validation of a virtual shaker testing approach for...

Validation of a Virtual Shaker Testing approach for improving environmental testing performance

S. Manzato1, F. Bucciarelli1,2, M. Arras2, G. Coppotelli2, B. Peeters1, A. Carrella1 1 Siemens Industry Sector, Siemens Industry Software N.V., Test Technology R&D, Interleuvenlaan 68, B-3001, Leuven, Belgium e-mail: [email protected] 2 Università degli Studi di Roma “La Sapienza”, Dipartimento di Ingegneria Meccanica e Aerospaziale, Via Eudossiana 18, 00184 Roma, Italy.

Abstract In the field of vibration testing, the interaction between the structure being tested and the instrumentation hardware used to perform the test is a critical issue. This is particularly true when testing massive structures (e.g. satellites), because due to physical design and manufacturing limits, the dynamics of the testing facility often couples with that of the test specimen in the frequency range of interest. Therefore it is of paramount importance to improve environmental testing performances by considering the dynamic coupling between the test specimen and the instrumentation hardware and take suitable countermeasures before running the actual program. In this context, a “Virtual Shaker Approach” is developed to run a multidisciplinary simulation which closely represents the real vibration test. For these reasons, models accurately replicating the behavior of the different hardware involved in the environmental test need to be developed and validated. Starting from these models, the Virtual Shaker approach can then be used to optimize test execution, improving controller performance and develop new methods to exploit the available experimental data.

1 Introduction

Laboratory vibration tests on (large spacecraft) structures essentially serve 2 goals: qualification of the structure by subjecting it to vibration environments which are representative for the operational conditions and validation of the Finite Element Model for a reliable simulation of the coupling of the structure with the launcher. However, in most cases the interaction between the structure being tested and the test facility hardware represents a critical issue. This is particularly true when massive structures are tested (e.g. satellites), because due to physical design and manufacturing limits, the dynamics of the testing facility often couples with that of the test specimen in the frequency range of interest. Specifically in the space business, test engineers pursue, for vibration testing, the following conditions:

• a theoretically fixed-free interface; • a desired (acceleration) profile with a certain frequency content within tight margins in terms of

amplitude deviation and minimum cross-coupling; • an unchanged damping for the spacecraft eigenmodes.

As these conditions are difficult to achieve in practice for large structures and unexpected behavior is observed, there is a high risk of overtesting and expensive delays in the test programs. Therefore, it is of paramount importance to be able to foresee these testing difficulties and take countermeasures before running the actual program. Hereto, a “Virtual Shaker Testing” approach is currently being developed that requires the integration of the following simulation blocks (Figure 1):

• the structural model (possibly including non-linearities);

767

• the test facility and shaker (electro-dynamic or hydraulic) model; • the vibration control system model.

(a) Physical

(b) Virtual

Figure 1: Vibration test facility (a) physical and (b) Virtual. The different blocks are also marked: (A) Closed-loop vibration controller; (B) Vibration exciter(s); (C) Unit Under Test.

By carrying out such a “virtual shaker test”, the test engineer can evaluate the test performance of the mentioned system prior to actually putting things in operation. Firstly, this will help defining the proper selection of all parameters involved in the experiment (location of control, measurement and notching sensors, controller settings such as sweep rate, number of periods and compression factor), thus accounting for a smoother test deployment. Also, sensitivity studies can be performed to quantify the importance of shaker-structure interaction and its effect on the controller. Finally, this process can lead to nimbly correlate the mathematical models with experimental results and gain deeper insight about the overall system physics. In the past, some works have already been presented about this virtual vibration testing subject. In [1], a clear and comprehensive view of the overall procedure in the case of satellites qualification testing, emphasizing the deployment of a specific facility used for such vibration tests in a major European test center is provided. [2] focuses on the coupled electrodynamic exciter lumped parameter model and the vibration controller. These results were taken as the starting point for this analysis, where additional steps towards a more accurate implementation will be taken. First, the methodology to identify the parameters of a shaker model installation will be applied and the models of the controller and shaker validated. After that, the attention will move to the UUT and an experimental campaign performed on a UAV helicopter will be presented together with the developed numerical models. Using then the table acceleration time histories measured during a controlled sine vibration test, the importance of properly modelling the connections will be demonstrated as they will influence the identified modal parameters as well as the test execution itself.

