sand--~b--a79lc characterizing electrodynamic shakers/67531/metadc684338/m2/1/high... ·...

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SAND 96-2791C

s A N D - - ~ b - - a 7 9 l C Characterizing Electrodynamic Shakers

David 0. Smallwood Sandia National Laboratories

Albuquerque, NM 87 185-0557

Authors' Biography

David Smallwood received his BSME degree from New Mexico State University in 1962 and his MSME degree from New York University in 1964. He has worked for Sandia National Laboratories since 1967 and is currently a Distinguished Member of the Technical Staff in the Environments Engineering Group of the Mechanical and Thermal Environments Department at Sandia. He is a fellow of the E S .

Abstract

An electrodynamic shaker is modeled as a mixed electricaVmechanica1 system with an experimentally derived two port network characterization. The model characterizes the shaker in a manner that the performance of the shaker with a mounted load (test item and fixture) can be predicted. The characterization depends on the measurements of shaker input voltage and current, and on the acceleration of the shaker armature with several mounted loads. The force into the load is also required, and can be measured directly or inferred from the load apparent mass.

Keywords

electrodynamic shakers, characterization, impedance, models, acceleration, force, two-port network

Introduction

The usual practice to characterize performance of electrodynamic shakers is to specify the displacement limits, the velocity limits, the force capability, and the acceleration capability. The usual physical characteristics like the mounting table size, and the armature weight are also given. The discussion in this paper will concentrate on the acceleration and force performance characteristics.

The usual practice is to specify the maximum acceleration which can be achieved on the bare (no fixtures or test item) armature of the shaker over a stated

frequency range. A typical specification might be 100 g from 10-2000 Hi except where limited by velocity or displacement. The maximum acceleration is often plotted as a function of frequency for a sine input. The force is usually specified as the maximum force which can be achieved. The force is seldom measured directly. A typical scenario is to multiply the bare table maximum acceleration by the armature mass to achieve a force rating. Sometimes a specified mass load will be added to the armature and the force will also be specified by the maximum acceleration of the armature + mass multiplied by the combined mass. These specifications are useful to the manufacturer because they are independent of the load which will be placed on the shaker and are easily measured. However, the specifications are dependent on the power amplifier used, which implies the shaker and power amplifier must be characterized as a unit. For random vibration, often only the rms acceleration and, velocity are specified. For transients vibration guidelines for peak acceleration, velocity, and displacements are given.

This characterization is of limited use to the user because the shaker is rarely'used bare table and more specific guidance is desired for random and transient testing. A rough idea of the shaker capability will result if the shaker armaturekest item can be assumed to be a rigid mass. This typically does not work for two reasons. First the test items and fixtures (the load) are seldom rigid masses. Second, the shaker manufacture actually takes advantage of the armature resonance at the higher frequencies to arrive at the force rating. The manufacture typically divides the maximum achievable acceleration by the static mass of the armature even though the apparent mass is decreasing as the first axial resonance of the armature is approached.

This leaves the user in the uncomfortable position of not being able to predict, except in the grossest of terms, the testability (does the shaker with a given power amplifier have the ability to meet a given test specification) of an item, without actually mounting the test item (or at least a mass mockup) on the shaker and running a trial run.

The purpose of this paper is to suggest a better way. The method is not without faults, but is much better than methods discussed above.

-aFt)llpI-s 1997 PROCEEDINGS-Institute of Environmental Sciences

DISCLAIMER

This report was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor any agency thereof, nor any of their employees, make any warranty, express or impiied, or asumes any legal liabili- ty or respom'biiity for the accuracy, completeness, or usefulness of any information, appa- ratus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Referewr! herein to any specific commem~al product, process, or service by trade name, trademark, maxmfacturer, or otherwise does not necesariiy constitute or imply its endorsement, recornmeadation, or favoring by the United States Government or any agency thereof. The views and opinions of authors exprrssed herein do not necessarily state or reflect those of the United States Government or any agency thereof.

DISCLAIMER

Portions of this document may be illegible in electronic image products. Images are pmduced from the best avaiiable original dOClXIIlent

The shaker is characterized in terms of an impedance matrix, which will be defined shortly. Two of the variables are force and acceleration at the interface to the test item. Or alternately the force and acceleration into the shaker armature. Strictly speaking velocity should be used as the motion variable for an impedance matrix. However, for this paper impedance will be used in the larger context of any motion measurement and acceleration in particular. Force can be measured directly with force gages or indirectly, since force gages are sometimes difficult to place in a test setup.

