EXPERIMENTALLY OBTAINED BIFURCATION PHENOMENON IN CHAOTIC TRACTOR VIBRATING IN TIME AND FREQUENCY DOMAIN

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1 International Journal of Bifurcation and Chaos, Vol. 15, No. 1 (2005) c World Scientific Publishing Company EXPERIMENTALLY OBTAINED BIFURCATION PHENOMENON IN CHAOTIC TRACTOR VIBRATING IN TIME AND FREQUENCY DOMAIN L. O. GARCIANO and K. SAKAI Faculty of Agriculture, Tokyo University of Agriculture and Technology, Saiwai-cho, Fuchu-shi, Tokyo , Japan leroy@cc.tuat.ac.jp R. TORISU Faculty of Agriculture, Iwate University, Morioka-shi, Iwate-ken, Japan Received January 26, 2004; Revised March 1, 2004 This paper investigates through experimental methods the dynamic characteristics of a farm tractor by changing the forward velocity from 0.63 m/s to 4.50 m/s. Nonlinear time series, frequency spectrum and continuous wavelet transform were used in the analysis. From the nonlinear time series analysis, a nonlinear resonance of the tractor vibration occurred at forward velocity of 2.15 m/s. The existence of a subharmonic frequency at 4.32 m/s indicated chaotic dynamics that was confirmed by the trend of the Lyapunov exponent analysis with a positive exponent indicating chaos. Continuous wavelet transform analysis results, presented graphically, called coefficient plots showed patterns composed of large and fine feature distribution in both time and scale. Quasiperiodic velocities of 1.95 m/s and 2.15 m/s were estimated by these coefficient plots. At forward velocity of 2.88 m/s, the coefficient plots showed dominant features that varied periodically and were estimated to be a period-doubling vibration. The coefficient plots during chaotic vibration at 3.52 m/s, 4.02 m/s and 4.32 m/s showed various feature distributions. At forward velocities of 0.95 m/s and 1.42 m/s, the existence of features with scale values of almost equal and half of the dominant feature was due to the influence of the gravel road surface and not to the artificial test track profile. Experimentally obtained bifurcation was observed clearly from the coefficient plots that showed three different patterns from quasi-periodic vibration at 2.15 m/s, period-doubling at 2.88 m/s and chaos at 3.52 m/s. Keywords: Farm tractors; nonlinear time series analysis; frequency spectrum analysis; continuous wavelet transform analysis; experimental bifurcation. 1. Introduction It is well known that tractor dynamics characteristic deteriorates at higher working speeds [Crolla & Horton, 1984; Sakai, 2000]. This deterioration condition presents a handling problem to the operator. Sudden changes in road conditions in combination with the forward velocity can produce a nonlinear resonance phenomenon or chaotic condition at higher velocities. This may result in tractor accidents that may lead to fatal injury to the operator. Previous works in the field of conventional tractor stability were conducted at slow speeds [Mitchell et al., 1972; Smith et al., 1974; Kim & Rehkugler, 1987] on the slope where handling behavior is not a 225

