Experimental dynamic load simulation by means of modal force combination

Aerospace Science and Technology, 1997, no 4, 267-275

Experimental Dynamic Load Simulation by Means of Modal Force Combination J. M. Sinapius DLR, Institute

of Aeroelasticity,

Bunsen&.

10, 37073

Goettingen,

Germany.

Manuscript received July 9, 1996, revised version September 27, 1996.

Sinapius J. M., Aerospace Science and Technology, Abstract

1997, no 4, 267-275.

The increase of dimensions and recent advances in the lightweight design of space systems have led to the requirement of new structural dynamic qualification techniques. The paper presents a new concept called the Modal Force Combination (MFC) technique. The method replacesthe base acceleration commonly applied during structural dynamic qualification by having a set of forces act directly in a selected number of structural points. The forces compensate for the d’Alembert inertia forces which arise due to transient multi-axial base acceleration. The transient forcing functions are calculated by means of modal deformations and related modal force distributions in order to obtain the same structural responsesof the structure as would occur with base acceleration. The MFC method offers two possibilities of performing a simulation: the energy method and the deflection method. Both versions were investigated experimentally. The essentialsof the verification tests are presented. The results of experimental dynamic load simulation by means of modal force combination are compared with measurements from base excitation performed on DLR’s multi-axis vibration simulator MAVIS in Goettingen, Germany. In conclusion, the state of the art of the development of the new qualification method is discussed. Keywords: structural dynamic qualification - modal force combination - multi-axis vibration test flight load simulation. Schlagwiirter: Strukturdynamische Qualifikation - Modalkraftkombination - Mehrachsenvibrationstest - Fluglastsimulation

List of symbols Matrices

Ml [nl

Viscous damping matrix Stiffness matrix Mass matrix Transfer function matrix Real part of transfer function Imaginary part of transfer function Geometric transformation matrix Eigenvector matrix Modal force vector matrix

Vectqrs {f(t)}, {F(w)}

d’Alembert

Aerospace

Science and Technology,

interia forces

0034.1223,

97/04/$

7.00/O

Gauthier-Villars

load simulation forces relative displacements absolute displacements 6 DOF displacement of base r-th eigenvector modal coordinates modal force vector participation frequency natural frequency of the r-tb mode generalized mass of the r-th mode viscous damping of the r-th mode eigenvalue

268

I - INTRODUCTION The past decade saw the development of several new structural dynamic qualification techniques. Dynamic qualification means the experimental simulation of the structural dynamic environment in order to check the dynamic behaviour of spacecraft. The load environment can be characterized by its nature and the sources by their time histories: l quasi-static loads (e.g. thrust, aerodynamic loads) a quasi harmonic loads (e.g. engine chugging, pogo instability) l transient loads (e.g. thrust build-up and decay, stage separation) l random loads (e.g. wind and vortex turbulence) Most of these loads occur during launching. Their significant characteristics are their multiaxial operation, transient nature, and superposition of the quasi-static and dynamic excitation. The structural dynamic qualification of space systems is presently accomplished by means of singleaxis electrodynamic shaker systems. During the last ten years, multi-axis hydraulic shaker systems have also been under investigation to supplement and possibly replace the current single-axis testing philosophy. A new multi-axis hydraulic test facility called HYDRA is being built at the ESTEC test centre in Noordwijk, The Netherlands. Both methods apply base acceleration at the interface between the satellite and launcher. Thus, they require high forcers in order to accelerate the total mass of the test structure. An alternative test philosophy was proposed and investigated in a study funded by ESA [l]. The new concept called the Modal Force Combination (MFC) technique replaces the base acceleration by having a set of forces act directly in a selected number of structural points. The forces compensate for the d’Alembert inertia forces which arise due to the transient multi-axial base acceleration. The transient forcing functions are calculated by means of modal deformations and related modal force distributions in order to obtain the same structural responses of the structure as would occur by base acceleration. The direct substitution of distributed forces by means of a set of single forces yielding the same responses is generally impossible. This would require too many exciters. However, the MFC method offers two possibilities of performing a simulation. The first version is based on the demand for the equality of the input energy expressed by means of the modes of vibration. The second version starts with the demand for the equality of the structural responses in the points of excitation. Both versions were investigated experimentally. The experimental verification was performed by testing structures at three levels of complexity. The essentials of the verification tests are presented. The results of experimental dynamic load simulation by means of modal force combination are compared

