VIBRATION TRANSFER PATH ANALYSIS BASED ON PSEUDO FORCE

Transcription

1 VIBRATION TRANSFER PATH ANALYSIS BASED ON PSEUDO FORCE Yan Weiguang, Wuhan University of Technology, School of Mechanical and Electrical Engineering, Wuhan, China Pan Yunping Wuhan University of Technology, School of Mechanical and Electrical Engineering, Wuhan, China Wu Xianjun* Institute of Acoustics, Chinese Academy of Sciences, Beijing, China The precision of classical transfer path analysis method is high, but it needs to dismantle the machine to measure the transfer function, this cannot be satisfied in most actual situation. So it is difficult to applying this method in engineering. The pseudo force, which can be measured at the measurable point, is used to replace the real excitation. In the pseudo force based transfer path analysis, the transfer function can be measurement without dismantling the machine and the accuracy of source identification is improved. The effectiveness of the proposed method is verified by test bench of a steel plate with two small motor sources. Keywords: vibration, transfer path analysis, pseudo force 1. Introduction Transfer path analysis (TPA) is an effective tool for vibration prediction and vibration source identification. The classical transfer path analysis method is of high precision, but it needs to dismantle the machine to measure the transfer function, which is difficult to satisfy. The operational transfer path analysis method is simple, but it has the fatal error source because of crosstalk. This paper studied on transfer path analysis based pseudo force. Some pseudo forces are acting at some places of the structure where the transfer function can be measured, and if this vibration cause by the pseudo force is equal to that caused by real excitation, it can be used in the transfer path analysis. Since this force is not real, it is called pseudo force. For the vibration problem of complex vibration structure and source, the pseudo force can be measured instead of the real exciting force. In this paper, the application process of the virtual force is verified by by test bench of a steel plate with two small motor sources. 2. The transfer path analysis based pseudo force 2.1 Traditional transfer path analysis and operational transfer path analysis *Corresponding author 1

2 The traditional transfer path analysis method is [1]: n a( ) H i ( ) F i ( ) (1) i1 where a is the response of the target point, ωis the frequency, i 1, 2,, n represents the id number of each sources, Hi is the transfer function for the i th load source to the target point, the load case can be a structural load or acoustic load. For traditional TPA, the following three steps are needed: first, the frequency response function of the machine is measured with the machine removed, and then the excitation is obtained by the measuring the response, Finally, the contribution of the vibration source is calculated according to the formula (1). In order to overcome the shortcomings of traditional transfer path analysis method, Operational Transfer Path (OTPA) is proposed in recent years [2]. In OTPA, response of the point near the excitation source is used to characterize the excitation source of, the formula is: n a( ) T i ( ) X i ( ) (2) i1 where a stand for the target response, ωis the frequency, i 1, 2,, n represents the id number of each sources, Ti is the transmission rate from reference point to the target point, of the ith source. X i is the response The basic steps of OTPA are follows: firstly, the machine is operated at several working condition and the response of target point and response points are measured. The transmission function between the reference points and the target points is calculated. In OTPA the excitation source does not need to be removed, so it is simple and efficient, but its precision is sometimes not good because of the crosstalk problem. 2.2 Transfer path analysis method based on pseudo force In order to overcome shortcomings of the traditional transfer path analysis method and the OTPA, the transfer path analysis based pseudo force is proposed. For each vibration source, pseudo force calculated by the measurable transfer function was using to characterize the vibration source, and then the steps similar to forecast source contribution in traditional transfer path analysis could be applied. It's easier and has higher precision. The steps of pseudo force based transfer path analysis are: selection of places for pseudo force near the source, where the transfer function can be easily measured; set more measured response points than the number of pseudo force points; artificial excite the selected points of pseudo force, measure the transfer functions from the pseudo force to the response points and target point. Then construct the admittance matrixes from the points of pseudo force to the response points and target point. Run the machine and determine the pseudo forces using this matrix, and then used the pseudo forces to calculate the contribution from each sources. Assume there are n vibration sources in the system, the j th vibration sources are represented by a number of k j pseudo forces, and the all numbers of pseudo forces are k1, k2,, kn respectively, the relationship between the target response and the pseudo forces is: k1 k2 a H F H F H F (3) 1i 1i 2i 2i ni ni i1 i1 i1 k n 2 ICSV24, London, July 2017

