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  1. 1. Interaction Between a Vibration Exciterand the Structure Under TestPaulo S. Varoto and Leopoldo P. R. de Oliveira, University of São Paulo, São Carlos, Brazil In order to obtain good quality data in modal and vibrationtesting, the experimentalist should pay attention to interac-tions that occur between the structure under test (SUT) and X1 F1 m1the instrumentation used in the test. Measurement errors can Tablearise from a number of possible causes, including transducer k2 c2mass loading effects, transverse stinger stiffness and exciter- ArmatureSUT interactions. This article aims to study one of these m2sources of error, namely, the interaction between the electro- X2 F2 Coildynamic exciter and the SUT. The effects caused by the shakerarmature mass on the dynamics of the SUT are assessed in dif- k1/2 c1/2 c1/2 k1/2ferent testing conditions by using theoretical models and ex-perimental analyses that include the exciter dynamic charac- Baseteristics. Results from these studies indicate that the shakerinteracts significantly with the structure under test and, un- Figure 1. Armature 2 DOF dynamic model.der some circumstances, exciter dynamic effects can be ac-counted for in the process of improving measured data. the fact that the excitation force must be directly measured during the test in order to obtain reliable FRF measurements. The electrodynamic vibration exciter has been intensively McConnell 2 developed an extensive study on exciter dynam-used in modal and vibration testing as a means to drive the SUT ics and its interaction with the SUT. In his work several ana-(Sructure Under Test). In experimental modal analysis, a com- lytical models were developed to explain mechanical interac-mon practice is to attach the exciter to the SUT through a flex- tions between the exciter and both free and grounded SUT. Inible stinger and a force transducer.1 The stinger is used to trans- addition, his work approached the electric characteristics ofmit excitation signals to the SUT in a single direction, reducing the exciter-power amplifier system during the test, showing thesecondary forms of excitation (e.g., bending moments) due to basic differences that occur when the power amplifier is usedpossible misalignments. The force transducer is used along either in its voltage or current modes of operation. Maia9 alsowith the exciter to measure the input force applied to the SUT. developed an interesting study on the subject by using simpleIn vibration testing, the SUT is attached to the exciter table dynamic models to model interactions between the exciter andthrough a test fixture. 2 In this case, the SUT is driven by base grounded structures. An insightful paper was published byexcitation signals that are transmitted to the SUT through the Lang 5 where several simple tests are performed in order totest fixture. In this type of test it is common practice to em- evaluate exciter dynamics.ploy a closed loop test procedure where the exciter’s table is The objective of this work is to perform an experimentalcontrolled so that a signal having a prescribed frequency con- study on two different vibration exciters, attempting to evalu-tent is applied to the SUT. In both cases, it is well known that ate some of their basic dynamic characteristics as well as theirthe exciter interacts with the SUT and that in some circum- interaction with the test environment. The results obtained instances, distortions due to the armature’s dynamics can signifi- this study and exhibited in this article constitute part of acantly alter measured data. project developed by one of the authors towards a degree in Exciter dynamics and its interactions with the SUT have Mechanical Engineering.been approached by several authors. Tomlinson 2 studied theinteraction between the exciter and the SUT, paying special at- Review of Theorytention to the force dropout that occurs when the structure is This section presents a review of important theoretical as-excited in the vicinity of a structural natural frequency. This pects that are concerned with the dynamics of a vibration ex-work also emphasizes the nonlinearity characteristics that arise citer. The theoretical development that is described in thisfrom the electromagnetic field that is generated during the section is based on the work by McConnell 2 that presents aexciter working cycle. Olsen 4 studied the effects of the comprehensive analytical development of exciter dynamics.armature’s mass and suspension stiffness on