Determination of Effective Masses and Modal Masses from Base-Driven Tests

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1 Determination of Effective Masses and Modal Masses from Base-Driven Tests U. Ftillekrug German Aerospace Research Establishment (DLR) Institute of Aeroelasticity Bunsenstr Gljttingen, Germany Abstract This paper deals with the identification of effective masses and modal masses from base-driven tests. When performing a base-driven test with an elastomechanical structure. the structural responses can be related to the base ac- &rations and a modal identification of the structure can be accomplished. If, in addition, the base forces are measured, it is possible to determine the effective and modal masses of the structure. Here, the required equations describing the dynamic behaviour are first developed and discussed. In the following, an analytical vibration system with simulated measurement errors is used for demonstrating the identification. In addition, a method to improve the accuracy of the identified parameters is shown. Based on the encouraging results. the application to real test data is envisaged for further investigations. Nomenclature Matrices. Vectors Mass matrix Viscous damping matrix [K]: Stiffness matrix Geometric transformation matrix for structure S tgsij Geometric transformation matrix for base B [+I,: FRF matrix Eigenvector matrix +%I: Displacement vector 6 DOF displacement vector of the base 14: Force vector hl: 6 DOF force vector of the base (4 : Relative displacement vector PI, : r eigenvector && 0: Frequency of vibration 0,: Natural frequency of r mode II,: Viscous damping ratio of r mode m,: Modal mass of r mode P,: Effective mass of i mode P: Static mass R,i : Modal participation coefficient r*jx: Modal residue PI: Number of modes NY Number of DOFs of structure S 64: Physical mass c: Physical viscous damping K: Physical stiffness SubscriDts S: structure n: Base 0 : 6 DOF coordinates of the base r: Current mode number j, k: Element numbers of vectors/matrices; general Integers Suwxscripts T: Transpose of matrix or vector /f: Complex conjugate (Hermitian) transpose of a matr,x -1: Inverse of a matrix 1. Introduction Base-driven tests are usually performed during the development and qualification process of spacecraft. Besides their main purpose of qualifying the structural design with respect to vibrational loads, these tests can be used for the modal identification of the tested structure. Once a structure has been instrumented with transducers and mounted on a vibration table, the structural responses can be related to the base excitation and the measured data can be used for modal identification. The main difference when compared to a classical modal identification test, where the vibrations are induced by point-force exciters, is the fact that here the inertia forces of the structural masses serve as excitation. This limits the controllability of the eigenmodes. In addition, the excitation forces cannot be measured directly and modal masses cannot be determined in the usual manner. However, if the forces between the vibration table and the tested structure are measured by means of an appropriate force measurement device, it is possible to determine the modal masses and effective masses. 671