The other variables are the voltage and current required to drive the shaker. The method requires that these variables be measured with a bandwidth equivalent to the frequency range of interest. Only recently has the measurement of current with this bandwidth been easily available.

The validity of the matrix will be limited in frequency range, but will typically extend from 10 to 1000 Hz or better, depending on the shaker and experimental methods used. However, methods to check the validity of the measurements are shown.

The output of the two port network can be can be the current and voltage requirements of the shaker. The capability of a power amplifier/ shaker system can be easily checked if the capabilities (available current and voltage) of the power amplifier are known.

This work grew out of two efforts. First was the effort to develop methods to run force controlled vibration tests without the requirement to measure force (Smallwood, 1996). The second was an effort at Sandia to develop a virtual test environment for vibration tests where concepts for vibration test design could be exercised, and where the testability of a test item could be checked before the test was conducted (Klenke, 1996.

Theory

An electrodynamic shaker can be characterized by viewing the system as a passive two port network (Baher, 1984 and Weinberg, 1962). Because we have a mixed mechanicaYelectrica1 system the terms used in the characterization will be a little different than for the strictly electrical two port system. A force-current analogy will be used for the characterization. Strictly, velocity should be used for the voltage analog, but acceleration will be used in this paper. The two port system is shown in Fig. 1

E z I

F

A

Figure 1 Two Port representation of an electrodynamic shaker

The input is the voltage, E, and current, I, from the power amplifier to the shaker. The output is the force, F, and the acceleration, A, of the shaker into the test item. All the variables are assumed to be complex functions of frequency. This system can be characterized with a two by two matrix of impedance frequency response functions (frf).

Note: The impedance G2 is not the load impedance even though it has the same dimensions. The load acceleration will be given by the sum of &,Z and G2F. G2 is a characteristic of the shaker.

All the terms in the impedance matrix cannot be easily measured directly (see Appendix A). However, the impedance matrix, Z, can be determined if the system is measured with a minimum of two known load conditions. The general case is given by a system of equations for the n measured load conditions, where the subscripts indicate the different loading conditions.

In short hand this equation will be written as

E = ZI (3)

Equations 1-3 are the appropriate forms for a sinusoidal input or the Fourier transform of a transient input. For a random input the equations can easily be written in the form of auto and cross spectra or frequency response functions (fro. For example, if the acceleration is picked as the reference, Eq. (1) is multiplied by the complex conjugate of the acceleration, the expected value is taken,

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and all terms are divided by the autospectrum of the acceleration the following equation results

{H;}=z{;} (4)

where HEA is the frf between the voltage and the acceleration, Hu is the frf between the current and the acceleration, and Z, is the apparent mass, , of the load.

The simplest loads are rigid masses. For this case the force is simply the mass multiplied by the acceleration and Eq. (2) becomes.

output can be effectively described by the single pair of output variables, force and acceleration. At frequencies where the table motion is not uniform, hence more than one acceleration is needed to describe the table motion, or where a single force is not representative of the shaker output, the method will break down. It is suggested that more than one accelerometer should always be measured on the shaker table top. The transmissibility of the ratio of response of these accelerometers should be unity over the frequency range where the model is valid. This will give a good indication of the useful frequency range of the characterization. Appendix B suggests a method when the motion must be described by more than a single acceleration and force.

The measurement of more than two loading conditions is also useful for determining the validity of the model. The predicted response of the load from the least squares impedance matrix model can be compared with the actual measurements giving an indication of the validity of the model.

The rank of E and I must be two at all frequencies. The If there are at least two loading conditions the impedance singular values of the E and I matrices plotted as matrix can be found from functions of frequency can offer insight about the

accuracy of the characterization. "Noisy" singular values or values much smaller than the largest singular value indicate possible errors in the characterization.

Z = EI-' (6)

where I-' indicates a Moore-Penrose pseudoinverse if the number of loading conditions is greater than two. In this case the pseudoinverse will give a least squares solution for the impedance matrix. Eq. (6) is solved for each frequency of interest.

The impedance matrix of the shaker is a characteristic of the shaker and is not dependent on the power amplifier or the load. This characterization has been made possible by the availability of inexpensive current transducers with a frequency response of several kHz. Once the impedance matrix of a shaker has been found, Eq. 1 can be used to solve for any two of the variables (E, A, I, and F) if at least two of the other variables are known (see Appendix A). This is a huge improvement over the simplistic ways of characterizing the performance of a shaker as discussed in the introduction. The requirement to characterize the shaker with four complex frequency response functions is more difficult than the simple characterizations usually used, but is easily accomplished in the current world of computers and electronic media.

