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Optimal Sensor

Placementfor MeasuringFloor Vibrations

Varun Kohli

Summer Internship

Civil Engineering Department

University of Auckland

Project Supervisor: Dr. PiotrOmenzetter

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Contents

1. Introduction.....................................................................................................61.1. Objective......................................................................................................8

2. Literature Review..............................................................................................9

2.1 Review of Vibrations Induced In Floors Due To Pedestrian Movement...................9

2.2 Parameters used to determine Vibration Response...........................................10

2.3 Optimal Sensor Location Decision...................................................................19

3. MATLAB Interfacing and Coding........................................................................24

3.1 MATLAB Code for EfI-DPR Technique..............................................................24

3.2 ANSYS Commands........................................................................................253.3 Validation of Code........................................................................................25

4. Model of Floor in ANSYS...................................................................................30

5. Results of Preliminary Model.............................................................................31

5.1 Use of Combitorial Technique.........................................................................32

5.2 Results of MATLAB Analysis...........................................................................33

6. Updated Model of Floor....................................................................................36

6.1 Optimal Sensor Locations..............................................................................42

7. OSP on Segmented Plan...................................................................................457.1 Discussion on Room Wise sensor Selection......................................................46

8. Model with Flexible Supports.............................................................................48

8.1 Discussion of Mode Shapes with Flexible Supports............................................52

8.2 Optimal Sensor Location for Flexible Supports..................................................53

9. Model with Flexible Columns.............................................................................56

9.1 OSP for Flexible Columns..............................................................................60

10. Conclusion......................................................................................................62

11. References.....................................................................................................6412. Appendix........................................................................................................65

Appendix

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List of Tables

Table 1 : Estimation of the floor damping [3]................................................................14Table 2 : Simple determination of the natural frequency [2]...........................................15Table 3 : Modal Frequencies of T-Beam from FEM Model.................................................26Table 4 : Optimal location of 4 sensors.........................................................................27Table 5 : Optimal location of 3 sensors.........................................................................27Table 6 : Modal Frequencies of Concrete Slabfrom FEM model.........................................27Table 7 : First 6 natural frequencies.............................................................................31Table 8 : Results of n=2 for Combitorial Technique........................................................33Table 9 : Results of n=3 for Combitorial Technique........................................................33Table 10 : Results of EfI-DPR technique using n=6........................................................34Table 11 : First 10 natural frequencies.........................................................................36Table 12 : Room 1.....................................................................................................38Table 13 : Room 2.....................................................................................................38Table 14 : Room 3.....................................................................................................40Table 15 : Room 4.....................................................................................................40Table 16 : Room 5.....................................................................................................41

Table 17 : OSP Room Wise.........................................................................................46Table 18 : Selected distributions (Room wise)...............................................................47Table 19 : Comparison of EFI-C with Room-wise method................................................47Table 20 : Natural frequencies with flexible supports.....................................................48Table 21 : Natural Frequencies with columns................................................................56Table 22 : 56 Combinations of 3 sensors......................................................................65Table 22: 56 Combinations of 3 sensors

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List of Figures

Figure 1: Timber frame of the building..........................................................................6Figure 2: Floor plan with cantilever concrete at the bottom end.......................................7Figure 3: Cross-section of the composite timber-concrete floor........................................8Figure 4: Typical walking forces in (a) vertical, (b) lateral and (c) longitudinal direction [5]..............................................................................................................................10Figure 5: Frequency distribution of step frequency [3]..................................................11Figure 6: Dependence of peak force and contact time on different pacing [5]...................11Figure 7: Normal distribution of pacing frequencies for normal walking [5]......................12Figure 8: Fourier spectrum of the walking force [10].....................................................12Figure 9: Relationship of DLF with walking frequency [2]...............................................13Figure 10: First Mode shape of simple and continuous spans [2]....................................15Figure 11: Floor Plan [2]...........................................................................................17Figure 12: Computational procedure of the EfI sensor placement [17]............................22Figure 13: T-Beam Section used for FEM modelling.......................................................25Figure 14: Displacement contour plots of first 5 mode shapes.......................................26

Figure 15: Displacement contour plots of first 5 mode shapes........................................28Figure 16: Optimal location of 3 sensors......................................................................29Figure 17; Optimal location of 2 sensors......................................................................29Figure 18: Representation of possible sensor locations..................................................34Figure 19: Representation of possible sensor locations (zoomed on the locations)............35Figure 20: Variation of Log(|Q|) with number of sensors used........................................42Figure 21: Floor Usage Plan.......................................................................................45Figure 22: Room Wise OSP........................................................................................46Figure 23: OSP for flexible supports............................................................................54Figure 24: Modal Displacements at 3 selected locations.................................................60Figure 25: OSP for full floor with flexible columns.........................................................62Figure 26: OSP for floor excluding galleries........................Error! Bookmark not defined.

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1. Introduction

The project is about permanent instrumentation of a multi-story timber building. Thebuilding is the Arts and Media building on the Nelson campus of the Nelson MarlboroughInstitute of Technology. The new building will be a world-first in commercial multi-storeywood construction.

The building is the result of a national design competition created in partnership betweenNMIT and Ministry of Agriculture and Forestry (MAF). The building had to be sustainable andsubstantially made of wood. Timber is sustainable, renewable, locally available and requiresless energy to manufacture than other building materials, like concrete and steel.

The building is also the worlds first for the innovative use of wood in the structure of amulti-storied building. Built using Laminated Veneer Lumber (LVL) beams and compositeLVLand concrete floors, the three-story building features steel-tensioned timber wallsdesigned to survive earthquakes. The structure features a new generation of seismic

engineering known as damage avoidance design that allows the building's main walls toabsorb seismic energy and reduce damage during an earthquake.

Figure 1: Timber frame of the building

It is planned to instrument the building with various sensors to measure and study thevariations in properties such as creep, deflection, temperature, humidity etc. Both static anddynamic sensors are planned to be used:

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Static Sensors for measuring slow deformations in timber : strain gauges,temperature gauges, deflection sensors

Dynamic Sensors for measuring responses of composite timber-concrete floor due towalking of people : accelerometers

This project is concerned with the measurement of dynamic response (vibration) of the floorunder walking loads. Measuring vibrations becomes important because:

Timber is light weight and has reduced stiffness as compared to concrete or steel.Thus it is more susceptible to excessive vibrations under walking loads.

The floor is composite, made of LVL and concrete with a cantilever concreteoverhang at one end. Thus it is interesting to study which part of floor experiencesmore vibrations.

Figure 2: Floor plan with cantilever concrete at the bottom end

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Figure 3: Cross-section of the composite timber-concrete floor

1.1.Objective

The primary objective of the project is to identify the optimal positions to place sensors(accelerometers) to record maximum vibration response of the floor. We are particularlyconcerned with level 3 of the building where we would place the sensors.

Vibration response of the floor depends on:

The vibration source (the frequency and force of walking vibration)

The vibration transmission path (structural characteristics of the floor)

Thus these parameters are discussed in the following sections. Section 2 provides a brief

literature review about vibration due to walking loads and methods to estimate optimalsensor locations. Section 3 describes the process and codes used to process modal dataobtained from ANSYS. Various models of the floor are presented in Sections 4, 5,6,7,8 and9 along with results for the optimal sensor locations. Section 10 gives a brief conclusion ofthe results thus obtained.

