verification of the vibration characteristics of a

VERIFICATION OF THE VIBRATION CHARACTERISTICS OF A RECIPROCATING COMPRESSOR IN OPERATION AND THE PROPOSAL OF THE MODEL

PARAMETERS ESTIMATION METHOD

Yoshifumi MORI Plant Diagnosis Section, Plant Maintenance

Dept., Tokuyama Corporation Shunan, Yamaguchi, Japan

Email: [email protected]

Takashi SAITO Systems Design and Engineering, Graduate

School of Science and Engineering Yamaguchi University

Ube, Yamaguchi, Japan Email: [email protected]

Katsuhide FUJITA

Ube National College of Technology Ube, Yamaguchi, Japan

ABSTRACT

Vibration characteristics of a reciprocating compressor are discussed. To investigate the frequency characteristics in the operation of unhealthy and healthy, we employed the proposed model, which includes stiffness characters about both the connecting and the sliding parts. Expressing the motion of the reciprocating compressor as a rigid body model with eleven degrees of freedom, we numerically investigated the frequency characteristics in operation and the natural vibration characteristics based on a locally linearized method. To examine the frequency characteristics, we carried out vibration tests using a small experimental machine and the eigenvalue analysis based on the proposed model. Comparing the analytical results with experimental results, we found that the proposed model could simulate the fundamental frequency characteristics in operation and the natural modes. Eigenvalue analysis shows that the natural frequencies and modes for a reciprocating compressor depend upon the angle of the crankshaft. In the proposed model, it can express the dominant frequencies occurring during operation and the natural vibration characteristics.

INTRODUCTION

Reciprocating compressors are widely used, particularly in petroleum refining and petrochemical processing, to compress low molecular weight gases such as hydrogen, because they

have very high compression ratios. From an economic point of view, there are considerable interests in management schemes, which ensure long-term continuous operation. Although the regular maintenance is performed every several years, it has been reported, during long-term continuous operation, serious breakdowns occur, which result in costly suspension of operation. Currently, the causes of these troubles have not yet been elucidated. Adding to the complexity of the phenomena, reciprocating compressors contain many components and driving and rotating mechanisms, which might have something troublesome during the operation. Because reciprocation produces a steady-state impact load, there are numerous cases for which conventional equipment diagnostics cannot predict the source of trouble [1].

Preceding studies aim at understanding the degree of breakdowns in the reciprocating compressor and establishing a new monitoring technique[1]-[3]. We proposed a mathematical model in which both the deterioration and damage can be expressed by changing model parameters. With the measuring data and the simulation results for real machines, we elucidated (1) the cause of the resonance peak that occurs during operation by examining the frequency characteristics of the reciprocating compressor and (2) the natural vibration characteristics by the use of numerical analysis [2][3].

In this study, to verify the relationship between the frequency characteristics during operation and the natural frequency of the system, we introduce a small experimental machine. Measuring the acceleration of the moving parts, we

1 Copyright © 2014 by ASME

Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition IMECE2014

November 14-20, 2014, Montreal, Quebec, Canada

IMECE2014-36812

carry out vibration tests and the eigenvalue analyses based on the proposed model. This experimental machine has frequency characteristics corresponding to those of actual compressors. Using this machine, we can examine the natural frequencies and modes by the vibration testing and the corresponding analysis. We confirm that the cross head – piston rod – piston system is dominant in the natural vibration characteristics for reciprocating compressors. In addition, we perform parameter identification using the natural frequency characteristics and examine the relationship between the natural frequency characteristics and the frequency characteristics during operation.

FREQUENCY CHARACTERISTICS OF A RECIPROCATING COMPRESSOR IN OPERATION

Implementation of experimental measurement in operation

To investigate the frequency characteristics of the real machine, we measured the vibrational acceleration for a reciprocating hydrogen compressor with two double-acting vertically oriented cylinders (Φ540 mm) operating at 440 rpm (brake shaft power, 488 kW) at a maximum piston speed of 3.7 m/s. We measured the accelerations for the crankcase vibration at the crosshead shoe to determine the internal propagation in the outer casing. As shown in Fig. 1, we set the measurement positions and the accelerations were simultaneously measured using three-axis accelerometers.

