STRUCTURAL OPTIMIZATION OF TRACTOR FRAME FOR NOISE REDUCTION

Transcription

1 STRUCTURAL OPTIMIZATION OF TRACTOR FRAME FOR NOISE REDUCTION Takayuki Koizumi, Nobutaka Tsujiuchi, Shigeyuki Sawabe Doshisha Univercity, Kyotanabe, Kyoto, Japan, Isamu Kubomoto, Eiichi Ishida Kubota Corp., Ishidu-kitamachi, Sakai, Osaka, Japan ABSTRACT This paper describes understanding and modeling of the dynamic characteristic of the tractor frame, and decrease of vibration and noise of the tractor by design modification of the frame. First of all, we measured the vibration characteristics each part of the tractor, and the noise characteristic in the cabin. As the result, we confirmed that the bending mode of the frame greatly influence the vibration characteristic of the tractor, and noise in cabin become very loud for resonance frequency of its mode coincide with resonance frequency in the cabin. Secondly, we separated the fullstructure of the frame into the sub-structure of a case and joint part, and modeled each one. Then we improved the model accuracy by the use of model tuning with sensitivity analysis, constructed a dynamic model for the frame. By comparing the analytical results with the experimental results, the validity of the dynamic model was verified. Finally, we carried out design change of the frame with the object of increasing stiffness while reducing weight. As the results of this modification, we could shift the objected natural frequency more than 13 Hz if it is needed. And also it could be said that noise level inside the cabin can be effectively suppressed at leased 4 db. INTRODUCTION In recent years, agricultural vehicle has come to require improved safety, comfort and functions due to the aging agricultural population and ergonomic requirement. Under these circumstances, medium and large sized agricultural tractors with the cabin is required decrease vibration and noise in order to improve operational environment. The monocoque-type frame occupying nearly tractor frame in which the engine, power transmission-case and differential-case are included, and they all are bolted to each other. The tractor possesses many excitation sources such as engine, drive system, exhaust system, etc. All kind of vibration caused from these sources travel into the cabin through the frame, in case when vibration from excitation source coincide with resonance frequency of the frame. Therefor, to decrease vibration and noise of the tractor, it is essential to improve the characteristic of the frame. This paper describes understanding and modeling of the dynamic characteristic of the tractor frame, and decrease of vibration and noise of the tractor by design modification of the frame. First of all, we measured the vibration characteristics each part of the tractor, and the noise characteristic in the cabin. Secondly, we separated the full-structure of the frame into the sub-structure of a case and joint part, and modeled each one. Then we improved the model accuracy by the use of model tuning with sensitivity analysis, constructed a dynamic model for the frame. Finally, we carried out design change of the frame with the object of increasing stiffness while reducing weight. As the results of this modification, we could shift the objected natural frequency if it is needed. And also it could be said that noise level inside the cabin can be effectively reduced. DYNAMIC CHARACTERISTIC OF TRACTOR In this chapter we apply experimental modal analysis to a practical frame, so as to identify the dynamic characteristic of the frame. Then we measure the noise inside of tractor cabin to make clear the relationship between vibration and noise. Vibration Characteristics of Tractor Based on the results of our preliminary experiments[1], we decided to carry out experimental modal analysis focusing on the frequency spectrum ranged from 0 Hz to 300 Hz in the vertical direction. In the frequency

