moment. Submitted to Acta Acustica United with Acustica, 2010.
This paper is the continuation of the paper 1 on decoupled structure. Here along with the reaction forces at the plate beam joints, an additional moment reaction is considered. Remaining force terms are the same as paper 1. Same Fourier transform solution technique is implemented. Fluid interactions are exactly same and calculated in a similar way. With the moment consideration the rigid boundary condition at the joint goes one step further. Now angular displacements, along with the lateral displacements of the plates and the attached beams need to be considered. Due to rigid connection, at each joint both lateral and angular displacements of plate are same as those of beams respectively. In paper 1 everything was given in a compressed format due to page restriction. Here in paper 2 things are given and explained comparatively in detail and results were verified with experimental data. The model agrees with the experimental result reasonably well. Accuracy of the predicted result was compared with that of the model for coupled structure in paper [4] which shows similar accuracy when moment is excluded from the decoupled model. With the inclusion of the moments a noticeable improvement is achieved in the higher frequencies. Low frequency region was dominated by mass air mass resonance which is directly related to the cavity depth. Predicted results show that the response is highly sensitive to the excitation location.
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Paper 3 Mosharrof, M. S. Ljunggren, F. Ågren, A. Parametric study of lightweight floors using a theoretical floor model. To be submitted to a journal.
Based on the models described on paper 2 a parametric study is done on lightweight decoupled structure. Reliability of the model was verified with some experimental data. However, with parametric study it is always possible to dig deep inside and relative importance of each element can be understood. In designing a proper efficient floor this understanding is very important. The key findings of this study are:
1. In general the effect of any element on the overall vibration is that the heavier the structure becomes due to any elemental change, the lower the vibration level gets and so does the sound pressure level (SPL). For example if the upper plate is made heavy by making it thick or dense the overall vibration will reduce. Or if the beam height is increased it will also decrease the vibration. Same is true for the lower plate as well. The effect is in all frequency. In general mass affects the low frequency region.
2. Stiffness on the other hand affects the high frequency region. For floor and ceiling the effect is different, for floors increasing stiffness makes SPL to reduce, while increasing ceiling stiffness increases SPL.
3. For impact sound, the key factor regarding the floor is how the vibration propagation through it. Since while propagating some frequency can go through and some gets stopped, mostly it is the upper plate which determines the pass bands and stop bands.
4. For ceiling propagation through it is not a key factor, since it is excite by the pressure wave coming through the cavity. As a result whole structure gets excited at a time rather than vibration being propagated from a point. Therefore lower plate mostly acts in order to reduce the vibration amplitude.
5. Mass air mass resonance is a dominating term for low frequencies. It has a direct relationship with the cavity depth.
Abstract Lightweight floor structure is widely used in building industries and to have better sound insulation builders come up with different ways of construction. Depending on the construction the floor structure could either be coupled (floor and ceiling coupled by beams) or decoupled (no mechanical connection between floor and ceiling). Although there are many models on coupled structure but for decoupled structure the number is not too many. Keeping that in mind the present thesis talks about lightweight floors: the construction, properties, behaviour etc with a focus on developing a model for decoupled floor structure where the core contribution being the decoupling and adding the moment effect at plate beam joints. The advantage of decoupled structure is that it disconnects the sound bridge through the beams. One consequence on the other hand is that cavity resonance dominates the low frequency region. A comparative analysis is also done with the coupled model. While developing the model this talks about different mathematical tools such as Fourier transform, Floquet principle, Poisson’s sum formula etc This also gives an overview of different types of modelling technique available such as analytical, Numerical, energy based approach, empirical method etc. A parametric study is also done here to find out the relative influence of different elements on sound pressure level.
Table of contents
1 Objective ............................................................................ 1
2 Introduction ........................................................................ 1 2.1 Overview of lightweight floor ...................................................................................... 1 2.2 Construction of lightweight floor structure .................................................................. 2
2.2.1 Coupled structure .............................................................................................. 2 2.2.2 Decoupled structure .......................................................................................... 3
2.3 Sound transmission in buildings .................................................................................. 3 2.3.1 Transmission through floor structure .............................................................. 4
2.4 Complexity in lightweight floor ................................................................................... 5 2.4.1 Material property ................................................................................................ 5 2.4.2 Structural property ............................................................................................. 5
2.5 Periodic property of lightweight structure ................................................................... 6 3 Modelling of lightweight floor ........................................... 7
3.1 Assumptions ................................................................................................................. 8 3.2 Mathematical tools ....................................................................................................... 8
3.2.1 Fourier transform ............................................................................................... 8 3.2.2 Poisson’s summation ........................................................................................ 9
4 Different solution methods ................................................. 9 4.1 Analytical ..................................................................................................................... 9
4.1.1 Modelling the excitation .................................................................................... 9 4.1.2 Modelling the vibration in the plates ............................................................. 10 4.1.3 Modelling the sound radiation ........................................................................ 14
4.2 Empirical method ....................................................................................................... 15 4.3 Fourier series solution ................................................................................................ 16 4.4 Statistical Energy Analysis SEA ................................................................................ 16 4.5 Finite Element Method FEM ...................................................................................... 17
5 Conclusions ...................................................................... 17
Referens ............................................................................... 18
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1 ObjectiveThe aim of the project is to better understand the lightweight construction and to develop prediction model for vibration and its propagation. At the preliminary stage infinite periodic floor structure is being dealt with. And gradually the complexity of the lightweight structure will be introduced. At this stage the model deals with
1. Decoupled structure. 2. Inclusion of moment reaction at plate beam joints.
2 Introduction
2.1 Overview of lightweight floorIn building construction the term lightweight in general distinguish from the heavy concrete construction and usually refers to steel or wooden constructions. They can be constructed only of wood or only of steel or sometimes combination of both. In many constructions however heavy and lightweight are implemented together. Because of their low weight, besides building industry they are widely used in aircraft and marine industries etc. as well. However, use of lightweight buildings is increasing day by day. The primary reason is their easy and quick installation and being very economic. Especially in Nordic countries it is an efficient way of utilizing the vast forest resource. Considering a building structure, volumes are built one over another. And floor structure in lightweight construction refers to floor part of the upper volume and the ceiling part of lower volume. The biggest advantage of these lightweight floors is low mass which makes it applicable in many different industries as mentioned, but one consequence of this low mass is their low stiffness. Therefore to regain stiffness beams are attached to a plate at regular interval. Beams can be attached along parallel to one axis or parallel to both axes making cross beam connections. This basic construction is the same regardless of where it is going to be used. Thus lightweight structures in general associate with the spatial periodicity regardless of its application and same mathematical model can be applicable for any type of lightweight structure. The draw back of these lightweight structures is their poor acoustic property. Being light they are vulnerable to vibration and thus are very effective sound transmitter. Both air borne and structure borne sound can transmit very easily through them. Since in the transport industry the primary goal is transportation and in building industry is providing comfort, building tenants seem to complaint much more regarding sound insulation. This makes the regulation and standards to be stricter regarding sound level for buildings especially in Sweden. As a result to better understand there have been many works of lightweight structure and many theoretical models [1-11] have been established. Evseev [1] is the earliest among these and proposed an analytical approach of modelling lightweight floor structure and others [1-11] later on modified it further. Other than this analytical approach there are many approaches to model lightweight structure. [12-16] considered SEA while [17-19] considered FEM and also there are some models [20, 21] based on empirical method. The thesis however is about the analytical model where the core contribution being decoupling of floor and ceiling, and consideration of moment at joints.
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2.2 Construction of lightweight floor structureAs mentioned before in all application lightweight structure is constructed by attaching beam at a fixed interval to plates. And if decoupled, two separate sets of beam are used, one for each plate. Construction of these could be done on site, where whole building is constructed on site. At first building the entire frame and then attaching each element to it. In this type of construction there remains mechanical connection between floor and ceiling through which vibration can propagate. An alternate approach to this is to construct each volume in the factory and joining them together as required while constructing whole building as described in [22]. In this type of construction there remains no rigid connection between two floors. The purpose of this is to break the sound bridge through floor-beam-ceiling. Indeed, at present there are many different kinds of lightweight floors other than these. In this thesis however, two types of floor structure are considered:
1. Coupled 2. Decoupled
2.2.1 Coupled structure
Figure 1: Coupled floor structure a), and free body diagram of coupled floor structure b). Figure 1 shows a typical coupled structure and the interacting forces, where floor structure is represented by two plates attached by beams placed at a fixed interval. For modelling purpose thicknesses of the upper and lower plate are considered by hp1 and hp2 respectively, d is the depth of the cavity formed due to coupling. l represents spacing between the beams. Most of the models mentioned before deals with the coupled floor and does not include the influence of the beam/frame to the cavity. In other words, air inside each cavity is assumed to flow between the cavities without being obstructed by the frames. Brunskog [5] however, proposed a modification considering this frame effect. Figure 1b shows the corresponding forces acting to the plates. pe is the excitation, this thesis was focused on impact sound pressure level representing foot fall or heavy things falling on the floor, and given to upper plate. pr and pt acting to upper and lower plate respectively are the interaction forces of the plates with the surrounding air (medium), while pc is the interaction with the cavity air. A plane wave is approximated inside the cavity and pc1 and pc2 acts toward plate 1 and 2 respectively. Finally pf1 and pf2 are the collective forces of the beams to the upper and lower plate respectively. The co-ordinate system is chosen as shown. Upper plate surface is considered to be the xy plane and thickness of the plate is neglected i.e. z = 0 at both upper and lower surface of the upper plate.
pt
pc(x,y,d)
pc(x,y,0)
pe pr
p´f2
l
x p´f1 y
z
hp2
hp1
d,hb
a b
3
2.2.2 Decoupled structure Figure 2: Decoupled floor structure a) and free body diagram of decoupled floor structure b).
In decoupled structure [10, 11] (see figure 2) there is no mechanical connection between the two plates. They are represented by two plates, each attached to different sets of beams. As in figure 2(a) one separate set of beams having height hb1, thickness b1 and spacing between them l1 is used for upper plate (floor). Another set of beams, with analogous notation as set 1, is attached to the lower plate 2 (ceiling) and the same coordinate system as for the coupled structure is used. Due to the described decoupling, the vibration transfer path through the beams is eliminated. The only way vibrations can be transferred from plate 1 to plate 2 is thus through the cavity as sound. Regarding the forces acting on the plates shown in Figure 2(b), interacting force terms with the air remains the same but two new reaction pressures pf1 and pf2 replace the previous ones p'f1,and p'f2, since two sets of beams are used. In addition to these there will be moment reaction at every joint, thus pm1 and pm2 the collective moment reactions from the joints should also be taken into consideration for more accurate modelling and considered in 2nd paper. The slippage between beam and plate at each joint as considered in [23], which was ignored here, might also be crucial. As pointed out earlier other than these types there might be different types of setup. Sometimes, plates are neither totally decoupled nor strongly coupled rather a week form of coupling via resilient joint are used.
2.3 Sound transmission in buildingsThe transmission of sound usually refers to how sound reaches from one place to another, e.g. from one room (sending room) to another room (receiving room). Sound can be transmitted in terms of pressure disturbance i.e. sound, through any fluid medium. That’s why we can hear one another. Another obvious example would be when we close or open the doors or windows of any room, music from the neighbouring room or noise from the street decrease or increase respectively. Besides these fluid paths sound can also be transmitted through the solid medium in terms of vibration. Structures that separates the rooms, e.g. sidewalls, floor structure etc. being excited by an impact or by pressure in the surrounding fluid medium, starts to vibrate which then propagates and ultimately radiate sound in the receiving room. However, it is obvious that before transmission, sound or vibration has to be initiated by a source. Depending on this source, sound can be categorised into two kinds:
Air borne: referring to sound originating from any kinds of pressure disturbance in the sending room, e.g. speech, music etc.
pm2
pm1 pf1
pt
pc(x,y,d)
pe pr
pf2
pc(x,y,0)
b
l1
hp1
hb1
hb2 l2
d
hp2
a
4
Structure borne: refers to any kinds of vibration given to the structure of the sending room, e.g. people walking, heavy object falling on the floor etc.
Whatever the source may be, for floor structure it is only the structure that propagates sound from the sending to the receiving room. The energy propagates in terms of vibration through different parts of the whole structure and ultimately radiates sound in the receiving room as illustrated schematically in Figure 3.
Figure 3: sound transmission through lightweight structures Depending on the path the vibration takes before ultimately radiating sound to the receiving end, sound transmission can be categorise into two kinds:
1. Direct sound transmission: some portion of the floor vibration energy radiates directly into the receiving room following the path AA´ in figure. This path is known as direct path and the transmission is called direct transmission.
2. Flanking sound transmission: the remaining energy other than transmitted through direct path propagates through the entire structure e.g. side walls and radiate sound from there. These paths are known as flanking path (AB) and the transmission is known as flanking transmission.
Sound from both these paths is equally important and in reality the total sound pressure in the receiving room is the sum from these two. In lab though the flanking transmission can be minimized or ignored. The thesis however, deals only with the direct transmission, i.e. vibration is not allowed to propagate through the structure, all the energy radiates through the direct path. Nevertheless, it would be interesting to introduce the flanking transmission which is kept for future contribution.
2.3.1 Transmission through floor structure As mentioned before lightweight floor structure is made of floor and ceiling and for decoupled structure these two are mechanically separated. Therefore the upper plate (ceiling) which is assumed to be a part of sending room can experience both air borne and structure borne excitation. For the case of air borne sound, excitation is given to the entire structure while in the case of structure borne excitation only the excitation point(s) are excited and vibration from these points propagate to the entire floor part. Thus, for impact sound, how vibration propagates through the floor is a key factor and the periodicity of the floor is of great importance. On the other hand, the ceiling part gets excited by pressure wave in the cavity generated due to sound radiated by the floor to the cavity. As a result, whole ceiling gets excited at a time and vibration propagation through the ceiling becomes less significant.
A
A´ B
Sending room
Receiving room
Floor structure
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This was seen in paper 3 by studying influence of the parameters of the ceiling and floor. More or less same peaks were observed when ceiling parameters are varied. Peaks changed frequency when the parameters of floor were varied, because that would change the structural property and thus the vibration propagation through the floor. Although, in decoupled structure the mechanical connection between floor and ceiling is broken, air inside cavity works as a spring. Thus provides a resonance frequency known as mass air mass frequency and influence the low frequency region.
2.4 Complexity in lightweight floor
2.4.1 Material property Lightweight floor is such a complex structure to deal with. The geometry and material property is as complex as is the construction. This made it so difficult to predict the vibration of such structures. Wood is made of fibres and has different properties in different directions. The property depends much on grain size, moisture content and other environmental factors, thus makes it so complicated to assign a single value for different properties. Besides material properties workmanship makes huge difference [24, 25], thus the total variation in lightweight structure measurements becomes quite noticeable. Johansson [26] made measurement on 170 nominally identical floor and the results were plotted in 1/3 octave band. The deviation in measurements seemed to be dependant on frequency and the magnitudes of these deviations were quite high. bellow 200 Hz the maximum deviation is 15 dB and above 1000 Hz it is around 20 dB, the mid frequencies (200 Hz to 1000 Hz) show more or less constant deviation of 10 dB. Even research [27-29] on variation in heavy concrete floors shows no exception. According to [27] the margin of safety due to uncertainty must be larger for lightweight floors.
2.4.2 Structural property Moreover beams are attached to the plate at a regular interval making the structure periodic in the space and the consequence being the addition of some new property to the structure which will be explained in later section. Trapped air inside the cavity also plays a significant role, especially for decoupled structure since this is the only means of sound propagation. Low frequency region is dominated by mass air mass resonance frequency. Another important factor is the connection between the plates and beams and plays significant role in sound generation. As mentioned earlier the plates are connected to number of beams, by different types of connection - point connection (screwed), line connection (glued). Craick [12] showed that sound radiation is influenced much on number of screws used and also on which type of connection method is implemented. If the screws are closely spaced the connection should be considered line connection and if they are widely spaced, connection should be considered as point connection. He [12] also mentioned that at low frequencies where the wave length is larger than the spacing between the screws, line theory is more appropriate. The transition frequency from line to point connection according to him [12] is where half bending wave length fits between the screws. The fact is, a slight change at anywhere in an element contributes to noticeable variation in overall sound radiation. Each and every element of the structure is important and should be taken into consideration with great care. Yet, in reality there are always some differences between the details of the identical structure, i.e. identical structure are not fully identical and sound level differs, as mentioned before. Therefore in general a simplified version of the structure is often modelled. Sometimes it is out of mathematical range to model every single detail or will add some more computation which will introduce computational error and will require longer time as well.