2 Controller implementation

Vibration testing of large and/or expensive structures requires the test to be carried out in a controlled manner. This differs significantly from classical modal testing approach, as the user is asked to specify a desired output (reference) level instead of an input one. This in turn means that some control application is required to close the loop, shaping the drive spectrum so that the excitation levels of the spectrum match

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the pre-defined ones verifying that the tested unit is not in danger and shutting the test down if it is. Among the different types of excitation that are applied, this paper will focus on the sine control excitation and the implementation in the software LMS Test.Lab Sine Control [3] has been chosen here as a basis for the modeling process. The measurement flow associated to the sine control application can be summarized as:

• a pure sine drive signal, with a certain amplitude and frequency, is provided to the shaker(s) power amplifier(s);

• a number of transducers mounted on the UUT measure its response at several locations, and some safety checks (overload, open loop, channel abort, etc.) are performed;

• a vibration amplitude (control) level is estimated and used to compute the next amplitude of the drive signal, depending on measurement state, amplitude reference profile, and a set of control parameters. Next, frequency and phase are also computed based on parameters defining the sine sweep;

• the drive signal is updated and a new measurement cycle is started. By sweeping throughout the specified frequency band the complete UUT response is measured.

The ultimate purpose of such a control system is to maintain the response of selected measurement point(s) at predetermined levels independent of structure dynamics. The reference profile levels for the various frequencies are set up by the test engineer to meet the particular test requirements. They are typically chosen to approximate those that will be encountered by the UUT during service. This application has been modeled using Matlab Simulink, keeping both software [3] and hardware [4] aspects into account to try to replicate as much as possible the controller behavior. Details on the controller implementation and the logic behind, as well as validation and sensitivity analysis, are discussed thoroughly in [2].

3 Shaker modeling

3.1 Electrodynamic shaker physics

Depending on the test installation, electrodynamic [1] or hydraulic [5] shakers are commonly used for environmental testing applications of aerospace structures. In this paper, we’ll focus on electrodynamic vibration exciters, which are extremely popular for use in vibration testing, due to the wide range of forces and frequencies they can obtain [6], the ease of use and the possibility to operate both in open-loop and closed-loop setups. The structure of such a device bears resemblance to a common loudspeaker, being more robust with respect to it [7]. Figure 2 (left) shows a schematic representation of the shaker structure. A coil (C) is suspended in a radial magnetic field acting in the normal plane with respect to the coil axis. Depending on the size of the shaker, this field is produced by building a magnetic permeable circuit to transmit flux from both poles of an axially magnetized permanent magnet (small shakers) or an electromagnet (large shakers). When a current is passed through the coil, an axial force is produced in proportion to this current and transmitted to a table structure (T) to which the coil is attached. An elastic suspension system (S) allows the coil to move axially while constraining all other motions. The table, suspension and coil assembly is called shaker armature (A) and represents the moving element of the device. The exciter base holding the magnetic elements, which is frequently isolated from the building floor by use of compliant mounts or pneumatic suspensions will be referred as shaker body (B).

DYNAMIC ENVIRONMENTAL TESTING 769

Figure 2: Electrodynamic shaker scheme (left) and lumped parameter model representation (right).

3.2 Lumped parameter model

Starting from the evidence of shaker operation, a number of lumped parameter electro-mechanical models has been developed in [7], [8] and [9]. The one chosen for this analysis is the one shown in Figure 2 (right), in which the mechanical model is simulated by a 3 DOF system. The compliant connection between the armature assembly and the shaker body form a first degree of freedom. Another DOF is introduced when considering the armature structure as being elastic and including the compliant suspension between the coil and the table. Finally, the shaker isolation system of the exciter, when present, can also be modeled as a spring and damper in between the body mass and the ground. These three DOFs mechanical system is then coupled to an electrical circuit which account for the complex impedance of the armature coil: the minimum (DC) impedance exhibited at the shaker input terminals defines the coil resistance, while the increase of such impedance with frequency holds for its inductive component. The interaction between the mechanical and electrical domain is two-sided. On one end, the force F [N] acting on the coil is proportional to the current flowing through it, as for Ampere’s law:

iKinBlF f== (1)