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1. Literature Review

1.1 Review of Vibrations Induced In Floors Due ToPedestrian Movement

In recent decades there has been a trend towards improved mechanical characteristicsofmaterials used in construction. It has enabled engineers to design lighter, moreslenderand more aesthetic structures. The current push towards stronger concrete materials andthe use of pre-stressing is resulting in increasing slenderness and liveliness of long-spanconcrete floors in buildings. Although concrete floors have had a good vibrationserviceability track record, this trend may lead to an increasing number of floors failing theirvibration serviceability[1].Modern floors with large spans are light-weight constructions witha low stiffness. The low stiffness leads to low natural frequencies. Therefore the assessmentof the vibration of floors may become essential.

Generally speaking, human induced footfall loading has proved to be the major source offloor vibration disturbance as it happens frequently and, in practice, cannot be isolated.Therefore, excessive floor vibrations due to human induced loading have been characterisedas probably the most persistent floor serviceability problem encountered by designers.

The analytical background of vibrations is well developed and understood. Detailed andrigorouscomputational tools are available to analyze vibration and response of both simpleand complexstructures. The difficulty in vibration design is the poor correlation between theoutcome of computations at the design stage, and the response of the floor constructed

accordingly. In addition tothe uncertainties inherent in material properties, dampingcharacteristics, and boundary conditions, thelevel of vibrations perceived by individuals, thevibration that is considered objectionable, and the forceand frequency of foot drop are allhighly subjective and prone to large variations[2].

The dynamic behaviour of floors depends basically on the mass, stiffness and damping ofthe floor.The ratio between stiffness and mass determines the natural frequency of thefloor. Naturalfrequencies in a range of step frequencies induced by pedestrians walking onthe floor can becomecritical[3].

Thus, typical parameters required to evaluate vibration response of a floor would be:

1. The vibration source (the frequency and force of walking vibration)2. The vibration transmission path (mass, stiffness, damping)

From these parameters, the vibration response of the floor may be predicted. But,perception of vibration and whether or not it is annoying or objectionable is highlysubjective andvaries from reference to reference

For the design of floor systems, two methods are frequently employed.

Frequency Tuning Method sets the natural frequency of a floor system abovefrequenciessusceptible to resonating when excited by the lower harmonic of walkingforces.

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Response Calculation Method, a performance-based method focuses ondetermining the likelyvibration response of a floor under the application of a dynamicforce from walking.

Here, we discuss a simplified procedure based on the Response Calculation Method.Due to

the variable nature of the parameters that determine the response of a floor system, suchasdamping, this procedure is valuable for a first design estimate. Damping cannot becalculated as such,and must always be assessed based on experience with floors of similarconstruction.[2]

1.1 Parameters used to determine Vibration Response

The following is a discussion about the parameters used for determining vibration response.Examples of typical functions used are also provided.

1.1.1 VIBRATION SOURCE

During walking, a pedestrian produces a dynamic time varying force which has componentsinall three directions: vertical, horizontal-lateral and horizontal-longitudinal. Thissinglepedestrian walking force, which is due to accelerating and decelerating of the mass oftheirbody, has been studied for many years. In particular, the vertical component of theforce hasbeen most investigated.

Andriacchi et al[4]measured single step walking forces in all three directions by means of aforce plate.Typical shapes are presented in Fig. 4[5].

Figure 4: Typical walking forces in (a) vertical, (b) lateral and (c) longitudinal direction[5]

Hechler et al[3] measured the step frequency of 200 persons passing the entrance hall ofthe TNO-administration building in Delft, which is shown in Fig. 5.

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Figure 5: Frequency distribution of step frequency[3]

Andriacchi[4]also reported that the dynamic effect of the forces was changing with thewalking speed.It was noted that all these parameters are different for differentpersons, butsome general conclusions can be drawn. For example, that with increasing stepfrequencythe peak amplitude, stride length and velocity increase while contact time decreases[5].

Figure 6: Dependence of peak force and contact time on different pacing[5]

A reliable statistical description of normal walking frequencies was first given by Matsumotoet al [6, 7]who investigated a sample of 505 persons. They concluded that the frequenciesfollowed a normal distribution with a mean pacing rate of 2.0 Hz and standard deviation of0.173 Hz (Fig. 7). Kerr and Bishop obtained a mean frequency of 1.9 Hz but from aninvestigation of only 40 subjects. It is also interesting that Leonard[8] concluded that thenormal walking frequency range is 1.72.3 Hz, which is in broad agreement with whatMatsumoto et al and successive researchers have found. Similar comprehensive statistically

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on the finishes and use of thepremises. Thus separation walls, ceilings under the floor, freefloating floors or swimming screedsaffect the damping properties significantly.

In the case of low-frequency floors, damping has the potential to reducesignificantly theresonant or near-resonant response due to walking excitation. On theother hand, in high-frequency floors, it increases the decay rate in the case of free vibrationresponse between

subsequent footsteps. In both cases the effects of dampingare beneficial and it is prudent tomodel it as accurately as possible.

Appropriate damping values can be taken from Table 1. The system damping D is obtainedbysumming up the appropriate values[3].

Table 1: Estimation of the floor damping[3]

Extent of Cracking

Cracking reduces floor stiffness and, consequently, lowers its natural frequency. For

conventionallyreinforced concrete it is important to allow for cracking. Otherwise, the resultsare likely to be on theunconservative side. For conventionally reinforced flat slabconstruction with span to depth ratio of 30or larger, a 30% reduction in stiffness isreasonable.

1.1.3 DETERMINATION OF VIBRATION CHARACTERISTICS OF A FLOOR

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Determining vibration characteristics requires finding (i) the natural frequency of a floorsystem and (ii)the associated peak acceleration. The peak acceleration is then compared tothe acceptable values based on perception of users for the given environment. Thesimplified method described herein is based on closedformulas for the first mode naturalfrequency of uniformly thick rectangular slabs with different boundaryconditions.

In Table 2 a selection of manual formulas for the determination of the first naturalfrequency (according to[2]) and the modal mass of isotropic plates for different supportingconditions are presented. The same are suggested by Hechlar[3].

Aalami[2] elaborates on the relevant boundary conditions for the slab. Observe Fig. 10-a.Note that for interior spans the displacement shape under self-weight is analogous tothat ofa single panel fixed at its supports (part d of Fig10). However, the first mode of vibrationasshown in part c of the figure implies that the vibration response of the floor is more affineto that of asingle panel simply supported along its four sides (part a). For this reason, forpanels bound bysimilarly sized spans, it is recommended to use a simply supportedboundary condition along the foursides of the panel. More specifically, the recommendedcondition is rigid supports, but rotationallyfree. This holds true for column supported panelstoo.

Where columns are organized on a regular orthogonal grid, the first natural frequency of atwo-way slabis likely to be in the form of a one-way slab deflecting in a cylindrical form.This alternative is describedin greater detail in the following.For panels that are bound bysmaller spans, in the limit as the size of the adjacent panels reduce, thevibration mode willbe analogous to a rotationally fixed condition, as indicated in part (b) of Fig 10

Figure 10: First Mode shape of simple and continuous spans[2]

Table 2: Simple determination of the natural frequency [2]

f=ca2 (1)

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wherec= Edynh312(1-v2)gq (2)

f = first natural frequency [Hz];a = span length in X-direction;E = dynamic modulus of elasticity [1.25 static E in psi; MPa];h = slab thickness [in; mm]; = Poissons ratio;

g = gravitational acceleration [9810 mm/sec2]; andq = weight per unit surface area of the slab. is used according to the following:

1.1.4 SAMPLE CALCULATION

The plan view of a typical level of a multi-story building is shown in Fig. 11. The floor slab ispost-tensioned.Evaluate the in-service vibration response of the floor from foot.Theparticulars of the floor are listed below:[2]

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Thickness of the slab = 8 inPoissons ration () = 0.2Superimposed dead load = 20 psffc = 5000 psiwc = 150 pcf

where,

fc = specified 28-day compressive strength of concrete, (lb/in2

(MPa))

Edyn = 1.2 EstEst= 33wc1.5f'clb/in2 in USCS units

Figure 11: Floor Plan[2]

Select a Critical Panel

For floors of uniform thickness the largest panel is typically selected.