2 With the knowledge of modal masses, the modal model of a structure can be completed and dynamic response calculations can be performed. If the effective masses of a structure are known, the eigenmodes can be judged with respect to the possible occurence of strong reaction forces which is particularly useful when dealing with the coupling of a launcher and its payload. Also, both quantities can be employed advantageously for the verification and improvement of analytical models. In contrast to publications where the analysis of basedriven vibration tests is based on the phase separation technique ISSPA [l,z], a different way of development, more similar to [3,4] is followed here. Interface forcing functions are derived and discussed on the basis of equations of motion in physical coordinates. The fact that effective masses can be identified from these functions is shown. Then the determination of modal masses is described. An example with an analytical vibration system is used for the illustration of the equations and the identificalion of the system. If, for a moment, the vibration table were substituted by a large seismic mass. the displacements of the base degrees of freedom would be zero: and the upper part of (2) would result in Once the structural displacements (u),are known, the forces at the base degrees of freedom would simply follow from the lower pari of (2) 2. Theoretical Development Figure 1 shows a typical set-up of a base-driven vibration test where base reaction forces are measured. The tested structure S is mounted to a force measurement device which is fixed to the table plate of a multi-axis vibration simulator. The interface forces between the vibration table and the structure are measured by force transducers which connect the upper and lower pal of the force measurement device. Due to the location where the forces are measured the structure B becomes part of the tested object. As the force measurement device has to be stiff in the tested frequency range - this is a prerequisite for adequate force measurements the displacements of the structure B can be considered to be composed of pure rigid-body motions. The goal of the following derivations is to first establish an analytical expression relating the interface forces and interface accelerations. 2.1 Basic Equations The dynamic behaviour of a linear, time-invariant, and viscously-damped etastomechanical structure consisting of parts Sand B can be described by [M] {ii) + [c] {ii} + [K] {!A] = {r}, (1) where [,M], [c] and [K] are the symmetric physical mass. damping, and stiffness matrices; {r] is the vector of the external forces and {u} is the displacement vector. Pariitioning equation (1) with respect to the degrees of freedom of the structure {CL}, and the degrees of freedom of the base {u}, delivers So the forces at the base would be determined by the structural motions {u), and by the coupling matrices of the structures Sand B. In the present case, however. motions at the base degrees of freedom are prescribed which excite structural vibrations. It is now appropriate to introduce new coordinates {uo], describing the 6 DOF motions of the base, and a displacement vector {v}, which contains the elastic deformations of the structure with respect to its rigid-body motion. This can be written as {I& = {v] + [Cs]{~t,} (3a) {IA), = [G,j]{+,] (3b) Matrices [cs] and [Gal transform the 6 DOF motions of the vibration table to the rigid-body motions of the structure S and 8. It should be noted that eq. (3) includes the transformation from absolute to relative coordinates which in general cases can become nonlinear. However, the linear relationship of equation (3) with time-independent matrices [c~] and [Go] can be used for most practical cases; see also [5]. The columns of matrices [Go] and [G,,] can be interpreted as the rigid body eigenvectors related to the respective translational or rotational axis. Premultiplying a force vector with the transposed matrices [Gsr and [Gel delivers a vector with resultant forces where all force components are summed up for the respective axis. The transformation (3) can be arranged as 6 7 2

3 base which are caused by the total mass of the structural parts Sand Band the structural vibrations. and inserted into (2). When performing the matrix multiplications it must be taken into account that a rigid-body displacement of the whole structure cannot cause any elastic forces, i.e. it must be valid that Transforming (&I) into the frequency domain leads to Likewise, a rigid-body velocity of the whole structure cannot cause any damping forces which leads to a corresponding equation for the damping submatrices. In the follwing, the mass coupling matrices [M]~~ and [MlsB are neglected when compared to the matrices [MIS3 and [Mlos. It should be noted that the equations remain precise for a lumped-mass discretization. The neglection is not necessary - it is introduced here just for the purpose of simplifying the equations. Premultiplying (2) with the transposed transformation matrix of (4) and taking equation (5) into account delivers The transfer function matrix [H(w)] can be determined by measuring the 6 DOF base accelerations and the resulting structural relative vibrations. Also, the transfer function matrix [H (co)] can be constructed by modal parameters and, in case of proportional damping, can be derived, where {v}, is the real normal mode shape.,)i, = {w}: [M]~~ (~1,~ the generalised mass, o, the eigenfrequency and q, the modal damping of the eigenmode r 2.2 Interface Forcing Functions Inserting eq. (10) into the transformed Fourier eq. (Eb), what follows is If no external forces act on the structure S. i.e. Id, = PI I (7) the structural vibrations are caused exclusively by the base excitation. The upper and lower parts of equation (6) then deliver [MISS [;I + [Cl,@} + [KISS 1~1 = W [GslT [MISS {; I + ([d [MISS [G.s] + [GE] [MI&R]) (41) = =hll ~~~ Equation (8a) describes the relative vibrations of the structure caused by the inertia forces of the base excitation. Equation (8b) delivers the resultant forces{r,] of the This equation relates the 6 DOF base accelerations {ijo (w)} to the resultant 6 DOF base forces{r, (a~)). The transfer function can therefore be called the interface apparent mass function.. In the low-frequency range where w << w,, the first part of (11) can be neglected and the dynamic forces at the interface are dominated by the moving rigid mass of the structures Sand B. - Around the resonances w, high interface forces can be produced due to structural resonances. The force amplitudes are limited only by the damping factors q,. 673