J. M. Sinapius

with measurements from base excitation performed on DLR’s multi-axis vibration simulator MAWS in Goettingen, Germany. In conclusion, the state of the art of the development of the new qualification method is discussed. II - REVIEW SPACECRAFT TECHNIQUES

OF EXPERIMENTAL QUALIFICATION

The analytical and experimental verification programme for spacecraft is performed in order to guarantee the reliability of the spacecraft structure, i.e. the spacecraft’s capability to survive not only l the analytically predicted flight load environment and l the experimental simulation of the predicted load environment, but, in particular, l the actual flight load environment. The consistency of these three load types allows for the construction of an optimal lightweight design. In the early days of spacecraft technology, qualification methods were developed as single-axis shaker tests (Fig. 1) using sinusoidal or random excitation. This was acceptable as long as spacecraft were simple and lightweight instrumentation receptacles which could be assumed to be rigid, No significant influence on the launch vehicle/spacecraft interface was expected. Two methods, the spectral method and the classical shock spectra method which differed in their excitation signals, were mainly applied. As spacecraft became larger, heavier, and more elastic, the influence of dynamic coupling between the launcher and spacecraft on the spacecraft/launcher interface motion could no longer be disregarded. Experiences from the single-axis testing methods led to the notching procedure in the spectral method and the generalized modal shock spectra method [2]. None of the single-axis shaker test methods are able to simulate static or quasi-static load types. Of exciter

Single-axis Shaker Test

Fig.

1. - Development

Multi-axis Vibration Simulator

of structural

Modal Force Combination Test

dynamic Aerospace

qualification. Science and Technology

Experimental

Dynamic Load Simulation

course, all of these single-axis approaches have to be separately applied to all six degrees of freedom, requiring six interface spectra, or shock spectra, respectively. How to superimpose the responses based on each degree of freedom excitation still remains a question. In order to overcome the inadequacies of single-axis testing, multi-axis transient tests by means of multiaxis vibration systems were proposed [3]. Most of the currently available facilities serve as earthquake simulators. Figure 1 sketches a Multi-Axis Vibration Simulator. Although these systems do not yet comply with all the demands of spacecraft qualification testing, they were used for initial investigations of multiaxis transient testing. A study [4] investigated the substitution of single-axis sweep excitation by multiaxis transient excitation by considering the extreme values of acceleration responses in both test methods. During single-axis sweep tests, response acceleration peaks were obtained with a range from undertesting up to strong overtesting at local structure parts with a factor. 3, depending on the notch criterion. Another objective of this study was to demonstrate that a multiaxis transient acceleration vector can be sufficiently simulated in a multi-axis transient test on realistic spacecraft hardware. The initial results of multi-axis transient testing are very promising. However, this qualification method requires considerably high forces due to the acceleration of the vibration table’s high mass (e.g. the vibration table mass of ESTEC’s HYDRA is about 20,000 kg). The application on larger spacecraft is limited by the dimensions of the test facility, e.g. the HYDRA octagonal vibration table has a span of 5Sm. These limitations led to the investigation of a test concept that is based on the approved method of modal survey test 171. Similarly, a multi-point excitation scheme is used in this case to simulate the dynamic loads which occur due to the base acceleration. Therefore, the forces have to be specially conditioned SO that they produce the same deformations and accelerations in the structure as would occur with base excitation. III - DYNAMIC LOAD SIMULATION COMBINING MODAL FORCES