2 response measurement, we employed the burst random excitation, and measured frequency response function at 46 measurement points on the bottom of the frame. After the measurement, we carried out curve-fitting all response functions, and calculated modal parameters. The results of modal analysis is shown in Table 1. In a mode of Hz, the front and rear of the tractor frame moves up and down out of phase. In a mode of Hz, the rear of tractor frame moves while twisting. A mode of Hz shows a behavior resembling the bending 1 st mode of the beam in the free-fixed end boundary condition. TABLE 1:Result of experimental modal analysis of tractor mode frequency[hz] 1 st nd rd Vibration Characteristic of Tractor Frame Tractor frame is removed all inside parts which are constructed from differential gears, drive shaft etc. and only outer frame is excited by impact hammer in the free-free boundary condition. The transfer response functions are measured with 76 measurement points focusing frequency ranged from 0 to 500 Hz in the vertical directions. We carried out experimental modal analysis and identified modal parameters. The results of modal analysis is shown in Table 2. A mode of Hz shows the bending 1 st mode. A mode of Hz shows a behavior resembling and deeply influenced the bending 1 st mode. A mode of Hz is the twisting mode. Compared the 1 st frame mode with the 3 rd tractor mode, these two modes are strikingly similar in mode shape and show very close frequency. From these results, it is found that the bending 1 st frame mode has strong influence on the dynamic characteristics of the tractor. TABLE 2:Result of experimental modal analysis of frame mode frequency[hz] 1 st nd rd Noise Characteristics in the Cabin The acoustic frequency response functions between the floor panel of cabin and the ear position of driver are measured under 500 Hz. Experimental configuration of the cabin adopts a loudspeaker mounted on the floor panel for exciting acoustic random input and microphones for measuring sound pressure is used. From the experimental results, it is found that there are two remarkable peaks around Hz and Hz. There are cavity resonance of tractor cabin and especially Hz is close to the 3 rd tractor mode. Noise Characteristics in the Tractor The noise frequency response functions between force input from rear mount and sound pressure at the ear position of driver are measured under 500 Hz. From the experimental results, it is found that there are two remarkable peaks around 265 Hz and 330 Hz. The noise around 330 Hz is caused by the resonance frequency of cabin structure. As mentioned above, loud noise around 265 Hz is caused by resonance frequency of tractor mode coincide with cavity resonance frequency of cabin. Therefore, if we could shift cavity resonance frequency of cabin, the noise of the tractor should be reduced. MODELING OF THE TRACTOT FRAME In this chapter we construct the numerical model of the tractor frame for design optimization. The full-structure of the frame is separated into the sub-structure of a case and joint parts. Then we improve the accuracy of the each sub-structure by using the model tuning method with sensitivity analysis. Monocoque-type Frame The monocoque-type tractor frame shown in Figure 1 consists of flywheel-case, clutch-housing-case, mid-case and transmission-case[2], and they are bolted to each other. In this paper, the influence of stiffness decrease caused by bolted joints is considered to construct a dynamic model of the frame. We apply experimental modal analysis to a practical frame, so as to identify dynamic characteristic of each case. In the frequency response measurement, we employed the impulse excitation with free-free boundary condition. The results of the experimental modal analysis are shown in Table 3 to Table 6. jointa jointc flywheel-case jointb mid-case clutch-housing-case transmission-case FIGURE 1:Monocoque-type frame TABLE 3:Result of finite element analysis of flywheel-case 1 st nd

3 TABLE 4:Result of finite element analysis of clutchhousing-case 1 st nd rd TABLE 5:Result of finite element analysis of mid-case 1 st nd rd TABLE 6:Result of finite element analysis of transmissioncase 1 st nd rd Finite Element Analysis Modeling of Each Case As an analytical tool, we used the finite element analysis software I-DEAS Master Series. The practical frame consist of a shell and has a very complex form. We decided to employ a exact box form divided by shell elements based on design blueprints to improve the model accuracy. As the thickness of each case is different at several areas, we classified into 4 kind of thickness groups; 5, 10, 15 and 20 [mm]. As the results, each case is separated from 20 to 40 areas. We used the material constants for numerical analysis based on FC20 (Japanese Industrial Standard), Young's modulus: 98 [GPa], Poison ratio 0.3 and density: 7850 [kg/m 3 ]. The results of numerical analysis are shown previous Table 3 to Table 6. Compared with the experimental results, all cases except for mid-case had over 10 % errors. As the principal reason of this difference is the variance of thickness of several areas, we must correct the variation of the thickness. Model Tuning Method We improved the model accuracy by the use of the model tuning based on the sensitivity analysis to change the design parameter by comparing the analytical results with the experimental ones. The relationship between the model parameters and experimental response is written by next equations. R } = { R } + [ S]({ P } { P }) (1) { e a u 0 { R} = [ S]{ P} (2) where [S] is sensitivity matrix, {R e } is experimental response vector, {R a } is initial experimental response vector, {P u } is predicted parameter vector and {P o } is initial parameter vector, respectively. The predicted parameters are given by { P} = [ S]{ R} (3) This equation is solved by the Pseudo-inverse Method (PIM) or the Bayesian Parameter Estimation Method (BPEM). The BPEM are used to take account of the uncertainties of both experimental responses and theoretical model parameters. In the PIM, the pseudoinverse matrix of [S] given by next equation is used as the matrix [G]. T 1 T [ G ] = ([ S] [ S]) [ S] (4) On the other hand in the BPEM, the weighting matrices [C P ] and [C R ]are used for both responses and parameters which have uncertainties, respectively. 1 T 1 1 T 1 [ G ] = [ C P ] [ S] ([ C R ] + [ S][ C P ] [ S] ) (5) In the model tuning method, thickness of each case was employed as design parameter, and the response parameters were both natural frequencies and MAC values. The BPEM were used to take account of the uncertainties[3]. These results of model tuning are shown in Table 7 to Table 10 and the parameter modification results are shown in Figure 2 to Figure 5. By comparing with the analytical results with experimental ones, it was found that the error of natural frequencies stayed within 0.2 % and MAC values were increased collectively. We improved the model accuracy by use of model tuning with sensitivity analysis, constructed a dynamic model for the each case. TABLE 7 : Result of model tuning of flywheel-case 1 st nd TABLE 8 : Result of model tuning of clutch-housing-case 1 st nd rd TABLE 9 : Result of model tuning of mid-case 1 st nd rd TABLE 10 : Result of model tuning of transmission-case 1 st nd rd