measured data and The theoretical development described here considers thepointed out that the armature’s mass effects on the measured exciter’s armature dynamics as well as the electrodynamic re-Frequency Response Functions (FRF) can be minimized by lationships that are needed to explain an exciter’s electric be-selecting an appropriate exciter for a given test. Rao 6 followed havior and the two modes of operation of the power amplifierthe basic development by Olsen, but the dropout of the excita- – the voltage and current modes.tion force was analyzed in more detail. According to his work Armature Dynamics. Figure 1 shows the armature dynamicthe dropout phenomenon is due primarily to mechanical in- model that consists of the table and the electromagnetic coil.teraction between the armature’s mass and the structure and In this case, the exciter base is considered to be rigidly attachedelectromagnetic characteristics of the exciter coil circuit. to the floor. Hence, we have the 2 DOF (Degree of Freedom) me- Exciter dynamics as well as its interactions with the SUT chanical system shown in Figure 1, where m 1, k1 and c1 repre-have been approached in various text books on modal and vi- sent the table mass, stiffness and viscous damping and m2 , k2bration testing. Ewins 1 formulated a simple dynamic model and c2 the spider mass, stiffness and viscous damping, respec-that explains the basic mechanical interaction between an tively.exciter’s armature and the SUT. The author draws attention to As described in McConnell,2 the frequency domain accelera- tions A1 (ω ) and A2 (ω ) exhibited by the table and the coil, re- Based on a paper presented at the 19th International Modal Analy- spectively, can be written in terms of the driving point andsis Conference, Kissimmee, FL, February 2001. transfer accelerance FRFs, A11, A12 and A 22, according to20 SOUND AND VIBRATION/OCTOBER 2002
  2. 2. A11(ω ) A (ω ) A1 = F1 + 12 F2 104 ma ma (1) A21(ω ) A (ω ) A2 = F1 + 22 F2 ma mawhere A11(ω ) is the table driving point FRF and A12(ω) = A21(ω) Table and Coil Accelerances 102is the transfer accelerance FRF. The armature’s FRFs present A22in Eq. (1) represent important quantities since they give someuseful information about armature behavior. The driving point and transfer accelerances A 11 ( ω ) and A11 100A22 (ω ) are expressed as 2 A12 mA −r 2(1 + M )(β2 − r 2 + iβη2 ) A11(ω ) = a 1 = (2) F1 ∆(r ) 10–2 ma A2 −r 2(1 + M ) (1 + M β2 ) − r 2 + i(η1 + βM η2 )   A22(ω ) = = (3) F2 ∆(r ) 10–1 100 101 102 103 Frequency Ratio [r] ma A2 −r (1 + M )(β + iβη2 ) 2 2 M-0,1 β-100 η1-0,05 η2-0,001 A12(ω ) = = (4) F2 ∆(r ) Figure 2. Dimensionless accelerance plot for an exciter armature.where the auxiliary variables appearing on Eqs. 2, 3 and 4 aredefined as ω m R L β = 22 M = 22 (5) ω11 m11 Xa Fc I(t) ∆(r ) = (1 + β2M ) − r 2 + j 2r (ζ1 + ζ2βM ) *   Ebemf   ka ca E(t) (6) β2 − r 2 + j 2rζ β  − β2M  β + j 2rζ 2   2       2 Figure 3. Exciter electromechanical model.These three dimensionless accelerance FRFs are shown in Fig-ure 2. They are identical as the dimensionless frequency ratio tions for both cases – voltage and current mode. The electro-r ranges from 0.1 to 10. In this frequency range it is observed dynamic phenomenon that occurs on the exciter’s circuits isthe existence of a natural frequency that is common to all FRFs mainly due to the interaction between the current and the ar-and that it is the first resonance of the two DOF system. Above mature motion in the exciter’s electromagnetic field.r = 10, the FRFs diverge and each one exhibits its own charac- The Ampere law relates the electromagnetic force F c(t) andteristics. the current I(t) through the coil, according to The coil’s accelerance A22(ω ) decreases after r = 10 up to r ≈ Fc = (nBl )I = K f I (10)31.6, where the table presents a dynamic absorber behavior forthe coil. This accelerance then increases up to r ≈ 105 (coil’s where B is the magnetic field intensity generated by the cur-resonance) and becomes constant. The table driving point rent I(t) through n coils, each one having length equal to l. Theaccelerance A 11 (ω ) exhibits an antiresonance at r = β = 100 constant Kf is the “force-current constant” and is equal to (nBl).followed by its resonance at r ≈ 105 and reaches a plateau of Tomlinson 3 developed a theoretical model establishing a1.1 (i.e., 1+M) for higher frequencies (r >> 100). relationship between the magnetic field B and the coil posi- Finally, considering the table transfer accelerance A12(ω), we tion x given as:have a resonance at r ≈ 