BY

Since the Modal Force Combination (MFC) method is an experimental technique, its fundamental relations are based on the assumption of a discretized structure. Under the condition of structural linearity, time invariant physical properties and viscous damping, the well-known equation of motion of a base-accelerated structure (Fig. 2) is given by WI@>

+ PIW

+ ~~1~~~ = 0,

(1)

where [Ml, [C] and [K] are the physical mass, damping and stiffness matrices. {u} is the displacement vector and {w} is the vector of the relative 1997, no 4

269

by Means of Modal Force Combination

PI =

Fig. 2. - Dynamic

-m(P)ii(t)

load due to base acceleration.

displacements with regard to the accelerated reference system. The relation between the absolute accelerations {ti}, and the relative accelerations {i;}, which in general is nonlinear, can be linearized by {ii} = {ti} + [G] . {Co}.

(2)

In most practical cases, the linear relation of Eq. (2) can be used where [G] is a constant transformation matrix. The columns of the matrix [G] are the rigid body movements of the structure related to the respective translational and rotational base excitation (60). In the relation (2) only the tangential and translational accelerations of the structural points are considered. Guiding accelerations evoked by the rotational velocity, i.e. centrifugal and Coriolis accelerations, can be neglected in cases where the vibration frequency is not too low - approx. less than 1 Hz - and where the angular amplitudes remain small. Inserting Eq. (2) into Eq. (1) yields [W+>

+ [Cl{G> + [K]@)

= -[Ml[Gl

. {Go)

where the right hand side are the d’Alembert of inertia {f (t)}.

-3 WI-= -WWl@o).

(3)

forces (4)

270

J. M. Sinapius

The main goal of the MFC technique is to simulate distributed forces like the d’hlembert inertia forces which arise due to the time-dependent multi-axial base acceleration of a given structure to be qualified. The simulation is performed by a set of single forces {p (t)} which yield the same structural responses of the structure as would occur by distributed forces, e.g. caused by base acceleration. As is well known, the dynamic structural response {U (t)} of a structure can be described by a series expansion by means of the eigenmodes of the structure [$] and time-dependent generalized coordinates { 4 (t)}.

How many exciters are necessary for a sufficient simulation of the dynamic loads? The determination of the real modal force vectors is most widely investigated within the development ‘of the phase resonance method [7, 81. A powerful tool for this goal is the minimization of the real part of the vibration energy compared with the total kinetic energy. By means of the transfer function matrix {H (w)} the minimization can be achieved by solving the eigenvalue problem [X]

(5)

- x ([Lq [M][H’]+ [H”]T[M][IT”]))[II] = 0

The basic idea [S] of the MFC method is to expand the force vector {p(t)} (6)

where {r (t)} are time-dependent expansion coefficients and [II] are so-called modal force vectors. Each modal force vector is related to a given mode and the work done on all other modes is assumed to be zero. The experience from modal survey tests by means of the phase resonance method shows that only few forces are required to excite a specific mode. The direct substitution of distributed forces {f (t)} by means of a set of single forces {p (t)} yielding the same responses is generally impossible. This would require too many exciters. However, the MFC method offers two possibilities of performing a simulation [6]. III.1

- Energy

l

(9) where the eigenvalues X form functions of mode indication [9]. The eigenvector related to the lowest eigenvalue gives under the condition of resonance (w = w,) the modal force vector {II}, which is best suited to excite the real normal mode {$}T by means of the given exciter configuration. The optimal exciter locations for the energy method of the MFC technique are the same as those for the normal modes which are specifically excited in order to simulate the dynamic loads. Numerous papers have been recently published on optimal exciter placement in normal mode testing [8, 11, 121. The exciter locations best suited for the load simulation are those structural points whose contribution to the kinetic energy of the modes considered is maximum. This can be directly calculated by the summarized share on the modal mass of each structural point k:

method

The first version is based on the demand for the equality of the input energy expressed by means of the modes of vibration [$I: (71 The expression in the frequency domain permits the consideration of complex modes. However, in experimental application, only complex structural responses can be measured which are a more or less high-quality approximation of the normal modes. By means of the series expansion of the simulating forces {P(w)} (Eq. (6)), the series expansion coefficients can easily be calculated, yielding the spectra of the simulating forces

{P (w>>= [nl(bbl’. II)-’ [ti]’ . {F b-4).