Fig.1 Scheme of the experimental set up

Results of frequency characteristics

To date, there is a rare study, which treats on the frequency characteristics of a reciprocating compressor in operation. Therefore, we examined frequency characteristics before and after the breakdown. As shown in Fig.2 the results were compared before and after the damage, which respectively correspond to the case for HEALTY and UNHEALTHY. From these results, during the normal operation, we find many resonance peaks that are not related to the fundamental frequency of the crank rotation speed. We presume that the dominant frequencies among those resonant peaks are related to the natural frequencies. Also, we found that the frequency characteristics notably change before and after a breakdown.

Fig.2 Comparison of the frequency responses about the acceleration for HEALTHY with that of UNHEALTHY on a reciprocating compressor in operation Mathematical model for reciprocating compressor

In preceding study, we proposed the mathematical model in which the dynamic characteristics of the system are expressed by varying the model parameters. This model is nonlinear and has eleven degrees of freedom. In the model, the moving parts of the reciprocating compressor are assumed to consist of four rigid body parts, and the characteristics of the connecting sections and sliding sections between each part are replaced by the characteristics of elastic members, which serve as model parameters.

To examine whether the proposed model can be express about the before and after frequency characteristics or not, we perform a numerical simulation on the change in rigidity that is considered to take place in the example of breakdown described in above. Fig.3 shows the result of the zoom from 0 to 500Hz of Fig.2. Fig.4 shows the corresponding results of the frequency response by the numerical analysis. We find that the resonant frequencies around 200Hz for both the results shifted upper side and the proposed model can be express the change of the frequency characteristics before and after. Further, from the results of the eigenvalue analysis based on the model and vibration experiments, we found that the dominant frequencies occurring during compressor operation are related to the natural frequencies corresponding to the bending modes.

Fig.3 Comparison of frequency response for HEALTHY and UNHEALTHY for the actual machine

Fig.4 Comparison of frequency response for stiffness change for the numerical simulation

480mm150m

m

480mm 150m

m0 200 400 600 800 1000 1200 1400 1600 1800 2000

0.0

2.0x10-4

4.0x10-4

6.0x10-4

8.0x10-4

1.0x10-3 HEALTHY

Acc

eler

atio

n of

exp

erim

enta

l [m

/s2 ]

Frequency [Hz]

UNHEALTHY

0 100 200 300 400 5000.0000

0.0005

0.0010

Acc

eler

atio

n [m

/s2 ]

Frequency [Hz]

UNHEALTHY HEALTHY

0 100 200 300 400 5000.01

0.1

1

10

100

1000

10000

Acc

eler

atio

n [m

/s2 ]

Frequency [Hz]

UNHEALTHY HEALTHY


VERIFICATION OF THE VIBRATION ANALYSIS MODEL USING THE SMALL EXPERIMENTAL MACHINE

Overview of the small experimental machine To verify the relationship between the frequency

characteristics during operation and the natural frequencies of the system, we have made a small experimental machine having a vibration characteristic similar to an existing machine. As the dominant frequency characteristics are related to bending natural frequencies during operation, the scale model is designed based on the bending natural frequencies for the crosshead-piston rod –piston system. As shown is Table 1, the bending natural frequencies were calculated by Bernoulli-Euler beam model with the rigid bodies at the both ends. In addition, Fig.5 shows the schematic view for the small experimental machine made, and Table 2 shows the model parameters for it.

Table 1 Natural frequencies of free vibration model

Mode order Natural frequency [Hz]

1st 245.6 2nd 964.5 3rd 2017.6

4th 3383.0

Fig.5 Computer aided drawing of the small experimental machine

Table 2 Parameters for the small experimental machine

Method of verification Here, we verified the vibration analysis model using the

small experimental machine. We showed that the proposed model can express the vibration characteristics in operation and the natural vibration characteristics corresponding to the vibration tests for real machines.