4 FIGURE 2:Parameter modification of flywheelcase FIGURE 3:Parameter modification of clutchhousing-case Dynamic Characteristics of Bolted Joint Parts When the stiffness of bolted joint part is evaluated for the numerical analysis, a method to identify the equivalent spring constant on the basis of experimental results[4] is used. In this study we propose a method to predict the equivalent spring constant for the numerical analysis. Forces working within the bolted joint part are shown in Figure 6. The forces supporting the stiffness of the bolted joint part include compressive force, friction force and stiffness of the bolt[5]. In this study we replaces these forces with a spring (shown in Figure 6), so as to model the bolted joint part. The spring is referred to as an equivalent spring. As shown in Figure 1, each joint part is called joint A, joint B and joint C. First of all, we employ an experimental modal analysis with impulse excitation and measure the frequency response functions of joint parts to identify the equivalent spring constant. We replaced the compressive force and the pull stiffness of the bolt with the x direction of equivalent spring constant, and the friction force and the shear stiffness of the bolt with the y and z direction of equivalent spring constant respectively. We identified equivalent spring constant by using above-mentioned model tuning method so that it coincided with natural frequency of the bolted joint model estimated from experimental results. The equivalent spring constant of each joint part are shown in Table 11. TABLE 11:Equivalent spring constant of each joint part Equivalent spring constant ( x 10 8 )[N/mm] x direction y direction z direction joint A joint B joint C FIGURE 4:Parameter modification of mid-case friction force equivalent spring compress force y z x FIGURE 6:Joint part of analysis model FIGURE 5:Parameter modification of transmission-case Total Modeling of Tractor Frame We applied these equivalent springs to each joint part of the tractor, and constructed the tractor frame model. The model of transmission-valve was added at the left side of mid-case to construct the same size of practical frame model. Transmission-valve was modeled by solid