(8)

The Fourier transformation of {P (w)} yields the driving signals for the exciters. This first MFC version is called the energy method. The experimental application of the energy method requires answers to three essential questions: l How can the modal force vectors be determined? l What are the optimal exciter locations?

r=l

where 4 is an estimation of the modal mass normalized modal matrix $. The ng maxima of this value provide optimal exciter locations for ng modes to be controlled simultaneously. Physically, this criterion means that the excitation locations should be selected as such that all the considered normal modes are excited as well as possible with regard to their maximum amplitudes. Of course, the criterion of Eq. (10) depends on the real normal mode approximation. The minimum number of exciters required for the simulation is determined by the fundamental relation of the energy method Eq. (7). It is identical to the number of normal modes to be controlled. Less exciters would yield linearly-dependent modal force vectors which lead to the uninvertible matrix ([$I” . [d> in Eq. (8). III.2

- Deflection

method

The second MFC version starts with the demand for the equality of the structural responses in the points of excitation. Expressed by means of the transfer function matrix [H(w)], this demand can be written by

PIT . {F c-4)= WIT. P Wh Aerospace

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Experimental

Dynamic Load Simulation

271

by Means of Modal Force Combination

Again, the simulating forces {P (w)} are substituted by their series expansion Eq. (6). Now the modal force vectors are used in an extended meaning, only fulfilling the demand of their linear independence. Finally, a frequency-dependent transformation of the distributed forces {F (w)} into the simulating forces {P (w)} is derived:

(12)

The Fourier transform of {P (w) } leads to the required force vector {p(t)} in the time domain. Optimal exciter locations of the deflection method are determined by its basic demand for the equality of the structural responses in the points of excitation. It follows that the structural points for which the highest accuracy is required within the load simulation have to be supplied by exciters. This demand also determines the number of exciters to be applied. This second MFC version is called the deflection method. The invertability of the complete part of the transfer function matrix ([H (w)]’ [II]) depends on the dynamic characteristics of the structure to be qualified. Based on the modally transformed equation of motion of the vibrating structure, the transfer function matrix [H] can be described by the sum of the weighted dyadic products of the normal modes [14]

In the vicinity of eigenfrequencies the matrix [H] becomes singular or nearly singular if the influence of the other modes is less due to gaps between This influence depends on the eigenfrequencies. magnitude of the modal damping. A second problem arises in areas of antiresonances. The point mobilities which are the diagonal elements of the matrix ([H (w)]’ [II]) in principle contain antiresonances between two resonances. For point mobilities the numerator of Eq. (13) is always positive, while the denominator changes its sign after each resonance. Especially for single point excitation, this fact yields very high simulation forces calculated by Eq. (12). Both invertability problems of the transfer function matrix can be discovered by means of the condition criterion according to Hadamard [ 131. This criterion is well-suited to assess the simulation forces calculated by Eq. (12). 1997, no 4