Therefore, we report on vibration tests performed with the small experimental device we made and compare with the analysis of eigenvalues based on the model. In addition, we perform parameter identification of connecting parts and sliding parts using the natural frequency characteristics to obtain more accessible model parameters.

VIBRATION TESTS AT THE SHUTDOWN PERIOD

Experimental method The cross head – piston rod – piston system of main parts to

measurement conditions were set to free support as shown in Fig.6. The measurement positions were seven points on the crosshead, piston rod and piston, which vibrate in the longitudinal direction and the vertical direction respectively. As in the eigenvalue analysis, measurement conditions were set to 0°, 45°, 90°, 135°, and 180° about the rotation angle of the crank, as shown in Fig. 7. The exciting positions were two points in the longitudinal direction of the piston and the vertical direction of the piston rod.

We investigated the natural frequencies by determining the accelerance from the acceleration response and excitation force from the impulse hammer. Furthermore, for each condition, we obtained arithmetic averages of eight measurements. The accelerance values obtained from the measurement points were converted to the barycentric position of the piston and piston rod for use in the vibration model.

Fig.6 Schematic view of crosshead－piston system in the small experimental machine

Fig.7 Excitation and measurement points for the vibration test

Revolution per minute N [rpm] 440 (Max 600)

Power [kW] 30

Inverter - 200V class

Mass mcr [kg] 44.9

Crank radius R [mm] 42.5

Mass mc [kg] 9.3

Length l c [mm] 230

Mass mch [kg] 7.28

Length l ch [mm] 150

Mass mpl [kg] 23.8

Length l pl [mm] 850 (full length; 1240)

Journal d pl [mm] 60

Mass mp [kg] 44.9

Piston diameter d p [mm] 197

Length l p [mm] 200

Piston stroke St [mm] 85

Average speed v p [m/s] 1.247

Crank shaft

Connecting rod

Crosshead

Piston

Piston rod

Motor

h3=5

5

Cross head h1=98.5

h2=3

0

Piston⑦ ⑥

⑤ ④ ③

② ①

Piston rod

180°

135°

90°

45°

Exciting Point Measurement Point


Results of vibration test by the cross head – piston rod – piston system

Table 3 shows the natural frequency results, related to the cross head-piston rod-piston system. By comparison with the results of calculation, it was confirmed that the bending natural frequency were the similar, as shown in Table 3.

Table 3 Results of vibration test by crosshead－piston system of small experimental machine

Results of vibration test for several rotation angle of the crank

Table 4 shows the natural frequency results for the different rotation angle of the crank. As the support conditions of the vibration test and eigenvalue analysis model are not identical, it is necessary to examine from both the natural frequencies and the modes. By comparing the results of the vibration test and the eigenvalue analysis, both show that natural frequencies change with the crank angle. Fig.8 shows the 2nd and 5th order mode. As a result, it is found that the 2nd order mode is the first bending mode of piston rod except for the 0 °. In addition, the 5th order mode is following corresponds to the 2nd bending mode, and it appears to reverse the phase of 0 ° and 45 °. From these results, we found that the natural vibration characteristics depend on the crank angle and dominant modes are caused by the bending mode of the piston rod. Table 4 Results of the natural frequency by the vibration test

Fig.8 Results of natural mode for 2nd and 5th by vibration test

Results of vibration test in operation We carried out the measurement of the acceleration for the

small machine in operation. The measurement points are the same with the static vibration tests.

Fig.9 and Table 5 shows the FFT analysis result of measurement of outside casing. From this result, we found that the natural frequency has appeared in operation. It is caused internal elastic characteristics in a reciprocating compressor, and frequency characteristics are influenced in operation. Then, it is supposed that these changes in frequency characteristics could be related to the failure. In particular, the dominant frequency occurring during compressor operation is related to the natural frequency corresponding to the bending and expansion/contraction mode. In addition, the dominant frequencies are confirmed near the natural frequencies.