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2.5 Periodic property of lightweight structurePeriodicity is an important phenomenon of the lightweight structure. The beams attached to the plates make the structure spatially periodic. This has a big impact on the overall vibration. The beams usually treated as rigid connection and a barrier to the vibration propagation. As a result vibration reduces significantly. For the wavelength whose nodal line matches the line corresponding to the beams i.e. which fits the bay length can easily propagate and beam does not offer any resistance. On the contrary for the wave length whose nodal line does not correspond to the beam locations, the propagation is disturbed. This disturbance becomes high when the lines corresponding to maximum displacements matches beam location. Beams act as a huge obstacle for this kind of waves. Thus for such structures a pass band and stop band can be seen. And this comes alternatively and repeatedly. Extensive study [30-33] has been made on this and here we look into an example given in Cremer [33]. Consider an infinitely large beam is resting on many supports each at a distance l apart. Consider also an impulse (harmonic pressure) is applied at a point on the beam. Free propagation of this (sinusoidal pressure) through the beam is disturbed due to reflections at the periodic supports. However, due to the periodicity in the structure displacement and pressure at two corresponding points in any adjacent bay can be written as [33]: geww 12 (1) Where, w1 and w2 are the displacements at two points (say at l distance apart), g is the propagation constant which is in general complex. Equation 1 says that the ratio of the corresponding points of two adjacent bays is constant. With this the displacements at any point of the plate can be expressed by that of any specific bay. However, biag . . Where, a is the real part of g signifies the attenuation or gain in amplitude and known as attenuation coefficient and b is the imaginary part of g signifies the phase and known as phase coefficient. The wave propagation is entirely free outside the excited bay and is dependent on the support and frequency rather than the excitation pressure, thus this propagation constant is a unique property of the structure and differs with the construction of the structure. Solving for this propagation constant it is possible to dig deep inside the fundamental behaviour of the vibration of the periodic structure. As mentioned before, researcher [30-33] worked a lot on understanding more about this propagation constant. Orris and Petty [19] used FEM to study the vibration propagation, where mass matrix M and stiffness matrix K were calculated using energy consideration and considering eigen value problem free wave propagation is derived. However all these research reached one fundamental conclusion about periodic structure having pass band and stop band, illustrated by Figure 3. Figure 3 is taken from Cremer [33] and shows the relationship of the real and imaginary part of propagation constant with wave number k. This corresponds to an infinite beam having periodic cross section at a fixed distance l as shown in the figure. Although the example ignores moment effect and deals with longitudinal wave propagation, this is equally valid for bending wave propagation through plate like structure stiffened via beams and also when moment is considered [30-32].
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Figure 4: Attenuation coefficient for longitudinal waves propagating along bar with spatially periodic blocking mass
Figure 4 shows that for a certain limit of kl attenuation constant (denoted by a here) is 0 and the phase coefficient (denoted by b here) is increasing, this simply means that without the decay of amplitude phase is changing i.e. the wave is propagating without any attenuation and this is known as propagation zone. And just after this limit there arise another limit of kl where attenuation coefficient becomes significantly large and the phase coefficient becomes 0, meaning high attenuation or near field of wave, thus wave can’t propagate in this region. This is known as attenuation zone. And frequency bands corresponding to these zones are known as pass band and stop band respectively. Figure also shows that these propagation and attenuation zones come alternatively and there exists infinite number of such zones. Another interesting thing is that propagation zone begins at kl = n , where n = 0, 1,2,3,4…i.e. at l = n. /2. When n=1 it corresponds to the case when wave length is twice the bay length, when n=2 corresponds to the case when bay length is equal to the wave length and so on. Therefore the propagation takes place if integer multiple of half of any wave length equals the bay length. Moreover form the figure it is evident that this phenomenon can be categorised into two regions in terms of propagation constant which are
1. |cosh(g)| < 1, resembles attenuation zone, 2. |cosh(g)| > 1, resembles propagation zone.
Nevertheless, this conclusion is only for this type of structure, not for all. Different structure has different relationship. The common phenomenon is that for any periodic structure there always exist propagation and attenuation zones corresponding to propagation constant. Number of propagation constant however, can be more than one. In that case there will be different set of propagation and attenuation zones similar to figure 3 each of which corresponds to a specific propagation constant. For example for plate the governing equation is a fourth order differential equation, thus gives rise to four of such propagation constants.
3 Modelling of lightweight floor Modelling usually refers to developing a method of predicting response to some excitation. It can be done analytically, numerically and by other methods. Analytical approach involves some governing equations and boundary conditions which are then solved directly. While in numerical approach solution is found by numerical iteration. FEM is an example of numerical method. SEA, an energy based approach is also a very widely used, sometimes modelling
Attenuati-on zone
Propagation Zone
a
b
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could also involve empirical method. Analytical approach is often preferable when there is a method of developing the model without going through numerical solution. In present paper, the analytical approach by means of spatial Fourier transformation is in focus.
3.1 Assumptions Modelling is often carried out for a simplified version of the problem with number of assumptions, which for floor modelling are:
1. Plates are assumed to be isotropic and obey Kirchhoff’s plate equation. 2. Beams are assumed to be Euler beams and are symmetric with respect to their neutral
axes so that flexural and torsional waves of the beams are uncoupled. Cross sectional area of beams remains constant throughout.
3. the system is assumed to be linear, meaning system obeys some phenomenon such as, a. superposition i.e. response of the plate w can be calculated by adding up the
responses due to each forces acting on the plate considered separately, or by adding up all the force terms and then calculating the response. For this paper later is considered.
b. Homogeneity i.e. the response is assumed to change linearly due to any change in the input. If the force is doubled the response is also expected to be doubled. Etc.
4. Time invariance: the response is assumed to have same time dependence as the excitation. Response at any point varies with time exactly the same way the excitation does. Thus time dependence of all pressures are taken to be equivalent to the time dependence of the excitation pressure, pe(x0,y0)ei t and therefore the term ei t is suppressed henceforth. The point (x0,y0) is the origin of the point excitation.
5. Rigid connections: Connections at plate beam junctions are considered to be rigid. Therefore at all joints linear and angular displacements are exactly the same for beam and plate.
6. Closed cavity: air trapped inside the cavity is considered to be completely isolated from the outside air having different density.
7. fully periodic: the structure is assumed to be totally periodic meaning the spacing between the beams is the same for all adjacent beams attached to a particular plate and beams are of exact same dimensions.
8. Infinite structure: the structure is assumed to be infinite and nothing reflects back from the edges (boundaries).
9. No flanking transmission: only the direct sound is considered in the model. Since it is an infinite structure, no coupling with the side walls is considered.
3.2 Mathematical tools
3.2.1 Fourier transform Fourier transform [book] is a powerful tool of analysing signals in frequency domain or wave number domain. For vibration, it is periodic in both time and space and thus along with temporal frequencies gives rise to spatial frequencies known as wave numbers. Therefore Fourier transform can be applied to both time and space and here this is applied to space. Usually here the system of complicated differential equations are transformed and solved, then transformed back to initial domain. Therefore there are two ways of transformations, one called forward transformation (spatial to wave number domain):
yxyxww yx dde),(),(~ )(i , (2)
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where, and are the transform wave numbers (spatial frequency) of the plate in x and y direction, respectively. And the other called inverse transform (from wave number to spatial domain) is
dde),(~4
1),( )i(2
yxii wyxw .
(3)
One of the advantages of this technique is that, the complicated differential equations after being transformed become algebraic equations and solution process becomes very simple.
3.2.2 Poisson’s summation It’s a mathematical tool to convert one infinite series to another, and one application is the transformation of exponential sum in to sum of Dirac functions as following:
.)/2(2i
nn
nl lnl
e (4)
And thus properties of Dirac function can be applied such that,
)./2(~)/2()(~ lnwdlnwn
(5)
4 Different solution methods
4.1 Analytical Analytical modelling of lightweight structure consists of three major parts. 1) Modelling the excitation, 2) Modelling the vibration in the plates and 3) Modelling the sound radiation. Part 2 could be treated as the main part of it. Different method can be applied to model the 2nd part.
4.1.1 Modelling the excitation Excitation can either be air borne or structure borne. In case of structure borne sound associates with people walking, something falls from a height or some other impact. Therefore unlike air borne case, here the excitation is given to the structure as an impact in a very small area and vibration propagates from that area. Thus impact from ISO taping machine well corresponds to the situation. An ISO tapping machine has 5 hammers of mass, M = 0.5 kg each falling from a height of 4 cm. Each hammer strikes the floor twice per second. One hammer makes impact at a time making the total hammering frequency fs = 10 per second. There has been many works on how to model this excitation process. Cremer [33] shows that the periodic impact force pulses can be expressed as Fourier series:
,cos)(1n
sn tnFtF (6)
where s = 2 fs and Fn is the Fourier coefficient given by,
.1,)cos()(2
0
ndttntFT
FT
sn
n (7)
This assumes a bare hard floor surface such as concrete. Also an elastic impact for a very short period is assumed leading to sharp frequency spectral lines separated by a length of fs. A resilient surface over the hard floor will make the impact less elastic as dealt by Lindbald [34], ver [35]. Lindbald modelled the impact with resilient covering on a hard floor by a connecting a resistance and spring in series. Brunskog [36] applied this concept in a bare lightweight floors i.e. modelling impact on lightweight floor using a resistance and spring
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constant representing floor characteristics, along with the mass of the hammer placed in series. Two extreme cases of impact can be imagined, one when the impact is fully elastic and the hammer bounce back with the same velocity but of opposite in sign and second is when the hammer after the impact, dissipates all energy and stay on the floor. In [34,36] model tries to find an in-between mode of impact. Assuming K and M to be frequency independent, the solution to this model is as following according to [34, 36]:
22/
2
2
0
22/
2
2
0
1
4,,)2/(/
)2/(/sin
4,,/)2/(
/)2/(sinh
)(RKMcalundercritie
RKMKRKMKtKv
RKMalovercriticeMKRK
MKRKtKv
tfRKt
RKt
The Fourier transform of this single impact is calculated as:
22
)2(0
22
0
14,,)1(
4,,/
RKMcalundercritiRKMiMK
eKMv
RKMalovercriticRKMiMK
KMv
F RKitcut
Therefore each component of the Fourier spectrum becomes:
rn
Tt
rn T
FttfT
F r 1/2i1 de)(1 (8)
4.1.2 Modelling the vibration in the plates A common way of modelling the floors is by assuming Kircchof’s plate equation to represent the plate and considering the interaction with the other elements as force acting to the plate and placed in the right side of the plate equation. Considering only the upper plate in figure 2 and the corresponding interacting forces the governing equation becomes:
amfe ppppwmwyx
D 22
2
2
2
2
(9)
Where, D, m´´ and w are the stiffness, mass per unit area and displacement of the plate respectively. pe is the excitation force and can be both air borne and structure borne, pa is the interacting force term with the surrounding air, pf is the total sum of the reaction forces from all joints and pm is the sum of all moment reaction forces. The idea is to calculate all the force terms in terms of the plate displacement and from that finding the solution for w. All the calculation done in wave number domain.
Calculation of pa: Surrounding air satisfies the Helmholtz equation:
020
2
aa pc
pzyx
, (10)
where, c0 and pa are velocity of sound in air and air pressure respectively. Applying the boundary condition at the plate surface that plate vibration and particle velocity close to the plate surface are equal, i.e.
, 02
0
wz
pz
a (11)
11
Assuming plane wave and transforming equation (10-11) the solution can be approximated as [2, 4, 10-11].
222
02
02 ),(
),(),()0,,(
kwwpa (12)
Where, is the wave number in z direction and the branch of is taken so that 0 ,
0 if 0 ,
As mentioned before the excitation can be both air borne and structure bone. The solution methods for both cases are given below:
Calculation of reactions from plate beam joints: depending on the excitation the calculation procedure of the reactions from the plate beam joint can be different.
1. Structure borne excitation: here excitation is given to some particular points and as a result, each joint will exert a reaction force and reaction moment and has no correlation among each other. Reactions from each joint have to be understood separately using the corresponding plate beam boundary condition. The Euler beam equation is considered to represent beam reaction which for nth beam together with the boundary condition at each joint is:
)()(
)( 1,2
1141,
4
11,1 yuAdy
yudIEyF nff
nffn (13)
)(),( yuynlw n (14) Where, E, I, , A, un are the Young’s modulus, moment of inertia, density, cross sectional area and displacement of nth beam. Combining equations (13) and (14) and summing up forces from all joints the total reaction force becomes:
)(),()()( 2114
4
111,11 nlxynlwAdydIEnlxyFp
nffff
nnf (15)
Transforming equation (15) into wave number domain:
n
nlf nlwZp ie),(~ (16)
Where, Z is the transformed linear operator of beams corresponding to force reaction and w is the partial Fourier transform with respect to y. In order to express fp~ in terms of total transform of displacement w~ equation (15) is written as
n
n
nlf dwZp )i(e),(~
2~ (17)
Where, is used to distinguish the integrating variable. Applying the poisons formula (4) and using the property of Dirac function as in equation (5) we get,
.,2~~n
f lnw
lZp (18)
Total moment reaction can also in similar way be derived. The moment equation is according to [8, 32] and the corresponding boundary condition:
,),( 22
22
2
2
nn yT
xynlw
yTM (19)
,),()()(x
ynlwyy nnn (20)
12
where, T and are the torsional stiffness and angular moment of inertia of the beam respectively, while n and n are the angular displacement of the nth beam and plate respectively. following the same procedure the total moment reaction is derived as,
,),/2(~)/2()(e)(~i~ 1i
nn
nln lnwln
lHMp
m (21)
Where H is the transformed linear operator of beams corresponding to moment reaction.The infinite force and moment terms are still in undetermined form. These need to be determined which is done by performing variable change and summing over all n and these is described in [2, 4]. With the infinite terms evaluated equation (9) can now be solved in wave number domain.
2. Air borne periodic excitation: in case of the air borne excitation the entire plate is excited by a pressure wave, thus excitation becomes periodic in both time and space domain which is of the form )(
0tykxki
eyxepp . Where, kx and ky are the wave numbers of the
excitation in the x and y direction respectively, represent spatial dependence of the excitation and are constant. While, is the angular frequency and defines the time dependence of the excitation. The fact that the response is governed by the periodic nature of the excitation simplifies the calculation a lot as considered in [2, 7]. Due to the spatial periodicity the displacement at two positions w(x1, y) and w(x2, y) only differs by a phase difference. i.e.
)(i12
12),(),( xxk xeyxwyxw . And same is true for the reaction forces from beams. Moreover, since the beams are periodic in space (along x axis) with a period l , the force acting on any two points l distance apart can be written according to Floquet principle as, nlikxexFnlxF )()( (22) Further modification would lead to nxik
nxeww 0 and nxik
nxeFF 0 . Where, the indices n and 0
corresponds to n’th and 0th beam, where x = 0. With this simplification forces and displacements from all beams are expressed in terms of F0 and the system is only to satisfy the boundary condition in the 0th beam. Therefore, the governing equation (9), if fluid loading and moment effect are ignored and force term is inserted, would look like:
)(02
2
2
2
2
2
nlxeFepwmwyx
D nlikxike
xx (23)
Here the both time and y dependences are suppressed. Fourier transform of this is,
n
n
nlinlikxe eeFkpwS x
0)()()( (24)
Where, 2''222 )( mDS the transformed linear plate operator, and are the wave number in x and y direction respectively. Poisson’s summation formula is applied and after some manipulation we get,
n
n
x
xe
Slnk
FkS
pw)(
2
)()(
)( 0 (25)
Only term unknown in equation (25) is the F0 which is calculated by inverting this equation and putting x = 0. Finally solving for F0 we get,
n
n xx
e
lnkSZkS
ZpF
)/2(11)(
0 (26)
13
Another thing noticeable in Rumerman [7] is the use of superposition method to determine the response, velocity. The velocities due to different forcing are linearly added to find the total velocity. The equation of the velocity of connecting line between the plate and the nth beam is
)()(0 kZ
MkZ
Fvvmn
m
mn
mnn (27)
Where, the first term in the right hand side of equation (27) corresponds to the velocity of the plate when no beam is attached, the second term represents the velocity component due to the forces exerted from the each beam and the third term represents the velocity component due to the moment effect of each beam and plate connection. Fm, Mm are the force and moment exerted from the joint, while Zmn and Z´mn are the transfer impedances corresponding to force and moment from nth beam to the mth beam respectively. All these velocities, forces and moments are expressed in terms of the velocity, force and moment at x=0 respectively using the Floquet principle stated before. Thus the equation becomes function only of v0, M0 and F0.
3. Free wave approach: This approach [37] is based on transfer function method. First the system is solved for homogeneous equation i.e. free response and from that the transfer functions is calculated. Desired function (impact function) is then multiplied to get the impact response. First free wave propagation is calculated using the homogeneous form of the equation (24) i.e.
n
nn nlxFSw )( (28)
Where, Fn is the force exerted from the nth beam and fluid loading is neglected. Since free waves are being considered, all bays differ only by phase and thus Floquet principle is applicable, i.e. displacement and force in all bays can be expressed in terms of the first one. After following more or less the same procedure as for the case of air borne excitation it is possible to get the dispersion relation and from that the possible propagation constants g:s described in section 2.6.1 are calculated. These g:s describes the wave propagates in the first bay, and for the other bays the Floquet principle is applicable. However, since the homogeneous equation is of the order 4, for each frequency there will be 4 different propagation constants ±g1, ±g2. Thus the general solution is: ),,,,,()( 21 ggCCCCfxw bbaa (29) Where C:s are the constants to be determined according to the loading and boundary conditions. For a point excitation the response is a Greens function. And also since the solution must be finite at infinity only two solutions are possible at each side of the excitation which according to [36, 37] is
lxxwheregxYCgxYCxxG bbaaor
0,.........),(),()|( .0,)........,(),()|( 00 xxwheregxYCgxYCxxG bbaa
l Four boundary conditions are needed to solve for these four constants. The boundary condition can be found by integrating the inhomogeneous equation (with point excitation term) which gives [36, 37],
0)()( 0xGxG lo
r
000 xx
l
xx
r
xG
xG
000
2
2
2
2
xx
l
xx
r
xG
xG
14
BxG
xG
xx
l
xx
r 1
00
3
3
3
3
Where B´ is the bending stiffness per unite length of the plate. Solving for this C:s the greens function can now be determined.