On the other hand, when the coil moves within the magnetic field, a voltage is generated across the coil in proportion to its velocity. According to Faraday’s law, such a back-electromotive force Ebemf [V] is given by:

xKxnBlE vbemf == (2)

The discussed electrical and mechanical models can be integrated and coupled in a single system of equations in the following form:

=

−

+

−

+

ei

RK

K

dtdi

LKKf

f

vv

0x

0

Kx0Cx0

0M 0000

(3)

The above equations where replicated in the LMS Imagine.Lab AMESim environment, where the mechanical and electrical part can be easily coupled together. By linearizing the mechanical part, the following shaker modes where found:

• the low-frequency mode, with the isolation mounts (or pneumatic suspension) allowing the shaker to translate as a whole with almost no relative motion between the component in the so called Isolation Mode;

Coil

N

S

Load table

Permanent magnetor electromagnet

Innerpole piece

Outer pole piece

Coil form

Elastic suspension

Isolation mounts

Coil

N

S

N

S

N

S

Load table

Permanent magnetor electromagnet

Innerpole piece

Outer pole piece

Coil form

Elastic suspension

Isolation mounts


• the Suspension Mode dominates the low end of the operating range, with the armature assembly (C+T) moving relatively to the shaker body.

• finally, at or beyond the higher frequency limit of operation the Coil Mode appears, being characterized by out of phase motion of the coil and the table.

3.3 Model parameters identification

A system identification methodology based on analytical formulation [2] has been applied to estimate the shaker model parameters of the ES-6-200/GT700M (Figure 3) Electrodynamic Shaker at the Aerospace and Mechanical Engineering Department of La Sapienza University in Rome. The first step in the identification procedure is to consider the mechanical acceleration-over-current FRF response of the shaker as a superposition of a number of components. To measure the current sent from the amplifier and applied to the coil, a FLUKE AC probe was used and connected to the LMS SCADAS acquisition system as an additional measurement channel (Figure 4). The acceleration was measured by an accelerometer placed on the table. According to [10], the armature can be considered as a rigid body as long as the system is excited at frequencies which are less than one half of the bare table coil resonance. In this frequency band, the armature accelerance transfer function can be expressed as:

( ) ( )asasa

f

analytical

a

mksmcss

mK

IX

++

= 2

2 (4)

Figure 3: The ES-6-200/GT700M shaker.

Figure 4: FLUKE AC i1000s Current Probe

measuring the amplifier output current.

By measuring the transfer function twice (each time with a different calibrated mass on the table), a modal identification algorithm (in this case PolyMAX) is applied to each FRF. By using the estimated modal parameters, a couple of accelerance in the form:

2,,,

2

2

, 2 isisis

i

idsynthesize

a

sssA

IX

Ω+Ω+=

ζ

(5)

can be synthesized. By following the procedure detailed in [1], the armature mass ma, the force-current coefficient Kf and the suspension stiffness and damping (ks, cs) can be estimated. In Figure 5, the FRFs measured using two different masses on the shaker table are compared. Figure 6, on the other hand, shows the stabilization diagram obtained from one of the FRFs. As can be seen, due to the additional masses (the added masses where not perfectly centered and the connection to the table possibly not rigid) some additional dynamic response was introduced. However, it was still possible to isolate the mode of interest (the one at lower frequency in Figure 6) and perform the identification correctly.


Figure 5: acceleration/current FRFs with two

different masses on the table

Figure 6: PolyMAX stabilization diagram for

identification. Due to mass unbalances, more poles than the 2 theoretical are found by the algorithm.