Fig.9 Results of frequency analysis of measurement of outside casing by a small experimental machine (Arithmetic in the longitudinal direction and vertical direction)

Table 5 Peaks list of the frequency analysis

PARAMETER IDENTIFICATION

Method of Parameter Identification As the model parameters used in the analysis are

determined by the theory of the equivalent spring constants, they have some uncertainty. To investigate more accessible phenomena in the reciprocating machine, we identify the model parameters using the natural frequencies and the modes.

Mode order Natural frequency [Hz] Mode

1st 104.0 First Expansion / Contraction mode

2nd 257.7 First Bending mode

3rd 346.0 Second Expansion / Contraction mode

4th 492.6 Third Expansion / Contraction mode

5th 931.4 Second Bending mode

θ 0 [°] 45 [°] 90 [°] 135 [°] 180 [°]

1 180.7 180.7 180.7 180.7 180.7

2 314.9 314.9 317.4 317.4 317.4

3 419.9 419.9 419.9 419.9 419.94 461.4 461.4 459.0 454.1 454.15 813.0 810.6 817.9 813.0 815.46 1164.6 1169.4 1184.1 1191.4 1191.47 1738.3 1740.7 1740.7 1740.7 1740.78 2541.5 2580.6 2617.2 2605.0 2602.59 3049.3 3034.7 3032.2 3039.6 3061.510 3454.6 3449.7 3471.7 3432.6 3413.111 3933.1 3918.5 3930.7 3906.3 3933.1

(x0, y0) (xc, yc) (xch, ych) (xpl, ypl) (xp, yp)

Initial 0 45 90 135 180

0 200 400 600 800 1000 1200 1400 1600 1800 20000

1

2

3

Acc

eler

atio

n [m

/s2 ]

Frequency [Hz]

Experimental

Dominant frequency [Hz]

81

176300432674791

120413351738

2nd order mode

5th order mode


Using the results of natural frequency of the crank angle for the real machine, we derive the following error function.

，，，，，，，

，，，，，，

（1）

To minimize it, we employ an optimization method. Present algorithm is composed of optimization techniques combined with Genetic Algorithms (GA) [4] and Down-hill Simplex Method (DHSM) [5], as shown in Fig.10.

In down-hill simplex method, we used Eqn.(1) as an error function, where shows experimental results and shows simulation results. In this regard, was obtained using Householder method.

P

（2）

To minimize the error in Eqn.(2), we determined the elastic springs.

Fig.10 Identification algorithm

Result of Identified parameters

Are shown in Table 6 the results obtained by performing the parameter identification according to the crank angle. The identified model parameters are the same tendency as physical amount calculated theoretically. To examine the vibration characteristics, we performed eigenvalue analysis using the average of the optimal solution, as shown in Table 7. As shown in Fig.11, the eigenmode was found to be the mode in which variations are present, similar to the case in the natural frequency mode. However, the position of the piston and the piston rod was the inverse phase. The reason for this is considered to bearing support conditions of a small experimental machine. From these results, we can determine that the proposed vibration analysis model is reasonable. Moreover, it is found that this identified method allowed quantitative estimation of model parameters.

Table 6 Identified parameters

Table 7 Results of natural frequency by identified parameters

Fig.11 Results of natural mode for 2nd and 5th by parameters estimation method

CONCLUSIONS

In this study, we examined the characteristics of the frequencies in operation and in natural vibration for reciprocating compressors using analysis based on the proposed model and the vibration tests and experimental measurements for a small experimental machine. Furthermore, we identified parameters of vibration analysis model based on experimental data. The conclusions can be summarized as follows:

(1) Comparing with the experimental results, we show that

the proposed model could be used to investigate the frequency characteristics in operation.

Start

Generation of initial parameter group

selection

Binary coding

Crossover

Mutation

It meets the constraints and fitness ?

Yes

No

Generate the simplex by the operation of the substraction or addition to most fitness gene

It meets the Evaluation value and Constrains ?

