Purdue University Purdue e-pubs International Compressor Engineering Conference School of Mechanical Engineering 1998 Indirect Measurement of Random Force Spectra on Fractional Horse Power Reciprocating Compressor Shell E. S. Kim LG Electronics Inc. K. Kang LG Electronics Inc. J. Sim LG Electronics Inc. Follow this and additional works at: http://docs.lib.purdue.edu/icec Kim, E. S.; Kang, K.; and Sim, J., "Indirect Measurement of Random Force Spectra on Fractional Horse Power Reciprocating Compressor Shell" (1998). International Compressor Engineering Conference. Paper 1295. http://docs.lib.purdue.edu/icec/1295 This document has been made available through Purdue e-pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu for additional information. Complete proceedings may be acquired in print and on CD-ROM directly from the Ray W. Herrick Laboratories at https://engineering.purdue.edu/ Herrick/Events/orderlit.html
INDIRECT MEASUREMENT OF RANDOM FORCE SPECTRA ON FRACTIONAL HORSE POWER RECIPROCATING COMPRESSOR SHELL Edward Sungjin Kim, Kyungseok Kang and Jaesool Sim Living System Research Laboratory (ChangWon) LG Electronics, ChangWon, KOREA ABSTRACT In the design of the small hermetic compressor, the forces acting on the compressor shell are important parameters to study for noise and vibration control. However, the input forces to the shell cannot be measured directly. In this paper, random force spectra on compressor shell has been measured indirectly with in-situ vibration responses and system information. Three existing techniques of indirect force measurement including direct inverse, principal component analysis and regularization have been compared. It has been shown that multiple vibration responses are essential for correct estimation of the random force spectra. It has also been shown that relatively higher forces are transmitted through spring paths to the compressor shell. 1. INTRODUCTION Most fractional horse power reciprocating compressors are spring mounted and the discharge pipes are directly connected on the enclosing shell. These springs and discharge pipes are major paths of vibration energy transfer from the compression mechanism to the shell. Therefore, measurements of these force transmission provide valuable infonnation for the design of quieter products. However, direct measurement of such forces are not easy to make while the compressors are in operation. The vibration resulting from the dynamic forces is generally easier to measure, although the information from all of the forces is mixed and altered by the respective transfer paths to any single vibration measurement location. If the transfer functions from force inputs to vibration outputs are known, then the input forces can be recovered by the solution of a general inverse problem. In this paper, three existing techniques of the general inverse problem are compared for the identification of the force inputs to the compressor shell. To overcome the difficulty associating with simultaneous measurement of multiple vibration responses, a reference method has been proposed in which the matrix of cross spectra for multiple vibration outputs can be measured even with two channel signal analyzers. 2. THEORY AND EXPERIMENTAL SET-UP For a multiple input, multiple output (MIMO) linear system, the matrix of vibration response cross spectra ( Saa ) is related to the matrix of unknown force cross spectra ( S if ) and the system frequency response function (FRF) matrix ( Hqf) by[l] where the superscript "H" denotes the Hermitian transpose. The unknown force matrix ( S if ) can be found by where the superscript "+" denotes the pseudo-inverse matrix operation which provides the least squares solution for the system of equations. This pseudo-inverse matrix operation often results large estimation errors since the FRF matrix ( Haf ) becomes ill-conditioned near system resonance or anti-resonance frequencies. The principal 537
component analysis (PCA) and the regularization method, both based on singular value decomposition of the FRF matrix in the frequency domain, are two common methods to solve this ill conditioned matrix problem.