The model of a two-port network for the shaker will be valid as long as the system is linear, and where the

Force measurements from acceleration, voltage, and current measurements.

If the acceleration at the interface to the load, the shaker current, and the shaker voltage are measured, the force into the load is given by

Or in terms of the reciprocal transmission matrix (see Appendix A)

F = & , E + % , I

In a previous paper equations were developed by Smallwood (1996) for this force. This equation was

F = K ( I - EY,) It is easy to see that in the current formulation

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Z,, =- 1 and z12 =-- Y, Y, K

Determination o f current and voltage requirements for a specified test

(9)

Assume we want to find the voltage, E, and current, I, requirements for a given load, Z,, and specified acceleration requirement, A . First the ‘required force is determined from

F = ZmA

The current requirement is given by

or in terms of the auto spectra for random

and the voltage requirement is given by

01

E = A[ 2 + Z, (4, - )) or in terms of the auto spectra for random

2

SEE = s, -+ zm z,, - - 1;: [ zp)/ Thus the power amplifier requirements for a given load with a complex mechanical impedance, Z, , (the load

does not have to be assumed to be a simple mass) can be predicted before a test is run. The load mechanical impedance used for the predictions can be measured, or calculated from a model.

Example

A Unholtz-Dickie TlOOO shaker was tested with five test runs and three different loading conditions as summarized in Table 1.

The frequency range was 20-3000 Hz. For example, Run 1 was at 0.01 g2/Hz from 20-3000 Hz, with a load consisting of the circular plate mounted on the shaker armature. In each test the acceleration was measured on two opposite sides of the armature. One of the accelerations was used as the reference. Other measurements included the voltage into the power amplifier, the voltage into the shaker (which was used for this report), and the current into the shaker.

Table 1 Test conditions for example problem

.001 4 .oo 1 A+C 5 .008 A+F

A- Bare Armature; 43 kg (95 lb) C- Circular plate, 14.5 kg (32 lb) F- Fixture, 30.4 kg (67 Ib)

The force was assumed to be the reference acceleration multiplied by the static mass of the load. The load did not include the armature mass.

The control and measurements were made with a GenRad 2552 control system. The shaker current was measured with a Ohio Semitronics, CTL-2000 HT current transducer. The measurements were transferred to a PC using the universal file format and the calculations were performed using MATLAB. The four measured impedance functions, Z, as derived from the five tests are shown as Fig. 2. The magnitude of the transmissibility between the two accelerometers is shown as Fig. 3. The results show that the acceleration at the two sides of the armature are the same to within 5% to about 1.1 kHz and within 30 ?h to about 2.3 kHz for Runs 1-4. The results were not quite as good for Run 5 for unknown reasons.


Frequency (Hz)

Frequency (Hz) -. .-

Frequency (Hz)

Frequency (Hz)

Figure 2a Magnitude of the impedance matrix, Z, for a TlOOO shaker

- - 1st SVof E 2nd SVof E -1st SVof I -2nd SV of I

' I 1 lo3 I I

I I

1 o2 1 o3 Frequency (Hz)

Figure 4 Singular values of E and I

Frequency (HI)

- o ~ r 2 -100

N c

-200.

1 o2 10' Frequency (Hz) Frequency (Hz) Frequency (Hi)

1 o2 1 0' 1 oz 10' Frequency (Hz) Frequency (HI)

Figure 2b Phase of the impedance matrix, Z, for a TlOOO shaker

Frequency (Hz) Frequency (Hz)

Figure 5 Magnitude of the admittance matrix, Y

._ Frequency (Hz)

Figure 3 Transmissibility from one side of the armature to the other side

Frequency (Hz) Frequency (Hz)

Freouency (Hzl Frequency (Hz)

Figure 6 Magnitude of the transmission matrix, T


I

The singular values (SV) of the E and I matrices are given in Fig. 4. The singular values of the I matrix values are "smooth" and about an order of magnitude apart below 2 kHz. The second singular value of E is more "noisy", is more than an order of magnitude below the first singular value, and has a "notch" at about 1600 Hz. This suggests that the characterization in terms of an impedance matrix, Z, which involves the inverse of I, will be better than a characterization in terms of an admittance matrix, Y, which involves the inverse of E. The magnitudes of the admittance matrix, Y, are plotted as Fig. 5. The magnitudes of the transmission matrix, T, and the reciprocal transmission matrix, R, are plotted as Figs. 6 and 7.