Select Boundary Conditions

The boundary condition 1 listed in Table 2 is the most conservative. It applies to a centralpanelbound on each side by one or more identical panels. An upper bound to the first

naturalfrequency is the boundary condition 6 noted in the same table. The full fixitysimulatesconnection to thicker core walls.

Find the First Natural Frequency

Using the recommended values for concrete parameters, calculate the first naturalfrequency ofthe panel using the formulas of Table 2.

g = 32.2 ft/sec2

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f=ca2

wherec= Edynh312(1-v2)gq

First natural frequency, fn:

Est = 33wc1.5f'clb/in2

= ( 33 x (150)1.55000 ) /1000 ksi= 4287 ksi

Edyn = 1.2 Est = 1.2 x 4287= 5144 ksi

q = 150x812x144self weight+ 20144superimposed dead load{lb/in2}

= 0.833 lb/in2

c= 5144 x 8312(1-0.22)32.2 x 120.833= 325,653 in2/sec

a = 30 x 12 = 360 in

= 1.57(1+2)

= 1.57 1+3026.252= 3.62

f = 3256533602 3.62= 9.1 Hz

1.1

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1.2 Optimal Sensor Location Decision

Deciding on an optimal sensor placement (OSP) isa common problem encountered in manyengineeringapplications and is a critical issue in the construction andimplementation of aneffective structural health monitoringsystem[12]. A great deal of research has beenconducted on optimal sensor placementusing a variety of placement techniques and criteria.

For example, usingmodal kinetic energy as a means of ranking the importance of candidatesensor locations.Another method, such as the geneticalgorithm can also be used to searchfor optimal sensor configurations based upon a number ofmeasures of goodness, such asthe determinant of the Fisher information matrix (FIM) or themodal assurance criterionmatrix.[13]

The Effective Independence (EfI) technique is the most widely used technique used foroptimal sensor placement (OSP)PEVuZE5vdGU+PENpdGU+PEF1dGhvcj5LYW1tZXI8L0F1dGhvcj48WWVhcj4yMDA0PC9ZZWFyPjxSZWNOdW0+MjQ8L1JlY051bT48RGlzcGxheVRleHQ+WzEyLTE2XTwvRGlzcGxheVRleHQ+PHJlY29y

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25/75

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WN0dXJhbCBkZXNpZ248L2tleXdvcmQ+PGtleXdvcmQ+VmlicmF0aW9ucyAobWVjaGFuaWNhbCk8L2tleXdvcmQ+PGtleXdvcmQ+Q3Jvd2QgY29uZmlndXJhdGlvbjwva2V5d29yZD48a2V5d29yZD5TdHJ1Y3R1cmFsIHNsZW5kZXJuZXNzPC9rZXl3b3JkPjwva2V5d29yZHM+PGRhdGVzPjx5ZWFyPjIwM

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

MaW5nZ29uZyBSRC4yLCBEYWxpYW4sIDExNjAyNCwgQ2hpbmEmI3hEO0luc3RpdHV0ZSBvZiBNZWNoYW5pY3MgYW5kIEF1dG9tYXRpYyBDb250cm9sLU1lY2hhdHJvbmljcywgVW5pdmVyc2l0eSBvZiBTaWVnZW4sIFBhdWwtIEJvbmF0eiBTdHJlZXQgOSAtMTEsIEQtNTcwNzYgU2llZ2VuLCBHZXJtYW55PC9hdXRoLWFkZHJlc3M+PHRpdGxlcz48dGl0bGU+QSBub3RlIG9uIGZhc3QgY29tcHV0YXRpb24gb2YgZWZmZWN0aXZlIGluZGVwZW5kZW5jZSB0aHJvdWdoIFFSIGRvd25kYXRpbmcgZm9yIHNlbnNvciBwbGFjZW1lbnQ8L3RpdGxlPjxzZWNvbmRhcnktdGl0bGU+TWVjaGFuaWNhbCBTeXN0ZW1zIGFuZCBT

aWduYWwgUHJvY2Vzc2luZzwvc2Vjb25kYXJ5LXRpdGxlPjwvdGl0bGVzPjxwZXJpb2RpY2FsPjxmdWxsLXRpdGxlPk1lY2hhbmljYWwgU3lzdGVtcyBhbmQgU2lnbmFsIFByb2Nlc3Npbmc8L2Z1bGwtdGl0bGU+PC9wZXJpb2RpY2FsPjxwYWdlcz4xMTYwLTExNjg8L3BhZ2VzPjx2b2x1bWU+MjM8L3ZvbHVtZT48bnVtYmVyPjQ8L251bWJlcj48a2V5d29yZHM+PGtleXdvcmQ+RWZmZWN0aXZlIGluZGVwZW5kZW5jZTwva2V5d29yZD48a2V5d29yZD5Nb2RhbCBraW5ldGljIGVuZXJneTwva2V5d29yZ

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D48a2V5d29yZD5RUiBkZWNvbXBvc2l0aW9uPC9rZXl3b3JkPjxrZXl3b3JkPlNlbnNvciBwbGFjZW1lbnQ8L2tleXdvcmQ+PGtleXdvcmQ+U3RydWN0dXJhbCBoZWFsdGggbW9uaXRvcmluZzwva2V5d29yZD48a2V5d29yZD5EZWNvbXBvc2l0aW9uPC9rZXl3b3JkPjxrZXl3b3JkPkVsZWN0cm9uIGVuZXJn

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

bj4wODg4MzI3MCAoSVNTTik8L2lzYm4+PHVybHM+PHJlbGF0ZWQtdXJscz48dXJsPmh0dHA6Ly93d3cuc2NvcHVzLmNvbS9pbndhcmQvcmVjb3JkLnVybD9laWQ9Mi1zMi4wLTU4OTQ5MDg2NjI3JmFtcDtwYXJ0bmVySUQ9NDAmYW1wO21kNT1iNzE5MjhhYzVlNjBhMDdlYzRkMTk5YzMxNTY5OGZjODwvdXJsPjwvcmVsYXRlZC11cmxzPjwvdXJscz48ZWxlY3Ryb25pYy1yZXNvdXJjZS1udW0+MTAuMTAxNi9qLnltc3NwLjIwMDguMDkuMDA3PC9lbGVjdHJvbmljLXJlc291cmNlLW51bT48L3JlY29yZD48L0NpdGU+PC9FbmROb3RlPn== [12-16]. The EfI-Driving Point Ratio technique is a variationof the EfI technique which aims to remove the drawbacks of the EfI technique.

Meo andZumpano[12] compared six different optimal sensor placement techniques, three

based on the maximisationof the Fisher information matrix (FIM), one on the properties ofthe covariance matrix coefficients, and two on energetic approaches and showed that theeffective independence driving-point residue (EFI-DPR) method provides an effectivemethod for optimalsensor placement to identify the vibration characteristics of the studiedbridge.