4 - In the high-frequency range where w >> wn, it can be written that and the static mass eq. (14) can be rewritten as with[y] as the modal matrix of the eigenvectors and the generalised mass matrix. Since it is defined as FTJ = [WIT [4ss [VI (12) 2.3 Effective Masses According to [6,7], the effective mass of a mode r and a degree of freedomk is defined as it follows that Then far w >> wa equation (11) can be expressed as It can be understood as an equivalent physical mass of the r modal degree of freedom which, in contrast to the modal mass, is independent of the eigenvector scaling. A characteristic of the effective mass is the fact that the summation over all modes I delivers the total mass (or the moment of inertia) of the structure S. This can be verified by { b (41 = [%lt r4obprl I% (41 (13) Now the forces of the masses of the structure S cancel out and the intedace forces are determined exclusively by the masses of the structure 8. Next, the transfer function relating the acceleration of the degree of freedomk to the resultant force degree of freedom j is extracted from (11) ;h = {Gs]: [Miss [ + I (h l% l, [VI)- h l [Ml.s.s {%I, = ={%I: [MISS ksl, (19) Using the definition (18) with equation (15). it follows that kw (14) I where the vectors {Gs}j, {Gs},, {Gs}i, and {Gs}, are the j-th and k-th column of matrices [Go] and [Gel, respectively. The effective masses of the structure S can therefore be determined by the measurement of the transfer function of a base acceleration to the corresponding base force and the following identification of the parameters of this transfer function. For the identification of the parameters P,~ and pjr and an appropriate phase separation technique can be employed. If the modal mass is known (or defined, e.g. to m, = l), the parameters R,i and I& can be calculated in addition. By defining modal participation coefficients (15) 674

5 2.4 Modal Identification and Modal Masses For the identification of modal parameters of the structure Seq. (IO) can be utilised.ajx. Since eq. (24) can be set up for each structural measurement pant j= 1, 2,, Nand each base excitation axis k = 1. 2,, 6. the value of the modal mass rn,~ can be estimated, e.g., by averaging. The absolute value of the modal mass depends, of course. on the scaling of the eigenvector {v},. which relates the 6 DOF base accelerations {tie (a~)} to the structural relative accelerations {G (ro)}. Since the transducers on the tested structure S usually measure the absolute accelerations (II (ro)},, the relative accelerations {ti (0)) need to be calculated previously from For the transfer function relating the structural acceleration of measurement point jto the base acceleration k, it can be written that When applying an appropriate phase separation technique, the modal parameters wr and II, can be determined from eq. (22). The mode shapes can be calculated when employing eq. (22) for all structural points j= 1, 2. N. In addition, the term can be determined from eq. (22). Using (15) and (20). it follows that and (23) Having determined the effective mass flrk based on measured interface forces, the modal mass m, follows from the eigenvector component yir and the parameter Illustrative Example The analytical vibration system employed as an illustrative example is shown in figure 2. It consists of 12 masses which are coupled by springs and dampers. Mass M, represents the base 6 whereas the remaining part forms the structure S. The masses M, to M,, are only allowed to move in the vertical direction. Their magnitudes are all defined to the same value. The damping parameters are set proporknal to the spring rates. The structure is therefore proportionally damped and the mode shapes are purely real. The modal parameters and the effective masses of the structure S are listed in table 1. The modal masses are given for the mode shapes with the largest component normalized to 1. The real normal mode shapes are shown in figure 3. The vibration system is excited by a base translational motion u and a (small) rotational motion a. which is linearized. With the vector of the base degrees of freedom the transformation matrices become I -n I -0.X< I -O.b< : I -0.4 I I 0.40 I 0.60 I 0.8a I * Measurement errors are simulated by adding random numbers with a magnitude of 10% to the true signals. For a translational base excitation the responses of the 11 masses are plotted in terms of the transmissibility functions Vx /U in figure 4. Figure 5 shows the summation of the 11 FRFs and in figure 6 the mode indicator function of DLR [6] is plotted. The functions indicate the presence of six modes which are the symmetric modes of the structure S. Due to the symmetric excitation and the symmetry of the structure S, no anti-symmetric mode can be excited by