A similar procedure is applied on the coil mode, where the equivalent system around resonance is a 2 DOFs one, found when considering the armature structure alone regardless of external (suspension) effects. In this case, coil (mc) and table (mt) masses, as well as the coil stiffness (kc) and damping (cc) are obtained. Finally, the same approach can also be used to estimate the “body” parameters using a single DOF equivalent model. To extract the isolation mode, sine sweeps at low frequency and high amplitude level were defined, and seismic DC accelerometers were used. However, large sine inputs means, at low frequency, large displacements and the validity of the linear relations in Equations (1) and (2) is not guaranteed anymore. This might erode the model validity in this frequency range, leading to potentially unreliable parameter estimates. Despite the additional measured dynamics introduced by the added masses, it was possible to correctly identify the model parameters to match the model behavior. After estimating the mechanical model properties, the remaining unknowns of the electrical part of the model will be identified. Assuming to focus on frequencies below half of the coil resonance, the coil mobility can be assumed equal to the table mobility. Moreover, we can expect the armature mobility to be much higher that the body one, especially around the suspension resonance, so that the latter could be neglected. Finally, assuming a constant power amplifier voltage gain values, the voltage over current transfer function can be expressed as:

IVG

IX

jK

LjR vav =

ω+ω+

(6)

where aX is the armature accelerance and Kv the voltage/velocity coefficient. By writing down this equation for each spectral line over the considered frequency band and solving the resulting least squares problem for the unknowns vGLR ,, , the set of estimated parameters necessary to characterize the treated lumped parameter model results complete. As an alternative to estimating vG , the voltage fed into the shaker could be measured instead of the DAC output signal of the front-end; however, these voltages at the amplifier output are typically too large to be measured with a typical data acquisition front-end having an input range of ±10V. The identified coefficients are summarized in Table 1.


Parameter Value

Coil mass 2.91 [kg] Table mass 6.67 [kg] Body Mass 204.73 [kg]

Coil Stiffness 8.91E+8 [N/m] Suspension Stiffness 2.26E+5 [N/m]

Body Stiffness 1.367E+5 [N/m] Coil Damping 519.45 [ N/(m/s)]

Parameter Value

Suspension damping 743.37 [ N/(m/s)] Body damping 135.89 [ N/(m/s)]

Inductance 1.56E+4 [H] Resistance 0.73 [Ohm]

Force/Current Coeff. -74 Voltage/Velocity

Coeff 74

Table 1: Identified model parameters values.

3.4 Model validation

3.4.1 Open-loop shaker model validation

As a preliminary model validation, the stand-alone shaker model results will be compared with the measured results. This is obtained by removing the controller sub-model and by feeding the shaker with the measured output voltage from the DAC. This will allow decoupling the two models and focusing on the validation of each of them separately. The same methodology will also be applied on the helicopter. Although the table acceleration and the current were measured together with the shaker, only the table acceleration will be validated, as it is in general the only measured signal in a standard environmental test. In Figure 9 the time and frequency domain simulated and measured accelerations are compared. Some differences can be already observed, but in general the predicted levels are in line with the simulated one. In test, of course, also noise is measured, which is not included in the simulations. Moreover, during test the accelerometer was measuring also the 50 Hz components and some of its harmonics, which are clearly visible in the frequency domain comparison. Another difference are the peaks around 100 and 350 Hz in the simulation model, which could not be explained considering the model parameters. As a result, the voltage signal used to drive the shaker simulation was analyzed and its spectrum is shown in Figure 10. These fluctuations, related also to the additional resonances observed in Figure 6, can explain the frequency and time domain fluctuations in the simulated data in Figure 7.

Figure 7: Time (left) and frequency (right) domain validation of the table acceleration considering only the shaker model

0 100 200 300 400-3

-2

-1

0

1

2

3

Time [s]

Acc

eler

atio

n [m

/s2 ]

TestSimulation

0 100 200 300 400 50010-7

10-6

10-5

10-4

10-3

10-2

10-1

Frequency [Hz]

PS

D [g

2 /Hz]

SimulationTest


500.000.00 Hz

3.48e-3

0.40e-9

LogV

F Spectrum Tension

Figure 8: DAC measured voltage spectrum used as a driver for the open-loop simulation

3.4.2 Closed-loop shaker and controller model validation

The next step in the validation of the virtual shaker approach is the comparison between the measured sine control test and the one simulated using the shaker model together with the controller described in Section 2. The same reference acceleration profile was defined as in the test, with the only difference that at lower frequencies a small ramp-up was defined to increase the stability of the solution. The comparison of the results is reported in Figure 9. The online estimated averaged spectrum of the table acceleration is shown against the reference profile defined. While in the test some fluctuations are observed, the simulated results perfectly match the reference profile, except at lower frequencies were some initial transient due to some numerical problems is observed. Besides the obvious absence of noise, the simulated shaker model assumes perfectly rigidly masses connected on the table, while, as already discussed, this was not the case during the test. The model is then able to correctly match the desired profile at least up to 300 Hz.