1.2.1 EFFECTIVE INDEPENDENCE DRIVING-POINT RESIDUE (EFI-DPR) METHOD

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The problem being addressed is that of placing Nsensors on a structure at locations whichwill allowthe computational software to give the best fit to a setof K targeted mode shapes.

The Fisher information is a way of measuring the amount of information that an observable

random variable X carries about an unknown parameter upon which the probability of Xdepends.

In civil engineering, each type of mechanical structure hasa specific pattern of vibration at aspecific frequency. Thisis called mode shape. Mode shape displacements are taken as themeasured data. Mathematically, the mode shapes of astructure form a mode informationmatrix:

MxK= 1, 2, ,K= a11a12 a1Ka21 a2K aM1aM2 aMK (3)

Generally, the number of mode shape K, the number of possible locations M forsensorplacement, and the final number of sensors N are fixed in advance(usually M > N >

K).Obviously, the actual target modes are notavailable to aid the test engineer in placement ofthesensors prior to the test. Instead, the next best thing is toplace sensors with the aid ofthe FEM representation ofthe target modes partitioned to the sensor locations.

EFI is aniterative method which begins bydesignating a large set of candidate sensorlocations fromwhich the smaller final sensor configuration will beselectedby removing onelocation in each iteration.The removal is carried out so that the determinant of asocalledFisher Information Matrix (FIM) is kept maximized.Intuitively a larger FIM indicatesa larger amount of the usefulinformation.

The vector of the measured structural responses denotedby y can be estimated as a

combination ofKmode shapesthrough the expression:

yMx1= MxKqKx1+= i=0Kqii+ (4)[12]

where:

is the matrix of FEM target mode shapes;qis the coefficient response vector;wis a is a stationary Gaussian white noise with zero mean and a variance of2

Evaluating the coefficient response vector using anefficient unbiased estimator and then,estimating thecovariance of the error results in:

J = E[(q -q)(q -q)T] =12T-1= Qo-1 (5)[12]

where:2representsthe variance of the stationary Gaussian measurement white noise in Eq.(4);Edenotestheexpectedvalue;qis thevectorofanefficientunbiasedestimatorof q

Thus,

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a o a

trix

Matrix

E

such thatmin(Eii)

j=N

ND

Update withreduced number

of locations

Yes

No

moveode p

Figure 12: Computational procedure of the EfI sensor placement[17]

A limitation of the EFI method is that sensor locationswith low energy content can beselected with a consequentpossible loss of information.The EFI-DPR (driving-pointresiduemethod eliminates this problem bymultiplying the candidate sensor contribution of the EFIbythe corresponding driving point residue (DPR) coefficient:

DPRj= j=1Mji2ji (10)[12]

Therefore,

EDi = Eii x DPRi (11)

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wherei is the ith target mode frequency. In essencethe DPR is a weighting factor of the EDvector. Thismethodology concentrates sensor positions in the highenergy content regionsresulting in sensors quasi-uniformlyspaced and symmetrically deployed[14].

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2. MATLAB Interfacing and Coding

For implementing the EFI-DPR technique, MATLAB has been used to process the data whichis obtained from modal analysis done in ANSYS. The process described in Fig 9 isimplemented and the steps are explained as follows:

2.1 MATLAB Code for EfI-DPR Technique

The inputs for the program are:

D -> Nodal Displacement Matrix (NxK)N -> Nodal Coordinate Matrix (Nx4)w -> Frequency Matrix (1xK)m -> Number of target locations (Scaler)

here:

N -> Total Candidate locationsK -> Frequency Modes

w2=(w.^(-1))' Calculates 1/wij which is later multiplied byij2 as in Eq.

9

while(length(N(:,1))>m) Loop until remaining nodes are equal to target number

DT=D'; Find Transpose of D

Q=DT*D; Find Q

Qi=inv(Q); Find Q inverse

E=D*Qi*DT; Find E as per Eq. 7

Ed=diag(E); Find Edas a diagonal Matrix

Ed=diag(Ed);

D2=D.^2; Find Matrix ofij2

DPR=diag(D2*w2); Get Diagonal Matrix of DPR

Ed=diag(Ed*DPR); Get Ed as per Eq. 10

[C,I] = min(Ed);

D(I,:)=[];N(I,:)=[];

end

Find node with minimum Ed and eliminate that node

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The following commands were also added to export the data to a spreadsheet:

s = xlswrite('Matlab Data.xlsx', N, 'Results', 'A5') ;

s = xlswrite('Matlab Data.xlsx', D, 'Results', 'A20') ;s = xlswrite('Matlab Data.xlsx', Q, 'Results', 'A30') ;

s = xlswrite('Matlab Data.xlsx', Ed, 'Results', 'A40') ;

1.1 ANSYS Commands

Since the initial data has to be obtained from FEM modelling done in ANSYS, the followingcommands were used to export data from ANSYS:

NLIST,ALL, , ,XYZ,NODE,NODE,NODE To list coordinates of all Nodes (Matrix N for MATLAB)

PRNSOL,U,Z/HEADER,OFF,OFF,ON,OFF

To obtain displacement in required direction per node(Matrix D for MATLAB)

1.1 Validation of Code

To validate the coded program, two test cases were taken:

Timber T-beam fixed at one end Concrete slab fixed at edges

1.1.1 TIMBER T-BEAM FIXED AT END

The material properties used for the beam were: Elastic Modulus (E) = 10.7 GPa Poissons Ratio () = 0.45 Density = 570 kg/m3

Figure 13: T-Beam Section used for FEM modelling

The model was created by using rectangular elements with a total of 2368 nodes.

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The modal frequencies obtained were:

Table 3: Modal Frequencies of T-Beam from FEM Model

ModeShape

Frequency (Hz)

1 8.65142 9.767

3 29.738

4 49.859

5 57.952

(i) (ii)

(iii)(iv)

(v)Figure 14: Displacement contour plots of first 5 mode shapes

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The model had 2368 nodes, all of which were taken as the candidate locations. The modaldisplacement matrix was obtained from ANSYS and the MATLAB code was run. The codewas run for n = 4 and n = 3. The optimal sensor locations for the two cases are shown inTables4 and 5 respectively.

Table 4: Optimal location of 4 sensorsNode Number x-Coordinate y-Coordinate z-Coordinate

106 -0.305 0.36 0.0589

178 -0.305 0.36 0

1493 0.305 0.36 0.0674

1566 0.305 0.396 0.0589

For n =3, node number 106 is dropped.

Table 5: Optimal location of 3 sensors

Node Number x-Coordinate y-Coordinate z-Coordinate

178 -0.305 0.36 0

1493 0.305 0.36 0.0674

1566 0.305 0.396 0.0589

The nodes suggested seem to be appropriate as they are the points that would experiencethe most displacement under both the flexural and torsion modes as can be seen in Fig. 14.

1.1.1 CONCRETE SLAB FIXED AT EDGES

A concrete slab of dimensions 8 m (x-direction) x 9.14 m (y-direction) with thickness 0.2 m(z-direction) was modelled. The material properties used were:

Elastic Modulus (E) = 35.46 GPa Poissons Ratio () = 0.2 Density = 2400 kg/m3

The modal frequencies obtained were:

Table 6: Modal Frequencies of Concrete Slabfrom FEM model

ModeShape Frequency (Hz)

1 17.964

2 33.621

3 39.556

4 53.845

5 58.786

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The model was created by using rectangular elements with a total of 1792 nodes.