4.1.3 Modelling the sound radiation In part 2 the vibration distribution is calculated for different frequencies must now be coupled to the surrounding air to calculate the radiated power. The general definition of power is how much energy is transferred through a surface per unit time.
T
a
dtdStyxvtyxpTAFvW
0
),,(),,( (30)
T
sa
dtdStyxvtyxpTAFvW
0
),,(),,( (31)
Where, p is the excitation pressure and v is the velocity distribution, A surface and Ta is the averaging time. Theoretically, for an infinite plate, surface being infinite the radiated power will also be infinite. Therefore, in practice while calculating, a partial area from the infinite area is considered. For point excitation, since the excitation is 0 out side the integral area equation (31) is valid and the limit can be extended to infinity as equation (30). Applying the Fourier transform to obtain the spatial distribution of the velocity [33] and using the relationship between the excitation pressure and velocity distribution [33] the following expression can be achieved:
dd),(~
81
222
22
2
2
2
22
k
Radk
wck (32)
Where, k, and c are the wave number, density and sound velocity of air respectively. This way the displacement is split as a sum of bending waves. Superposition of radiation from each bending mode is considered. The term
222kk represents the radiation efficiency
which depends on frequency. For wavelengths shorter than that of sound in air, radiation will be a near field because of hydrodynamic short circuit having very low radiation efficiency and provide no radiation. For Wavelengths greater than that of ambient air, sound will be radiated at an angle so that the component of the bending wave at that angle equals the wavelength of the air. This is known as trace matching. Frequency where the wavelength equals that of air is known as critical frequency. Radiation efficiency at this frequency is maximum and above this frequency it is unity.
15
Figure 5: radiation efficiency of infinite plate
Figure 5, taken from [33] shows the relationship between radiation efficiency and the wave number for an infinite plate. k0 and kB are the wave numbers of sound in air and the bending wave of plate respectively. At k0/kB = 1 i.e. at critical frequency, radiation efficiency goes to infinity and so does the pressure level. In reality pressure level remains finite since the vibrating area is finite. Nevertheless at this frequency the radiation efficiency becomes very high and above this that radiation efficiency remains at unity, while bellow this frequency the radiation efficiency is very low. Therefore it is sufficient only to calculate sound radiation from the bending waves having wavelength greater than that of air which in regard to
equation (32) is done by limiting integration limit to 12
22
k.
4.2 Empirical method Empirical method is also a very useful way of modelling lightweight floors. This is a combination of theoretical and experimental approach. Here some equations are set for modelling and some terms with in the equations are determined by experiments. Model by Bradley and Bradley and Brita [20] is one example which talks about developing an empirical model to analyse the effect of resilient joint between the plates and beams. In real construction it is often the case that the rigid connection between the plates and beams are broken, and a thin resilient metal connection is employed instead. This is mainly to break the transmission path from plate 1 to plate 2 and also to isolate the vibration of the plate 2. However, this paper describes a simple model to study the performance and effect of such connections. The approach is not to include all complexity of double leaf construction rather the idea is only to study the effect of adding the resilient connection to the structure. The key parameters are derived from the lab measurements empirically for various resilient connections and for different cavity depth. The main concern of this paper is to study the performance at low frequency because practical out door sounds are mostly of low frequency.
16
4.3 Fourier series solution This is an analytical approach as well but the difference is here instead of Fourier transform method, Fourier series is used although the governing equations remain the same as mentioned. A very recent paper on this method is ref. [23] by Chung and Emms. Here plate and beam displacements are expanded as Fourier sine series up to a finite number considering sine function as basis function,
xAmxm sin)( ; y
Bmym sin)( For m, n = 1,2,3…….N.
A and B are the length and width of the structure. For plates displacements in both axis is considered while for beams displacement in the direction along the beams is considered. The system of equations is then solved for the constants of Fourier series. After inserting the Fourier series of displacements into the governing equations, orthogonally property of the basis functions is implemented which states,
A
mnnm dxxx0
)()( ;B
mnnm dyyy0
)()(
Where, nmwhennmwhen
mn
mn
,0,1
.
With these it is possible to construct one equation corresponding to each constant. Therefore number of equation being equal to number of constants, the system of equation can be solved for all the constants and Fourier series can thus be constructed.
4.4 Statistical Energy Analysis SEA SEA is an energy based method. Energy flow through different systems is the primary concern. Moreover, it is a kind of approach based on average value, i.e. to take average over frequency band and over whole element into consideration. System is analysed and understood in terms of averages rather than trying to consider each and every detail. Here the whole structure is subdivided into many small subsystems. The choice of the subsystems is arbitrary. For lightweight floor the sub systems could be plates, beams, cavity etc. It is possible to combine more than one element to make one subsystem. After any excitation is given to any subsystem some portion of energy is lost within that subsystem due to internal damping and the remaining flows through the other subsystems. Same thing happens in each subsystem that is, some part gets lost and the remaining flows through the neighbouring subsystems. Therefore the key parameters are:
1. The losses in each subsystem, described in terms of the internal loss factor 2. Coupling between subsystems, described by coupling loss factor.
With these loss factors a power balance equation is set for each subsystem and this system of equations are then solved. J. C. Sun et al. and H. B. Sun et al. in the series of their papers [39-41], described the calculating procedure of the loss factors for various conditions. Uniformity of the energy over the entire element, i.e. diffuse energy field is the primary prerequisite for SEA. Therefore, it is better approximated at high frequency where modal density is high and uniform and system act more independently without being influenced much by the material property and boundary conditions. While, it fails at low frequency, since at low frequency number of resonance frequencies is few and are widely spread, the system here is strongly influenced by the boundary condition. Nevertheless SEA is a powerful tool and so widely used for sound prediction.
17
4.5 Finite Element Method FEM FEM is an example of numerical method of solving. Here the entire structure is subdivided into small elements which are different from the structural elements. These elements could be of different size and shape and dimension for example straight lines, quadratic lines, squares, cubes etc can be used as elements. Each element is then treated separately and separate equations are extracted from each element to form separate elemental matrices, mass Me and stiffness Ke matrices. Number of element equals the number of such elemental matrices which are then combined using inter-elemental boundary condition and the global mass M and global stiffness K matrices for the entire system are constructed. Solving for this global matrix gives the response. One way of forming such elemental equation is by making use of the energy equations, e.g. potential energy (related to mass) and potential energy (related to stiffness). Expressions of elemental mass and elemental stiffness, in terms of the field variable are derived. Element size is very crucial in FEM. For linear elements, largest element size must be six times smaller than the minimum wave length, for quadratic elements it should be 12 times. Therefore, for acoustics where frequency goes very high causing the wave length to be very small, FEM does not work reasonably well at high frequencies. Nevertheless, it is indeed a powerful and very useful tool for modelling low frequencies. While, at mid frequencies there always remains a possibility for crossing the element size - wavelength limit. Paper [42] deals with such limitations while modelling with FEM.
5 Conclusions In this thesis paper a general overview of the lightweight floor structure is presented. Construction, complexity, periodic phenomenon as well as the modelling approaches are discussed. Attached papers deals with modelling and parametric studies of lightweight decoupled structure. Some mathematical tools used in the modelling are discussed here. With this, the attached paper will be much easier to understand. The thesis includes understanding different modelling approaches and their corresponding models and it is noticeable that there are many. At this state the focus restricted only to analytical approach not because it is the best one rather it seemed convenient to the author. Nevertheless, no such approach could claim to be the best one. System has different characteristics in different frequencies and different approaches are appropriate for different frequency bands. For example at low frequencies, number of resonance frequency is few and are widely spread and the system shows strong modal behaviour. At mid frequencies response is quite irregular, shows irregular modal densities and strongly influenced by boundary conditions, material properties, geometry etc. At high frequencies on the other hand, modal density is high and uniform, the system is not influence much by boundaries, geometry, material properties etc. Therefore, SEA can be a good method for modelling high frequencies, where as FEM is suitable at Low frequencies and with some tolerances at mid as well. Nevertheless, for FEM computational time and element size is of big concern. For analytical approach the difficult thing is to handle governing equation and boundary condition. Sometimes the equation is too complicated to handle and some sort of oversimplification is made. Therefore, the best thing would be to combine different approaches for different frequency range. However, the assumptions and solution method considered in the analytical approach of present thesis seems to be quite reasonable and acceptable. The model can be modified further to introduce real life complexity in more detail, such as:
1. Introducing finiteness of the floor. 2. Modelling the side walls and couple it to the floor structure in order to model the
flanking transmission.
18
3. Modifying the joints. For example introducing slippage, 4. Introducing variable loss factor. 5. Using different approaches for different frequency range.
Referens [1] V. N. Evseev: Sound radiation from an infinite plate with periodic inhomogeneities. Soviet Physics –Acoustics 19(1973) 226-229 [2] B. R. Mace: Sound radiation from a plate reinforced by two sets of parallel stiffeners. Journal of Sound and Vibration 71(1980) 435-441 [3] G. F. Lin and J. M. Garrelick: Sound transmission through periodically framed parallel plates. Journal of Acoustics. Society of America 61(1977) 1014–1018. [4] J. Brunskog and P. Hammer: Prediction model for the impact sound level of lightweight floors. Acta Acustica United with Acustica 89(2003) 309–322. [5] J. Brunskog: The influence of finite cavities on the sound insulation of double-plate structures. Journal of Acoustic Society of America 117(2005) 3727–3739. [6] D. Takahashi: Sound radiated from periodically connected double-plate structures. Journal of Sound and Vibration 90(1983) 541–557. [7] M. L. Rumerman: Vibration and wave propagation in ribbed plates. Journal of Acoustical Society of America. 57(1975) 370–373. [8] L. G. Sjökvist: Structural sound transmission and attenuation in lightweight structures. Doctoral thesis, paper 3, Chalmers University of Technology, Göteborg, Sweden, 2008. [9] J. Wang, T. J. LU, J. Woodhouse, R.S. Langley, J. Evans: Sound transmission through lightweight double-leaf partitions: Theoretical modelling. Journal of Sound and Vibration 286(2005), 817-847. [10] M. S. Mosharrof, F. Ljunggren, A. Ågren, J. Brunskog: Prediction model for the impact sound pressure level of decoupled lightweight floors; Proceedings of internoise 2009, Ottawa, 2009. [11] M. S.Mosharrof, J. Brunskog, F. Ljunggren, A. Ågren: Improved prediction model for the impact sound level of lightweight floors - introducing decoupled floor-ceiling and beam-plate moment. Submitted to Acta Acustica United with Acustica, 2010. [12] Craik RJM, Smith RS. Sound transmission through double leaf lightweight partitions part I: airborne sound. Applied Acoustics, 2000; 61(2): 223-245 [13] Craik RJM, Smith RS. Sound transmission through lightweight parallel plates, part II: structure-borne sound. Appl Acoust, 2000; 61(2): 247–269. [14] Elmallawany A. Criticism of statistical energy analysis for the calculation of sound insulation Ð part 2: double partitions. Applied Acoustics 1980; 13:33±41. [15] Price AJ, Crocker MJ. Sound transmission through double panels using statistical energy analysis. Journal of the Acoustical Society of America, 1970;47:683±93. [16] Crocker M. J., Bhattacharya M. C., Price A. J. Sound and vibration transmission through panels and tie beams using statistical energy analysis. Transactions of ASME, 971;93:775±82. [17] A. Rabold, A. Duster, E. Rank: FEM based prediction model for the impact sound level of floors. J. Acoust. Soc. Am. 123 (2008) 3356. [18] Sound transmission loss analysis through a multilayer lightweight concrete hollow brick wall by FEM and experimental validation. Building and Environment, Volume 45, Issue 11, November 2010, Pages 2373-2386 J. J. del Coz Díaz, F. P. Álvarez Rabanal, P. J. García Nieto, M. A. Serrano López. [19] Orris Ruth M. and Petyt, M.: A Finite element study of harmonic wave propagation in periodic structure. Journal of Sound and Vibration 33(1974), Issue 2, 223-236. [20] J. S. Bradley, J. A. Brita: A simple model of sound insulation of gypsum board on resilient supports. Noise control Engineering 49(2001) 216-223.
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[21] Sharp, B. H. Prediction methods for the sound transmission of building elements. Noise Control Engineering 11(1978) 53-63. [22] Ljunggren, F. Using elastic layers to improve sound insulation in volume based multi-storey lightweight buildings. Proceedings of InterNoise, Ottawa, Canada 2009. [23] Chung, H. Emms, G. Fourier series solutions to the vibration of rectangular lightweight floor/ceiling structures. Acta Acustica united with Acustica, 94(2008) 401-409. [24] Ökvist, R. Ljunggren, F. Ågren, A.: Variations in sound insulation in nominally identical prefabricated light weight timber volume constructions. Submitted to applied Acoustics. 2010. [25] Trevathan, J.W. and Pearse, J.R., The Effect of Workmanship on the Transmission of Airborne Sound through Light Framed Walls, Applied Acoustics, 2008, 69, 127-131 [26] Johansson, C., Field measurements of 170 nominally identical timber floors – a statistical analysis, Proceedings of InterNoise 2000, Nice, France, 27-30 August 2000, 4072-4075 [27] Simmons, C., Uncertainty of measured and calculated sound insulation in buildings – Results of a Round Robin Test, Noise Control Eng. J., 2007 Jan-Feb, 55 (1), 67-75 [28] Wittstock, V., On the Uncertainty of Single-Number Quantities for Rating Airborne Sound Insulation, Acta Acustica united with Acustica, 2007, 93, 375-386 [29] Craik, J.M. and Evans, D.I., The Effect of Workmanship on Sound Transmission through Buildings: Part 2 - Structure Borne Sound, Applied Acoustics, 1989, 27, 137-145 [30] G. Sen Gupta: Natural flexural waves and the normal modes of periodically supported beams and plates. Journal of Sound and Vibration 13 (1971) 89-101. [31] D. J. Mead: Wave propagation and natural modes in periodic systems: I. mono coupled systems. Journal of Sound and Vibration 40(1975) 1-18,. [32] D. J. Mead and K. K. Pujara: Space harmonic analysis of periodically supported beams: response to convected random loading. Journal of Sound and Vibration 14(1971), 525-541. [33] L. Cremer, M. Heckl, and E. E. Ungar: Structure-Borne Sound. 1988 ISBN 0-387-18241-1 [34] S. Lindblad: Impact sound characteristics of resilient floor coverings. A study on linear and nonlinear dissipative compliance. Dissertation. Division of Building Technology, Lund Institute of Technology, Lund, Sweden, 1968. [35] I. L. Ver: Impact noise isolation of composite floors. Journal of the Acoustical Society of America 50(1971) 1043–1050. [36] J. Brunskog and P. Hammer: The interaction between the ISO tapping machine and lightweight floors. Acta Acustica United with Acustica 89(2003) 296-308. [37] Nordborg, A. Vehiical rail vibratios: Point force excitation. Acustica United with Acta Acustica, 84(1998) 280-288. [38] Jonas Brunskog. ‘Acoustic excitation and transmission of lightweight structures’. Doctoral thesis. Lund University of Technology, 2002. [39] J. C. Sun and E. J. Richards Prediction of total loss factors of structures, Part I: Theory and experiments. Journal of Sound and Vibration 103(1985) 109-117. [40] J. C. Sun, H. B. Sun, L. C. Chow and E. J. Richards: Predictions of total loss factors of structures, part II: Loss factors of sand-filled. Journal of Sound and Vibration 104(1986) 243-257. [41] H. B. Sun, J. C. SUN and E. J. Richards: Prediction of total loss factors of structures, Part III: Effective loss factor in quasi transient condition. Journal of Sound and Vibration 106(1986) 465-479 [42] T. odygowski, W. Sumelka: Limitations in application of finite element method in acoustic numerical simulation. Journal of Theoretical and Applied Mechanics 44(2006) 849-865.
