Vibration analysis of compressor piping system with fluid pulsation

Transcription

1 Journal of Mechanical Science and Technology 6 () () 393~399 DOI.7/s Vibration analysis of compressor piping system with fluid pulsation Seong-Hyeon Lee, Sang-Mo Ryu and Weui-Bong Jeong * School of Mechanical Engineering, Pusan National University, 3 Jangeon-dong, eumjeong-gu, Seoul, 69-73, Korea (Manuscript Received December, ; Revised July 3, ; Accepted August 4, ) Abstract A piping system connected to a rotary compressor is an essential component for refrigerant transport in air-conditioning systems. The vibration of the pipes has been thought to be generated only by the mechanical forces due to the compressor operation. In this study, the fluid pulsation in the pipe is considered to be a source of the vibration, as well as the mechanical forces by the compressor operation. The mechanical force was first identified experimentally using measured acceleration signals over the shell. The calculation of the fluid force resulting from the pulsating fluid in the pipe was then derived theoretically. The estimation used the pressure pulsation signal in the pipe measured by a pressure transducer. Both sources of the vibration were finally applied to the finite element model of the piping system. Conclusively, the prediction of the vibration response to both sources showed better agreement with the experimental results than prediction considering only the mechanical force. Therefore the theoretical process deriving the fluid force was valid. Keywords: Fluid pulsation; Force identification; Piping system; Rotary compressor; Vibration Introduction * Corresponding author. Tel.: , Fax.: address: wbjeong@pusan.ac.kr Recommended by Associate Editor Cheolung Cheong KSME & Springer The compressor is an important part of an air-conditioning system. A piping system is attached to the compressor to carry the refrigerant. The design of the shape of the pipes is a difficult problem for the designer because pipe vibrations of the pipe are not easy to predict at the design stage. When severe vibrations occur in prototype pipe models, damping materials are often used instead of changing the shape of the pipe. A study to predict the vibrations of the pipes attached to a compressor is therefore necessary at the design stage. Yanagisawa et al. [] studied the vibrations of a rollingpiston-type rotary compressor for air-conditioners and suggested a simple method for predicting the vibration under different operating conditions. Etemad and Nieter [] discussed the basic theory of the scroll-compressor concept from a design point of view and proposed an optimization approach. Cohen [3] surveyed the technologies of various types of compressor. Padhy [4] studied the dynamics of rotary compressors in detail and presented experimental validations. Han [] studied the vibration of rolling-piston-type rotary compressors for air-conditioning both numerically and experimentally. He also investigated several variables that affect compressor vibrations. Kim [6] performed a dynamic analysis of the reciprocating compression mechanism of a small refrigeration compressor, with respect to the hydrodynamic forces of the journal bearings. Some research has been carried out on force identification, and several methods have been suggested. Law [7] derived the vertical dynamic interaction forces between a moving vehicle and a bridge deck analytically and verified the results experimentally. Law [8] also provided bounds to the ill-conditioned results in an identification problem using the Tikhonov regularization technique. Liu [9] suggested enhanced least square schemes to overcome the inversion instability of an illconditioned frequency response function (FRF) matrix at frequencies near the structural resonances during the identification of dynamic forces from structural responses. Lee [] solved the impact load on the plate theoretically, using reens function and the acoustic waveform. Kim [] verified Lee s [] research experimentally. Previous studies have assumed that the source of pipe vibrations is the force generated by the compressor. However, the refrigerant discharged from the cylinder will pulsate periodically. The pulsating fluid will also generate pipe vibrations. In this paper, two sources of pipe vibrations are considered. One is the mechanical force generated by the compressor. The other is the fluid force generated by the pulsating refrigerant. An experimental method is presented for identifying the mechanical force using the acceleration measured at the compressor shell. The identified force was validated by comparing the predicted acceleration with that measured at the shell. The fluid force acting on a curved pipe as a result of pulsating refrigerant was also derived theoretically. The pulsating pressure in a pipe was obtained experimentally using pressure