IV - THE EXPERIMENTAL

INVESTIGATION

The experimental verification of the MFC technique was performed by testing structures at three levels of complexity. First, the MFC method was applied on beam-type structures. The related tests investigate the effect of damping and closely-spaced eigenfrequencies. Second, the method was applied on a more complex three-dimensional test structure characterized by the dynamic behaviour of real space structures like high modal density, various degrees of damping, and damping coupling. Third, the applicability of the MFC method on real space structures was investigated. The test results of the application on the three-dimensional test structure are presented here. They are quoted from an ESTEC study [l]. The three-dimensional test structure used in the second verification step consists of four rectangular blades of different lengths mounted perpendicularly on a hollow central mast. The blades are clamped at their centers, two of them to the tip, and the other two to the middle of the mast. The lower pair of beams is rotated 45” in relation to the upper pair. The overall structure is manufactured from steel. The total mass is approximately 50 kg. A non-uniform damping is realized by a thin layer of silicone placed in the neutral axis of two blades. Thus, the structure is dynamically characterized by o clusters of eigenfrequencies (11 modes from 5 to 15 Hz) o damping coupling o a wide range of damping (from 0.5 to 5%). The MFC simulation requires the calculation of the inertia forces due to the base acceleration. Therefore, the mass distribution of the test structure was approximated by a lumped mass model related to the measurement points. The structural responses were measured by means of 37 accelerometers in 30 nodal points. For the MFC simulation tests the structure was clamped on a seismic block. The dynamic loads were simulated by electrodynamic shakers. The exciters were located at the tips of the upper pair of blades. Figuve 3 depicts the test rig. Verification tests were carried out on DLR’s MAVIS in Goettingen. Figure 4 depicts the test on MAVIS. A three-axial base acceleration was applied. The test signals were typically measured transient loads taken from a study carried out by ESTEC [15]. Their time histories are shown in Figure 5. The results taken from two tests are presented here, one applying the energy method the other applying the deflection method. The results are shown by comparing the acceleration spectra on one blade tip of the test structure and by comparing all transient responses by means of a transient error measure TEM [ 161. The latter compares the structural responses of the MFC simulation with the responses due to base acceleration. The error measure is based on

J. M. Sinapius

Fig. 3.

- Test rig of MFC

simulation.

the correlation function and gives the deviation of two transients with respect to amplitude and phase. In order to assess the global quality of the MFC simulation, the weighted average and adjusted standard deviation of the error distribution is given. IV.1 - Energy

method

The application of the energy method starts with the measurement of frequency response functions in order to obtain an a priori data base. After the calculation of the multivariate mode indicator function (MIF) according to Eq. (9), the modes used to describe the input energy of the dynamic loads are selected from these MIF. In the current tests, the four lowest resonant peaks were used, representing the fundamental bending modes of the upper blades. On the basis of this a priori data base, the optimization of exciter placements was performed by means of the SSMM criterion described in Eq. (10). The modal forces appropriated to the estimated fundamental blade modes were calculated according to Eq. (9). A final tuning of the phase purity, which is required by the classical phase resonance method, was performed by hand. Table 1 presents

Fig. 4. - Test rig on MAVIS.

the measured eigenfrequencies, modal forces, and the phase resonance criterion, indicating the phase purity achieved. The related mode shapes are the antisymmetric bending of Blade 2, the antisymmetric and symmetric bending of Blade 1, and the symmetric bending of Blade 2. The qualification measurement was preceded by a safety calculation which is a prediction of the structural responses to the load simulation based on the measured transfer functions. Figure 6 shows the measured structural responses of one blade tip, the driven base measurements with a dotted line, and the MFC simulation results with a solid line. The real and imaginary parts of their frequency spectra agree very well up to 7 Hz, the frequency range of the four lowest modes considered in the calculations. The main discrepancy is near 7 Hz, the resonance of the mast. This mode was not controlled by a modal force vector. Figure 7 gives an overview of the error distribution of the entire structure. Whereas the errors on the upper blades are slight, the magnitude error as well as the phase error on the lower blades are considerable. This discrepancy is due to the fact that the lower blades are not suspended by exciters. Thus, their dynamic behaviour described by Aerospace

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Experimental

Dynamic Load Simulation

by Means of Modal Force Combination

273

n/s2-0’5 0.5 $ a 2

0 Fig.

1

5. - Transient

2

5

base acceleration

3

(time

4

0

5

histories).

-0.5 0

Table

1. - Modal

I

force

I

vectors

I

[mN].