Paper
1
Prediction model for the impact sound pressure level of decoupled lightweight floors Mohammad Sazzad MosharrofaFredrik Ljunggrenb
Anders Ågrenc
Division of Sound and Vibration Luleå University of Technology 97187 Luleå Sweden
Jonas Brunskogd
Department of Electrical Engineering Denmark Technical University Copenhagen Denmark
ABSTRACT Prediction of impact sound insulation in timber buildings is difficult since the constructions are complex, non isotropic and show large variances. There are earlier publications on precise models of simplified structures and simplified models of complex structures, where so far, none is considered as an acceptable simplified engineering model tool. The goal of our research is to develop methodology for such a simplified engineering tool. The starting focus is to model impact sound in a simple floor model of two plates that are realistically coupled via beams. The floor structure in Nordic countries are mostly decoupled, meaning there is either no structural connection or an elastic connection between the plates and the beams. In this paper, an improved floor model is proposed where as a first step there is no physical connection between the plates and beams, and where each plate is stiffened by separate beams. Hence the construction becomes more realistic as only coupling between the plates via air trapped inside the cavity is taken into account. Next step will be to introduce resilient coupling between the respective beams. The modelling shows promising results with reasonable sound insulation improvements.
a Email address: [email protected] b Email address: Fredrik.ljunggren@ltu,se c Email address: anders.agren@ltu,se d Email address: [email protected]
2
1. INTRODUCTION Predicting the acoustic characteristics of light weight floor structure is very difficult and it is almost impossible to know exactly what the real scenario is. This is due to the complexity in the construction. It is not always possible to include all complexity in the model. Models are often based on some assumption and represents simplified conditions where as in real practise the assumptions and simplified conditions are no longer valid. Many issues come into account which is often excluded from the assumptions. Still theoretical models provide useful information and show ways to draw conclusion. On the other hand experimental results are more reliable, they resembles reality more precisely than calculated results. Despite this, the drawback of the experimental approach is that it requires heavy installations. And also any little change in the design would require a totally new construction. To better understand the characteristics both theoretical and experimental approaches are important.
There has been many works on developing Theoretical models for lightweight structure. For example London1 investigated transmission of reverberant sound of a single wall both experimentally and theoretically. And later on a modification of this was proposed by the same author2 where the analysis was extended further from single wall to double wall. Later researchers like Evseev, Lin and Gerralik introduced the concept of stiffening the plate using periodic set of parallel beams. Evseev3 studied single plate stiffened by beams where Lin and Gerralik4 studied two plates coupled by a set of parallel beams. Further Brunskog5 introduced the excitation from an impact hammer and some absorbing material inside the cavity although the basic concept was same two plates coupled via beams. Here in this paper am model is developed to deal with decoupled floor structure where no rigid connection between the plates by any means is considered.
2. OVERVIEW OF THE MODEL A. Floor structure for existing model: y x pr pe z pf1 pc(x,y,0) hp1 d hb, d hp2 pf1 pc(x,y,d) l pt (a) (b)
Figure 1 : (a) coupled floor structure (b) force diagram of coupled structure
The theoretical model4,5 so far has been developed for the case of two parallel infinite plates with thickness hp1, hp2 placed along the x axis and coupled via a set of beams placed parallel along the y axis as show in figure 1(a). As a result Cavities are formed between the plates. Depth of cavities d is same as the beam height hb. Air trapped inside the cavity and the surrounding airs are taken into consideration. Both the acoustic field in the cavity (0<z<d) and the semi infinite fluid in contact of plate 1 (z<0) and plate 2 (z>d) follow the Helmholtz’s equation. The plates are considered as a classical thin plate and beams are considered as Euler beams. In Lin Garrelik4 a plane wave is considered to be incident on plate 1, where in Brunskog5 excitation is given to the upper plate using an ISO tapping machine. This excitation is transferred to the lower plate through the beams and the cavity giving rise to a standing wave inside the cavity. So the total force acting on the plate 1 are excitation pressure pe, reaction pressure from beams pf1, cavity
3
pressure due to coupling with the plate 2, interaction with the surrounding air (radiated pressure pr for plate 1) where as for plate 2 the forces acting are reaction pressure from beams pf2, cavity pressure due to coupling with the plate 1, interaction with the surrounding air (transmitted pressure) pt. this is shown in figure 1(b).
B. Floor structure for modified model pr pe pf1 pc(x,y,0) pf2 pc(x,y,0) pt a b
Figure 2: a) decoupled floor structure, b) force diagram of decoupled structure In this paper the model is modified for decoupled condition meaning, there is no rigid connection between the plates by any means. As in figure 2(a) two separate sets of beams are used each of which are attached to each plate separately. Beam set 1 is attached to upper plate (plate 1). Each beam in this set has height hb1 thickness b and spacing between each beam is l. Beams are infinitely long in y direction. On the other hand beam set 2 is attached to lower plate (plate 2). Each beam in this set has height hb2, width b and spacing between each beam is also l. Due to this decoupling the transfer path of the vibration through beams is cut. Only way vibration can be transferred to plate 2 is through the cavity. From figure 1(b) and figure 2(b) it is noticeable that all the force terms acting on the plates are the same. And derivation of these terms is expressed in details in the previous papers. Here also in this paper the same expressions are used for all of them except for the reaction forces from the beams pf1 and pf2. these two are not the same as before. Previously the forces to the plates were coming from the same set of beams. But now since two sets of beams are used and each set is attached to one plate only, the reaction forces in this case to any plate are from the beams attached to it.
3. GOVERNING EQUATIONS AND SOLUTION METHOD
A. Governing equations Equations for two plates are considered and solved for two variable w1 and w1 representing the displacement of plate 1 and plate 2 respectively. These displacements are considered positive in positive z direction. Two plate equations are as following:
cfre ppppwmwyx
D 112''
112
2
2
2
2
1 )( (1)
222''
222
2
2
2
2
2 )( ftc pppwmwyx
D (2)
Where, m1 and m2 are the mass per unit length of plate 1 and plate 2 respectively, is the angular frequency, terms in the right hand side of the equations are the force terms mentioned before, D1 and D2 are the flexural rigidity of plate 1 and plate 2 respectively which are calculated as follows.
hp2
hp1
beam set 1
hb1 l
l
beam set 2
hb2
4
)1(12 2
3EhD (3)
Where E is the Young’s modulus of the plate, h is plate thickness, is the poisons ratio.
B. Solution method Fourier transform method is used here. Fourier transform of the equations (1) and (2) gives:
cfre ppppwSwmD ~~~~~~))(( 11112''
1222
1 (4)
22222''
2222
2~~~~~))(( ftc pppwSwmD (5)
Where ‘~’ denotes the transformed form, and are the spatial transform of wave number in x and y direction respectively. With this transform differential equations are converted into algebraic equations. Component in the right hand sides are derived in known forms and then equations (4) and (5) are solved for w1 and w2.
4. DERIVATION OF THE FORCE TERMS AND SOLUTION
A. Excitation pressure Excitation here is considered as a point excitation given from an impact hammer machine at x0=0, y0=0. Same expression used in Brunskog1 is considered here.
),(~00 yyxxFp Re (6)
Where, FR is the time to frequency Fourier transform of the impact under consideration. The corresponding spatial Fourier transform is
)( 00),(~ yxiRe eFp (7)
FR can be found in Brunskog6.
B. Fluid interaction Air in all three regions described before follows Helmholtz equation. Sound speed (c0) and the
wave number (k0= 2/c0) are same in region 1 and 3 while in region 2 sound speed is cc and wave number is kc= 2/cc.
0,,20
2
,, trctrc pc
pzyx
(8)
Here p is the acoustic pressure and index c,r,t represents cavity, radiated and transmitted respectively. Boundary conditions for these three acoustic pressures are:
; 202,
102
0
, wz
pw
zp
dz
ct
z
cr (9)
Inside the cavity reflected wave from plate 2 is also present, so both these boundary conditions in equation (9) are required. After solving for transformed cavity pressure pc and putting z=0and z=d we get cavity pressure acting on two plates:
),(2
),(122211211*
2
),(2
),(1)cot()csc()csc()cot(
*2
),,(
)0,,(
ww
JJJJ
dkc
ww
ddkddkddkddk
dkc
dcpcp
(10)
5
Where, kd= (kc2- 2
2- 22). In region 1 and 3 no reflected wave is considered, so one boundary
condition each is required to solve for each. The solution to transformed radiated and transmitted pressure acting on plate 1 and plate 2 respectively are:
),(),()0,,( 122210
2
Rwk
wpr Where, 222
02
kR
),(),(),,( 222220
2
Twk
wdpt Where, 222
102 ),(
kwT
An elaborate calculation can be found in Brunskog1.
C. Forces from the beams The floor structure here consists of two plates, each attached to a different set of beams. Therefore each plate experiences reaction forces only from the set of beams attached to it as shown in figure 2(b). pf1 is the reaction force between plate 1 and beam set 1 while pf2 is the reaction force between plate 2 and beam set 2. Each force is calculated separately. Each beam of set 1 is considered to have parameters as height hb1, width (b), Young’s modulus (Ef1), moment of inertia (If1), density ( f1), area (Af1). The spacing between any two beams is equal to l. Connection between the set 1 beams and plate 1 is rigid and reaction forces act along the lines x=nl. Where n is an integer no. representing position of any beam in the set and have a span of ‘–
’ to ‘+ ’ Equation for nth beam is:
)()()(
,11,2
1141,
4
11 yFyuAdy
yudIE nnff
nff (11)
Where, u1,n is the displacement of nth beam of set 1 and F1,n is the reaction force acting on it. Therefore the total force (pf1) acting on the plate 1 from infinite number of beams is,
n
nnf nlxyFp )()(,11 (12)
From the continuity of beam and plate we get: )(),( ,11 yuynlw n (13)
Combining equations (11-13) gives n
nffff
n
nnf nlxynlwA
dydIEnlxyFp )(),()()()( 1
2114
4
11,11 (14)
The Fourier transform of the equation (14) is,
n
nffff
n
n
nlinf nlxynlwTAIEeFp )(),(.).()(~~
12
114
11,11 (15)
Where, T. stands for Fourier transform. Employing Poisson’s sum formula the force term in equation (13) can be expressed in terms of w1. Similar kind of derivation can be found in Mace7.
n
n
n
n
nlinf lnw
lGeFp ),/2(~)(~~
11
,11 (16)
Where, G1=Ef1If14- f1Af1
2. Following the same procedure reaction force between plate 2 and beam set 2 can be derived as:
n
n
n
n
nlinf lnw
lGeFp ),/2(~)(~~
22
,22 (17)
Where, the transformed beam operator, G2=Ef2If24- f2Af2
2, Ef2 is Young’s modulus, If2 is moment of inertia, f2 is density, Af2 is the cross sectional area of beams in set 2.
6
D. Solution Substituting the expressions for the force terms derived so far in equations (4),(5) and solving for w1 and w2 we get the following expression. Here and dependences are suppressed.
n
n
nlin
n
n
nlin
eF
eFPe
SSSS
Sww
,2
,1
2
1
0.
11211222
)det(1
~~ (18)
Where, 22221
1211122211211
TJSJJJRS
SSSS
S . To solve for w1 and w2, F1,nei nl and F2,nei nl need
to be evaluated. This is done by changing the variable of equation (18) to lm /2 and then summing over all m. The periodicity of the beam i.e. F1,nei( -2m /l)nl = F1,nei nl is considered and finally letting m=n we have:
n
n
nlin
n
n
nlin
n
n
n
n
eF
eFYP
lnw
lnw
,2
,1
2
1
~
~
))/2((~
)/2(~
(19)
Where, n
n lnSlnSlnSlnS
lnSYYYY
Y)/2(11)/2(21)/2(12)/2(22
))/2(det(1
22211211
, and
n
ne lnP
lnSlnS
lnSP )/2(*
)/2(21)/2(22
))/2(det(1 .
Substituting w1( -2n /l) and w2( -2n /l) from equation (19) by F1,nei nland F2,nei nl according to equations (16) and (17) and then solving for F1,nei nl and F2,nei nl we get:
Y22)Y11GGY11GY22GY12Y21G/(-GG1Y11P(2))P(1)Y21G-(lP(2)G-
Y22)Y11GGY11GY22GY12Y21GY12)/(-GP(2)G-Y22P(1)GP(1)(G
2112
21212
2112
212221
,2
,1
lll
llll
eF
eF
n
n
nlin
n
n
nlin (20)
Finally, substituting the values of F1,nei n land F2,nei nl in equation (18) w1 and w2 are solved. The infinite force expression here is used for the case where two sets of beam are attached to two plates separately. If we put F2,n=0 i.e. G2=0, the force terms then resembles the case where one set of beam attached to plate 1 only. Force terms in that case becomes:
0Y11)*/GP(1)/(1*/G 11
,2
,1 ll
eF
eF
n
n
nlin
n
n
nlin (21)
Furthermore, If we put F1,n= F2,n=0, the force terms vanishes and it resembles the case of two plates without any beam.
5. CALCULATE RADIATED SOUND PRESSURE LEVEL
A. Radiated sound power and pressure level Radiated sound power due to the vibration of plate 2 can be derived following Cremer, M. Heckl8.
7
ddk
wckrad
1222
222
2
222
),(
8 (22)
For simplicity let, =krsin( ), =krcos( ) and d d =krdkrd . Equation (26) then becomes: k
rr
r
rad ddkkkk
wck
0
2
022
222
2
2
),(
8 (23)
Equation (23) is integrated numerically as in Brunskog1. Impact sound pressure level at the receiving room can be expressed as:
dBA
cp
Lref
radn )
4log(10
0
00 (24)
Where, pref is the reference pressure and A0 (10 m2) is the reference area, 0 is the density of the air.
B. Calculation Using the above equations a Matlab code is generated to calculate for sound pressure level in the receiving room. The code calculates plate displacements at each wave no. corresponding to frequencies from 10 Hz to 5000 Hz with a resolution of 10 Hz and at each phase angel from – to - with a resolution of (2 /25). With these displacements of ceiling plate radiated sound power is then calculated for each frequency which is then used to calculate sound pressure level for corresponding frequency. Later on these pressure levels are summed over third octave frequency band.
6. RESULTS AND DISCUSSION Transmitted sound pressure level in the receiving are calculated for many different cases. Here some comparative results of coupled and decoupled conditions are shown. For decoupled condition beams are considered to be attached only to upper plate. Beam height in both conditions is set equal to cavity depth. Spacing between any two beams for both conditions is considered to be 60cm.
10 20 31.5 40 50 63 80 100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 500010
20
30
40
50
60
70
80
90
frequency
Ln
Comparison of different coupling condition at different hp2 at and at hp1=0.022hp2=0.05d=0.22
hp2=11 mm, f res=68.4301+0.034215i,f crt=1526.329-22.889786ihp2= 22mm, f res=55.8729+0.0279364i,f crt=763.1645-11.44489ihp2= 32mm, f res=51.3225+0.0256613i,f crt=524.6756-7.868364ihp2= 80mm, f res=47.4097+0.0237049i,f crt=335.7924-5.035753i
Figure 3: Comparison of calculated SPL for coupled and decoupled condition at different hp2. Dash dotted
lines (-.-) represent coupled condition, while solid lines (-) represent decoupled condition.
8
Figure 3 shows sound pressure level (SPL) at different hp2 for coupled and decoupled condition. hp1 is set equal to 22 mm, while hb and hb1 were set equal to 22 cm. The resonance frequencies and the critical frequencies for the corresponding plate thicknesses are also shown. Result shows significant transmitted sound pressure level reduction for decoupled condition in higher frequencies, above (150 Hz). Below 150 Hz transmitted sound pressure level is higher at all hp1 compare to coupled condition. Also it is clearly seen from the figure that there are peaks around the resonance while for coupled condition there is no peak at resonance. This is because, for coupled condition both plates receives extra stiffness as a result of their mutual connection. It is no longer mass air mass system. The whole structure acts as a single unit. Therefore, mass air mass resonance is not visible.
On the other hand for decoupled condition both plate vibrate separately resembling somewhat a mass air mass system. At low frequency two plates vibrates more or less in phase. This continues as it approaches resonance, and after resonance the lower plate tends to vibrate out of phase and becomes more isolated as frequency increases. Another noticeable thing is that, there is no effect of critical frequency of the lower plate for coupled condition. This is also because of the mutual connection between the plates. Where as for decoupled condition transmitted sound pressure level is influenced by critical frequency of the lower plate. Above the critical frequency transmitted sound pressure level graphs don’t follow any pattern even though the pressure level is lower above critical frequency compares to coupled condition. Furthermore below critical frequency transmitted sound pressure level reduces with the increase of thickness of the lower plate. Because with the increase of thickness lower plate gets heavier thus the vibration is reduced.
10 20 31.5 40 50 63 80 100 125 160 200 250 315 400 500 630 800 1000 1250 1600 2000 2500 3150 4000 500020
30
40
50
60
70
80
90
frequency
Ln
Comparison of SPL at different d and at hp1=0.022 hp2=0.011 hb=d
hb1=11 cm, f res=96.7747+0.0483873i,f crt=1526.329-22.889786ihb1= 22cm, f res=68.4301+0.034215i,f crt=1526.329-22.889786ihb1= 36cm, f res=53.4942+0.0267471i,f crt=1526.329-22.889786i
Figure 4: Comparison of calculated SPL for coupled and decoupled condition at different d. Dash dotted lines
(-.-) represent coupled condition, while solid lines (-) represent decoupled condition. Figure (4) shows transmitted sound pressure level at different cavity depth for two coupling conditions as before. For decoupled condition transmitted sound pressure level reduces with increase of cavity depth. It is evident once more that transmitted sound pressure level for decouple structure is strongly dependant on resonance frequency. The frequency span where transmitted sound pressure level for decoupled condition is more compare to that for coupled condition, is related to the resonance frequency. From the graph we see this span (up to 200 Hz) is higher when resonance frequency (96.77 Hz) is higher. Although there is a big difference between 96.77Hz and 200 Hz, still it gives an impression that something is happening in lower frequencies which is related to resonance. Moreover the resonance frequencies shown in the figure were calculated with out considering the beams.