(i) (ii)

(iii) (iv)

(v)Figure 15: Displacement contour plots of first 5 mode shapes

The model had 1792 nodes, all of which were taken as the candidate locations. The modaldisplacement matrix was obtained from ANSYS and the MATLAB code was run. The codewas run for n = 3 and n = 2. The optimal sensor locations for the two cases are shown inFig 16 and 17 respectively.

NodeNumber

x-Coordinate

y-Coordinate

z-Coordinate

998 5.071276 2.625342 0

1005 2.971391 2.614834 0

1180 5.048302 4.725233 0

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Figure 16: Optimal location of 3 sensors

For n=2, node number 998 is dropped.


1005 2.971391 2.614834 0

1180 5.048302 4.725233 0

Figure 17; Optimal location of 2 sensors

1.

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2. Model of Floor in ANSYS

The floor model created in ANSYS had the following characteristics:

Three types of materials were used:

Type 1 (Timber) : Elastic Modulus = 10.7 GPa Poissons Ratio = .45 Density = 570 kg/m3

Type 2 (Concrete) : Elastic Modulus = 30 GPa Poissons Ratio = .2 Density =2400 kg/m3

Type 3 (Steel) : Elastic Modulus = 200 GPa Poissons Ratio = 0.285 Density = 7750 kg/m3

The frame of the building was considered to be fixed. Hence, all contacts withframing i.e. contact between concrete and double beams, concrete and columns, LVLjoists and beams were taken as fixed restraints at nodes.

Reinforcement was taken into account by entering it as a volume ratio of concreteand entering the direction of reinforcement.

Due to holes in the volumes (due to presence of ducts and stairs), the model wasmeshed with tetrahedral elements. Different mesh sizes were used for concrete layerand LVL timber.

Timber was meshed with a division factor of 3 while for concrete a maximumelement length of 1m was used.

The meshed model had the following nodes:

6643 nodes for concrete 15932 nodes for Timber The number of nodes in Timber was higher to account for the irregularity in

the shape Timber joists and their small dimensions (thickness).

The origin of the coordinate system was taken at the centre of column at E9 with x-axis running along length 21 m, y-axis along length 30 m and z-axis being the

depth. The units were taken as mm.

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1. Results of Preliminary Model

The model described in Section 3 was solved using Modal solver in ANSYS and first 6 naturalfrequencies were obtained which are listed in Table 7.

Table 7: First 6 natural frequencies

ModeShape

Frequency (Hz)

1 84.172 85.654

3 91.0164 111.625 117.726 144.18

(i) (ii)

(iii) (iv)

(v) (vi)

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The following inferences can be made from the above mode shapes:

The analysis suggests maximum vibrations in the cantilever part of concrete whichcan be the case as that part lacks the extra stiffness provided by the timber joists.

The modal frequencies are in the high range starting at 84 Hz.

For deciding the optimal sensor location, nodes lying on the lower face of timber joist andlower face of cantilever part of concrete were selected. This selection was made taking intoaccount aesthetic considerations. Thus a total of 5328 nodes were selected to be candidatesout of which 4636 were on Timber and remaining 692 were on concrete.

The optimal sensor locations were found in 3 ways:

Using EfI-DPR Technique to investigate the best 6 locations

Using EfI-DPR technique to identify the best 75 locations out of which best 2 were

found using combitorial technique by selecting the two locations which form the Qmatrix with the highest determinant.

Using EfI-DPR technique to identify the best 25 locations out of which best 3 wereselected using combitorial technique.

1.1 Use of Combitorial Technique

Combitorial technique was introduced as EfI-DPR technique was found to be historydependent i.e. elimination at each step was dependent on the sensors already deleted. Onthe other hand combitorial technique is completely independent as all possible combinationsare investigated. Still EfI-DPR technique was used to reduce the candidate set so as tooptimize computational time.

The following code was used in MATLAB to find the best locations using combitorial method:

c=1;i=1;

NC=[];

Initializing count and output variables

while(i


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end

1.1 Results of MATLAB Analysis

The following is a discussion of the results obtained from the three analyses run in MATLAB.

1.1.1 RESULTS OF COMBITORIAL METHOD WITH 75 CANDIDATE NODES (75C2 CASE)

The initial set of 5328 locations was reduced to 75 using the EfI-DPR technique. It isinteresting to note that all the shortlisted 75 positions were on the cantilever part ofconcrete as was expected since that is the region experiencing the most vibration as seen inTable 7.

Combitorial method was then used to identify the best set of 2 sensors out of a possible2775 combinations. The results are shown in Table 8.

Table 8: Results of n=2 for Combitorial Technique


5985 21550 21400 -99

6340 21123.77 3910.94 -99

1.1.2 RESULTS OF COMBITORIAL METHOD WITH 25 CANDIDATE NODES (25C3 CASE)

EfI-DPR algorithm was then run on the set of 75 nodes to reduce it to a set of 25 nodeswhich can be used to apply Combitorial method with n=3. The best set of 3 sensors wasfound out of a possible 2300 combinations. The results are shown in Table 9.

Table 9: Results of n=3 for Combitorial Technique


5963 21550 27009.65 -99

6011 21550 9007.69 -99

6033 21550 4120.01 -99

1.1.3 RESULTS OF EFI-DPR TECHNIQUE

The set of 25 locations was the reduced to 6 nodes. It should be noted that 6 is theminimum number of nodes that can be suggested by EfI-DPR technique due to thelimitation that target number of nodes should be greater than or equal to the target numberof mode shapes. The 6 suggested locations are shown in Table 10.

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Figure 19: Representation of possible sensor locations (zoomed on the locations)

1.1.4 DISCUSSION OF RESULTS

On running the EfI-DPR method for n=3, the set of sensors produced had a value ofdeterminant of Q equal to 5.09E-43 as compared to a value of 4.50E-38 for the 25C3 case.Similarly, for the 75C2 case the value of determinant of Q was 7.38E-57 as compared to avalue of 1.9565e-062 for EfI-DPR method with n=2.

Thus it can be seen that EfI-DPR technique is good for shortlisting the locations withmaximum contribution but for selection of the best set which contribute the best in

combination, the combitorial method is better.

Thus the decision on the best sensor location should be taken based on EfI-DPR shortlistingfollowed by Combitorial selection process.

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2. Updated Model of Floor

An updated model was created to improve the results which showed very high natural

frequencies. Changes made were:

The model was made using beam elements for timber joists and shell elements forconcrete whereas previously brick elements were used for both.

Support conditions were changed from fixed to simply supported at all points ofcontact.

The same physical properties were used for the materials. The model was meshed separately based on element type. Two separate models

were compared.

The model with the mesh with less nodes (small mesh) had the following characteristics: Beam elements (timber joists) were divided into 4 elements along their length

Shell elements (concrete layer) were meshed based on element length of 1m. The model consisted a total of 1385 nodes with 6 degrees of freedom at eachnode.

The model with the finer mesh (Large Mesh) had the following characteristics: Beam elements (timber joists) were divided into 10 elements along their

length Shell elements (concrete layer) were meshed based on element length of

0.5m. The model consisted a total of 3795 nodes with 6 degrees of freedom at each

node.

The above described models gave the following mode shapes on analysis:

Table 11: First 10 natural frequencies

NaturalFreq.