Figure 9: Reference versus estimated table acceleration spectrum; test (left) and simulation(right).

4 UAV helicopter

The last element in the Virtual Shaker approach is the numerical model of the structure under test. As typically environmental tests involve testing the structure under transient loads, they require implementing the structure in a flexible Multibody environment, so that the response to the table acceleration can be simulated. Within the multibody environment, the shaking table, the structure under test and their connections need to be implemented. If the UUT must always be implemented as a flexible body to capture its dynamic response, the user can choose whether to consider the table and the connections as flexibles to verify possible interactions with the response of the structure. These choices are quite critical to replicate the actual test correctly and it is important to understand their influence on the global response. For these purposes, a numerical model of a UAV helicopter has been developed and tests

0 50 100 150 200 250 300

0.05

0.1

0.15

0.2

0.25

Frequency [Hz]

Acc

eler

atio

n am

plitu

de [g

]

Spectrum ReferenceSpectrum AcgControl


performed in the structural dynamics lab of the Mechanical and Aerospace Engineering Department of La Sapienza University in Rome.

4.1 Experimental Modal Analysis test campaign

The UAV helicopter is shown in Figure 10. The main rotor blades where removed, both to simplify the test but also to avoid to include all the flexible rotor modes which usually dominate the lower frequency structural response. As it was not possible to remove also the tail rotor, the transmission shaft was locked to avoid undesired rotation during the test. The experimental campaign was designed to obtain first modal models in free-free conditions of the different sub-components and then of the complete assembled helicopter to perform model validation and updating. Also, different configurations were tested, both at component and assembled level, to replicate different flight conditions for future analysis (with and without LMS SCADAS Recorder DAQ, with different fuel levels).

Figure 10: The tested UAV helicopter.

The following components and configurations were tested:

• Landing pad with and without the data acquisition system; • Left and right tail rods; • Tail boom; • Fuselage with and without fuel; • Assembled helicopter with/without LMS Scadas and empty fuel tank; • Assembled helicopter with/without LMS Scadas and approximately half a liter of fuel.

The different components were analyzed in free-free conditions by suspending them with soft bungees to try to isolate as much as possible the rigid body modes from the flexible ones. A combination of the roving hammer and roving accelerometers approaches was used, to excite the structures in different directions and obtain modal models in as many points as possible. Modal models of the different components were then obtained by processing the acquired Frequency Response Functions with the Polymax identification method. The modal identification of the different components focused on the more global modes below 200 Hz.

4.2 Numerical modelling

Finite element models of the different components were then developed after the test. While for some of the components the implementation was relatively straightforward, for others, such as the fuselage, some strong approximation needed to be made. In particular, no detailed models for the engine and gearbox and all the other mechanical components were implemented, but they were modeled only as rigid stiffeners


with lumped masses connected to the main frame. Before assembling the components, a correlation analysis was performed. While for the tail substructure a good correlation was obtained, as can be observed in Figure 11, the fuselage model didn’t correspond well with the experimental results. In this analysis however, the numerical model was developed to obtain results representative of the structure under test to perform some preliminary investigation on the effects of flexible connection in the response of the system. A more accurate validation and updating analysis is currently still being performed to try to match as much as possible the experimental and numerical results. The assembled numerical model of the helicopter is shown in Figure 12.

Figure 11: MAC matrix and relative frequency errors for the tail model up to 240 Hz.

Figure 12: Assembled helicopter model including the fuselage and the tail components

4.3 Sine control vibration test

The assembled helicopter without the landing component was placed on the shaker table described in Section 3.3 and an environmental test under controlled sine excitation was performed. The structure mounted on the shaker installation is shown in Figure 13. The control accelerometer was positioned on the table while the remaining 11 available accelerometers were used to acquire the vibration response on the different components (nose, fuselage, tail) in locations that were also used during the modal test. The test was performed with the shaker in the upright position, exciting the structure with a linear sine sweep of 2 Hz/s with amplitude of 1g from 10 to 256 Hz. The results of the test in terms of reference against measured table acceleration profile are shown in Figure 14, showing that the controller was able to properly excite the structure with the required profile.