9
7. CONCLUSIONSThe model developed in this paper is the simplest form of the decoupled structure. While the real floor structure includes some other things as well. For example in real structure there is damping layer on top of the upper plate, there is absorbing material inside the cavity etc. also the Moment effect of beam plate connection is not considered. In spite of excluding all these real structure phenomenon it provides useful information regarding decoupled structures. It draws the starting point of modelling complex decoupled floor structures. And this model will be modified further to including other real structure phenomenon. Nevertheless, proposed model in this paper gives an overall idea of what happens in such structures. For example, now we know that this structure is strongly influenced by resonance frequency and plate critical frequency. Proper designing is necessary to take care of these frequencies.
REFERENCES 1 A. London, ‘‘Transmission of reverberant sound through single walls,’’ J. Research Nat. Bur. Of Stand. 42,605 (1949) RP1998. 2 A. London, ‘‘Transmission of reverberant sound through double walls,’’ J. Acoust. Soc. Am. 22, 270–279 (1950). 3 V.N.Evseev, ‘’Sound radiation from an infinite Plate with Periodic inhomogeneoties,’’ Sov. Phys. –Acoust. 19,226-229 (1973). 4 G. F. Lin and J. M. Garrelick, ‘‘Sound transmission through periodically framed parallel plates,’’ J. Acoust. Soc. Am. 61, 1014–1018 (1977). 5 J. Brunskog1 and P. Hammer, ‘‘Prediction model for the impact sound level of lightweight floors,’’ Acust. Acta Acust. 89, 309–322 (2003). 6 J. Brunskog1 and P. Hammer, ‘‘The interaction between the ISO tapping machine and lightweight floors,’’ Acustica/ Acta Acustica. 7. B. R. Mace, ‘‘Sound radiation from a plate reinforced by two sets of parallel stiffeners,’’ J. of Sound and Vibration 71, 435-441(1980). 8 L. Cremer, M. Heckl, and E. E. Ungar, Structure-Borne Sound (2nd edition)
Paper
1
Improved prediction model for the impact sound level of lightweight floors – introducing decoupled floor-ceiling and beam-plate moment Mohammad Sazzad Mosharrofa Division of Sound and Vibration Luleå University of Technology 97187 Luleå Sweden
Jonas Brunskogb Department of Electrical Engineering Technical University of Denmark Copenhagen. Denmark Fredrik Ljunggrenc, Anders Ågrend Luleå University of Technology Division of Sound and Vibration Luleå University of Technology 97187 Luleå Sweden
Abstract In order to better understand the complex acoustic behaviour of lightweight building structures, a mixture of experimental and theoretical approaches is necessary. An accurate mathematical model is of great importance for the physical understanding and the scope is here to further develop an existing method to predict the impact sound pressure level in a receiving room. An analytical method has been implemented, where spatial Fourier transform as well as the Poisson’s sum formula are applied to solve for transformed plate displacements. Radiated sound power was calculated from these displacements and finally normalized sound pressure levels were calculated in one-third octave frequency bands. Decoupled floor structures, having separate ceilings with no mechanical connection to the floor frame, are in focus and the result from the model is compared with experiments as well as with previous results indicating similar accuracy. In addition, the effect of introducing beam-plate moment in the model is studied and found to be dependent on frequency, showing significant improvement in high frequency region.
1. Introduction The use of lightweight buildings is increasing and so is the need for understanding its complex acoustical behaviour. At the same time the complaint regarding poor sound insulation is a reality; especially the impact sound is often inadequate in lightweight tenements. Due to high complexity no reliable acoustical prediction models exist concerning the actual framed building technique. For given reasons, such engineered models would be
2
appreciated for the building industry and several researchers have been working with that topic over the past. Modelling can be performed using different approaches such as analytical techniques, FEM, SEA, and sometimes an empirical foundation might be an option. In the present paper, the analytical approach, by means of spatial Fourier transformation, is in focus. The technique was first introduced by Evseev [1] who calculated the response of a plate stiffened by beams. Later on, other researchers used this approach. Mace [2] considered single plates stiffened via two parallel sets of beam. He considered point excitation and calculated the far field pressure using a stationary phase approximation. Lin and Garelick [3] considered two plates coupled by a set of parallel beams. Air was trapped inside the cavity but the frames were ignored. Further, they described the significance of each component regarding the overall acoustic behaviour of the structure. Brunskog and Hammer [4] added absorbing material inside the cavity but did not, in similarity with Lin and Garrelick [3], consider the effect of the frames inside the cavity. Later Brunskog came up with another paper [7] where the effect of frame in the cavity was considered in the same coupled structure. Each cavity formed due to the frames was considered separately. Nevertheless, in Ref. [4] impact from ISO tapping machine was introduced, which was derived in paper [5] by the same authors. In all these papers [2-4, 7] far field sound pressure level were predicted in the receiving room using radiated power calculated using spatial transformation of displacement according to Cremer et al. [6] However, some important limitations occur in all the papers mentioned so far: the floor plates are considered infinite in size and neither moment nor slippage between plates and beams were treated. In addition only forces positioned exactly at an intersection point of the plate and beam was considered. Several of these limitations are still not considered in the models found in the literature. However, Rumerman [8] took moment effect of plate-beam connection into account when studying single plates stiffened by beams. Takahashi [9] considered double plates coupled by beams and including the moment. Different conditions for beam-plate connections were analysed; point-, line- and point-line connection. Sjökvist [10] did consider the moment as well when studying the vibration propagation along different directions in relation to a structure’s periodicity. In the present study is the moment coupling included. The influence of the periodicity on the vibration propagation in lightweight structures has also been analysed by other researchers, e.g. Sen Gupta [11], Mead [12] and Mead and Pujara [13] using a slightly different technique. As output of these works (together with several others, not explicitly mentioned) we do have a decent understanding of how different parts in a lightweight floor affect the vibro-acoustical behaviour. Thereby models, although complicated in nature and with severe limitations, might be generated. So far, only the condition of two plates being rigidly connected via mutual beams has been considered, However, in modern industrial building technique it is increasingly common to construct volume elements which after assembling form floors which possess no mechanical connection through the periodic load bearing beams, e.g., described by Ljunggren [14] The main scope of the present paper is therefore to comprehend existing models for the case of such a decupled structure for increased application. The second scope is to include the moment effect between plates and beams for increased accuracy.
2. Method and theory
2.1 Floor structures For convenience, floors structures are hereafter divided into two groups, both representing simplified structures suitable for modelling.
3
1. Coupled structure: Two plates with mutual beams in between connecting both plates mechanically. 2. Decoupled structure: Two plates with no mechanical connection, due to individual set of beams.
2.1.1 Coupled structure
Figure 1: Coupled floor structure a), and free body diagram of coupled floor structure b).
The floor structure is made of two isotropic plates with a thickness of hp1 and hp2 respectively as shown in figure 1. The upper plate represents the floor of the upper room while the lower plate represents the ceiling of the lower room. The two plates are connected via parallel equidistant beams with height hb, also equal to the cavity depth d, and spacing between adjacent beams l. In terms of the coordinate system, beams are placed parallel to the y-axis, the structure’s periodicity along the x-axis and cavity depth along the z-axis. The bottom of the upper plate constitutes the xy-plane, where the cavity depth is zero while the lower plate is located at z = d. Since the structure is infinite in x- and y-directions, any beam can be considered as the origin of coordinates. In the mathematical models so far, air trapped inside the cavity as well as the surrounding air are considered. Both the acoustic field in the cavity (0<z<d) and the semi-infinite fluid in contact with plate 1 (z<0) and plate 2 (z>d) follow the Helmholtz equation. The plates and beams follow the theory of classical thin plates and Euler beams respectively. In Lin and Garrelik [3] a plane wave is considered to be incident on plate 1. Excitation is given to the upper plate by from an ISO tapping machine, as in Brunskog and Hammer [4]. The vibration energy is transferred to the lower plate through the beams and through the cavity giving rise to standing waves inside the cavity. The total pressures acting on plate 1 are then excitation pressure pe, reaction pressure from beams p'f1, cavity pressure acting on plate 1 pc(x,y,0) and interaction with the surrounding air (radiated pressure) pr for plate 1. Pressures acting on plate 2 are reaction pressure from beams p'f2, cavity pressure acting on plate 2 pc(x,y,d) and interaction with the surrounding air (transmitted pressure) pt, all shown in figure 1(b).
2.1.2 Decoupled structure
4
Figure 2: Decoupled floor structure a) and free body diagram of decoupled floor structure b). The model is here modified to deal with the situation where plates are decoupled, i.e., there is no mechanical connection between the two plates. As in figure 2(a) one separate set of beams is used for each plate. Beam set 1, infinitely long in y-direction, is attached to the upper plate (plate 1) in which each beam has the height hb1, thickness b1 and spacing between adjacent beams l1. Beam set 2, with analogous notation as set 1, is attached to the lower plate 2 and the same coordinate system as for the coupled structure is used. Note that the spacing between the beams is different between the two sets of beams, which complicate the mathematical treatment, as there are two spatial periodicities in the structure. Due to the described decoupling, the vibration transfer path through the beams is eliminated. The only way vibrations can be transferred from plate 1 to plate 2 is thus through the cavity as sound. Note that according to figure 1(b) and 2(b) all the air pressure terms drawn for the coupled case remain in the decoupled model and two new reaction pressures pf1 and pf2 replace the previous ones p'f1,and p'f2, since two sets of beams now are used. In addition to normal pressures, due to moment acting at plate-beam joints gives rise to two additional pressure terms, pm1 acting on the floor and pm2 acting on the ceiling, according to Sjökvist [10]. Moment equations along with boundary conditions concerning angular displacement are described by Cremer et al. [6]
2.2 General assumption Any modelling is carried out for a simplified version of the problem with number of assumptions, such as:
Plates are assumed to be isotropic and obey Kirchhoff’s plate equation. The system is assumed to be in the linear range. Time invariance: the response is assumed to have same time dependence as the excitation. Thus, time dependence of all pressures are taken to be equivalent to the time dependence of the excitation pressure, pe(x0,y0)ei t and therefore the term ei t is suppressed henceforth. The point (x0,y0) is the origin of the point excitation.
2.3 Fourier transforms The spatial Fourier transform will be used as a tool within this paper. The form of the transform used is
yxyxww yxii dde),(),(~ )(i ,
(1)
5
where, i = 1,2 and and are the transform wave numbers in the plate in x and y direction, respectively. The corresponding inverse transform is
dde),(~4
1),( )i(2
yxii wyxw .
(2)
2.4 Solution method The governing equations for the plates according to Kirchhoff’s plate equation shown bellow represent the upper (3) and lower plate (4). The terms to the right are the exerted pressures. m is the plate mass per unit length, =2 f is the angular frequency; D is the plates’ flexural rigidity according to equation (3) and the indexes 1 and 2 refer to the plates. E is the Young’s modulus plate, h the thickness and the poisons ratio of actual plate.
1112
11
2
2
2
2
2
1 )0,,( mcfre pyxppppwmwyx
D (3)
tmfc pppdyxpwmwyx
D 2222
22
2
2
2
2
2
2 ),,( (4)
)1(12 2
3
i
iii
hED ; iii hm , where i =1,2.
(5)
The above differential equations (3) and (4) are spatially Fourier transformed; cmfre pppppwS ~~~~~~
1111 , (6)
tmfc ppppwS ~~~~~2222 ,
(7)
where, 2
1222
11 )( mDS and 22
22212 )( mDS are the transformed operators
of plate 1 and plate 2 respectively. With the linearity assumption all the pressures can be considered to be related to displacement by linear operators. Therefore, after transformation pressure operators become algebraic expressions times the transformed displacements like the plate operators. Having these pressure operators evaluated, equations (6) and (7) can be used to solve for 1
~w and 2~w , from which radiated sound power and sound pressure level are
calculated according to section 2.12.
2.5 Excitation pressure
Here, point excitations from an ISO tapping machine is considered where five individual hammers of mass M = 0.5 kg are dropped freely from a height h = 4cm. Each hammers strikes the floor twice a second and only one hammer at a time makes the impact at a single point (x0,y0). Thus the overall hammering frequency fr = 10 per second and time period Tr = 0.1 second. The impact for each hammering f1(t) is considered to be the same and repeats after every Tr, thus the time history of the repeated force is,
n
Tntn
nrR
rFnTtftf /2i1 e)()( ,
(8)
6
where, Fn is the amplitude of infinite discrete Fourier series components of fR. The force spectrum in frequency domain can be found by making a Fourier transform of equation (8).
nrnR nffFfF )()( .
(9)
Each component Fn can be calculated as,
n
Tt
rn ttf
TF r de)(1 /2i
1 . (10)
The term inside the integral in equation (10) is the Fourier transform of single impact f1 i.e. F1. The single impact f1 and thus F1 depends on the floor characteristics. Ref. [4] is modelling the hammer impact using mass spring damper system placed in series, where mass represents mass of the hammer M, and spring stiffness K and damper resistance R comes from the floor properties. The values of K and R are in the general case frequency dependent. Brunskog and Hammer [5] describes methods to calculate F1 considering constant K and R as well as a method considering the frequency dependence by introducing frequency dependant mobility calculated from the force and velocity measurement at the excitation point. In the present paper, however, the one using constant K = 4.7. 107 and R = 1.68.103 are used. Having the value of FR, the excitation pressure can be derived as5:
),(),(~00 yyxxFyxp Re , (11)
and corresponding spatial Fourier transform is )i( 00e),(~ yx
Re Fp . (12)
2.6 Fluid interaction Three air regions can be defined: 1) above plate 1 (z <0); 2) between the plates (0 < z < d) and 3) below plate 2. The regions 1) and 3) are semi infinite while standing waves are formed inside the cavity, region 2. According to Helmholtz equation,
0,,2,0
2
,, trcc
trc pc
pzyx
, (13)
where p stands for air pressure and the indexes c, r, t correspond to cavity, radiated and transmitted air pressure, respectively. To solve for this boundary conditions are taken from two plate-air boundaries. At the plate-air intersections, the vibration velocity of the plate surface is equal to the particle velocity in the air:
1 plate ; 1,02
0
wzp
cz
r (14)
2 plate ; 2,02 w
zp
cdz
t (15)
Velocity of sound and density of air in region 1) and 3) is taken as c0 and 0, corresponding wave number as k= /c0. Velocity of air inside the cavity cc varies due to air density change, a loss factor c is introduced within the cavity so that density, c= 0(1+i c) and cc cc i10 . The wave number in side the cavity kc= /cc. The pressure terms are then transformed into wave number domain and the transformed fluid pressure is as follows, (Brunskog and Hammer [4]).
7
),(),(
)0,,( 1222
102
Rwk
wpr , (16)
where 2220
2 kR and the branch of is taken so that, 222 k and 0 and 0 if 0 , and thus condition for outgoing wave is met. Similarly,
),(),(),,( 222220
2
Twk
wdpt , (17)
where, 22210
2 ),( kwT and
),(),(
)cot()csc()csc()cot(
),,()0,,(
2
12
ww
dkdkdkdk
kdpp
dd
dd
d
c
c
c , (18)
where, dcd kk i222 is taken so that 00,0 ddd if . The matrix notation in equation (18) shows the cavity pressures acting on both plates, pc( , ,0) and pc( , ,d) for plate 1 and plate 2, respectively.
2.7 Force reactions Each beam exerts a reaction force and moment, thus the reaction pressures acting on the plates due to reaction force are the total contribution from attached beams’ reaction forces
nnf nlxyFp )()( 1,11 ,
(19)
nnf nlxyFp )()( 2,22 ,
(20)
where F1,n and F2,n represent the reaction force of the nth beam of set 1 and set 2, respectively. As the plate-beam connections are assumed to be rigid, displacements at each beam plate connection are considered to be the same. Therefore boundary conditions at each junction with respect of displacement are,
)(),( ,111 yuynlw n , (21)
)(),( ,222 yuynlw n , (22)where u1,n and u2,n represent displacement of nth beam of set 1 and set 2 respectively. The beam parameters of set 1 are height hb1, width b1, Young’s modulus Ef1, moment of inertia 12/3
111 bf hbI , density f1 area Af1= b1hb1 and spacing between adjacent beams l1. The reaction forces act along the lines x = nl1, where n is an integer and represents the beam position. Equation for the nth beam is:
),()(
)()(
)(
112
1141,
4
11
1,2
1141,
4
11,1
ynlwAdy
yudIE
yuAdy
yudIEyF
ffn
ff
nffn
ffn
.
(23)
Here, u1,n is replaced by w1(nl1,y) according to equation (21). Transforming equations (19, 21, 23) into wave number domain and applying Poisson’s sum formula, following expression can be achieved [2-4]:
(24)
8
nn
nl lnwl
nlw ),/2(~1e),(~11
1
i11
1 .
And the transformed reaction pressure acting on plate 1 becomes,
nn
nlnf lnw
lZFp ),/2(~e)(~~
111
1i,11
1 , (25)
where 211
4111 ffff AIEZ is the transformed beam operator of plate 1.
Following the same procedure, the reaction pressure between plate 2 and beam set 2, pf2, can be derived as:
nn
nlnf lnw
lZFp ),/2(~e)(~
222
2i,22
2 , (26)
where the transformed beam operator, 222
4222 ffff AIEZ . Moreover, Ef2,
12/3222 bf hbI , Af2= b2hb2, b2, hb2 are Young’s modulus, moment of inertia and cross sectional
area, thickness and height of the beams attached to plate 2, respectively.