2 394 S.-H. Lee et al. / Journal of Mechanical Science and Technology 6 () () 393~399 transducers. The piping system was modeled using a finiteelement method to determine the characteristics of the pipe. Using an experimental set-up as an air-conditioning system, the vibrations at several points in the piping system are measured to verify the predicted results. The experiments showed that predictions of the pipe vibrations that considered the mechanical force and the fluid force showed better agreement with the experimental data than do those considering only the mechanical force. It was shown that the effects of the pulsating fluid on the pipe vibrations should be taken into consideration in the prediction of the pipe vibrations.. Mechanical force of a compressor. Experimental identification The highest vibration of a compressor occur at a frequency of Hz, which is the operating frequency of the motor. Since the first natural frequency of the flexible mode of the compressor is about 6 Hz, the compressor can be assumed to be a rigid-body at the operating frequency. A rigid compressor supported by rubber mounts can be modeled as a six- degreesof-freedom system with three translational motions and three rotational motions. The equation of motion of the rigid-compressor model can be given as follows: { ( )} ( ) { } { ( )} M X ɺɺ t K X t F t () + = { X( t) } T = { x y z θx θy θz} { F( t) } T { Fx Fy Fz Mx My Mz} =. The vector {X(t)} and {F(t)} respectively represent the displacement and the force vector generated at the mass center of the compressor. The 6 6 mass matrix [M] consists of the mass and mass moment of inertia of the rigid compressor, and the 6 6 stiffness matrix [K] consists of the stiffnesses and positions of the three rubber mounts as following equations. M M M M = Ixx Ixy Ixz Ixy Iyy I yz Ixz Iyz I zz () 3 k xi T K = T kyi T i i i= kzi (3) In [M], M is mass of compressor ; I xx, I yy, I zz, I xy, I yz and I zx are mass moments of inertia of the rigid compressor. In [K], k xi, k yi and k zi are the stiffness of i-th mount in x, y and z direction, respectively. [T] i is the matrix which represents the relation between mount-contact position (x i, y i, z i ) and mass center (x, y, z ) as given in Eq. (4). ( zi z) ( yi y) ( i ) ( i ) ( y y ) ( x x ) T = z z x x i i i (4) The theoretical derivation of the force vector {F(t)} generated in a compressor is very complicated. Also, the displacement vector {X(t)} cannot be measured directly using sensors. In this paper, the displacement vector {X(t)} at the mass center was identified from acceleration data measured at points on the compressor shell. The force vector {F(t)} was identified from the identified displacement vector {X(t)}. The procedure for the experimental identification is as follows. The relation between the velocities at two points in a rigid body is given as follows: x ɺ p = x ɺ + ω rp. () Here, x ɺ P and x ɺ denote the velocity vector at points P and. ω is the angular velocity of the rigid body and r P is the position vector from point to point P. From this relation, a vibration measured at the shell of the rigid compressor has the following relation with the vibration at the mass center of the compressor. { ( )} ( ) Xɺɺ f = T P( i) P i Xɺɺ f i = (6) ( ){ },(, ) { X ɺɺ ( f )} = { x ɺɺ ( ) ( ) ( )} T P i y ɺɺ P i ɺɺ zp i,( i=, ) P( i) { Xɺɺ ( f )} = { xɺɺ } T ɺɺ y ɺɺ z ɺɺ θ ɺɺ x θ ɺɺ y θz T = P( i) ( i= ),,. ( zp( i) z) ( yp( i) y) ( zp( i) z) ( xp( i) x) ( yp( i) y) ( xp( i) x) Here, (x, y, z ) denote the position of the mass center and (x P(i), y P(i), z P(i) ) denote the positions the accelerometers are attached to the shell. { Xɺɺ ( f )} P ( i ) denotes the Fouriertransformed data of the acceleration vector at the shell in the