The transmission matrix (Fig. 6) illustrates a couple of characteristics about the shaker. TZ2 shows that the blocked force for the shaker is almost a flat line as a function of current, which implies the blocked force is proportional to current almost independent of frequency, a result we would expect. However, if the armature moves the force should be calculated from the reciprocal transmission matrix (Eq. 7b) which involves both voltage and current. Figure 6 suggests that the force measurements should be valid, perhaps a little noisy, up to about 2 kHz. Tl1 indicates that at low frequencies the bare table (zero load) voltage to acceleration frequency response function has a slope of almost -1 on a log-log plot. This implies the voltage is proportional to velocity, another expected result. However, this result is valid only to about 200 Hz. The results degenerate partly because force is needed to move the armature, and hence the force generated at the driver coil is not zero even for the zero load case. RI1 has a slope of +1 to almost 1 kHz indicating the voltage generated across and open shaker, when the armature is moved, is proportional to velocity.

The load apparent mass can be derived from the impedance matrix, the current, voltage, and acceleration measurements using Eq. 4 in two ways.

Two equations are available because there is only one unknown, the force. The first uses Zlz and Zll. The second uses Z& and &. Eq. 13a can also be written as

Frequency (Hz)

Frequency (Hz)

Frequency (Hz)

Frequency [Ez)

Figure 7 Magnitude of the reciprocal transmission matrix, R.

TO1 I I 1 o2 10'

Frequency (Hz) Apparent mass -Test 2 - - --Test 3

lo2 I

I I i 1 o2 10'

Frequency (Hz)

Figure 8 Estimated load apparent mass, Geest) .--magnitude of test (tern aopa:ent mass -achievable acceleration

10%:

n I \ / - - - - t - -- - 1 i i i;

1 O2 10'

Voltage and current at max acceleration: ... current limit -.-.current ---voltage limit -Voltage I . . . . . . . . . . . . . I i

lo', . . I ~

-.. . . . . . " . . . - .

i - 1 I

\ ,

5 1 -

I I I 10'

I 10' 10'

freauency (Hz) Figure 9 Testability of a 2-mass 1-spring test item


The apparent load mass, estimated as the average of the two methods (Eq. 13), for the tests are shown as Fig. 8. As the load mass decreases to zero the two terms on the right side of Eq. 13 will approach the same number, and the estimated mass will be the difference of two numbers nearly the same, which limits the accuracy of the estimate for masses small relative to the mass of the armature.

The load mass for Tests 2 and 3 should be zero. The values calculated indicated the magnitude of the errors which can be expected from these tests. The magnitude of the mass for Tests 2 and 3 are from .05 to 5 kg (0.1 to 10 lb) over the frequency range of 30 to 1000 Hz. This is a few percent of the armature mass (43 kg, (95 lb)) and indicates the approximate accuracy of the measurements.

The load mass for Tests 1,4, and 5 should be 14.5 kg (32 lb), 14.5 kg (32 lb), and 30.4 kg (67 lb) respectively . As can be seen the predictions are within a few percent to above 2 kHz.

To further illustrate the use of the characterization. Assume the test item is a two mass one spring system where each mass is 23 kg (50 lb) with a free-free natural frequency of 300 Hz and 5% of critical damping. Assume the shaker has a displacement limit of 1.26 cm (0.5 in), a velocity limit of 178 c d s (70 ids), a voltage limit of 132 peak volts, and current limit of 1414 peak amp. The magnitude of the apparent mass is plotted as Fig. 9a. The transmission matrix, T, can be used to calculate the achievable acceleration of this test item on the specified shaker. The voltage and current required at the maximum achievable acceleration is plotted as Fig. 9b, and the maximum achievable acceleration is plotted as Fig. 9a. Examination of Fig. 9 reveals that the system is displacement or velocity limited below 70 Hz, is current limited from about 70 Hz to 1000 Hz, and is voltage limited from 1000 Hz to 3000 Hz except for a narrow band near 2000 Hz. The maximum achievable acceleration is just a little over 10 g near 200 Hz.

Conclusions

A practical characterization of a shaker in terms of a two by two impedance matrix is developed. The impedance matrix is a characteristic of the shaker and is independent of the power amplifier and load on the shaker. The measured impedance matrix can be used for a variety of uses, including; the prediction of voltage and current requirements for a specified test, and estimation of the force into a test item. These predictions can be

made with a model of the test item and do not necessarily require the actual hardware.

The next step would be to fit the measured impedance frfs with transfer functions as rational polynomials in s or z, or as state space models. The frfs are quite well behaved so this process would require a relatively small number of poles and zeros. This would result in acceptable models of the shaker in a very compact form.