105 that decreases at a ratio of 40 dB/decade. This behavior shown by A12(ω ) clearly indicates that dψ   x + x 2 B= = B0 1 −  0  (11)it is quite impossible to control the table after r = 300, since its dx   x max     dynamic response is more affected by external forces than bythe coil’s input. Therefore, the armature-coil system has an where B0 is the highest intensity that the field B reaches, x0 isupper frequency limit for effective use of the exciter. In this the armature initial position, x is its oscillation amplitude andcase this frequency limit is given by2 : xmax is the maximum amplitude. It can be verified yet by Equa- tions 10 and 11 that the relationship between the force and the ω22 k2(m1 + m2 ) current is nonlinear: r2 = β 1 + M = 1+ M = (7) ω11 k1m2   x + x 2  Fc = nlB0 1 −  0  I (12)   x max   Electromechanical Model. Figure 3 shows the electrome-  chanical model used by Olsem4 and McConnell2 to describe In addition, Lenz’s law gives the relationship between the Ebemfthe electromagnetic coupling on the armature-coil system. This and the armature’s velocity as:electromechanical coupling is governed by several parameters:   x + x 2 the coil resistance R, inductance L , input voltage signal E(t) Ebemf = nBlx = K v x = K v 1 −  0  x (13)and the back electromagnetic voltage Ebemf . The equations that   x max    govern the mechanical and electrical systems shown in Figure3 are, respectively, given as: It could be noticed from Eq. 13 that the E bemf also presents a ma x + ca x + ka x = Fc (t ) (8) nonlinear factor proportional to the relationship between ex- citation amplitude and armature maximum amplitude and its RI + LI + Ebemf = E (t ) (9) velocity, i.e., the excitation frequency. 3where Eq. 9 was obtained using standard electric circuit rela- Figure 4 shows the E bemf behavior in terms of excitation am-tionships. However, the electromagnetic force F c(t), as well as plitude and frequency. When the frequency is half of the origi-the E bemf voltage are hardly dependent on the exciter’s mode nal excitation frequency, the E bemf drops to a new value that isof operation. 2 The subsequent sections show the basic equa- half of the original one. The effects of the amplitude of oscil-DYNAMIC TESTING REFERENCE ISSUE 21
  3. 3. lation are related to the nonlinear behavior. The smaller theamplitude, the more linear the E bemf variation.2 Power Amplifier – Modes of Operation. As stated earlier, thepower amplifier has two modes of operation, the current andvoltage modes, respectively. These modes establish the volt- ω = ω1age versus current relationships during exciter operation. The xmax = 1 Ebemfbasics of each mode of operation will be described. ω = 1/2 ω1 In the current mode of operation, the relationship betweenthe input voltage to the amplifier and its output current is given xmax = 0.5by an equation of the type I (ω ) = Gi (ω )V (ω ) (14)The frequency domain versions of Equations 8 and 9 are given –1 –0.5 0 0.5 1.0as X / Xmax (ka − maω 2 + jcaω )X = K f I 0 (15) Figure 4. E bemf behavior due to frequency and amplitude variation. (R + jLω )I 0 + jK v ω X = E0 (16) 101where I0 and E 0 are reference amplitudes for the current andvoltage, respectively. Notice that the simpler relationship be- Armature Accelerance (dimensionless)tween the E bemf and the table’s velocity (Eq. 13) is used in Eq. 10016. The dimensionless armature accelerance in the current modeis given by: 10–1 2 ma (−ω X ) 2 −r A(ω ) = = (17) K f I0 1 − r 2 + j 2ζa r 10–2where r is the dimensionless frequency ratio, now based on thearmature natural frequency, and ζ a is the armature viscousdamping ratio. 10–3 Current Mode Thus, the voltage needed to maintain the current magnitude Voltage Modeis given as E  2ζe  10–4 E (ω ) = 0 = 1 + j β1 +  (18) 10–2 10–1 100 101 102 RI 0   1 - r 2 + j 2ζa r   Frequency Ratio (r)where β1 = ω a / ω e and ω a is the armature’s natural frequencyand ω e is the break frequency: Figure 5. Dimensionless bare table accelerance. R ωe = (19) LThe electromagnetic damping ratio ζe is given by: Accelerometer Force Transducer Cm Kv K f ζe = = (20) 2 ka ma 2R ka ma Signal Conditioning SUTwhich is the result of the back emf current being dissipated by Spectral Analyzerthe coil circuit. It is important to remember that these equa-tions are valid for low table amplitudes, which make the non- 1 2 3 4linear terms vanish, as described by Tomlinson. in out Similar to the current mode, the voltage mode of poweramplifier operation can be modeled as a gain, constant in fre-quency up to a given cutoff frequency, 2 according to the fol-lowing equation Shaker Power Amplifier CPU E (ω ) = G (ω )V (ω ) (21) v Figure 6. Test on free-free steel beam.where G ν ( ω ) is the amplifier gain, V( ω ) and E( ω ) are theamplifier’s input and output voltages, respectively. are shown below and correspond to a MB Dynamics Modal 50 Following a similar procedure, the dimensionless armature vibration exciter. These parameters were obtained from theaccelerance in the voltage mode of operation is give as: exciter’s operating manual. ma (ω 2X ) −r 2 A(ω ) = = (22) Force .................................................................................................... 