Deform the Simplex

Yes

NoFinish

0 [°] 45 [°] 90 [°] 135 [°] 180 [°]

k xc 3.11×109

3.11×109

3.27×109

3.27×109

3.28×109

3.21×109

1.57×109

k yc 4.73×108

4.63×108

4.83×108

4.66×108

4.54×108

4.68×108

1.80×108

k xch 3.77×109

3.87×109

3.66×109

3.80×109

3.80×109

3.78×109

1.57×109

k ych 3.80×107

3.94×107

3.68×107

3.83×107

3.90×107

3.83×107

2.08×107

k xpl 6.96×108

6.86×108

7.06×108

6.89×108

6.97×108

6.95×108

1.16×108

k ypl 6.95×107

6.85×107

7.05×107

6.88×107

6.96×107

6.94×107

1.15×107

k xp 4.87×109

4.75×109

5.07×109

4.94×109

4.88×109

4.90×109

2.94×1010

k yp 1.06×108

1.06×108

1.08×108

1.07×108

1.07×108

1.07×108

1.15×107

k zch 7.21×107

7.14×107

7.34×107

7.24×107

7.24×107

7.23×107

3.17×107

k zpl 2.76×107

2.67×107

2.87×107

2.70×107

2.75×107

2.75×107

2.51×107

k zbpl 4.54×107

4.63×107

4.44×107

4.57×107

4.58×107

4.55×107

2.51×107

k zcpl 7.68×107

7.54×107

7.80×107

7.63×107

7.47×107

7.62×107

2.51×107

k zbp 4.07×107

4.15×107

3.94×107

4.06×107

4.05×107

4.06×107

2.87×107

k zcp 5.30×107

5.30×107

5.49×107

5.48×107

5.49×107

5.41×107

2.87×107

Average[N/m]

Calculation[N/m]

Identified parameters [N/m]Spring

θ 0 [°] 45 [°] 90 [°] 135 [°] 180 [°]

1 172.96 173.11 173.25 173.40 173.472 315.19 315.63 316.51 317.23 317.473 416.21 415.67 415.06 415.64 416.214 583.91 582.82 580.42 578.46 577.585 813.40 813.56 814.03 814.34 814.496 1176.46 1176.13 1175.81 1175.70 1175.817 1579.97 1581.25 1584.77 1588.76 1590.438 2317.88 2312.41 2306.92 2311.86 2317.889 2947.16 2947.16 2946.73 2947.16 2947.1610 5394.86 5259.34 5102.90 5154.75 5359.5311 5678.51 5620.22 5608.94 5524.76 5394.86

(x0, y0) (xc, yc) (xch, ych) (xpl, ypl) (xp, yp)

Initial 0 45 90 135 180

2nd order mode

5th order mode


(2) It is found that natural vibration characteristics changed depending on the rotation angle of the crankshaft.

(3) It is found that frequencies related with the natural vibration characteristics, especially corresponding to the bending natural frequency appear.

(4) It is found that vibration analysis model proposed is reasonable.

(5) It was possible to allow the quantitative estimation of the model parameters.

REFERENCES

[1] Yoshifumi MORI, Study on a Vibration Analysis Model for Reciprocating Compressors and the Application to Technology of Monitoring, Yamaguchi University, 2013, Ph.D. thesis.

[2] Yoshifumi MORI, Takashi SAITO, Katsuhide FUJITA, Takehisa AOKI, RESEARCH ON FLUCTUATING FORCES OCCURRED IN COMPONENTS OF RECIPROCATING HYDROGEN COMPRESSORS, Proceedings of 2012 ASME International Mechanical Engineering Congress And Exposition, IMECE2012-86554

[3] Y. Mori, T. Saito, K. Fujita, Study on the characteristics of the natural frequency during operation in a reciprocating compressor, The 20th International Congress on Sound and Vibration, 2013, No.419.

[4] Matthew Wall, GAlib: A C++ Library of Genetic Algorithm Components, Mechanical Engineering Department Massachusetts Institute of Technology.

[5] J. A. Nelder, R. Mead, A simplex method for function minimization, The Computer Journal, Volume 7, Issue 4, 1965, pp.308-313.


verification of the vibration characteristics of a

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