2.8 Moment reaction The pressures from moment forces are derived in a similar way as the pressure due to the reaction forces. The total pressure from the beam due to moment reaction can be written as
nn nlxyMmp )()( 1,11 ,
(27)
where M1,n is the moment corresponding to the nth beam of set 1 acting on plate 1 and ´= / x is the derivative of the Dirac delta function. Similar to the linear displacement there
is also continuity in rotational displacement between plate and beam at each connection,
xynlw
yy nnn
),()()( 1,1
,1,1 , (28)
where 1,n and 1,n are the angular displacement of beam and plate 1, respectively, at x = nl1. The moment M1,n in terms of angular displacement of plates can be written according to Sjökvist [10] as:
nn yT
xynlw
yTM ,1
212
2
1112
12
2
1,1),(
, (29)
where 112
12
111 / bf hbAGCT is the torsional stiffness of the beam and 2
2
2
12
111 12 b
bfb
hb
bhA
is
the rotational mass moment of inertia of the beam per unit length [6], where )1(2 111 bfEG is the shear modulus, b1 is the Poisson’s ratio, b1 is the density of the
beams attached to plate 1 and C1 depends on the ratio b1/hb1 according to Cremer et al. [6] as described in section 2.13. Transformation of (26) into the wave number domain gives
n
nln
n
nln HMp
m
11
1
i,11
i,1 e)(~)(ie)(~i~ ,
(30)
9
where H1( ) =(T12+ 1
2) is the transformed torsional wave operator for the beams attached to plate 1. Following the same procedure as mentioned before while deriving pf1, i.e., transforming equations (27, 29 and 30) and applying Poisson’s sum formula, it can be shown that
nn
nln lnwln
lnl ),/2(~)/2(ie),(~
1111
i1,1
1 . (31)
From equation (29) and (30) the moment reaction becomes
nn
nln lnwln
lHMp
m),/2(~)/2()(e)(~i~
1111
1i,1
1
1.
(32)
Following the same procedure as for pm1, pm2 can be derived
nn
nln lnwln
lHMp
m),/2(~)/2()(e)(~i~
2222
2i,2
2
2.
(33)
where H2( ) =(T22+ 2
2), and T2 and 2 are the tortional stiffness and rotational mass moment of inertia respectively, calculated as T1 and 1 but with the corresponding parameters of beams attached to plate 2.
2.9 Solution From equations (6), (7), (16-18), (25-26), (32-33), after suppressing the and dependence, we get
n n
nln
nlne wJwJMFwRpwS 212111
i,1
i,1111
~~e~)i(e~~~ 11 , (34)
n
nln
n
nln MFwTwJwJwS 22 i
,2i
,2222212122 e~)i(e~~~~~ . (35)
Combining equations (34) and (35) to be represent in matrix form
n
nln
n
nln
n
nln
n
nln
e
M
M
F
FP
ww
TJSJJJRS
2
1
2
1
i,2
i,1
i,2
i,1
2
1
22221
12111
e~
e~
ie~
e~
0
~
~~
n
nln
n
nln
n
nln
n
nln
e
M
M
F
FP
SSSS
Sww
2
1
2
1
i,2
i,1
i,2
i,1
1121
1222
2
1
e~
e~
ie~
e~
0
~.
)det(1
~~
,
(36)
where
22221
12111
2221
1211
TJSJJJRS
SSSS
S
and
)cot()csc()csc()cot(
2221
1211
dkdkdkdk
JJJJ
dd
ddJ .
Calculation of the terms with infinite sums are give in the appendix. Having these infinite sums calculated 1
~w and 2~w can now be solved according to equation (36), which has 3 terms
in the right hand side. The first term is the solution for structure having no beams attached to the plates, i.e., two plates only. The 2nd and 3rd terms comes into the scene when beams are attached. The 2nd term is the correction for reaction forces from the attached beams and the
10
third term corresponds to the moment reactions. This equation can be used to solve for any type of decoupled structure by limiting the values of infinite sums. For example selecting the
infinite terms related to upper plate n
nlnF 1i
,1 e~ and n
nlnM 1i
,1 e~ i.e. Z1 and H1 equal to zero,
effect of beams can be ignored. System then corresponds to decoupled structure where beams are attached to lower plate only. Similar way beams attached to lower plate can be ignored by selecting Z2 and H2 equal to zero. And by selecting Z1, H1, Z2 and H2 equal to zero structure with 2 plates only can be simulated. Similarly consideration of moments or forces can also be controlled.
2.10 Radiated sound power Radiated sound power form an infinite structure can be calculated according to Cremer et al. [6] which were used in previous papers [4]:
dd),(~
81
222
22
2
222 k
wckRad
(37)
Making the substitution as )sin(rk and )cos(rk , so that ddkkdd rr .. and changing the limits accordingly, equation (37) becomes:
dd))cos(),sin((~
8
2
0 022
22
2
2 rr
k
r
rrRad kk
kkkkwck
(38)
The integration is made numerically in Matlab as described in section 2.12.
2.11 Normalized impact sound level Normalized sound pressure level, Ln, is defined according to [19] as
0
1log10AALL pn ,
(39)
where Lp is pressure level in the receiving room, A1 is the room absorption area and A0=10 m2 is the reference area. Power dissipated from the receiving room can be expressed as
00
02
4 cApdis .
(40)
Making an power balance between radiated and dissipated sound power the normalized sound pressure level can be derived [5,17]
0
002
4log10A
cp
Lref
Radn ,
(41)
where pref = 2.10-5 Pa is the reference pressure. These sound pressure levels are then summed over one-third octave frequency band.
2.12 Numerical calculations The whole calculation process was done in Matlab. The calculation can be divided into 3 sections.
11
1. Defining appropriate values for the parameters and selecting range and resolution for frequency f, wave number parameters, k and . The frequency range is the full 1/3 octave bands from 10 Hz to 5000 Hz, with a resolution equal to that of the tapping machine, i.e,
10rff Hz. For the wave number parameters the range was as , and 1,0/ kkr , where a very small number =1.10-8 is introduced in order to avoid singularity. Their corresponding resolution were = 2 and kr/k=1/100.
2. Based on the values of parameter selected the code defines appropriate coupling condition, i.e. whether one or both plates have beams, etc. The force spectrum of the tapping machine is also calculated. After that it goes for calculation and runs over 3 for-loop, starting with the inner most one over kr/k , then one over and finally over the outer most one – frequency f. During each loop operators for each pressure term are calculated as well as the infinite sum terms. Depending on the frequency, value of impact force is selected from the spectrum. Plate displacements are then calculated and 2
~w is integrated as equation (38) to calculate for the radiated power at each frequency. Trapezoidal rule is used to integrate over both kr/k and loop.
3. Normalized sound pressure levels calculated from radiated sound pressures are then summed over third octave band. Where band frequencies are chosen in an ascending order as:{10, 20, 31.5, 40, 50, 63, 80, 100, 125, 160, 200, 250, 315, 400, 500, 630, 800, 1000, 1250, 1600, 2000, 2500, 3150, 4000 and 5000} Hz.
2.13 Material properties Material properties for the real floor structure described in chapter 4 serve as input to the model. Plate 1, (22 mm particle board); weight 151m kg/m2, Poison’s ratio 1 = 0.3. Material damping is considered by a loss factor of = 0.07 and thus Young’s modulus E1 = 3(1+i ) GPa. Properties for plate 2 (13 mm gypsum board); Young’s modulus E2 = 2.2 (1+i ) GPa, weight 92m kg/m2, Poison’s ratio 2 = 0.08. For the pinewood beams, f1 = 500 kg/m3 and loss factor, b =0.03, Ef1 = 9.8 (1+i b). Similar beams are attached to both upper and lower plate. For upper plate; height, hb1 = 22.5 cm, b1 = 42 mm, l1 =60 cm are set. While for the lower plate; hb2 = 9.5 cm, b2 = 45 mm, l2 =30 cm are set. The properties of air are set as, c0 = 340 m/sec, 0 = 1.29 kg/m3, c = 1.10-3. 11
21111 / hf bbAGTC and 22
22222 / hf bbAGTC
were calculated by interpolating the data given by equation (II-64) in Cremer et al.6(page-93) for bh1/b1 and bh2/b2 respectively.
2.14 Experimental setup For verification purpose, measurements have been carried out with a decoupled construction having separate floor and ceiling, both simply supported.
12
2.14.1 Floor construction
Figure 3: Floor structure used for experiment.
The floor structure is shown in figure 3. The floor part consists of one layer 22 mm particle board, glued and screwed to 6 pinewood glulam beams, 42x225 mm. The beam spacing is 600mm except for the two spaces closest to the borders where 390 mm are used for symmetry reason. The floor rests on a separate isolated concrete frame to avoid flanking transmission The floor plate measures 3180x3780 mm. The ceiling part is made of 13 mm gypsum board in conjunction with framed pinewood joists, 45x95 mm, screwed together. Each beam is placed parallel to y axis. The joist spacing is 300mm except for the two spaces closest to borders where 290 mm are used for symmetry reason. The ceiling is hanged into place by two additional joists positioned vertically at two sides. The ceiling plate measures 2980 x 3580 mm.
2.14.2 Measurement conditions The impact sound level measurement was carried out in the acoustic laboratory of Luleå University of Technology, where the room underneath the floor structure is 63 m3 with an overall reverberation time of about two seconds and fulfils the minimum requirement for impact sound measurement according to ISO 140 [19]. The floor was placed over a stiff concrete frame keeping soft absorbing material in between the frames and the floor structure, thus very efficient in obstructing any flanking transmission. Four microphone positions were used for the measurements where the average is taken for further comparisons. The ISO tapping machine was used for excitation and it was placed at the centre of the floor which corresponds to an exact bay between two adjacent beams.
3 Results and discussion
3.1 Force excitation Figure 4 shows the modelled response from 11 different excitation positions together with the average, concerning decoupled condition including moment effect. One excitation position is exactly at the beam location and a second one is exactly in the centre of the bay between two adjacent beams (x0=0.3m). Between these two points, nine complementary excitations points were randomly selected (x0=[0.0554m, 0.07m, 0.113m, 0.15m, 0.197m, 0.21m, 0.23m, 0.25m, 0.275m]). It is not possible to represent the impacts from an ISO tapping machine by a single point force since each hammer affects an area, 30 mm, rather than a point, and the five hammers are separated a few centimetres. Also, any two of the five hammers, never strike the floor simultaneously but the hammering frequency corresponds to the case where a single hammer is repeatedly exciting five distinct areas of the floor within a certain time interval. Thus, an energy average response from several excitation positions is a way to
13
represent the actual input from the tapping machine which originally was suggested by Brunskog and Hammer [4], although they used 15 excitation points. Since the structure in the model is infinite in y-direction, no variation along this axis is considered.
Figure 4: Response from 11 excitation positions and the corresponding average of predicted
results. It is noticed that the lowest sound pressure level is obtained when excitation is given just above the beam location. As soon as the excitation is slightly moved the pressure increases significantly and simultaneously, the frequency dependent curve shows a different pattern. The explanation is that when excitation is given directly above the beam, most of the energy is consumed in getting the proportionally stiff and heavy beam in to motion, resulting in low vibration amplitude. Reaction forces and moment forces exist from each beam towards the plate at each plate-beam junction causing reduction of the inserted net energy. A great amount of energy is then converted into non-radiating in plane waves generated at the first beam-plate junction. As a result, the overall bending wave vibration of the plate is reduced and so is the pressure level. This is evident for mid and high frequencies. When the excitation on the other hand is moved slightly away from the beam, the same energy is given to the surface of the thinner and softer plate, resulting in higher vibration level over the entire bay, thus sound is radiated to a higher extent.
3.2 Validation Figure 5 shows the experimental result together with the predictions with and without the moment consideration represented by averaging 11 excitation points.
14
Figure 5: Comparison of modelled impact sound level with and without moment
consideration compared to measurement on a finite floor. ( ) represents the measurement, (-·-) the proposed decoupled model when moment is not considered and (·····) when moment
is considered.
The average curves agree reasonably well with the measured one from 80 to 200 Hz while the deviation is generally larger from 200 up to 1000 Hz. In this mid frequency region the model performs better when moment is not included. Above 1000 Hz the difference between model and measurement is somewhat larger when moment is neglected and above 1250 Hz the moment inclusion is performing significantly better. This indicates that moment can be a key factor along with other factors such as choice of excitation model as concluded in paper [4] for accurate modelling of high frequencies but not at mid frequencies. For the frequencies below 80 Hz one possible reason for mismatch can be the non diffusive sound field. The Schröder cut-on frequency, above which the diffuse field assumption is reasonable, of the receiving room is 356 Hz according to [15]:
VTfcuton
602000 , (42)
15
where, T60 is the reverberation time (about 2 Sec) and V is the volume of the receiving room (63 m3). In terms of the single number grading Ln,w+CI,50-2500 (according to [20]) the experimental result was 75.2 dB while the predicted was 71.2 and 74.4 dB respectively for the cases with and without moment consideration.
Figure 6: Average graph of figure 5 (decoupled and moment considered) spread over the entire frequency range with a 10 Hz resolution. The narrow band sound pressure level (SPL), instead of 1/3 octave bands, is plotted in figure 6. Many peaks and dips can be noticed. Similar peaks and dips can be seen in previous papers [9,16], which were alluded to be originating from the structure’s periodicity. In a periodic structure beams are attached periodically to the plate and offer an additional stiffness which give obstruction to the vibration propagation through the plate. For the vibration where two adjacent bays vibrate symmetrically, beam stiffness imposes strong constraints thus the vibration being obstructed by the beams, is reflected back and forth within the bay and eventually dies out. While on the other hand, for the vibration where two adjacent bays vibrate anti-symmetrically, beams being located at the nodal lines do not add in stiffness. They rather act as added masses which being stationary do not offer any inertia. Therefore vibration wave can propagate very easily without being obstructed, and are then able to radiate sound. Whether vibration mode is symmetric or anti-symmetric depends on the wavelength of a particular wave. Nevertheless, the band of frequency where wave can propagate with out any interference or little interference from beams are known as pass bands and for the band where waves are obstructed by the beams are known as stop band. It is well known that such pass and stop bands will always occur in a periodic structure. Pass band and stop band comes alternatively, which are seen as peaks and dips in figure 6. The peaks correspond to the pass bands and the dips to the stop bands. A similar discussion can be found in paper [16]. Figure 7 shows the low frequency characteristics of the model. Sound pressure level is calculated for the frequency range from 10 Hz to 100 Hz with a resolution of 1 Hz at four different cavity depths 0.345m, 0.69m, 1.035m and 1.48m. The same resolution was considered for excitation force. This increased frequency resolution as compared to the 10 Hz corresponding to the tapping machine was used in order to give a better insight to the lower frequencies. Each graph shows how the first peak decreases in frequency with the increase of the cavity depth, indicating a strong correlation between the cavity depth and the resonance frequency. Moreover, the mass-air-mass resonance frequency for double wall structure can be calculated as following6:
16
21
21
2 mmmm
dc
f ccc .
(43)
The frequencies corresponding to the four cavity depths above are approximately 44 Hz, 31 Hz, 25Hz, and 22 Hz respectively, while the peaks observed in figure 7 for same cavity depths are seen at 37 Hz, 27 Hz, 22 Hz and 19 Hz respectively. This indicates that the peak in the model assumes the mass air mass resonance frequency of the structure even though there is some deviation in the predicted frequency with that found from the simplified double wall mass-air-mass resonance frequency. Lin and Garrelick [3] also concluded similarly by showing that for a coupled structure the first dip in transmission loss corresponds to a structural resonance. As soon as the structure is decoupled another peak arises near the cavity resonance frequency but at a lower frequency than the one found for the double wall structure. The reason for deviation in frequency is found in the effect of the attached beams. Therefore equation (43) must be modified in order to have accurate prediction of this frequency for periodic structure. Bradley and Brita [18] proposed an empirical method of calculating the mass air mass resonance frequency considering the effect of attached beams as an added stiffness. Although they restricted their study to a coupled structure via resilient joints, the effect of beams to the resonance frequency is quite evident from their study. Since the frequency range for the experiment is kept over 50 Hz, the mass-air-mass resonance can not be experimentally verified. Nevertheless a tendency of approaching the peak is observed.
Figure 7: SPL at four different cavity depth. ( ) represents 0.345m, (-·-) 0.69m, (····)
1.035m and (---) 1.48m of cavity depth. Figure 7 also more clearly shows the appeared peak in figure 6 at the 63 Hz band which is not at all visible in figure 7. At a 10 Hz resolution the band 63.5 Hz has 3 frequencies while the 50 Hz and 80 Hz bands have one frequency each within their respective bands. Therefore, since the SPL is almost flat within the range from 50 Hz to 100 Hz, a peak appear in the 63.5 Hz band only because of summing up 2 more frequency components than the 50 Hz and 80 Hz bands.