[2,3,4] In general, the number of vibration outputs to be measured is greater than that of force inputs since the estimation errors could be effectively reduced by making the system of equations over determined. However, arbitrary increase of vibration output number is often limited in practice since all vibration responses should be measured simultaneously to have the matrix of response cross spectra ( Saa ). The cross spectrum (S 11 ) between i and j measurement points can be expressed by (3) where Hr represents the transfer function between a measurement point i and a reference point r, and Sr:,; represeii.ts the cross spectrum between the reference point r and the measurement point j. Therefore, the matrix of cross spectra ( Saa) for multiple outputs can be obtained even with two channel signal analyzers by having a reference signal and measuring the required transfer function Htr and the cross spectrum St:J of equation (3). However, care should be taken that sufficient coherence between the reference signal and the vibration responses should be maintained during the entire measurement. The experimental set up of this study is shown in Figure L Two forces on the lower half compressor shell via mechanical shakers are measured directly and compared with the indirectly measured ones. To estimate the five force inputs to the lower half compressor shell under air compression, the suction and the discharge pressure (gauge pressure) are maintained as 0 and 15 Kgf/cm 2, respectively. B&K 3550, 48 and 8200 have been used as signal analyzer, mini shaker and force sensor, respectively. The vibrations are measured with PCB JM353 accelerometer. A Pentium PC has been interfaced with the analyzer through IEEE488 bus to control the entire measurement process. 3. RESULTS Figure 2 shows the comparison of the estimated force spectra from two vibration responses with the directly measured force spectra when two mini shakers provide force inputs on the lower half compressor shell. It is clearly shown that both spectra are in good agreement for most frequency range of interests except relatively large errors near resonance frequencies. Figure 3 and 4 show the simulated force spectra using PCA and regularization method, respectively. In these figures, it is shown that the estimation errors near resonance frequencies can be reduced with these advanced numerical techniques. However, in Figure 2 and 3, relatively large errors are occurred near 2200Hz and 4800Hz off 1 estimation. When the number ofvibration responses is identical to the number of force inputs, the transfer matrix can be ilkonditioned easily.[4] Figure 5 shows the estimation result when four vibration responses are measured. It is clearly shown that the accuracy of the estimation has been greatly improved near resonance and anti resonance frequencies. It is also shown that the estimation errors near 2200 Hz and 4800 Hz are largely reduced. In comparison with Figure 3 and 4, it is clearly demonstrated that better estimation performance is expected with more vibration measurements than the advanced numerical. techniques in the inverse problems such as PCA and regularization method. Figure 6 and 7 show the comparison of the estimation results using PCA and regularization method with measured ones when four vibration responses are measured. It is shown that the estimations are not improved much with these techniques but rather large errors for certain frequencies. In conclusion of this prinuuy test results, it is clearly demonstrated that the force inputs on the lower half compressor shell can be effectively estimated from the vibration response measurements. There might be five major force inputs on the compressor shell while in operation since there are four mounting springs and one discharge pipe coimection. Unlike shaker excitations, direct measurement of these forces, especially force input through piping, are not possible. Therefore, indirect force estimation scheme of this 538
study has been applied to estimate five input force spectra on the shell. The compressor has been operated in air with open lower half shell, and their suction and discharge pressures have been controlled to be 0 and 15 Kgf/cm 2 (gauge pressure). Ten vibration responses are measured to estimate five force spectra. Prior to the vibration measurement in operation, the system transfer functions between four spring and one pipe connection points and ten vibration measurement points have been measured. Figure 8 shows those five input force spectra on the compressor shell. In this figure, FP represents force through pipe connection and F,i represents force through springs. It is shown that overall trends of five force spectra are almost identical, decreasing force level as the frequency increases. Unexpectedly, in Figure 8 (b), it is shown that spring forces are higher than pipe forces. Springs 3 and 4 are located below cylinder head, hence experience heavier weight than springs I and 2. Interestingly, it is shown that F, 3 and F, 4 are higher than F, 1 and Fs2 Even though force transmissions through the springs are higher than pipe, it does not tell that spring forces should be reduced to make compressor quiet since modal participation factors of spring excitations were measured to be less than that of pipe