3 S.-H. Lee et al. / Journal of Mechanical Science and Technology 6 () () 393~ x-, y- and z- directions. { Xɺɺ ( f )} denotes the Fouriertransformed data for the acceleration at the mass center, consisting of three translational motions and three rotational motions. Based on the least squares estimation, the acceleration at the mass center { Xɺɺ ( f )} is identified to minimize the cost function J given in Eq. (7). { ( )} ( ) { } J = Xɺɺ f T Xɺɺ f (7) P P 3 If the number of acceleration channels N p is selected to be larger than six, the { Xɺɺ ( f )}, which minimizes J is given by Eq. (8). T T { X( f) } = ( T T ) T X( f) P P P { } ɺɺ ɺɺ (8) T T T = P T T P( ) P( N p /3). Here, { Xɺɺ ( f )} P is an N p column vector, { Xɺɺ ( f )} is a 6 column vector and [T] P is an N p 6 matrix. The force vector {F} which is defined in Eq. (), can be expressed as follows: F f = M K X f ( π f) { ( )} P { ( )} ɺɺ. (9) Here, [M] and [K] represent the 6 6 mass and stiffness matrices of the rigid compressor with three supporting mounts and f denotes the frequency in the Fourier-transformed signal. A piping system is generally connected to a compressor. The responses of compressor-piping system can be predicted using the identified force {F(f)} or {F(t)}.. Verification Three-directional accelerometers were established at three points on the shell of the rotary compressor, as shown in Fig.. This compressor has 4.7 cc/cycle flowrate,.86 MPa discharge pressure, suction pressure.34 MPa and 46.4 cc/cycle suction volume. The compressor is supported by three rubber mounts and has an operating frequency of Hz. A typical example of the acceleration measured at point 3 is shown in Fig.. From Fig., we can see that the tangential acceleration is higher than the acceleration in the other directions, this is characteristic of a rotary compressor. Fig. 3 shows the estimation of the compressor forces and moments at the center of mass. Fig.. Acceleration-measurement points of compressor. acceleration(m/s ) acceleration(m/s ) acceleration(m/s ) time [s] 4 - (a) Tangential component time [s] 4 - (b) Vertical component time [s] (c) Radial component Fig.. Steady-state acceleration responses of point 3 in time domain.

4 396 S.-H. Lee et al. / Journal of Mechanical Science and Technology 6 () () 393~399 force [N] Fx Fy Fz acc magnitude[m/s ] experiemnt analysis (a) Force (a) Point Moment [Nm] Mx My Mz acc magnitude[m/s ] experiemnt analysis (b) Moment (b) Point Fig. 3. Identified peak value of force and moment generated by compressor. The forces and moments were identified using the method described in section.. According to Fig. 3(b), M z is higher than the other moments, and both the forces and moments have about Hz (48.6 Hz to be precise) components and their multiples, because Hz is the operating frequency of the motor in the rotary compressor. The accelerations, predicted from the identified forces and moments were compared with the measured values at three points in Fig.. The results showed good agreement, as shown in Fig. 4. The experimentally identified compressor force is therefore valid enough to use. 3. Fluid force in a pipe 3. Theory Eq. () is the momentum equation for a fluid in an arbitrary control-volume. d ρudv + ρu ( Un) da ρu ( Un) da dt V A out Ain = Fpressure Ffluid () Here, F pressure is the force caused by the pressure and F fluid is the fluid force acting on the wall. V is the volume, ρ is the fluid density, A out is the outflow area, A in is the inflow area and U n is the fluid velocity normal to the cross-section. acc magnitude[m/s ] experiemnt analysis (c) Point 3 Fig. 4. Comparison of peak value of the acceleration at the compressor shell. y P ( U( t) Fig.. Curved pipe. x θ ( ) ( t) P t U Let us consider a curved pipe of radius R and curve angle θ, as shown in Fig.. R F fluid F wall

5 S.-H. Lee et al. / Journal of Mechanical Science and Technology 6 () () 393~ If the pressure P(t) and fluid velocity U(t) are composed of both static components and dynamic components, these terms can be formulated as in Eq. (). ( ) ( ) ( ) ( ) P t = P + P t (-) U t = U + U t (-) Here, P and U are the static terms of the pressure and flow, respectively, and P (t) and U (t) are the dynamic terms of the pressure and flow, respectively. If the fluid is pulsating with a frequency ω, the pressure and velocity of the fluid will be given by Eq. (). ( ) sin ( ) sin P t = P + P ωt (-) U t = U + U ωt (-) The relations d ρudv dt V d θ = ρ( U+ U( t) )( cosφ i + sinφ j) ARdφ dt = ρrauɺ t θ i + θ j Aout Ain ( )( sin ( cos ) ) ( n) = ( )( ( )cos + ( )sin ) ρu U da ρ AU t U t θ i U t θ j = ρ A U + U t θ i + θ j ( ( )) ( cos sin ) ρu U da ρ AU t i ρ A U U t i ( ) = ( ) = + ( ) n [ ] F A P t P t i AP t sinθ j pressure = ( ( ) ( ) cosθ) ( ) (( cosθ) sinθ ) = AP i j + AP t i j ( cos sinθ ) ( ) ( θ) F fluid act- and substituting into Eq. () give the fluid force ing on the pipe as follows: Ffluid