111 N E   2  K f  0  1 − (1 + M L )r + j 2(ζa + ζe ) + β(1 − r ) r 2 Useful Displacement ..................................................................... 25.4 mm  R Maximum Displacement .............................................................. 27.9 mm Shaker Mass ...................................................................................... 24.9 kgwhere the dimensionless mass ratio ML = me/ma and the induc- Armature Mass ................................................................................ 0.227 kgtive mass m e is given by: Armature Axial Stiffness ............................................................ 2312 N/m Lc Coil Current (Max) .............................................. 8.5 A (Low Impedance) me = a (23) 4.2 A (High Impedance) R Coil Resistance .................................................... 1.3 Ω (Low Impedance)where it can be noticed from this last equation that the induc- 5.2 Ω (High Impedance)tive mass is dependent on armature damping. In order to gain additional insight to an exciter’s dynamic Figure 5 shows the bare armature accelerance FRF behaviorbehavior when operated in either the current or voltage modes, for the current and voltage modes, according to Eqs. 17 and 22,a simple simulation was performed with the physical param- where it shows the differences between the two modes of op-eters of an available vibration exciter. The exciter parameters eration. In the current mode FRF (solid line), once the table22 SOUND AND VIBRATION/OCTOBER 2002
  4. 4. 10–1 a 10–3 FRF Magnitude, V Magnitude 10–5 Force 10–7 Force Acceleration Voltage 10–9 0 200 400 600 800 1K Frequency, Hz 0 40 80 120 160 200Figure 7. Force and current measurements for beam test. Frequency, Hzpasses through its mechanical resonance, the FRF ampli- 100 btude becomes constant. In principle, this implies a reli-able condition for exciter operation as stated byMcConnell.2 The other two FRFs depicted on Figure 5 cor- 10–2 Accelerance, g/Nrespond to the bare table FRF in the voltage mode of opera-tion. There is a clear distinction between these FRFs when 10–4compared to the current mode FRF. The bare table reso- SUTnance is severely damped in the voltage mode FRF, mostly 10–6 Accelerationdue to high electrodynamic damping. Also, the two volt- Forceage mode FRFs shown in Figure 5 are different in the sense 10–8that they use different values for the coil resistance. It is 0 20 40 60 80 100seen that the smaller resistance yielded magnitudes closer Frequency, Hzto the current FRF for higher frequencies. Figure 8. (a) FRF, accelerance and force measurements of an aircraftExperimental Results wing structure. (b) Force dropout in shaker testing. This section presents some experimental results obtainedfrom tests performed using two different exciters, a B&K 4812 celeration. Rather, they occur where the input force dropswith Power Amplifier B&K 2707 and a MB Dynamics Modal out to a minimum value. Again, the same observations were50 with Power Amplifier SL500VCF. These tests were per- made by other authors. 1,2,6 Figure 8b shows essentially theformed in order to get some practical understanding of the same behavior observed in Figure 8a. In this case, a sim-exciter’s dynamic behavior as well as the interaction between pler structure was used that contains a single natural fre-the exciter and the SUT. quency in the 0-100 Hz frequency range. Once again, the Results for Exciter-SUT Interaction. This section shows a value of the structure’s natural frequency and the frequencysample among many results obtained in the developed project. where the force dropouts occur are very close.The results shown here contain important features concerned Figure 9 shows an accelerance FRF that was obtainedwith the exciter-SUT interaction. using the B&K 4812 exciter with random excitation and the Figure 6 shows the experimental setup used in one of the PCB impact