excitation. Since the direct measurement of pipe force is impossible, the results of the above indirect force measurement have been verified by the comparison of the vibration response variation with force input variation. Figure 9 (a) shows effects of discharge pipe on measured vibration spectra. The dashed line represents vibration spectra at a point of the compressor shell when the discharge pipe is connected to the shell. It is shown that vibration level has its peak values near 3200 Hz. In Figure 9 (a), the solid line represents the vibration level at the same point without discharge pipe connection. It is clearly shown that vibration levels near 3200 Hz are greatly reduced by disconnecting the discharge pipe. Figure 9 (b) shows simulated vibration response spectra for the variation of pipe connection. It is clearly shown that simulated vibration responses with the estimated force input spectra are in good agreement with those of measured spectra, demonstrating correct indirect force measurement of pipe force in this study. 4. CONCLUSIONS In this study, three existing techniques of the general inverse problem were compared to identify the force inputs to the compressor shell. To overcome the difficulties associating with simultaneous measurement of multiple vibration responses, the reference method has been proposed in which the matrix of cross spectra for multiple vibration outputs can be measured with two channel signal analyzers. It has been shown that the existing indirect force measurement techniques could provide plausible results for the estimation of input forces on. compressor shells. It has been demonstrated also that better estimation performance is expected with more vibration measurements than the advanced numerical techniques in the inverse problems such as PCA and regularization method. Ten vibration responses were measured to estimate five force spectra on the half compressor shell. It has been shown that overall trends of five force spectra are almost identical, decreasing force level as the frequency increases. It also has been shown that spring forces are higher than pipe forces. Finally, it has been shown that the simulated vibration responses with input force variations are in good agreement with measured ones, demonstrating correct indirect force measurement of this study. REFERENCES 1. T. J. Roggenkamp and R. J. Bernhard, 1993, "Indirect Measurement of Multiple Random Force Spectra", Proceedings of Inter-Noise 93, pp. 881-883. 2. J. A. Fabunmi, 1986, "Effects of Structural Modes on Vibratory Force Determination by the Pseudoinverse Teclurique", AIAA Journal, Vol. 24 (3), pp. 504-509. 3. R E. Powell and W. Seering, 1984, "Multichannel Structural Inverse Filtering'', Transactions of ASME, Vol. 6, pp. 22-28. 4. J. K. Lee, 1993, "A Study on Indirect Force Measurement in Structure", Ph. D. Dissertation, KAIST, Korea. 539
Force Tranducer Half Comp. Shell Power Amp PC GPIB Signal Analyzer B&K 3550 Fig 1. Schematic Diagram of Experimental Set up z l!! :!:!.20... ll\o-k\j.,..., -30!'l &-40 z... l!! :!:!..20 ru > -30!'l &-40 ow0---16oo-oo-32oooo-oooo 800 1600 2400 3200 4000 4800 00 6400 (b)f2 Fig 2. Comparison of Simulated Force Inputs by Direct Inverse with Measured Force Inputs (2 In-2 Out) z 0 l!! - 'E.-20 ]!'l. --40 g l!! - :E. 20 ]-30-40 & ----L--- 0 800 1600 00 3200.woo <1800 00 00 800 1600 2<0 3200.woo 4800 5600 6400 (b) F2 Fig 3. Comparison of Simulated Force Inputs by PCA with Measured Force Inputs (2 In-2 Out) 540
z... /!'_ m 20 0-20 -30._j B & -40-50 z 2D 0 /!'_ - m -20-30 B & -40 -al -60 D 800 1600 2400 3200 4000 800 1600 2400 3200 4000 4800 5600 6400 (b) F2 Fig4. Comparison of Simulated Force Inputs by Regularization with Measured Force Inputs (2 In-2 Out) 0 /!'_ - m :E..-20 -- _5-30 B &-40 - ().....;.::1--..,!!! - -50-60--4------- 0 600 1600 2400 3200 o4000-4800 5600 6400 (b)f2 Fig 5. Comparison of Simulated Force Inputs by Direct Inverse with Measured Force Inputs (2 In-4 Out) -600600---1600240000-4000-480056006400 (b)f2 Fig 6. Comparison of Simulated Force Inputs by PCA with Measured Force Inputs (2 In-4 Out) 541
20 20 z 0 z.,... e - e ro :::. -20 ] _;;() 1l J: -' 1l -50-50 -50-50 :::. (]] J: 0 BOO 1600 2400 3200 4000 4800 5600 6400 0 BOO 1600 2400 3200 4000 4800 5600 6400 (b) F2 Fig 7. Comparison of Simulated Force Inputs by Regularization with Measured Force Inputs (2 In-4 Out) 0 20.-----------...,.----.,------, 00 2000 3000 4000 5000 6000 1750 2000 20 2000 3000 (a) Total Range (b) Zoom Range Fig 8. Estimation of 5 input Force Spectra to the Half Compressor Shell (P.,JP dg = 0 I 15 Kgf/Cm 2 ) e e -.; <( ro :::. "@ (a) Measurement (b) Simulation Fig 9. Effects of Pipe Connection on Vibration Response Spectra for Measurement Point 1 542