Fig.

I

I

normal modes and appropriate force vectors can not be controlled. The significant error on the bottom of the mast is due to the bad signal-to-noise ratio near the tailing of the test structure. Usually, the displacements due to dynamic load simulation by means of the energy version are less than driven base test measurements. This fact is indicated by the weighted mean error which is about -18% in magnitude. IV.2 - Deflection

method

The deflection of the frequency

version starts with the selection range for the FRF measurement,

1997. Ilo 4

10

6. - Structural

responses

to energy

HZ

20

method

I

100 x I

Fig. 7. - Error

distribution

(energy

method).

which is determined by the main frequency content of the base acceleration. The FRF’s were measured by applying a fast sinusoidal sweep. The simulation was also preceded by the safety calculation predicting the structural responses to the load simulation. Thus,

274

J. M. Sinapius

the qualification could be performed by means of the calculated simulation forces. Figure 8 compares the measured acceleration spectra of the test structure in one blade tip, the driven base measurements shown by a dotted line and the MFC simulation results by a solid line. The real and imaginary parts of their frequency spectra agree very well. Figure 9 gives an overview of the error distribution of the entire structure.

ioo x )

100 x 1

Fig. 9. - Error Table I

-0.5

I

I

0

Fig.

8. - Structural

responses

I

I 10

to deflection

,

I HZ

20

method.

Whereas the errors on the upper blades are modest, the magnitude error as well as the phase error on the other parts of the structure are considerable. This discrepancy is due to the basic demand of this version to express the identity of the responses in the excitation points. Usually, the displacements due to dynamic load simulation by means of the deflection version are a little bit, higher than driven base test measurements. This fact is indicated by the weighted mean error, which is about +5% in magnitude. On the other hand, the adjusted standard deviation of about 43% indicates that structural parts which are not effected by exciters are overtested by the deflection version, while other parts show considerably lower displacements than those in the driven base tests. For comparison of both MFC versions, Table 2 gives the weighted mean value and adjusted standard deviation of the magnitude and phase errors. Whereas

distribution

2. - Comparison

(deflection

method).

of both MCF

versions.

I

I

I

I

I

the deflection version produces a lower mean error in the magnitude, the standard deviation is significantly higher than in the other versions. When comparing the mean phase error, no significant differences between the different versions can be seen, whereas the deviation in phase is higher for the deflection version, just as with the magnitude deviation. Normally, the energy version which requires the identity of input energy produces less structural displacements than driven base tests do. This fact is due to the limited number of modes used to describe the input energy. V - CONCLUSIONS The state of the art of the development of the MFC method can be summarized as follows: l A simulation of multi-axial transient loads by means of the MFC technique is possible. l The MFC technique is experimentally developed and verified in two different versions. First, the load simulation by means of the energy version is possible, which demands the equality of the input energies of the base acceleration and multi-point shaker excitation. Second, the deflection version Aerospace

Science and Technology

Experimental

Dynamic Load Simulation

by Means of Modal Force Combination

is applicable which requires the equality of the responses in the points of excitation by comparing base acceleration and shaker excitation. l For both variants of the MFC technique, criteria of optimal exciter location are developed, tried, and tested. l The advantages and disadvantages of both MFC versions are brought out within the verification tests: - The deflection method is the most accurate in the points of excitation. Deviation in other structural parts is significantly higher. - Load simulation by means of the energy version generally yields lower structural responses than driven base tests do due to the limitation of the considered modes. Compared with the deflection version, the energy version simulates the loads more evenly. - An appropriation of the modal force distributions to be combined in MFC calculations is necessary, especially for structures characterized by coupled modes. - Comparing both MFC versions, the energy version is more favourable for structures with a low modal density because well-decoupled modes can cause singularities in the frequency response function matrix to be inverted iyithin the deflection method. The deflection method is favourable for structures with a high modal density because the number of modes determines the number of exciters to be applied within the energy method. Based on experience gathered from the application of the MFC technique, the following improvements of this method and topics for further investigation can be pointed out: l Within the energy version of MFC simulation, the number of modes to be considered in the input energy description is limited by the number of applied exciters which, in general, leads to lower structural responses than those of driven base tests. The influence of higher-order resonances which have not been explicitly considered by means of modal forces still remains to be analysed. o The influence of structural nonlinearities have not yet been considered. Proposals for the analysis of structural nonlinearities within the modal force combination method [6] have to be experimentally verified.