Hz (SmallMesh)

Hz (LargeMesh)

PercentageDifference

1 5.491 6.2909 12.722 7.5215 6.5588 14.683 9.0724 8.2084 10.534 9.6494 8.2506 16.955 9.7042 8.9684 8.206 10.451 9.3834 11.387 12.149 10.346 17.438 12.339 10.485 17.689 13.523 10.868 24.4310 13.738 11.121 23.5311 14.083 11.163 26.1612 14.928 11.548 29.2713 15.066 13.058 15.3814 15.119 13.174 14.76

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15 15.642 13.444 16.3516 15.739 13.841 13.7117 16.292 14.656 11.1618 17.816 15.079 18.1519 18.158 16.121 12.6420 19.469 16.215 20.0721 20.139 16.509 21.9922 20.185 18.593 8.5623 20.254 18.657 8.5624 21.18 19.214 10.2325 21.971 19.672 11.6926 22.592 20.012 12.8927 22.921 20.603 11.2528 23.265 21.038 10.5929 23.636 21.18 11.6030 24.252 22.028 10.10

Thus we saw that the larger mesh shows better results. This was also observed from themodal shapes which were finer for larger mesh whereas for the smaller mesh, interactionbetween concrete and timber was not satisfactory.

From the above figures, it can be seen that the model is better than the previous as thefrequencies are in the acceptable range and also the interaction between panels andconcrete is better.

Thus the new model is far more accurate than the previous one while at the same timeusing less computation due to less number of nodes.

As we can see, the modal frequencies start at around 6 Hz which is higher than 2.1 Hz, theprescribed upper limit for walking frequency for open office space (discussed in Section1.2.1, Page 6). Still, the higher harmonics of the walking force can excite the fundamentalfrequencies of the floor. It is suggested that a walking frequency between 1.5-2.1 Hz beassumed and higher harmonics up to the 6th be taken into account.

Looking at the frequencies for the finer mesh, we take into account the first 12 naturalfrequencies till 11.5 Hz after which we see a jump to 13 Hz.

It is important to note that since a lot of modes are localised, it is better to compare theresults from the full floor with results derived from single panel. The first 21 mode shapesare shown below, grouped by their localised positions.

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Table 12: Room 1

1 - 6.29 Hz 11 11.16 Hz

10 11.12 Hz

Table 13: Room 2

2 6.55 Hz 4 8.25 Hz

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5 8.96 6 9.38 Hz

8 10.48 Hz 12 11.54 Hz

15 13.44 Hz 17 14.65

20 16.21 Hz

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Table 14: Room 3

3 8.2 Hz 13 13.05 Hz

Table 15: Room 4

4 8.25 Hz 8 10.45 Hz

12 11.54 Hz 15 13.44 Hz

16 13.8 Hz 20 16.21 Hz

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Table 16: Room 5

7 10.34 Hz 9 10.86 Hz

10 11.12 Hz 14 13.17 Hz

19 16.12 Hz 21 16.5 Hz

Listing these shapes and truncating till the 12th mode:

Room 1: 6.29, 11.12, 11.16Room 2: 6.55, 8.25, 8.96, 9.38, 10.48, 11.54Room 3: 8.2Room 4: 8.25, 10.45, 11.54Room 5: 10.34, 10.86, 11.12

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From the above figures, it can be seen that the model is better than the previous as thefrequencies are in the acceptable range and also the interaction between panels andconcrete is better.

Thus the new model is far more accurate than the previous one while at the same time

using less computation due to less number of nodes.

1.1 Optimal Sensor Locations

For deciding the optimal sensor locations, 2464 candidate locations were used spread overthe bottom layer of concrete.

The best sensor locations given by Combitorial method (25C3) and EFI-DPR method werecompared.

-200

-150

-100

-50

0

50

3 4 5 6 7 8 9 10111213141516171819202122232425

Value of Log(|Q|

Figure 20: Variation of Log(|Q|) with number of sensors used

The value of Log(|Q|) for 25C3 method was -120 as compared to a value of -168 for 3sensors through EFI method. Also it can be seen that 2 3 sensors placed according to 25C3method is better than 5 sensors according to EFI method.

The positions suggested by 25C3 method are:


1920 8859.96 1620 0

3825 21195.39 13432.81 0

3838 27261.11 5524.172 0

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In comparison the positions suggested by EFI technique are:


1907 2360.04 1620 0

1908 2859.96 1620 0

1909 3360 1620 0

As can be seen, the positions through 25C3 method are better distributed rather thanclubbed around a single position. Thus the have a better chance of picking up all modalfrequencies.

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2. OSP on Segmented Plan

We can now try a different approach by segmenting the floor by taking different roomsindividually. We can find the optimal placement of sensor room-wise.

As is seen in the floor plan (Fig. 21), the floor was divided into 8 sections (rooms). Theoptimal location for each room was then calculated using all 12 mode shapes as input butonly locations falling in that room as the candidate locations. The Optimal location of 1sensor was in each room was found using combitorial technique (NC1). The numbers ofcandidate locations in each room are shown in Table 17.

Figure 21: Floor Usage Plan

Thus, if we just select the rooms with maximum individual |Q| value, we would select room

7,1 and 3. But this is not appropriate as we would end up missing various mode shapeswhich are localised specifically in rooms 5 and 6. Moreover, selecting both 1 and 7 is notsuggested as they both have complimentary mode shapes and placing one sensor in thatregion would be able to get the response of both rooms.

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Table 17: OSP Room Wise

RoomNumber

CandidateLocations

NodeNo.

x y z |Q| Log|Q|

7 (Gallery) 146 1921 9360 1620 0 1.81E-190 -189.7421 222 1908 2859.96 1620 0 7.04E-199 -198.1523 314 3807 14904.5 13936.26 0 1.66E-199 -198.784 314 3779 14992.27 13498.76 0 8.04E-200 -199.095

8 (Corridor) 96 2573 21340.12 5792.047 0 2.90E-210 -209.5382 287 3505 7323.02 14885.1 0 3.14E-211 -210.5035 315 3730 22577.62 13425.05 0 6.45E-216 -215.1916 144 1970 27829.02 1620 0 2.24E-227 -226.649

Figure 22: Room Wise OSP

Thus, we use combitorial method to find the best combination of 3 locations out of these 8possible locations. The results are listed in Appendix.

2.1 Discussion on Room Wise sensor Selection

It can be seen from the results that although room 6 has low individual Q value, it has ahigh priority in combitorial distribution. This can be attributed to the fact that room 6 hashighly localized mode shapes which cant be measured by sensors in any other room.Moreover, room 6 is marked as a room for meetings, an activity which can be disturbed dueto high vibrations. Thus placing a sensor in room 6 becomes important.

It can also be seen that room 1 and room 7 are interchangeable. This can be attributed tothe fact that they have complimentary mode shapes and a sensor in either room may pickup vibrations of both rooms. Moreover, we would be inclined to place a sensor in room 1

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rather than 7 as room 7 is a gallery where vibrations might not be felt explicitly whereasroom 1 is marked as a staff room where even low vibrations may be troublesome for theusers.

Moreover, since rooms 2, 3, 4 and 5 have similar structure and usage, we could place 1sensor in those rooms and record the representative response of those types of rooms.

Thus, a possible distribution could be a sensor in room 6, one in room 1 and 1 in either 2, 3,4 or 5.

Some combinations are shown with their ranking out of 56 in Table 18.