hammer. The experimental results are com-tests. This test used a cold-rolled steel beam (1000×25.4×6.25 pared with results obtained from an analytical model ofmm) mounted directly on the MB Dynamics exciter table. A the SUT. The main feature of this test is that even thoughKistler 912 force transducer was used to measure the input two different excitation mechanisms were employed toforce to the beam and a B&K 4371 accelerometer was used to drive the test structure, the same value is obtained for themeasure the beam’s output acceleration. Hanning windows natural frequency (about 27.3 Hz), as shown in Figure 9.were used in both the input and output signals. The beam was While not shown here, recall that in impact testing theexcited with a random signal in the 0-1000 Hz frequency range. input force auto spectral density is constant (or almostThe SL500VCF power amplifier was adjusted to operate in the constant) in the tested frequency range. Figure 9 stillvoltage mode. Figure 7 shows the results obtained for the in- shows a rigid body natural frequency (about 5 Hz) due toput force to the beam and the voltage for the voltage mode the SUT suspension system that is not present in the idealof the power amplifier. Note that a dropout on the voltage simulated FRF. Although the natural frequencies are es-values occurs at frequencies in the vicinity of the beam’s sentially the same for both testing conditions, there is aungrounded natural frequencies. These voltage dropouts mismatch in the anti-resonance that occurs at a frequencycoincide with the force dropout for the lower natural fre- close to 20 Hz. An interesting fact can be observed in thequency, but deviate as frequency increases. Similar behav- anti-resonance obtained in the exciter test. The anti-reso-ior was observed by McConnell 2 in numerically simulated nance (dotted line) occurs at approximately 18 Hz, a valueresults. that is slightly lower than the anti-resonance value for the Figure 8a simultaneously shows a plot of a FRF of an aircraft hammer test. The armature’s suspension stiffness and masswing structure and plots of the output acceleration and input values for the B&K exciter are about Ka = 21 kN/m and ma =force. These quantities were normalized so that they could be 0.454 kg, as stated in the operating manual. On the otherplotted on the same graph. These experimental results were ob- hand, the SUT has a mass of m = 1.046 kg. These valuestained using the B&K exciter and power amplifier. This figure allow us to get the frequency ω = [ka/(m+ma )] 1/2 = 118.3 rad/clearly shows that a given structural natural frequency (notice s = 18.83 Hz!the dashed lines on Figure 8a) does not necessarily occur where Hence, the anti-resonance observed in Figure 9 is essentiallythe structure presents maximum values for the output ac- the natural frequency of the armature-SUT system, but inDYNAMIC TESTING REFERENCE ISSUE 23
  5. 5. 102 102 Off 101 Accelerance, g/N 100 101 Accelerance, g/N Impact 10–1 Shaker Simulation F(t) 10–2 a(t) 100 0 20 40 60 80 100 On Frequency, HzFigure 9. Analytical and experimental FRFs. 10–1 10 0 100 200 300 400 500 a(t) Frequency, Hz 8 V(t) Figure 11. Electromagnetic damping. 6 102 Magnitude, g/V 4 H31 2 101 Accelerance, g/N H22 0 –2 1 100 –4 2 0 2 4 6 8 10 Frequency, kHz Table 3Figure 10. B&K 4812 bare table FRF. 10–1the SUT response it appears as a dynamic absorber action. 0 100 200 300 400 500Finally, the anti-resonance amplitude obtained with the Frequency (Hz)exciter testing is higher than that obtained with impact Figure 12. Effects of rocking motion on table.testing. This result, along with the frequency mismatch al-ready discussed, indicates a clear exciter-SUT interaction. Figure 12 shows results that were obtained by impacting Results for Armature’s Dynamics. This section shows a the B&K 4812 bare table. The aim of this test was to inves-sample of experimental results that were obtained in tests tigate eventual rocking motions presented by the exciter’sthat aimed to determine an exciter’s basic dynamic behav- table. For this purpose, two accelerance FRFs were gath-ior. ered, as shown in Figure 12. Accelerance H 22 (ω ) is nearly a Figure 10 depicts the B&K 4812 exciter’s bare table accelera- driving point FRF, where the exciter’s table was impactedtion FRF. This result was obtained by measuring the exciter’s at location 2 (Figure 12) and the response was measuredtable acceleration using the B&K 4371 accelerometer while the close to the table center. The plot of H 22( ω ) shows a singleexcitation frequency was