6 translational base excitation. Figure 7 shows the magnitude and phase of the translational base force which is divided here by the translational base acceleration, thus forming the apparent mass function. No rotational base force (moment) occurs for translational base excitation. For a rotational base excitation the transmissibilities Vk / ri of the 11 masses are plotted in figure 6. the sum transmissibilities in figure 9, and the mode indicator function in figure 10. From these functions the resonances of only three modes can be detected which are the first antisymmetric modes of the structure S. The magnitude and phase of the rotational base force (moment) which is scaled by the rotational base acceleration is plotted in figure Il. First, the apparent mass functions of figure 7 and 11 are analysed. According to eq. (17) and (20), it should be possible to identify the eigenfrequencies, damping values and effective masses from these functions. The Frequency Direct Parameter Identification algorithm [9] of the CADA-X modal analysis software (101 is employed for the analysis. The results are listed in table 2. Modes , 5, 7, and 11 could be identified. The accuracy of the identified frequecies is rather high. However, the damping values. and identified effective masses show larger deviations. The total mass and moment of interia, obtained by the sumation of the effective masses are in good agreement with the true values. Modes 4, 6, 6, 9. and 10 could not be identified at all. The reason for this is the fact that these modes do not cause any significant base lorces. In the following step. the structural responses induced by base acceleration are analysed with the FDPI algorithm. The results are listed in table 3. The agreement of the identified frequency and damping values with the true values is very good. The improvement is due to the more significant data utilised in the identification process. Modes 6 and 10 could not be identified at all since they cannot be excited strongly enough by translational or rotational base excitation. This fact is also pointed out by the mode indicator function which shows no peak for these modes. In order to improve the identification of the effective masses. the following procedure is carried out: The identified eigenfrequencies and damping values of table 3 are inserted into the equation for the base forces (17) and the effective masses are determined from the measured forces and eq. (17) by solving an overdetermined system of linear equations (see appendix for details). The identified effective masses as well as the identified eigenvector components and modal residues are then used for the calculation of modal masses according to eq. (24). The results of this identification are listed in table 4. The relative errors of the identified effective masses are very small in all cases where the value of the effective mass is large. The relative error becomes higher only for the small effective masses. The summations of the effective masses deliver the total mass and the total moment of inedia with good accuracy. The modal ma?.ses are in good agreement 676 with the true values in all cases where the effective masses are large. Only in cases where the effective masses are small the modal masses are flawed by higher f?rrors. 4. Summary and Conclusions The dynamic behaviour of an etastomechanical structure undergoing a base-driven vibration test is considered. The analytical expressions for the base forces and the relative structural vibrations which are caused by base accleerations are set up. Based on this, the equations for the identification of effective masses and modal masses are developed and discussed. An analytical vibration system is used for demonstration and identification. The main results of the simulated experiments are: Effective masses can be determined from measured base forces with satisfactory accuracy. The accuracy can be improved if the results of a previous modal analysis of the structure are used additionally. The accuracy of effective and modal masses is generally good. It decreases only in those cases where the magnitudes of the effective masses are low. Since the identification of effective and modal masses provides valuable additional data on a structure tested under base excitation, experimental investigations with a suitable test structure are planned and prepared. Special attention will be placed on the multi-axis base excitation case. 5. Appendix: Determination of effective masses from measured base forces using identified eigenfrequencies and damping values. Eq. (17) used far j = k delivers with eq. (20): Rearranging this yields: W

where Now the identified parameters o, and 7, can be inserted and eq. (A2) can be set up for each frequency o at which the base force and the base acceleration wwe measured.

(A3) can be solved in the sense of least squares {ox} = ([A]"[.$ [A]" (h) b44) and {x) yields the identified effective masses 6. References [II Link, M. :.