REFERENCES [I] Sinapius M. - Experimental Dynamic Load Simulation by Means of Modal Force Vectors, ESA- Contract

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No. 7509/87iNL/pP, Final Report, DLR-FB, 93-35, 1993. PI Trnbert M., Salama M. - A Generalized Modal Shock Spectra Method for Spacecraft Load Analysis, JPL Publications 79-2, 1979. [31 &y H., Homung E., Eckhardt K. - The Hydraulic Multi-Axis Vibrator - Its capability to Simulate Real Mechanical Environments and to Verify Aerospace Strnctures, Proc. of Workshop on Spacecraft Vibration Testing, ESA-SP 197, Noordwijk, 1983, 5-10. [41 Hohung, E. et al. - Design Verification of Large Spacecraft, MBB/ERNO - Final Report, INTELSATContract No INTEL 276, Bremen 1985. [5] Breitbach, E. - Experimentelle Simulation dynamischer Lasten an Raumfahrtsystemen mittels modaler Erregerkraftkombinationen, Habilitationschrift, RWTH Aachen, 1988. [6] Sinapius J.M. - Die experimentelle Umsetzung der Modalkraftsimulation verteilter dynamischer Lasten unter besonderer Beticksichtigung struktnreller Nichtlinearit%en, PhD-Thesis, RWTH Aachen, 1994. [7] Degener M. - Die experimentelle Modalanalyse groper Systeme, VFI (1988), Baltz-Verlag, Miinchen, 39-41, 64-66. [8] Niedbal N., Klusowski E. - Optimal Exciter Placement and Force Vector Tuning Required for Experimental Modal Analysis, Proc. of AIAA Dynamic Specialist Conference, Long Beach, CA, USA, 1990, 130-141. [9] Hunt D. L. et al. - Optimal Selection of Excitation Methods for Enhanced Modal Testing, Proc. of Structures, Structural Dynamics and Materials Conference, Palm Springs, CA, USA, 1984, Part 2, 549-553.

[lo] Breitbach, E. - Recent Developments in Multiple Input Modal Analysis, Journal of Vibration, Stress,and Reliability Design, 1988, 10, 478-484. [ 1 l] Holmes P.S., Wright J. R., Cooper J. E. Optimum Exciter Placement for Normal Mode Force Appropriation Using an A Priori Model, Proc. of XIV Int’l Modal Analysis Conference, Dearborn, MI, USA, 1996, 1-7. [12] Schedlinski C., Link M. - An Approach to Optimal Pick-Up and Exciter Placement, Proc. of XIV Int’l Modal Analysis Conference, Dearborn, MI, USA, 1996, 376-382.

[13] Engeln-Miillges G., Reuter F. - Formelsammlung zur numerischen Mathematik, Mannheim 1988. [14] Ewins D.J. - Modal Testing, Theory and Practice Research Studies Press Ltd., Letchworth, England, 1984.

[15] Ftillekmg U., Hiiners H. - Study of Interaction of Shaker Table on Test Specimen for Different Mass Ratios and Assessment of Hydraulic Power Requirements for the Different Load Cases, DLR IB 232-91 CO9, 1991. [16] Sinapius M., Fiillekrug U. - Advancements in Structural Dynamic Qualification Methods, Proc of Int’l Conf. on Spacecraft Structures and Mechanical Testing, Paris, 1994, 1057-1065.