t Fstatic Fdynamic t ( ) = + ( ) ( )( ) ( ) (3) (4) () (6) (7) Fstatic = AP + ρ AU cosθ i AP + ρ AU sinθ j AP ( t) ρ A U U( t) ( U( t + + )) Fdynamic = i ( cosθ) ρrauɺ ( t) sinθ AP ( t) ρ A U U( t) ( U( t + + )) sin + ρrauɺ ( t)( cosθ) θ j. Table. Acceleration magnitudes measured at the point in Fig. 6 at harmonic frequencies Hz ( st order) Here, F static represents the static fluid force acting on the pipe and Fdynamic( t) represents the dynamic fluid force acting on the pipe structure as a result of pressure and velocity fluctuations. In the finite-element formation, the force in Eq. (7) is derived for local coordinates of each element. Therefore it is necessary to transform the local coordinates into global coordinates for the finite-element application. 3. Experiments In order to verify the theory presented in this paper, experiments were performed on real compressor-piping systems. Fig. 6 shows the measurement point in a discharge pipe. This point is selected because it is the point the most severe vibrations occur. Table shows the magnitudes of the acceleration at the harmonic components of the operating frequency. Fig. 7 shows the experimental set-up used to measure the pulsation pressure of the fluid in a pipe using a pressure transducer. Table shows the dynamic pressure measured at a distance of m from the compressor. 3.3 Verification 97. Hz ( nd order) 4.87 Hz (3 rd order) 9.7 m/s m/s.8 m/s : measurement point compressor Fig. 6. Acceleration measurement point of compressor pipe. It was assumed that the static discharge pressure was P =.46 MPa and the static flow-rate was AU = m 3 /s. It can be seen that the dynamic pressure given in Table is very small compared to the static pressure. A finite-element beam-model of this compressor pipe was made for CAE analysis, as shown in Fig. 8 with 34 nodes and 339 elements. Young's modulus is.6 [N/m], Poisson ratio is.3 and the density is 94 [kg/m 3 ]. The pipe it has uniform cross-section with outer diameter.9 m

6 398 S.-H. Lee et al. / Journal of Mechanical Science and Technology 6 () () 393~399 Table. Dynamic pressure magnitudes of harmonic frequencies at distance of m from discharging nozzle Hz ( st order) 97. Hz ( nd order) 4.87 Hz (3 rd order) 44 Pa 6 Pa 463 Pa Table 3. Estimation of fluid forces acting on curved part of pipe Hz ( st order) 97. Hz ( nd order) 4.87 Hz (3 rd order) Point F x.979 N N.46 - N, ' F y N N N Point F x.98 N.77 N N, ' F y Table 4. Estimation of acceleration in Fig. 9. m Order Experiments [m/s ] Compressor force (error [%]) Estimation [m/s ] Compressor + Fluid force (error [%]) (4.89) (-3.83) (88.7) 3.33 (4.4) (86.98) (9.4) Pressure measuring point Fig. 7. Pressure measurement at distance of m from compressor. ' ' acceleration magnitude [m/s ] 8 4 experiment compressor force compressor force + fluid force 3 order Fig. 9. Acceleration magnitudes of rotating frequency and its multiples. z compressor connected here y x Fig. 8. Compressor pipe (FEM model). and inner diameter.8 m. This pipe is combined with a compressor using rigid constraints. Since the pipe model has uniform cross-section, and little pressure change, the Poisson coupling which causes axial stress by rapid pressure fluctuation was neglected. Therefore it was assumed that the fluid force was generated at the curved point, ',, and ' in Fig. 8. The fluid force caused by fluid pulsation acting on these points can be calculated from Eq. (7). Table 3 shows the estimated fluid forces. The acceleration magnitudes at the rotating frequency and its multiples are shown in Fig. 9. The measurement point is that given in Fig. 6 or point in Fig. 8. The symbol denotes experimentally determined acceleration magnitudes. The symbol denotes the accelerations predicted using only the mechanical force generated by the compressor. The symbol denotes the accelerations predicted using both the mechanical force and the fluid force generated by fluid pulsation. When only the mechanical force is considered, the calculated pipe accelerations are less than the experimental values as shown in Fig. 9. However, when both the mechanical force and the fluid force are considered simultaneously, the predicted value is closer to the experimental value. The values of Fig. 9 is also shown in Table 4 with the errors in respect of Experimental results. Therefore, as well as the mechanical force generated by the compressor, the fluid force generated by the pulsating fluid in a pipe should be considered in predicting the vibrations of a piping system.