3 where a is the vibration response of the target point, Fji is the i th pseudo force of jth source, H ji is the corresponding transfer function from the pseudo force to the target point. Each summation represents the contribution from each source. To write a formula (3) as a matrix: A Htar F (4) where Htar is the target point admittance matrix. The relationship between the vibration response of reference point and the pseudo force can also be obtained: X Href F (5) where is the admittance at the reference point, X is the response of reference point. Href Then, at working condition the response force can be calculated by the formula (6): X ope of reference point is measured, and the pseudo F H X (6) ope ref ope where + stand for pseudo inverse. The calculation process pseudo inverse can be done by singular value decomposition [3,4]: m ref i i i i1 H UV u v (7) U (u where 1,,u n) V (v are 1,, v n) are unit orthogonal matrix,"t" stand for conjugate transpose, 0 is a diagonal matrix, the singular values on the diagonal are 1 2 n. Use least squares method to solve the virtual force, the result is: r ui X ope F ope vi (8) i1 i Fope where is the pseudo force under working condition and r is the number of positive singular values being retained, according to the PCA technique, the smallest eigen value is considered as measurement noise and is omitted. Then the contribution is calculated according to (3). 3. Experimental verification In order to verify the pseudo force based transfer path analysis, a comparison is made for the OTPA and the pseudo force based transfer path analysis. 3.1 The experiment procedures The experimental platform is composed of 2 DC motors and a 1000mm 1000mm 3mm steel plates. The motors are rigidly mounted on the back of the plate, as shown in Figure 1 (b). The acceleration measuring points are arranged in front of the steel plate, as shown in Figure 1 (a). The two motors are respectively source 1 and source 2. 5 points near each motor is used to excite pseudo forces point, namely use 5 pseudo forces to characterize each motor. A total 15 randomly selected points are selected as the reference points. The target point is at the lower right corner of the plate, and experiment steps are as follows: ICSV24, London, July

4 (a) Front of the plate (b) Back of the plate Figure 1 Experiment plate (1) A force hammer is used to measure the transfer function from the 10 pseudo force points to the 15 reference points; (2) The motor speed is regulated, and four operational conditions are measured as listed in table 1. The fifth work condition is measured is measured to solve the source contribution. The sixth and seventh operational condition is the used to get the true contribution from each source with source running separately. 3.2 Comparison of the results of pseudo force TPA and OTPA The specific calculation processes are as follows: (1) Pseudo force TPA: working condition 5 is analyzed, 13 reference points are used to calculate the pseudo forces using formula (6), and formula (3) is used to calculate the contribution from the two sources. (2)OTPA method: working condition one to four are used for the train working condition, the transmission function is calculated used two response points. The contribution from the two source is calculated by equation (2). Comparison between the two methods and the true value are shown in Figure 1 to 2. 4 ICSV24, London, July 2017

5 Figure 1. The calculated contribution from source 1 to the target Figure 2. The calculated contribution from source 1 to the target Analyzing the graph, it shows that the precision of the results of pseudo force TPA is higher than OTPA. 4. Conclusion This paper discusses the pseudo force based TPA, it is not necessary to dismantle the machine to measure the transfer function from the pseudo force to response and target point. A model of two motors platform was done to verify the results of OTPA and pseudo force TPA, it is found that the ICSV24, London, July

6 pseudo force can accurately characterize vibration source, eliminate the crosstalk, it has more advantages than traditional TPA and OTPA. ACKNOWLEDGEMENT This work was supported by NSFC (No ). REFERENCES 1 F.X. Magrans. Method of measuring transmission paths, Journal of Sound and Vibration, 73 (3), ,(1981). 2 D. deklerk, A.Ossipov. Operational transfer path analysis: Theory, guidelines and tire noise application, Mechanical Systems and Signal Processing, 24,1950~1962,( 2010). 3 Xie Dongxiu, Lei Jigang,Chen Guizhichen. Matrix theory and method[in Chinese].China Science Press, Beijing(2012). 4 StrangeG. Linear algebra and its applications. 2nd ed. Academic Press, ICSV24, London, July 2017