varied in the 0-10,000 Hz frequency natural frequency that corresponds to the armature’s sus-range. The acceleration values were normalized by the input pension natural frequency. The plot for accelerance H 31 ( ω )voltage from the signal generator. The result shown in Figure was obtained by impacting the table at location 1 and10 exhibits a peak frequency at 6885 Hz, while the exciter measuring the acceleration at location 3. Both acceleranceoperating manual gives 7200 Hz as the bare table natural FRFs show essentially the same behavior from 0-120 Hz,frequency. Hence, although there is a difference of about after which they differ significantly. While H 22 ( ω ) is nearly4%, this test was considered effective in obtaining the bare constant up to the end of the frequency bandwidth, H 31( ω )table natural frequency. exhibits a second resonant peak at approximately 280 Hz Figure 11 shows results obtained when the shaker table followed by a valley at 330 Hz. This second natural fre-is impacted by an instrumented hammer and the table ac- quency indicates that a strong rocking motion is takingceleration is measured in two different conditions. The place on the shaker’s table when the impact is applied at asolid line shows the FRF that was obtained when the power point that does not coincide with the armature’s verticalamplifier is turned off and the dashed line shows the same axis. Some other interesting results not reported here onFRF with the power amplifier turned on. In this condition damping differences were also observed. This test tells usno excitation signals were sent to the shaker. The only dif- something about running base excitation tests in the SUTference between these test conditions is that in the first that are not symmetrical with respect to the armature’scase the exciter’s internal circuits are not electrically en- vertical axis!ergized while in the second the power amplifier is turned A simple but interesting test was performed with the MBon and thus electrical energy is flowing through the coil Dynamics Modal 50 exciter in order to get an estimate ofcircuit. The results shown in Figure 11 are suitable for ob- the armature’s suspension damping ratio. A small massserving the effects of the additional damping and possibly with a miniature accelerometer was mounted on the top ofstiffness induced by the coil electromagnetic field that is the exciter table. With the power amplifier off, the arma-established during exciter operation. ture was plucked and the free decay acceleration was mea-24 SOUND AND VIBRATION/OCTOBER 2002
  6. 6. sured. These data were used to estimate the damping fac-tor through logarithmic decrement, resulting in a value of2.6% for the armature’s damping ratio.Summary and Conclusions This article presents an experimental study on exciterdynamic behavior as well as on its interaction with thestructure under test. Several tests were performed and in-teresting characteristics were observed. The major conclu-sion from this work is that the exciter represents an effec-tive excitation mechanism. However, it should be used withcare, since it interacts with the test environment. Also, thetwo power amplifier modes can distort test results.Acknowledgements The authors kindly acknowledge the financial supportprovided by FAPESP (grant # 99/0229-6) to Mr. Leopoldo P.R. de Oliveira during the development of the project andto EESC-USP for providing the necessary laboratory facili-ties.References1. Ewins, D. J., Modal Testing: Theory and Practice, RPS, London, 1984.2. McConnell, K. G., Vibration Testing: Theory and Practice, John Wiley & Sons, NY, 1995.3. Tomlinson, G. R., “Force Distortion in Resonance Testing of Struc- tures with Electrodynamic Vibration Exciters,” Journal of Sound and Vibration, Vol. 63, No. 3, 1979, pp. 337-350.4. Olsem, N. L., “Using and Understanding Electrodynamic Shakers in Modal Applications,” Proceedings of the 4th International Modal Analysis Conference, IMAC 1986, Vol. 2, pp. 1160-1167.5. Lang, G. F., “ElectroDynamic Shaker Fundamentals,” Sound and Vi- bration, April, 1997. pp. 14-25.6. Rao, D. K., “Electrodynamic Interaction Between a Resonating Struc- ture and an Exciter,” Proceedings of the 5th International Modal Analysis Conference, IMAC 1987, Vol. 2, pp. 1142-1150.7. Instruction Manual, B&K Vibration Exciter System V, 1974.8. Instruction Manual, MB Dynamics Modal 50A Vibration Exciter, 2000.9. Maia, N. M. M., Silva, J., Theoretical and Experimental Modal Analy- sis, RSP, 1997.The authors may be contacted at: and TESTING REFERENCE ISSUE 25