7 S.-H. Lee et al. / Journal of Mechanical Science and Technology 6 () () 393~ Conclusion The forces by the pressure and velocity of a pulsating fluid acting on a curved pipe were derived theoretically and applied to compressor pipe system. The mechanical force generated by the compressor and the fluid force generated by the pulsating fluid in the pipe were considered simultaneously as the source of vibration in the pipe. The estimated error of acceleration of pipe by the simultaneous use of both the fluid force and the mechanical force showed 3.83% in st order, 4.4% in nd order and 9.4% in 3 rd order while those by use of only mechanical force showed 4.89% in st order, 88.7% in nd order and 86.98% in 3 rd order. Both the fluid and mechanical forces should be considered simultaneously to get better estimation of the vibration of the pipe. In conclusion, this study shows that the fluid force in the pipe, which was derived in this paper should be considered to improve the reliability of the prediction of the vibration of compressor-pipe system. References [] T. Yanagisawa, M. Mori and Y, Ogi, Vibration of a rolling piston type rotary compressor, International Journal of Refrigeration, 7 (4) (984) [] S. Etemad and J. Nieter, Design optimization of the scroll compressor, International Journal of Refrigeration, (3) (989) 46-. [3] R. Cohen, Advances in compressor technology, International Journal of Refrigeration, 3 (4) (99) [4] S. K. Padly, Dynamic analysis of a rotary compressor, ASME Journal of Mechanical Design, 66 (994) [] H. S. Han, S. W. Hwang and J. S. Koo, Vibration analysis of a rotary compressor, International Journal of Precision Engineering and Manufacturing, (3) (4) [6] T. J. Kim, Dynamic analysis of a reciprocating compression mechanism considering hydrodynamic forces, Journal of Mechanical Science and Technology, 7 (6) (3) [7] S. S. Law, T. H. T. Chan and Q. H. Zeng, Moving force identification: A time domain method, Journal of Sound and Vibration, () (997) -. [8] S. S. Law and X. Q. Zhu, Study on different beam models in moving force identification, Journal of Sound and Vibration, 34 (4) () [9] Y. Liu and W. Steve Shepard Jr., Dynamic force identification based on enhanced least squares and total least-squares schemes in the frequency domain, Journal of Sound and Vibration, 8 (-) () [] S. K. Lee, Identification of impact force in thick plates based on the elastodynamics and time-frequency method (I) - Theoretical approach for identification the impact force based on elastodynamics, Journal of Mechanical Science and Technology, (7) (8) [] S. J. Kim and S K. Lee, Identification of impact force in thick plates based on the elastodynamics and time-frequency method (II) - Experimental approach for identification of the impact force based on time frequency methods, Journal of Mechanical Science and Technology, (7) (8) Ulsan, Korea. Seong Hyeon Lee received a B.S. degree in Mechanical Engineering from Pusan National University in 6. He then went on to receive his M.S. and Ph.D. degrees from Pusan National University in 8 and, respectively. He is currently researcher at Hyundai Heavy Industries Co., Ltd., Sang Mo Ryu received a B.S. degree in Mechanical Engineering from Pusan National University in. He then went on to receive his M.S. degree from Pusan National University in. He is currently a researcher at Hyundai Motors Company, Hwaseong, Korea. Weui Bong Jeong received a B.S degree from Seoul National University in 978. He then went on to receive his M.S. and Ph.D. degrees from KAIST in 98 and from Tokyo Institute of Technology in 99, respectively. Dr. Jeong is currently a Professor at the Mechanical Engineering at Pusan National University in Busan, Korea.