It is suggested that:

1- The ability to measure the voltage and current with a bandwidth over the useful range of the shaker should be a part of every shaker system. This will be accomplished if new shaker orders require this feature.

2- The impedance matrix of a shaker should be part of the published characteristics of a shaker. If not supplied by the manufacturer, the user should make the required measurements for the shakers. If possible the conditions for the measurements for a given shaker should be standardized. The conditions for the measurements can be a subject of future debate.

3- The repeated measurements of the impedance matrix of a shaker, under the same conditions, over time, might serve as a measure of the state of health of a shaker and be used as a preventive maintenance tool.

References

Baher, H., 1984, Synthesis of Electrical Networks, Wiley and Sons, New York, NY

Klenke, et al, 1996, "The Vibration Virtual Environment for Test Optimization (VETO)," Proc. of the 67th Shock and Vibration Symposium, Vol. 1, pp 13-22, SAVIAC, 2231 Crystal Drive - Suite 711, Arlington, VA 22202.

Smallwood, D. O., 1996, "Shaker Force Measurements Using Voltage and Current," Proc. of the 67th Shock and Vibration Symposium, Vol. 1, pp 31-37, SAVIAC, 2231 Crystal Drive - Suite 711, Arlington, VA 22202.

Weinberg, L., 1962, Network Analysis and Synthesis, McGraw-Hill, New York, NY.


Appendix A

Other Forms of the Two-Porf Network Analogy

The body of the report described how a shaker can be described as an impedance matrix.

The terms in the impedance matrix are defined as

(A-1)

A few of the terms are well known. For example; Zll is the unloaded impedance of the shaker, and &l is the ratio of the unloaded acceleration to current response of the shaker. The direct measurement of 212 and G2 would require that an external force be applied to shaker and the resulting open circuit voltage and acceleration be measured. A feat difficult in practice. G2 is the accelerance looking into the shaker with the shaker electrical input open (but with the field on). Z12 is the ratio of voltage, generated at the open electrical shaker input, to a driving force applied at the armature.

The two-port network can also be defined in terms of an admittance matrix.

The terms of the admittance matrix are defined as

(A-3)

(A-4)

Yl, is the blocked (zero motion) admittance of the shaker, and Y21 is the blocked ratio of force to voltage of the shaker. Both would be difficult to measure directly, since the blocked condition is very difficult.

A transmission matrix can also be used

{ ;} = [ "]{A} T21 T22 F

or if F = ZmA

{ ;} = A[" ".I{ j m } T 2 1 T22

The terms of the transmission matrix are defined as

A=O

(A-5a)

F=O A=O (A-6)

T11 is the unloaded ratio of voltage to acceleration of the shaker, and TZ1 is the unloaded ratio of current to acceleration. Both can be measured directly. T12 and require a blocked armature, which is very difficult.

Note that this form together with Eq. A-9 is essentially Eqs. 11 an412 which solve for the voltage and current in terms of the acceleration and force.

A reciprocal transmission matrix can also be used

64-71

The terms of the reciprocal transmission matrix are defined as

1 4 =q E I=O E=O (A-8)

These terms are not easy to measure directly. For example, R l l would require that the shaker be open circuit, an acceleration applied to the shaker top, and the resulting voltage measured.

These matrices are all related by the equations


. r

Appendix B

Generalization to a Multiple-Shaker System with Multiple-Outputs

('4-9) E1 ... EIN

...

. ... A,, ... A,,

I

or in short hand notation

E = Z I

...

...

...

...

...

(B-lb)

The method can be generalized to a system with one or more shaker inputs and one or more shaker outputs. Assume there are n inputs and m outputs. The system can then be characterized as

where N 2 n + rn and the rank of I must be n+m. We can then solve for Z as

Z = EI-' (B-2) I

For Z to be of full rank and hence invertible to Y, the rank of E must also be n+m. A good plan would be to find the singular values of E and I to judge the condition of the problem. Singular values much smaller than the largest singular value will indicate poor conditioning and possible errors in the computed Z matrix.

(B-l)

If N > n + rn , a least squares solution is found where 1- is a Moore-Penrose pseudoinverse.

where Z is a square n+m by n+m matrix. Of course the equations could also be written in one of the alternate forms listed in Appendix A. To measure Z the system must be tested under at least N conditions as indicated by

----This work performed at Sandia National Laboratories is supported by the United States Department of Energy (DOE) under Contract DE-AC04-94AL85000. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the DOE.


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