Figure 13: Helicopter mounted on the shaker for the

sine control test.

Figure 14: Since control test reference vs. measured table acceleration profile.

During the test, time histories of the accelerations at the different measurement locations were also acquired to allow further data analysis, in particular with the objective of deriving a modal model for the helicopter installed on the shaking table. In [11], a comparison of the different modal parameters that can be obtained during a vibration test is shown. [12] then demonstrates the possibility of extracting from a base excitation test the modal parameters of the structure in fixed-based configuration by using transmissibility functions instead of standard Frequency Response Functions. However, the main assumption of this method is that the measured table acceleration is perfectly transferred to the structure, meaning the table and the connections between the table and the UUT are perfectly rigid. Only within this assumption, the poles identified from transmissibilities functions corresponds to those of the structure in fixed-based boundary conditions. In this case, the helicopter was mounted on the shaker table using small steel blocks, which might however introduce some additional flexibility. As it is practically impossible to realize a perfectly rigid connection, an accelerometer was placed on top of this block to verify how flexible the connection in the frequency range of interest is.

Figure 15: Examples of the calculated transmissibility functions between the reference accelerometer on

the table and other interesting measurement points

The transmissibilities between the table acceleration and those acquired in the structure where computed and some of them are shown in Figure 13.

• The red curve, showing the transmissibility across one of the mount between the table and helicopter, shows some dynamic response, although limited, that might have effects on the fixed-base assumption.

• The blue curve, which is computed using one of the fuselage acceleration, shows that the frame can be considered rigid in the vertical direction in the frequency range of interest.


• Finally, the two green curves show the acceleration on the two rods connecting the fuselage to the tail. Even though these two rods are supposedly the same, their dynamic response is completely different and most of the peaks are shifted, as was actually also observed during the modal test.

Modal parameter identification using the standard Polymax method was then performed on the computed transmissibilities and the modes up to 200 Hz were extracted. Some of the results are shown in Figure 16. On the left, the quality of the synthesized transmissibility using the estimated modal model is compared with the measured one, showing that the dynamics of the structure is properly described. Then, the autoMAC for the identified modal matrix is computed and shown. From this matrix we see that with the sensor configuration defined for this test some cross-correlation exists between modes. Moreover, no sensors measured the lateral acceleration so some of the correlating modes, in particular in the big cluster in the center of the matrix, are probably associated to coupled vertical and lateral modes and are then difficult to distinguish with the implemented configuration. As no modes were extracted with the helicopter in fixed-based configuration, a virtual testing was then performed to understand the influence of connection flexibility on the identified poles.

Figure 16: Example of synthesized transmissibility (left) and autoMAC mode correlation matrix (right)for

the data extracted from the transmissibilities.

4.4 Simulated results under base excitation

A preliminary analysis to understand the influence of connection flexibility on the poles estimated using transmissibility functions was performed by running a simulated experiment. First, some of the nodes at the bottom of the numerical model of the helicopter in Figure 12 were clamped and the modal solution computed to obtain the theoretical fixed-base modes. Then, in the LMS Virtual.Lab Motion multibody simulation environment, a rigid body model of the table was created and the Finite Element model of the helicopter was connected to it at the 4 physical connection points using a Craig-Bampton reduction technique including the first 50 flexible modes. The helicopter and table where then connected using both perfectly rigid constraints restraining all degrees of freedom as well as flexible bushing elements with the lower stiffness set in the vertical direction to 2 and 0.5 kN/mm and a fixed damping of 500kg/s. The model is shown in Figure 17. The vertical acceleration on the shaker table measured during the test was then defined also in the simulation model and the equivalent displacement profile to obtain the same acceleration was then obtained using a Time Waveform Replication approach. After performing the simulations, the simulated accelerations at the same locations measured during the real test, both with rigid and flexible connections, were exported to the LMS Test.Lab platform and the transmissibility computed using the same approach as described in Section 4.3.