Table 18: Selected distributions (Room wise)

Ranking Room No. Room No. Room No. |Q| Log|Q|

1 5 6 7 3.73E-123 -122.428

2 1 6 7 6.16E-124 -123.211

3 1 3 6 6.53E-125 -124.185

As we see, the best combination includes room 7 whereas we prefer using room 1.Combination 2 includes both room 1 and 7 and excludes locations in room 2,3,4 or 5 whichis not appropriate.

Combination 3 seems acceptable as it places one sensor in room 1, one in room 6 and onein 3, which was what we aimed for.

Table 19 provides a comparison between the two methods.

Table 19: Comparison of EFI-C with Room-wise method

Method Room No. Room No. Room No. |Q| Log|Q|EFI-C

(Full Floor)7 4 6 1.07E-121 -120.971

Combitorial(Room Wise)

1 3 6 6.53E-125 -124.185

It should be noted that although EFI-C method gives better Q value, it places a sensor inroom 7 where vibration may not be considered as troublesome as compared to room 1.

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3. Model with Flexible Supports

To further analyse the response of the structure, another model was proposed in which thesupport conditions were relaxed. The support beams were considered as flexible, pinned atcolumns while displacements at the columns were still restricted to zero. This wasimplemented by creating new elements for the support beams which were bonded to thejoists and concrete. Upon modal analysis, the following mode shapes were obtained:

Table 20: Natural frequencies with flexible supports

Natural Freq. Hz

1 2.2234

2 2.6945

3 3.7664

4 4.7397

5 5.123

6 5.1751

7 5.5815

8 6.3242

9 7.018

10 7.1357

11 7.5539

12 7.7335

13 7.9357

14 8.1476

15 8.385

16 8.6642

17 8.9722

18 9.9943

19 10.245

20 10.319

21 10.553

22 10.852

23 11.016

24 11.62

25 11.731

26 12.12927 12.252

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1: 2.22 Hz 2: 2.69 Hz

3: 3.76 Hz 4: 4.74 Hz

5: 5.12 Hz (Previously 6.29 Hz) 6: 5.175 Hz (Previously 6.55 Hz)

7: 5.58 Hz8: 6.32 Hz(Previously 8.96 Hz)

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9: 7.01 Hz Hz 10: 7.13 Hz

11: 7.55 Hz 12: 7.73 Hz

13: 7.93 Hz 14: 8.14 Hz

15:8.38 Hz 16: 8.66 Hz

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17: 8.972 Hz 18: 9.994 Hz

19: 10.245 Hz 20: 10.319 Hz

21: 10.553 Hz 22: 10.852 Hz

23: 11.01 Hz 24: 11.62 Hz

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25: 11.73 hz 26: 12.12 Hz

27: 12.25 Hz

3.1 Discussion of Mode Shapes with Flexible Supports

As can be seen, the initial mode shapes are quite different from those obtained from fixedrestraints. The following are the points that can be noted:

Due to added flexibility, inter-panel interaction is high

The initial mode shapes are concentrated in the lower part due to longer span of thesupport beams providing for high deflection in them.

Higher mode shapes are quite similar to those obtained in the previous analysis. Thefrequency of mode shapes is lower than the previous case. Eg: Mode shape 1 forprevious case had freq. 6.29 Hz where it now has freq. 5.12 Hz.

Higher mode shapes have different frequency due to added flexibility and inter-panelinteraction. Now they are not totally localized but have effect on neighbouringpanels.

To correlate the two models, the flexible beams were made stronger to reduce deformationand to simulate zero displacement condition. The following is a comparison of modalfrequencies thus obtained with frequencies from analysis with fixed supports.

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ModalFrequency

Fixed Supports(Hz)

Flexible Supports-Strong Beams

(Hz)

PercentageChange

1 6.2909 6.4441 2.432 6.5588 6.6372 1.193 8.2084 8.0774 1.594 8.2506 8.2038 0.565 8.9684 8.6597 3.446 9.3834 8.9247 4.887 10.346 9.0833 12.208 10.485 9.3741 10.599 10.868 9.8408 9.4510 11.121 10.507 5.5211 11.163 10.776 3.4612 11.548 10.92 5.4313 13.058 11.909 8.7914 13.174 12.218 7.25

15 13.444 12.731 5.30

Thus, the values from two analyses seem to converge indicating validity of the methodsused.

1.1 Optimal Sensor Location for Flexible Supports

Three cases were taken to evaluate OSP:

1. Full floor plan2. Floor plan excluding galleries3. Room wise OSP

OSP FOR FULL FLOOR INPUT

The OSP using EFI+C method by calculating 25C3 case were:


1908 2859 1620 0

1972 26829 1620 0

2078 360 13405 0

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Figure 23: OSP for flexible supports

OSP FOR FLOOR INPUT EXCLUDING GALLERIES

242 locations were eliminated when we excluded galleries and possible locations werereduced from 2464 to 2222.

The OSP using EFI+C method by calculating 25C3 case were:


1908 2859.96 1620 0

1972 26829 1620 0

2078 360 13405.41 0

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ROOM-WISE OSP

When doing room-wise OSP, the results were not acceptable as due to large number ofmode shapes, Q value for all nodes reduced to zero and thus nodes became incomparable.

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1. Model with Flexible Columns

In this model, the columns were also modelled as solids of height 6 meters with thefoundation being fixed and floor connections being pinned. The mode shapes thus obtainedwere:

Table 21: Natural Frequencies with columns

Natural Freq. Hz

1 4.762

2 5.2706

3 5.8405

4 6.2746

5 6.3432

6 6.4291

7 6.6198

8 7.0119

9 7.9986

10 8.4477

11 9.0455

12 9.6051

13 9.7518

14 9.9393

15 10.015

16 10.38517 11.016

18 11.46

19 11.656

20 11.72

21 12.156

22 12.38

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1: 4.762 Hz 2: 5.271 Hz

3: 5.84 Hz 4: 6.275 Hz

5: 6.343 Hz 6: 6.429 Hz

7: 6.62 Hz 8: 7.01 Hz

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9: 7.99 Hz 10: 8.44 Hz

11: 9.045 Hz 12: 9.605 Hz

13: 9.752 Hz 14: 9.939 Hz

15: 10.01 Hz 16: 10.38 Hz

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17: 11.01 Hz 18: 11.46 Hz

19: 11.65 Hz 20: 11.72 Hz

21: 12.15 Hz 22: 12.38 Hz

Although the mode shapes are quite similar to those obtained without columns, thefrequency values are different because of the change in interaction between columns andbeams. While previously, the beams were only pinned in translational at places of columns,now due to surface to line contact; the connection is more like a fixed connection. Thus,frequency values increase due to increased stiffness.

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1.1 OSP for Flexible Columns

We used the EfI-DPR+Combinatorial technique to obtain the optimal sensor locations whichmaximize the value of |Q| with the constraint of only 3 sensor locations. For this, first wetake a total of 2464 candidate locations. Using the EfI-DPR technique, we eliminatelocations which have least contribution until we have 25 candidate locations left. Then we

use the Combinatorial method to find the best combination of 3 locations out of these 25options. Thus we select the combination with the highest value of |Q| out of the 2300possibilities.

Thus, on finding the OSP for this model, the following locations were obtained:


1921 9360 1620 0

1953 21335.75 5300 0

3755 9001.61 12905.09 0

Fig. 24 shows the modal displacements at the various senor locations. These give arepresentation the mode shapes being measured by each sensor.