5 element, and an aluminum diecasting alloy was adopted as the material parameters (Young's modulus: 71 [GPa], Poison ratio: 0.3, density 2700 [kg/m3]). The number of grid points on the constructed model is 6382, and the number of elements on the model is 6207 shown in Figure 7. The results of eigenvalue analysis is shown in Table 12. By comparing the analytical results with the experimental results, it was found that the error of the natural frequencies stayed within 5 %. This results confirmed that the model accuracy was improved by the use of model tuning method. TABLE 12:Result of finite element analysis of frame 1 st nd rd FIGURE 7:Finite element model of full-structure of frame OPTIMIZATION OF FRAME In this chapter we conduct optimization of the tractor frame by applying sensitivity analysis to the presented dynamic model, with the object of increasing stiffness while reducing weight. Sensitivity Analysis Using sensitivity analysis, we can know how the dynamic characteristic of the model changes when design data on the shape, size, material etc. is changed, and how much the design data should be changed to achieve the target dynamic characteristic[6]. In the sensitivity analysis, thickness of each case on the frame was employed as design data, and the design objective was to increase the natural frequency. The results for the 1 st natural frequency is shown in Figure 8. For all orders of sensitivity 1 st,2 nd and 3 rd, the mid-case is high, on the contrary, the flywheel-case and the transmission-caserear are low. In general, the natural frequency decrease as thickness reduces, because the influence of natural frequency decrease caused by stiffness reduction is greater than that of natural frequency increase caused by weight decrease. However in the part of low sensitivity, the influence of natural frequency increase caused by weight decrease is greater than that of natural frequency decrease caused by stiffness reduction. sensitivity [Hz/mm] FIGURE 8:Sensitivity of thickness to 1st natural frequency Optimization We carry out design optimization of the frame with the objective of increasing stiffness while reducing weight. The responses of optimization are set both mass and natural frequency, and the parameter is the thickness of each area of the each case. From the results of sensitivity analysis, we set the upper and lower boundary thickness of each area, and then the optimization analysis is carried out to reduce the frame weight and to increase natural frequency. We use the PIM for optimization analysis as these responses do not contain the uncertainties. Next, the frequency response analysis is carried to evaluate the effect of optimization results by the maximum response for 1 st mode. The 1 st and 2 nd modal damping factor are used and respectively from the experimental results. The results of the optimization is in Table 13, and the thickness of each case after the modification in Figure 9. After the optimization, the thickness of sensitive area was increased several percents, especially the thickness of mid-case which was most sensitive was increased 30 %. On the contrary, the thickness of non-sensitive area was decreased from 5 to 10 %. The weight of the frame as a whole was reduced by 2.8 [kg] and the first natural frequency increased 13 [Hz]. The result of the frequency response analysis is shown in Figure 10. As shown in Figure 10, the results of our study has proved that the maximum response of 1 st mode mostly reduced 8.74 % (0.80[dB]) compared with that of before optimization. We already confirmed experimentally that if the natural frequency of tractor frame was shifted 10.6 [Hz] to the resonance frequency of the cabin, the noise of the tractor cabin should be reduced 3.64 [db]. As the results of this modification, we could shift the objected natural frequency more than 13 [Hz]. Therefore, it could be said that the noise level inside the cabin can be effectively suppressed at least 4 [db].

6 TABLE 13:Result of optimization data after change original data change rate[%] 1 st [Hz] nd [Hz] rd [Hz] mass[kg] % up FIGURE 9:Thickness modification by optimization the use of proposed model tuning with sensitivity analysis. (3) After the optimization of the tractor frame, the weight of the frame as a whole was reduced by 2.8 [kg] and the first natural frequency increased 13 [Hz]. The results of our study has proved that the maximum response of 1 st mode reduced 0.80 [db] compared with that of before optimization. REFERENCES 1. T.Koizumi, S.Sawabe, I.Kubomoto and E.Ishida : Optimization for tractor frame including dynamic characteristics of joint part, Proc. of the 15 th IMAC Japan, pp.40-46, (1997) 2. Y.Yasuda : A hand book for the frame and the agricultural machinery, Koronasha, pp.3-8, (1975) 3. J.Collins, G.Hart, T.Hasselman and B.Kenndey, Statistical identification of structures, AIAA Journal, pp , (1974) 4. J.H.Wang and C.M.Lion : Experimental identification of mechanical joint parameters, ASME, pp.28-36, (1991) 5. I.Yoshimoto : Point for design of treaded connection, The Society for the Standard of Japan, pp , (1992) 6. A.Nagamatu : Modal analysis, Baihuukan, pp , (1993) original after optimization Frequency [Hz] FIGURE 10:Frequency response function of frame model CONCLUSION In this paper, we proposed the modeling technique of the dynamic characteristics of the monocoque-type tractor frame, and reduction technique of noise and vibration of the tractor by design modification of the frame. Consequently following conclusive remarks could be proposed. (1) We improved the model accuracy of the each part of tractor frame by the use of model tuning with sensitivity analysis. (2) We separated the full-structure of the frame into the sub-structure of a case and joint part, and modeled each one. The model accuracy was improved by