3.3 Comparison with the previous study The accuracy of the results obtained from the modified decoupled model in figure 6 is compared with the accuracy from Brunskog and Hammer’s [4] study who dealt with a coupled structure. The difference between the experimental and predicted values for the model presented here, both with and with out moment consideration, and the difference in the
17
experimental and predicted values taken from previous paper [4] for the case where no absorbing material is placed inside the cavity, is shown in figure 8. At frequencies up to 100 Hz the coupled model [4] greatly underestimates the sound pressure, while the present study dealing with decoupled condition shows better accuracy, although the model to some extent overestimates the level at mid frequencies. The maximum deviation within the frequency range 100 to 1000 Hz was about 7 -10 dB in all cases.
Figure 8: Difference between experimental (finite floor) and predicted results (infinite floor). Coupled model in [4] ( ) and decoupled model of this paper- with and without moment (····)
and (-·-)respectively. Increasing deviation is noticed for higher frequencies in both Brunskog and Hammers model and the model of this paper when moment is excluded. Concerning Brunskog and Hammer [4] it starts from 800 Hz while for the new model it appears around 1600 Hz. By including moment in the model of this paper, the prediction error for higher frequencies is decreased significantly.
4 Conclusions The analytical method for predicting the impact sound pressure level of an infinite structure was here further developed to be applicable for decoupled floors, which is an extension from the previously treated coupled plate-beam-plate structures. The predicted Ln,w deviated 2-3 dB compared to measurements on a finite floor which is within the same order as reported for coupled floors. One additional modification in terms of adding the effect of beam-plate moment was introduced. The moment significantly improved the results at high frequencies and it is suggested that this moment is of vital importance for models of the type used here. On the other hand, the accuracy dropped in the mid frequency region. Furthermore, a couple of physical phenomena were observed. A periodic effect of the structure was noticed in terms of peaks in the frequency spectra indicating local effects originating from the periodicity of the structure. The model showed also to be highly sensitive to excitation positions for frequencies above 100 Hz, while below 100 Hz the mass air mass resonance is the dominating phenomenon.
A. Appendix Changing the variable in equation (36) as = -2n /l1 and then summing for all n we get
18
n
nln
n
nln
n
nln
n
nln
ne
n
n
n
M
Mln
F
FlnP
lnSlnSlnSlnS
lnlnw
lnw
2
1
2
1
i,2
i,1
1i
,2
i,1
1
111121
112122
112
11
e~
e~
))/2i((e~
e~
0
)/2(~
)/2()/2()/2()/2(
))/2(det(1
))/2((~
)/2(~
S
(A1)
The periodicity of the infinite sums n
nlnF 1i
,1 e~ and n
nlnM 1i
,1 e~ were here made use of.
Again multiplying the factor ( -2n /l1) each time the variable is changed to = -2n /l1 and then summing for all n we get
n
nln
n
nln
n
nln
n
nln
ne
n
n
n
M
Mln
F
Flnp
lnSlnSlnSlnS
lnln
lnwln
lnwln
2
1
2
1
i,2
i,1
1i
,2
i,1
1
111121
112122
1
1
121
111
e~
e~
))/2i((e~
e~
0
)/2(~
)/2()/2()/2()/2(
))/2(det()/2(
))/2((~)/2(
)/2(~)/2(
S
(A2)
19
For simplification let use the following auxiliary functions:
n lnSlnSlnSlnS
lnXXXX
)/2()/2()/2()/2(
))/2(det(1
111121
112122
12221
1211
SX
n lnSlnSlnSlnS
lnln
AAAA
)/2()/2()/2()/2(
))/2(det()/2(
111121
112122
1
1
2221
1211
SA
n lnSlnSlnSlnS
lnln
UUUU
)/2()/2()/2()/2(
))/2(det()/2(
111121
112122
1
21
2221
1211
SU
)/2(~)/2(
)/2())/2(det(
10
)/2(~1
121
122
1
1
2
1 lnplnS
lnSln
lnpPP
en
e
SXP
)/2(~)/2(
)/2())/2(det(
)/2()/2( 1121
122
1
11
2
1 lnplnS
lnSln
lnlnRR
en S
PR
Now the first terms of the matrices in the left hand sides of equations (A1) and (A2) can be
converted to infinite terms n
nlnF 1i
,1 e~ and )e~i( 1i,1
n
nlnM respectively according to equations
(25) and (32) respectively
)e~(i)e~i(
)e~(e~e~
21
211
i,212
1
1i,111
1
1
i,212
1
1i,111
1
11
1
1i,1
n
nln
n
nln
n
nln
n
nln
n
nln
MAlZMA
lZ
FXlZFX
lZP
lZF
(A3)
)e~(i)e~i(
)e~(e~e~
21
211
i,212
1
1i,111
1
1
i,212
1
1i,111
1
11
1
1i,1
n
nln
n
nln
n
nln
n
nln
n
nln
MUlHMU
lH
FAlHFA
lHR
lHMi
(A4)
It is interesting to notice that the 2nd terms are unaffected and can not be converted as 1st terms because they are related to plate 2 and has different periodicity depending on l2. therefore to convert them like 1st term the whole process of variable changing and summing has to be repeated by changing the variable in equation (36) as = -2n /l2. Having done so and making the following auxiliary functions
n lnSlnSlnSlnS
lnYYYY
)/2()/2()/2()/2(
))/2(det(1
211221
212222
22221
1211
SY
n lnSlnSlnSlnS
lnln
BBBB
)/2()/2()/2()/2(
))/2(det()/2(
211221
212222
2
2
2221
1211
SB
n lnSlnSlnSlnS
lnln
VVVV
)/2()/2()/2()/2(
))/2(det()/2(
211221
212222
2
22
2221
1211
SV
)/2(~)/2(
)/2())/2(det(
10
)/2(~2
221
222
2
2
2
1 lnplnS
lnSln
lnpQQ
en
e
SYQ
)/2(~)/2(
)/2())/2(det(
)/2()/2( 2221
222
2
22
2
1 lnplnS
lnSln
lnlnOO
en S
QO
20
two more equations having the infinite terms can be obtained
)e~(i)e~i(
)e~(e~)2(e~
21
212
i,222
2
2i,121
2
2
i,222
2
2i,121
2
2
2
2i,2
n
nln
n
nln
n
nln
n
nln
n
nln
MBlZMB
lZ
FYlZFY
lZQ
lZF
(A5)
and
)e~(i)e~i(
)e~(e~e~i
21
212
i,222
2
2i,121
2
2
i,222
2
2i,121
2
22
2
2i,2
n
nln
n
nln
n
nln
n
nln
n
nln
MVlHMV
lH
FBlHFB
lHO
lHM
(A6)
Combining equation (A3-A6) in matrix form
22
2
11
1
22
2
11
1
i,2
i,1
i,2
i,1
222
221
2
222
2
221
2
2
121
111
1
112
1
111
1
1
222
221
2
222
2
221
2
2
111
111
1
112
1
111
1
1
2
1
2
1
e~i
e~i
e~
e~
1
1
1
1
OlH
RlH
QlZ
PlZ
M
M
F
F
VlHV
lHB
lHB
lH
UlHU
lHA
lHA
lH
BlZB
lZY
lZY
lZ
AlZA
lZX
lZX
lZ
n
nln
n
nln
n
nln
n
nln
(A7)
Solution of this infinite terms then becomes K= C-1L,where,
222
221
2
222
2
221
2
2
121
111
1
112
1
111
1
1
222
221
2
222
2
221
2
2
111
111
1
112
1
111
1
1
1
1
1
1
VlHV
lHB
lHB
lH
UlHU
lHA
lHA
lH
BlZB
lZY
lZY
lZ
AlZA
lZX
lZX
lZ
C ,
22
2
11
1
22
2
11
1
OlH
RlH
QlZ
PlZ
L and
n
nln
n
nln
n
nln
n
nln
M
M
F
F
2
1
2
1
i,2
i,1
i,2
i,1
e~i
e~i
e~
e~
K .
References [1] V. N. Evseev: Sound radiation from an infinite plate with periodic inhomogeneities. Soviet Physics –Acoustics 19(1973) 226-229 [2] B. R. Mace: Sound radiation from a plate reinforced by two sets of parallel stiffeners. Journal of Sound and Vibration 71(1980) 435-441
21
[3] G. F. Lin and J. M. Garrelick: Sound transmission through periodically framed parallel plates. Journal of Acoustics. Society of America 61(1977) 1014–1018. [4] J. Brunskog and P. Hammer: Prediction model for the impact sound level of lightweight floors. Acta Acustica United with Acustica 89(2003) 309–322.
[5] J. Brunskog and P. Hammer: The interaction between the ISO tapping machine and lightweight floors. Acta Acustica United with Acustica 89( 2003) 296-308.
[6] L. Cremer, M. Heckl, and E. E. Ungar: Structure-Borne Sound. 1988 ISBN 0-387-18241-1 [7] J. Brunskog: The influence of finite cavities on the sound insulation of double-plate structures. Journal of Acoustic Society of America 117(2005) 3727–3739. [8] M. L. Rumerman: Vibration and wave propagation in ribbed plates. Journal of Acoustical Society of America. 57(1975) 370–373,. [9] D. Takahashi: Sound radiated from periodically connected double-plate structures. Journal of Sound and Vibration 90(1983) 541–557.
[10] L. G. Sjökvist: Structural sound transmission and attenuation in lightweight structures. Doctoral thesis, paper 3 ,Chalmers University of Technology, Göteborg, Sweden, 2008. [11] G. Sen Gupta: Natural flexural waves and the normal modes of periodically supported beams and plates. Journal of Sound and Vibration 13 (1971) 89-101. [12] D. J. Mead: Wave propagation and natural modes in periodic systems: I. mono coupled systems. Journal of Sound and Vibration 40(1975) 1-18,. [13] D. J. Mead and K. K. Pujara: Space harmonic analysis of periodically supported beams: response to convected random loading. Journal of Sound and Vibration 14(1971), 525-541. [14] F. Ljunggren: Using elastic layers to improve sound insulation in volume based multi-storey lightweight buildings. Proceedings of InterNoise, Ottawa, Canada 2009. [15] H. Kuttruff: Room acoustics. Elsevier applied science, London, 1973, Third edition 1991. [16] J. Wang, T. J. LU, J. Woodhouse, R.S. Langley, J. Evans: Sound transmission through lightweight double-leaf partitions: Theoretical modelling. Journal of Sound and Vibration 286(2005), 817-847. [17] I. L. Ver: Impact noise isolation of composite floors. Journal of the Acoustical Society of America 50(1971) 1043–1050. [18] J. S. Bradley, J. A. Brita: A simple model of sound insulation of gypsum board on resilient supports. Noise control Engineering 49(2001) 216-223. [19] IS0 140-6: – Acoustics – Measurement of sound insulation in buildings and building element – part 6: Laboratory measurements of impact sound insulation of floors.
22
[20] ISO 717-2: – Acoustics – Rating of sound insulation in buildings and of building elements –part 2: Impact sound insulation.
Paper
1
Parametric study of lightweight floors using a theoretical floor model.
Mohammad Sazzad Mosharrof1, Fredrik Ljunggren2, Anders Ågren3. Div. of Sound and Vibration Luleå University of Technology
Abstract Lightweight floor structures are constructed by attaching many components, each affecting the overall vibration of the structure and the sound radiation in a different way. For efficient floor design the understanding of the relative effect of each element is important. And with this objective a parametric study is made here in this paper by using the theoretical model by the same author which models a decoupled floor structure, i.e. mechanical coupling of floor and ceiling via beams are broken. The parameters considered are mass of both plates, heights of the beams, spacing between beams and the cavity depth. Prior to parametric analysis the model’s reliability is further examined here. Some results are compared with the experiments and the conventional ideas to justify the conclusions.
1. Introduction A complete light weight timber floor structure is complex and consists of many elements, where each of them influences the overall sound insulation of the structure. Both the internal properties of each element and how they are connected contributes to the overall vibration of the structure. For an efficient design of a floor, knowledge of the relative influence of each element is necessary. From theoretical point of view to have model for the floor structure is the prerequisite for this, and there has been many work on this. Theoretical model of the impact sound caused by an impact hammer are developed by many others [1-5] most of these considered coupled floor structure, i.e. two plate rigidly attached via beams. In [6, 7] a model for decoupled structure, i.e. rigid connection between plates are broken and two separate sets are used for both plates. In this paper the functionality of the model is tested through modelling of some realistic parameter changes. The changes are compared to previous knowledge and experiments. The objective is to investigate the reliability of the model and at the same time to study the relative importance of the lightweight floor parameters. The modelled floor is constructed according to figure 1 below, where the floor and ceiling are mechanically separated. Both the upper floor particle board and the lower ceiling plaster board are stiffened by separate sets of timber beams. A difference with the experiments though is that the model is infinite while the experimental results are made in lab on a 10 m2 floor where the flanking beams are lying on isolated strips on a concrete frame.
2
Figure 1: Floor structure corresponding to the model, used for experiment.
The study is done by predicting the sound pressure level (SPL) in the receiving room under the structure using the theoretical model for this type of structure as in [7]. The model is considered to be reliable for this parametric study and reliability is further examined in this paper. A model is only an approximation of a physical structure and differences are therefore expected. One reason is that the complexity of the real situation is overlooked and parameters are simplified. In this case one big difference is as mentioned that the model is based on an infinite floor while measurements are made on finite floors. Despite these absolute differences the relative difference between the results due to a small parametric changes in the model are assumed to be realistic and similar to the relative differences for experimental data due to the same parametric changes, i.e. the trends in the model and in experiments should be similar if the model is reliable. Each element of lightweight structure influences the pressure level, therefore by parametric study it is possible to find out the relative importance of each element. Similar to this paper there exists some others concerning parametric study, Lin and Garrelik in [8] showed the relative influence of the number of frames attached to plates on sound pressure level and Brunskog [9] in his doctoral thesis made parametric study in a coupled structure,
1.1 Material properties of the components in the floor model The components that have been used to construct the floor structure are: two plates, the upper plate that belongs to the floor part and a lower plate that belongs to the ceiling part. Two sets of beams, beams attached to the upper plate and to the lower plate. Thicknesses of these plates and the height of the beams as well as their internal properties (Young’s modulus, density etc) are varied in order to understand their relative influences as mentioned earlier. Assuming a set of values for each parameter as default values each parameter is then varied by multiplication factors {0.5, 1, 1.5, and 2}. Only one parameter is varied at a time and all others take the default values. Floor plate density, i.e. mass 681.82 kg/m3 Floor plate thickness, hp1 22 mm Floor plate- Young’s modulus, Ep1 3.23 GPa Floor plate- Poisson’s ratio 0.3 Floor plate- loss factor 0.07 Ceiling plate density, i.e. mass 692.31 kg/m3 Ceiling plate thickness 13 mm Ceiling plate- Young’s modulus, Ep2 2.2 GPa Ceiling plate- Poisson’s ratio 0.08 Ceiling plate- loss factor 0.07 Thickness of beams attached to floor plate , 225 mm Young’s modulus of beams attached to floor plate , 9.8 GPa. Poisson’s ratio of beams attached to floor plate , 0.3 loss factor of beams attached to floor plate , 0.03
3580
390 990 3390 3780
290 590 3290
3
Thickness of beams attached to ceiling plate , 95 mm Young’s modulus of beams attached to ceiling plate , 9.8 GPa. Poisson’s ratio of beams attached to ceiling plate , 0.3 loss factor of beams attached to ceiling plate , 0.03 Cavity depth, 345 mm Height of the beams attached to upper plate, 225 mm Thickness of the beams attached to upper plate 42 mm Height of the beams attached to lower plate, 95 mm Thickness of the beams attached to lower plate 45 mm
Table 1: The floor parameters and their default values.
2 Checking for the reliability of the model
2.1 Vibration propagation across the beams: Any periodic structure involves some specific periodic properties and reliability of the model is examined by checking for the periodic effects. For plates with out any beams, vibrations of all wave length would propagate very easily, as soon as beams are attached to the plate, waves experience some sort of obstruction and starts to reflect back and forth and also gives rise to structural resonance. The amount of obstruction and reflection depend on the wave length of the propagating wave. Each bay can be compared as plates supported on two beams and resonance frequency for this type is according to Cremer [10]:
,.21 2k
mDfn (1)
where, fn is resonance frequency, )1(12 2
3EhD is the bending stiffness , E is the Young’s
modulus, is the Poisson’s ratio, hp is the plate thickness, m’’ is the mass per unit length of the plate respectively. Therefore, with the change in plate thickness, m’’ and D changes and so does the wave length. This resonance takes place when the wave length exactly fits the bay length, and nodal lines match the beam locations as shown in figure 2. As a result vibration can propagate through the entire structure. wave length expression can be written by rearranging equation (1) as:
,.24
1
mD
fnb (2)
where b is the bending wave length. Figure 2 shows propagation of three wave length, but in fact this happens when an integer multiple of half wave length or even integer or the quarter wave length match the bay length. While on the other hand, for wave lengths not matching the bay length, beams seem to act as a barrier to vibration propagation thus weakens the vibration and makes it to reflect back. This obstruction becomes maximum when the peak points of a wave match the beam locations, thus gives rise to anti-resonance.
Figure 2: Wave length matching at different frequencies for efficient vibration propagation across the floor beams.
Obviously, the resonance frequencies come in terms of peaks in the sound level, and by tracking the peaks it is supposed to be possible to identify the resonance frequencies.