Figure 24: Modal Displacements at 3 selected locations

Thus, we see that using the combinatorial technique, we get a better result than EfI-DPRtechnique. Using combinatorial technique, we get a log(|Q|) value with 3 sensors which ishigher than that obtained with 4 sensors usingEfI-DPR technique.

Fig. 25 shows a representation of the sensor placement.

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Figure 25: OSP for full floor with flexible columns

2. ConclusionThis in this project, we have been able to study the methods used for optimal sensorplacement to measure floor vibrations. EFI-DPR was found to be the most widely usedmethod. Due to our limitation of sensors, we have proposed a modification to the EFI-DPRtechnique by using a direct combitorial method along with EFI-DPR technique, the EFI-Ctechnique.

Results from the EFI-C technique have been found to be better than the EFI-DPR technique.Example: For the initial model without frame elements, we see that by using combitorialmethod with 3, we can get a |Q| value better than that given by EFI with 5 sensors.

The optimal sensor locations were similar for all the models. With one sensor in room 6, onein either room 1 or 7 and 1 in the lower half (rooms 2,3,4 or 5).

Depending on the usage, room wise OSP was also calculated by excluding galleries (rooms 7and 8) from the input candidate locations.

Apon changing the model by adding flexibility of beams, a reduction in natural frequencieswas observed which was expected due to reduction in stiffness of the system.

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On modelling the columns, the change was different as the change in stiffness of columnswas not much whereas stiffness was increased as the interaction between columns andbeams changed from purely pinned to partial fixity.

Thus, a total of 5 possible solution sets are provided:

1. Model without beams Full floor

2. Model without beams Room wise

3. Model with beams Full floor

4. Model with beams and columns Full floor

Out of these 4 options, option 4 is the most realistic as it models all the columns, beams

and floor elements and their interactions. Moreover, the results from this correlate with

results from option 2 and 3.

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1. References

1. Pavic, A. and P. Reynolds, Vibration serviceability of long-span concrete buildingfloors. Part 1: Review of background information. Shock and Vibration Digest, 2002.34(3): p. 191-211.

2. Aalami, B.O. (2008) Vibration Design of Concrete Floors for Servicability.3. Oliver Hechler, M.F., Christoph Heinemeyer, Flavio Galanti, Design Guide For Floor

Vibrations. 2008, Eurosteel 2008: Graz, Austria. p. 6.4. Andriacchi, T.P., J.A. Ogle, and J.O. Galante, Walking speed as a basis for normal

and abnormal gait measurements. Journal of Biomechanics, 1977. 10(4): p. 261-268.

5. ivanovi, S., Pavi, A. and Reynolds, P., Vibration serviceability of footbridges underhuman-induced excitation: a literature review. Journal of Sound and Vibration, 2005.279: p. 1-74.

6. Matsumoto, Y., et al., A study on dynamic design of pedestrian over-bridges.Transactions of the Japan Society of Civil Engineers, 1972. 4: p. 50-51.

7. Matsumoto, Y., et al., Dynamic design of footbridges. IABSE Proceedings, 1978. P-

17-78(78): p. 1-15.8. Leonard, D.R., Human tolerance levels for bridge vibrations. TRRL Report No. 34,1966. 34.

9. Bachmann, H., A.J. Pretlove, and H. Rainer, Dynamic forces from rhythmical humanbody motions. Vibration Problems in Structures: Practical Guidelines, 1995.

10. Sachse, R., A. Pavic, and P. Reynolds, Human-structure dynamic interaction in civilengineering dynamics: A literature review. Shock and Vibration Digest, 2003. 35(1):p. 3-18.

11. Pavic, A. and P. Reynolds, Vibration serviceability of long-span concrete buildingfloors. Part 2: Review of mathematical modelling approaches. Shock and VibrationDigest, 2002. 34(4): p. 279-297.

12. Meo, M. and G. Zumpano, On the optimal sensor placement techniques for a bridgestructure. Engineering Structures, 2005. 27(10): p. 1488-1497.

13. Kammer, D.C. and M.L. Tinker, Optimal placement of triaxial accelerometers formodal vibration tests. Mechanical Systems and Signal Processing, 2004. 18(1): p.29-41.

14. Meo, M. and G. Zumpano. Optimal sensor placement on a large scale civil structure.2004. San Diego, CA.

15. Reynolds, P. and A. Pavic, Vibration performance of a large cantilever grandstandduring an international football match. Journal of Performance of ConstructedFacilities, 2006. 20(3): p. 202-212.

16. Li, D.S., H.N. Li, and C.P. Fritzen, A note on fast computation of effectiveindependence through QR downdating for sensor placement. Mechanical Systemsand Signal Processing, 2009. 23(4): p. 1160-1168.

17. Camelio, J.A., S.J. Hu, and H. Yim, Sensor placement for effective diagnosis of

multiple faults in fixturing of compliant parts. Journal of Manufacturing Science andEngineering, Transactions of the ASME, 2005. 127(1): p. 68-74.

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2. Appendix

Table 22: 56 Combinations of 3 sensors

Room No. Room No. Room No. |Q| Log|Q|

5 6 7 3.73E-123 -122.428

1 6 7 6.16E-124 -123.211

1 3 6 6.53E-125 -124.185

1 4 6 1.83E-125 -124.736

6 7 8 5.65E-126 -125.248

1 6 8 5.47E-126 -125.262

3 6 7 5.04E-126 -125.298

4 6 7 1.02E-126 -125.99

1 5 6 5.33E-127 -126.273

1 7 8 4.98E-127 -126.303

5 7 8 8.10E-128 -127.091

1 5 8 3.89E-128 -127.41

1 4 8 3.26E-128 -127.486

1 3 8 5.80E-129 -128.236

1 2 8 4.39E-129 -128.357

2 6 7 2.61E-129 -128.5841 5 7 1.30E-129 -128.887

4 7 8 3.61E-130 -129.443

3 7 8 1.60E-130 -129.796

1 2 6 1.39E-131 -130.858

2 7 8 2.50E-132 -131.603

1 3 5 2.73E-133 -132.564

1 4 5 1.56E-133 -132.807

1 2 5 1.68E-134 -133.775

4 5 7 1.22E-135 -134.912

3 5 7 6.79E-136 -135.168

2 5 7 1.11E-136 -135.955

1 3 7 1.25E-137 -136.904

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2 6 8 9.23E-140 -139.035

1 4 7 9.08E-140 -139.042

2 4 6 6.18E-140 -139.209

2 3 6 1.23E-140 -139.912

1 3 4 6.98E-141 -140.156

1 2 4 9.82E-142 -141.008

1 2 3 5.70E-142 -141.244

2 5 8 3.91E-142 -141.408

2 5 6 1.07E-142 -141.97

3 4 7 8.25E-143 -142.083

2 4 8 4.40E-143 -142.356

2 3 8 1.04E-143 -142.9842 4 7 5.03E-144 -143.299

2 3 7 1.47E-144 -143.832

3 5 6 5.76E-146 -145.24

3 6 8 1.69E-147 -146.773

4 5 6 1.41E-147 -146.852

4 6 8 1.15E-147 -146.941

2 3 5 1.06E-148 -147.974

2 4 5 7.25E-149 -148.144 5 8 5.01E-150 -149.3

3 5 8 3.68E-150 -149.435

3 4 6 2.46E-150 -149.609

1 2 7 1.10E-151 -150.959

3 4 8 2.83E-152 -151.548

2 3 4 4.18E-156 -155.379

3 4 5 5.75E-160 -159.24

report_updated-section-2.3

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