7 where Now the identified parameters o, and 7, can be inserted and eq. (A2) can be set up for each frequency o at which the base force and the base acceleration wwe measured. The combination of all of these equations leads to a system of linear equations 1 ) = [AlId (A3) If the number of measured frequencies w is greater than the number of eigenfrequencies n, eq. (A3) can be solved in the sense of least squares {ox} = ([A]"[.$ [A]" (h) b44) and {x) yields the identified effective masses 6. References [II Link, M. :..Structural System Identification Using Single and Multi-Axial Vibration Test Data, Proc. of Conference on Spacecraft Structures, CNES. Toulouse. ESA SP 238, pp , Link, M.; Qian, G.: Identification of Dynamic Models Using Base Excitation and Measured Reaction Forces, Revue Franc&e de Mechanique n , pp [31 [41 [51 El [71 PI PI [loi Girard, A.; lmberf J.F.:,,Modal Effective Parameters and Truncation Effects in Structural Dynamics, Proc. of the 5th International Conference on Modal Analysis, London, pp , Meurat, A.; Girard, A.; Bugeat, L.P.:.Improvement of Dynamic Force Measurement for Vibration Tests on Spacecraft Structures, Proc. of Conference on Spacecraft Structures, CNES, Paris, pp Ftillekrug, U.:,,Time Domain Analysis of Base- Driven Transient Tests, Int l Journal of Analytical and Experimental Modal Analysis, ~8. 3, pp ,1993. Wada. B.K.; Chen, J.C.: Test and Analysis Correlation for Structural Dynamic Systems, 2nd Int l Symposium on Aeroelasticity and Structural Dynamics, Aachen, DGLR Report 85-02, pp , Wada, B.K.; Branford, R.; Garba. P.A.:,,Equivak?nt Sprung Mass: A Physical Interpretation, Shock and Vibration Bulletin. Bull. 42, pp , Breitbach, E.: A Semi-Automatic Modal Survey Test Technique for Complex Aircraft and Spacecraft Structures, Proc. of the lllrd Testing Symposium, Frascati, Italy, pp , Lembregts. F.; Leuridan. J.; van Brussel, H.; Frequency Domain Direct Parameter Identification for Multiple Input Analysis: State Space Formulation, Mechanical Systems and Signal Processing, 4 (I), pp , LMS: CADA-X Modal Analysis Manual, Revision 3.3, Leuven, Table 1: Modal Data of the Analytical Vibration System 677

I I I I 9 10 11 1 28.579 1 (0.4%) 1 7.063 1 (-21.3%) 0.248 (-8.6%) Table 2: Results of the Analysis of the Base Forces 5% 10.

463 1 (0.0%) 1 8.787 1 (-2.1%) 1 97.3 Table 3: Resulls of the Modal Analysis of the Structural Responses I I t I 9 0.094 (7.0%) 3.

8 I I I I (0.4%) (-21.3%) (-8.6%) Table 2: Results of the Analysis of the Base Forces 5% (-4.9%) (0.2%),=I I (error) MAC [ A] t 8 I I I I I (-0.2%) (-5.7%) I I I I I (0.0%) (-2.1%) Table 3: Resulls of the Modal Analysis of the Structural Responses I I t I (7.0%) % (2.9%) (7.1%),I 1 & j (-0.4%) (0.2%) I I=/ I I Table 4: Identified Effective and Modal Masses 678

9 Figure 1. Test Set-Up of a Base-Driven Vibration Test r=ll Figure 2. Analytical Vibration System Figure 3. Real Normal Mode Shapes 679

10 Figure 4. Structural Responses for Translational Base Excitation (Transmissibility Functions) Figure 6. Mode Indicator Function for Translational Base Excitation Figure 5. Summation of Transmissibility Functions for Figure 7. Translational Base Force (Apparent Mass) Translational Base Excitation 680

11 r 1.0 Mode Indicator Function mfalimal bare excitalon Figure 13. Structural Responses for Rotational Base Figure 10. Mode fndicator Function for Rotational Excitation (Transmissibility Functions) Base Excitation I - Figure 9. Summation of Transmissibility Functions for Rotational Base Excitation Figure 11. Rotational Base Force (Apparent Mass) 681

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations.

Structural Dynamics Lecture 4. Outline of Lecture 4. Multi-Degree-of-Freedom Systems. Formulation of Equations of Motions. Undamped Eigenvibrations. Outline of Multi-Degree-of-Freedom Systems Formulation of Equations of Motions. Newton s 2 nd Law Applied to Free Masses. D Alembert s Principle. Basic Equations of Motion for Forced Vibrations of Linear