Wave length equal to two third of the bay length Wave length equal to half the bay length
Wave length equal to the bay length
4
a
b
Figure 3: Sound pressure level at default setup in 1/3 octave band a), in narrow band b). Figure 3 shows the sound pressure level for default set up, where a) and b) represent graph in 1/3 octave band and in narrow band respectively. The first peak in the narrow band graph is due the mass air mass resonance as mentioned in [7, 8] and the followings are supposedly due to the structural resonances as explained, which are at around 180 Hz, 524 Hz, 862 Hz etc. corresponding wave lengths according to equation (2) are 0.69 m (about equal size as bay length), 0.41 m (wavelength equals two third of the bay length) and 0.32m (Wave length equal to half the bay length) respectively. In all these three frequencies the wave length matches the bay length such that the beam location is kept at a nodal point of those waves as in figure 2. This strengthens the argument about peaks corresponding to the structural resonances, and shows that the model is capable of identifying the peaks quite accurately. Moreover, measurements were done with the objective to see wave length interactions with the propagation across the floor beams.
2.2 Experiment setup: The tested decoupled floor was manufactured according to figure 1. The floor part consisted of one layer 22 mm particle board, glued and screwed to 6 pinewood glulam beams, 42x225 mm. The beam spacing was c/c 600mm except for the two spaces closest to the borders where 390 mm were used for symmetry reason. The floor rested on a separate isolated concrete frame to avoid flanking transmission The floor plate measures 3180x3780 mm. The ceiling
5
part was made of 13 mm gypsum board in conjunction with framed pinewood joists, 45x95 mm, screwed together. Each beam was placed parallel to the y axis. The joist spacing was 300mm except for the two spaces closest to borders where 290 mm were used for symmetry reason. The ceiling was hung into place by two additional joists positioned vertically at two sides. The ceiling plate measured 2980 x 3580 mm. The excitation was given at the middle which lies at the middle of a bay as well and 12 measurement points were selected on a line across the beams, each placed 150 mm apart. First point was located at 150 mm distance from the excitation position which was excluded in figure 4. Therefore the first points in figure 3 are 300 mm from the middle and placed on a beam, and following 3 points are on the plate while the 5th ones are again on a beam. Same pattern continues for the remaining points.
a
b
c
Figure 4: Vibration propagation across the beam at 3 different frequencies; a) at 180 Hz, b) at 524 Hz and c) at 862 Hz
6
Figure 4 show the propagation pattern of waves at these three frequencies 180 Hz, 524 Hz and 862 Hz respectively. It is evident from this figure that in these frequencies vibrations in the bay area experience strong vibration while at the beam locations vibration is minimum, especially for the case of 180 Hz. For other 2 frequencies vibration profile could not be identified accurately because of having few measurement points. However, this confirms the argument presented before regarding vibration propagation through the periodically stiffened plates. Furthermore, figure 4 also shows the decay in amplitude while propagating. Since beams in the real floor structure has finite thickness vibration always experience some sort of obstruction while passing through and eventually decrease in amplitude. Besides this, material damping also causes the amplitude to get reduced. The discussion presented above show that the model is quite reliable and provides quite accurate prediction. Going back to figure 3, it is interesting to notice that the peaks in the third octave band slightly differ in frequencies with those in the narrow band and some peaks even disappear. While considering frequency band such as 1/3 octave band, peaks can pear simply because of having more frequency lines with in the band, that is summing up more pressures components. Thus peaks in the narrow band don’t always correspond to any structural phenomenon, e.g. resonance. Moreover, when peaks appear quite frequently and uniformly, due to summing up to bands, graph would straighten up and peaks would disappear as in the high frequency region of figure 3(a) Nevertheless analysis in the third octave band is easier when the primary objective is not to identify these frequency points exactly rather to have an overall idea, e.g. what influences the pressure level to change or peaks to moves along the frequency etc. therefore the remaining studies is made on 1/3 octave band. It is true that although peaks in the 1/3 octave bands don’t always appear at the right frequencies but indicate the existence of peaks in the narrow band.
3 Parametric analysis The influence of each parameter on the impact sound is tested below by changing only that particular parameter and keeping others constant. Four values are selected by multiplying the default value with four multiplying factor [0.5 1 1.5 2] represented by four different colours [‘blue’ ‘green’ ‘red’ ‘cyan’] respectively. All predictions are done in third octave bands.
Figure 5: Predicted impact sound pressure level at four different floor plate densities, in 1/3
octave bands. Where, p1 = 681.82 kg/m3 In figure 5, sound level for four different upper plate masses are plotted. Mass are varied by varying the density using four multiplying factor as mentioned, to the default value for floor density. Figure 5 shows that sound pressure level decreases with increasing floor plate mass in
7
a perfect agreement with mass law bellow 2 kHz. reduction of approximately 4 dB can be seen in the frequency range 40-80 Hz, with mass doubling and in the frequency range 200-500 Hz the reduction is 5-6dB with mass doubling. The reason is quite common that the floor being heavy vibrates less with the addition of mass. While, above 2 kHz no changes are seen, probably because at this point the upper plate switches to stiffness controlled region. The critical frequency of the lower plate is 2.7 kHz which indicates a correlation of the sound level at high frequencies to this frequency. Also the apparent peaks in the high frequency region reduce in frequency with the increase of mass. As explained before peaks are due to structural resonances and change in material property such as mass, change these resonance frequencies. According to equation (1) with the increase of mass the resonance frequency decreases and Figure 5 agrees with this.
Figure 6: Predicted impact sound pressure level at different ceiling plate densities, in 1/3
octave bands. Where, p2 = 692.31 kg/m3 Figure 6 shows variation of sound pressure level with lower plate mass change which is done by changing the lower plate density. the general trend is same as that with upper plate mass change. In agreement with mass law, sound level reduces with the increase of lower plate mass and this reduction increases with frequency. Reason is the same that the floor being heavy vibrates less and thus less sound is radiated. Moreover, here sound levels reduction does not coincide after a specific frequency which was noticed earlier. In fact here the critical frequency of the lower plate is not constant rather changes with lower plate mass change. This further strengthens the claim that the corresponding frequency in Figure 5, represent critical frequency of the lower plate. Unlike figure 5 here the peaks in high frequency do not seem to vary with lower plate mass variation. This shows the strong influence of the structural resonances, i.e. periodicity of the floor to the overall vibration of the entire floor structure. Nevertheless, mass of both floor and ceiling influence positively in sound reduction, i.e. increasing the mass increases sound reduction.
8
Figure 7: Predicted impact sound pressure level at four different floor plate stiffness, in 1/3
octave bands. Where, Ep1 = 3.23 GPa. Changing the stiffness of the upper plate does not affect the low frequency region known as mass controlled region, since mass was kept constant, see figure 7. Bellow 40 Hz there is absolutely no change in SPL and between 40 Hz to 200 Hz band sound pressure level decreases with a maximum reduction of 2 dB for mass doubling. Above 200 Hz stiffness seem to start affecting and from 200 Hz to 1600 Hz band the graphs seem to loose the pattern, because resonance peaks change frequency due to stiffness change. In this range things seem very chaotic and it is very difficult to make any conclusion other than stiffness affecting resonance peaks. Increasing in frequency with increment of the stiffness agrees with equation (1). Above 1600 Hz band graphs smoothens up and reduction of SPL with the increase of stiffness is seen with an average reduction of 2 dB with mass doubling.
Figure 8: Predicted impact sound pressure level at four different ceiling plate stiffness, in 1/3
octave bands. Where, Ep2 = 2.2 GPa
Changing stiffness of the lower plate does not make any difference at all bellow 100 Hz and up to 250 Hz the difference is negligible (0.5 dB). See figure 8 Therefore from figure 7 and 8 it can be said that the low frequency region is entirely or to be more precise, strongly controlled by mass and agrees quite well with the conventional term of ‘mass controlled low frequency’. However, unlike figure 7 above 200 Hz band, SPL increases with the increase of lower plate stiffness. The overall pattern of the graphs is the same that with the change of ceiling stiffness, sound level only changes the magnitude but the peaks always appear at the same frequencies.
9
Figure 9: Predicted impact sound pressure level at four different floor plate thicknesses, in
1/3 octave bands. Where, hp1 = 22 mm. Figure 9 shows the sound pressure levels for four floor plate thicknesses. Since thickness changes both mass and stiffness, the overall effect is the combined effect of both mass and stiffness as seen in figure 5 and 7, mass according to figure 5 has a positive affect in sound reduction up to 2 kHz and above this the effect seems not to be present. While stiffness on the other hand as figure 7 seems to have a positive effect on sound reduction at high frequencies, above 1600 Hz band. Therefore, floor thickness has positive effect on sound level reduction at all frequencies, see figure 9. Like all other floor parameter, thickness also affects the structural resonance frequencies. According to equation (1) this depends on the ration of bending stiffness to the mass per unit area of the floor.
Figure 10: Predicted impact sound pressure level at four different ceiling plate thicknesses, in 1/3 octave bands.
Where, hp2 = 13 mm Figure 10 shows the influence of lower plate thickness on sound pressure level. Like upper plate thickness, increasing the ceiling thickness increases both mass and stiffness of ceiling thus has effects due to both mass and stiffness, i.e. combined effects as seen in figure 6 and 8 therefore, low frequencies and up to 2 kHz the effect is mass controlled and has positive effect on sound reduction. While above 2 kHz the graphs are scattered, because mass at this range affects positively and stiffness does negatively, making the combined effect to be a bit irregular. Again no influence of the ceiling parameters to the resonance peaks is noticed.
10
Figure 11: Predicted impact sound pressure level at four different (floor) beam heights, in 1/3 octave bands.
Where, hb1 = 225 mm.
Figure 11 shows the sound pressure level at different upper beam height. Increased height means increased mass and stiffness of the floor, thus should affect both high and low frequencies as floor thickness. In agreement to this, upper beam height has a clear impact on the low frequency region, below 100 Hz, where, a 2x increase in beam heights is predicted to reduce the sound pressure level by about 3-4 dB. While at high frequencies the effect is negligible. In the range 125-315 Hz something happens with the prediction for the beams of half the original height. However, this shows increasing beam heights although increases the mass but does not add up to the overall stiffness of the floor significantly. Similar reductions at low frequencies were noticed in Carin [11], where the author studied the effect of stiffness of the attached beams. She made experiments on two similar floor structures having joists of two (plywood and glulam) different stiffnesses and found that the floor having beams with higher stiffness (glulam) had 3–5 dB lower impact sound level in the frequency range below 100 Hz.
Figure 12: Predicted impact sound pressure level at four different (ceiling) beam heights, in 1/3 octave bands.
Where, hp2 = 95 mm.
Figure 12 shows that the beam height attached to the lower plate has similar influence as beams attached to the upper plate as in figure 11. The same tendency, i.e. SPL reduces with increasing beam height at low frequencies and very minor differences at high frequencies are noticed.
11
Figure 13: Predicted impact sound pressure level at four different spacing between beams attached to floor
plate, in 1/3 octave bands. Where, l1 = 60 cm Figure 13 shows the sound pressure level at different spacing between two beams attached to upper plate. The peaks are scattered due to the change in beam spacing. Because, similar to the floor properties, spacing between the beams also influences the structural resonance. When beam spacing is changed, waves having different lengths now match the spacing and eventually structural resonance takes place at different frequencies. One result is that the predicted peaks around 800 Hz is shifted towards 1600Hz with doubling the spacing. Physically, by reducing the spacing makes the floor part to have more beams and as a result becomes heavier and stiffer. In addition to this, more beams mean more obstacles and much dissipation of energy. Therefore the graph having lowest value for beam spacing corresponds to lowest sound pressure level.
Figure 14: Predicted impact sound pressure level at four different spacing between beams attached to ceiling
plate, in 1/3 octave bands. Where, l2 = 30 cm Figure 14 shows the sound pressure level for various lower beam spacing. It seems very difficult to draw any conclusion from this. One thing can be seen is that same peaks are seen, around 200Hz and 1000Hz. The reason for this is as explained, nothing has been varied in the floor part.
12
Figure 15: Predicted impact sound pressure level at four different cavity depth in 1/3 octave bands.
Where, d = 345 mm
Figure 15 shows the influence of the cavity depth to the sound pressure level. The low frequency region is influenced largely by the mass air mass resonance frequency as mentioned in [3, 7]. With an increased resolution this influence can be understood more clearly as seen in [7]. Since air inside the cavity is the only means of sound propagation, the overall trend is that the pressure level decreases with the increase of the cavity depth. Improvement in the pressure level seem be maximum within frequencies, from 80 Hz to 630 Hz, about 5 dB. At higher frequencies (2000 Hz to 5000 Hz) this reduces to around 3 dB.
4 ConclusionsWith this parametric study it was possible to have some sort of in sight of the lightweight floor structure. Although most of the study is done on theoretical basis, i.e. without any experimental validation, still the findings are reliable. Further analysing through the experiment will definitely provide more insight. However some key findings are mentioned bellow:
1 In general the effect of any element on the overall vibration is that, the heavier the structure becomes due to any element, the lower it vibrates and thus the sound level gets reduced. For example if the upper plate is made heavy by making it thick or dense the overall vibration will reduce. Or if the beam height is increased, the result would be increase of mass and thus reduction in sound level. Same is true for the lower plate as well. And this mass effect governs the low frequency region, both for floor and ceiling.
2 Stiffness generally influences the high frequency region and the general behaviour of the stiffness differs between ceiling and floor. For floor it influences positively in sound reduction, therefore any change that adds up to the stiffness of the floor, e.g. increasing the material properties like Young’s modulus, thickness, beam height etc, would reduce the sound level. While, for ceiling the effect is opposite, therefore increasing those of ceiling would increase the sound level. Some of the parameters such as thickness, beam height etc would increase both mass and stiffness. In that case effects are seen in all frequencies, where low frequency region behaves as it should with the change of mass and the high frequency region behaves as it should with the change of stiffness.
3 Apart from the general behaviour the study shows that upper plate’s elements are the key factor controlling the overall vibration pattern. In periodic structures there always occur pass band and stop band giving rise to some peaks. And this study
13
shows that a little change in any of the parameters on the upper plate would change the peaks in frequencies, since it would change the banding wave length of the floor. While with the change in ceiling parameters the peaks remains at the same frequencies, thus maintains the same pattern. Therefore, it can be concluded that the periodic nature of the floor dominates the periodicity of the decoupled floor structure. The influence of the ceiling is mainly in determining the amplitude of the sound level.
4 From previous conclusion one could through more lights, that for impact sound, since the excitation is given to some points on the floor, the key factor regarding the upper plate is how the vibration propagates through it. Thus periodicity of the floor comes in to concern and eventually this is reflected in the radiated sound pressure level. While for ceiling the excitation is given to the entire structure by the pressure wave coming through the cavity. As a result whole structure gets excited at a time rather than vibration being propagated from a point. Therefore vibration propagation through the ceiling, i.e. periodicity of the ceiling is less of a concern.
5 Increasing beam heights, both for floor and ceiling have influences mostly in the low frequency region, indicating that increasing the beam heights although makes the floor heavy, does not seem to add more to stiffness. Nevertheless attachment of beams definitely makes the floor stiffer compared to a single plate.
6 Mass air mass resonance is a dominating term for low frequencies. It has a direct relationship with the cavity depth. The air density inside cavity should also influence and so does the absorbing material placed inside the cavity, which was excluded from present study. However, by proper designing of floor parameter such as, cavity depth, absorbing material etc this resonance effect can be controlled, i.e. be kept out of the operating frequency range.
5 References [1] V. N. Evseev: Sound radiation from an infinite plate with periodic inhomogeneities. Soviet Physics –Acoustics 19(1973) 226-229 [2] B. R. Mace: Sound radiation from a plate reinforced by two sets of parallel stiffeners. Journal of Sound and Vibration 71(1980) 435-441 [3] G. F. Lin and J. M. Garrelick: Sound transmission through periodically framed parallel plates. Journal of Acoustics. Society of America 61(1977) 1014–1018. [4] J. Brunskog and P. Hammer: Prediction model for the impact sound level of lightweight floors. Acta Acustica United with Acustica 89(2003) 309–322. [5] D. Takahashi: Sound radiated from periodically connected double-plate structures. Journal of Sound and Vibration 90(1983) 541–557. [6] Mosharrof, M. S. Ljunggren, F. Ågren, A. Brunskog, J. Prediction model for the impact sound pressure level of decoupled lightweight floors; Proceedings of internoise 2009, Ottawa, 2009. [7]Mosharrof, M. S. Brunskog, J. Ljunggren, F. Ågren, A. Improved prediction model for the impact sound level of lightweight floors - introducing decoupled floor-ceiling and beam-plate moment. Submitted to Acta Acustica United with Acustica, 2010. [8]Lin and Garrelik ‘Effect of the number of frames on the sound radiated by fluid-loaded, frame-stiffened plates’, J. Acoust. Soc. Am. Volume 58, Issue 2, pp. 499-500 (August 1975) [9] Jonas Brunskog. ‘Acoustic excitation and transmission of lightweight structures’. Doctoral thesis. Lund University of Technology, 2002. [10] L. Cremer, M. Heckl, and E. E. Ungar: Structure-Borne Sound. 1988 ISBN 0-387-18241-
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[11] Carin Johansson,‘The Behaviour of Lightweight Wooden Joist Floor at Low Frequencies’, Licentiate thesis. Luleå University of Technology,1994.