Figure 17: Multibody model of the UAV helicopter

on the shaking table

Figure 18:Relative bushing displacement for the simulation with flexible mounts with different

stiffness

First, the relative displacement of the flexible mounts was analyzed and the results are shown in Figure 18. In both the analyzed cases, the relative displacement at the mounts was less than 0.15 mm, but with the lower stiffness value the effect of the relative higher damping can be clearly observed as the peaks are generally higher but less sharp. However, to get a clearer idea on the effects of having flexible mounts, the time histories and transmissibility functions in one of the virtually measured points are compared in Figure 19. The effect of the flexible connections is clearly visible; beside the softening effect visible in the lower frequency peak, the added damping is almost filtering out all the dynamic response at higher frequencies. Moreover, the softer the connection (and thus the higher the relative damping) the more different the transmissibilities looks and the more different the estimated poles are going to be.

Figure 19: Comparison of acceleration (top) and transmissibilities (bottom) at the same point using rigid

or flexible connections.

The Polymax method was then used to identify the modal parameters for the 3 simulated cases and the real mode shapes were compared to those obtained with the fixed-based Finite Element Model. Figure 20 left shows the autoMAC of the identified modes with the rigid connections, while on the right the correlation with the fixed based modes in shown. The strong correlation between the identified modes shown by the autoMAC demonstrates that the measured degrees of freedom are not sufficient to properly describe the tested structure, so the sensor configuration for the real test should be revisited. On the other hand, the MAC matrix with the true modes shows that some of the modes correlate very well, in particular those which show a dominant vertical component. However, the modes for which the lateral dynamic is dominant are, as expected, not identified at all during the test under vertical excitation.


Figure 20: (left) autoMAC of the identified modes from simulated transmissibilities with rigid connection

and (right) MAC matrix with the true fixed-based modes

The same considerations can be done comparing all identified mode sets in the different conditions to understand how much the stiffness influences the identified poles and mode shapes. The evolution of the poles compared to the true model due to variations of the connection properties is shown in Figure 21.

Figure 21: Relative variations of some of the structural poles for the 3 different connection properties.

Some of the poles are shifting in frequency as the stiffness properties of the connections change. This confirms also the difference between the transmissibilities shown in Figure 19, where not only the amplitude of the transmissibilities is changing but also the peaks show shifts in frequency. Because of the poor spatial resolution of the sensors, investigating the variations of the mode shapes is very complicated, in particular because most of the dynamic in the range between 50 and 120 Hz is dominated by the lateral component. However, this analysis shows already the importance of properly modeling the connections, firstly to validate the assumption of fixed-base modes when using transmissibility functions, but also in the broader context of the Virtual Shaker to be able to correctly replicate the structural response during the test and the additional dynamic introduced by the compliant supports.

5 Conclusions and future works

In this paper, some practical aspects for the development of a Virtual Shaker Testing approach to reliably simulate environmental testing were presented. The development of such a tool is of paramount importance for many aspects, the most important being the possibility to predict the outcomes of the test and the response of the structure of the test, optimize the plan execution by properly selecting control and measurement sensors, optimization of the control law and the prediction of the structural response for further validating the numerical model. The methodology for identifying the shaker parameters and implement a model able to replicate the behavior of an installation was presented and validated against experimental results, showing the importance of including also the controller model in the validation process. After this, a numerical model of a UAV helicopter was developed based on the results of an experimental modal analysis campaign. The model was then use to demonstrate how much an adequate identification and modeling of the connection between the table and the UUT are important to correctly

0.000

2.000

4.000

29.3 32.1 73.9 84.8 88 129

Rela

tive

Pole

va

riatio

n [%

]

True Fixed-base frequency [Hz]

RIGID

STIFF

SOFT


characterize the response of the structure. Moreover, it was demonstrated how the results of an environmental test can be used, under specific assumptions, for validation purposes, bearing in mind that an adequate sensor setup is needed to correctly identify and characterize the structural mode shapes in fixed-base condition. These preliminary considerations will be extended in the future by developing specific methodologies to experimentally characterize and model the connections, as well as integrating the controller, shaker and UUT model together to fully simulate and predict the execution of an environmental testing.

Acknowledgements

The research was conducted in the frame of the project IWT 130936 ADVENT (Advanced Vibration Environmental Testing). The financial support of the IWT (Flemish Agency for Innovation by Science and Technology) is gratefully acknowledged.

References

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