2 226 L. O. Garciano et al. problem or during turning motion at high speeds on level ground [Rehkugler, 1982]. As the gradual trend for off-road vehicle has been to increase speed, technical problems have developed. Crolla and Horton [1984] reported that the problems associated with higher working speeds are the general deterioration of tractor dynamics due to increased shock loading and reduced controllability. A paradigm shift in these analyses was shown by Sakai [2000] regarding violent bouncing condition of tractor using nonlinear time series analysis. In his paper, chaos dynamics of the bouncing tractor were experimentally observed. Various researches reported the use of wavelets in the analysis of vibration phenomena. Newland [1994a, 1994b] reported wavelets as a new tool for the analysis of vibration and by way of wavelet maps that provide a graphical picture of the frequency compositions of a vibration signal. The objectives of this paper are to investigate the experimentally obtained time series data of tractor vibrations without a violent bouncing condition using nonlinear time series, frequency and continuous wavelet transform analysis. 2. Design of Experiments 2.1. Tested tractor, test track and instrumentation A John Deere Werke Mannheim model 2850D tractor (65 kw) equipped with single front and single rear tires was used in the experiment [Fig. 1(a)]. The total mass was 3690 kg with 2280 kg on the rear axle. The center of gravity (c.g.) was located m in front of the rear axle center and 0.93 m above the ground [Garciano et al., 2001]. Tire inflation pressures were set to 200 kpa and 150 kpa for the front and rear tires, respectively. Tire inflation pressure setting at the front was greater than the recommended 40 km/h maximum speed [Goodyear Tire and Rubber Company, 2001], and at the rear was according to the handbook. These inflation pressures setting were used during normal farm work operation in the experimental farm. The wheel base of the tractor was 2.29 m. The artificial test track consisted of angle irons of profile height m spaced at 0.5 m apart and installed on a gravel farm road surface with a total of 25 m thus giving 50 excitation steps. Tractor vibration was measured by two strain gauge type linear accelerometers installed on the accelerometer c.g. (a) (b) Fig. 1. (a) Tested tractor. (b) Center of gravity and vertical accelerometer location and artificial test track profile. front and rear of the tractor chassis and were 2 g and 1 g respectively, where g means gravity and is equal to 9.81 m/s 2. These accelerometers were installed approximately along the longitudinal axis. The accelerometers were oriented to measure accelerations in the vertical directions. The signal obtained was processed with a low pass filter of 30 Hz. 3. Results and Discussion 3.1. Frequency response test A frequency response test was conducted at different forward velocities using an artificial test track as input frequency to the tractor [Fig. 1(b)]. The frequency f of the forced vibration is expressed as, f =(v/l) (1)

3 Experimentally Obtained Bifurcation Phenomenon in Chaotic Tractor 227 where v is the forward velocity and l the spacing between the angle irons. In the experiment, the forward velocity was varied from m/s. The speed was increased at intervals in the range of m/s for a total of 27 vibration experiments Nonlinear resonance curve From the experimentally obtained time series data, the first and the last few seconds of the data were not considered. In the making of the nonlinear resonance curve, the acceleration values were converted into root-mean-square (RMS) values. Figure 2 shows the nonlinear resonance curve of tractor vibrations for all velocities. The x-axis is the frequency and the forward velocity. The y-axis is the vertical acceleration in RMS. Nonlinear resonance occurred at 2.15 m/s with a peak acceleration of m/s 2. A closer investigation revealed local peaks at 2.40 m/s and 1.62 m/s. These local peaks are the bounce and pitch natural frequencies of the tractor [Garciano et al., 2001] Trend of the largest Lyapunov exponents analysis Figure 3 shows the trend of the largest Lyapunov exponents [Wolf et al., 1985] versus the forcing frequencies and the forward velocity. Spurious Lyapunov exponents in the low speed range of 0.63 to 1.42 were observed which is due to the low signal to noise ratio that was caused by the gravel road condition; therefore at this velocity range they were not included in Fig. 3. The largest Lyapunov exponent occurred at Hz with a corresponding velocity range of m/s [Garciano et al., 2002] Nonlinear time series and frequency spectrum analysis Figures 4(a) and 4(b) show selected experimentally obtained time series data, its frequency spectrum x" [m/s 2 rms] Forcing frequency [Hz] Forward velocity [m/s] Fig. 2. Nonlinear resonance curve of the tractor vibration. 0.4 Lyapunov exponents λ i Forcing frequency [Hz] Forward speed [m/s] Fig. 3. Trend of the largest Lyapunov exponents.

4 228 L. O. Garciano et al. (a) Fig. 4. (a and b) Time series, frequency spectrum and wavelet coefficient plots of tractor vibrations at various forward velocities.

5 Experimentally Obtained Bifurcation Phenomenon in Chaotic Tractor (b) Fig. 4. (Continued ) 229

6 230 L. O. Garciano et al. and the results of the wavelet analysis presented in a graphical figure called coefficient plots. From the time series data, as the forward velocity was increased, the amplitude of vibration increased. With a velocity of 1.95 m/s, a beat wave can be observed that indicated approaching nonlinear resonance condition. At velocity 2.15 m/s, a maximum peak of 23.4 m/s 2 indicated nonlinear resonance. The bifurcation structure was observed from the change of forcing frequencies in the frequency spectrum with the peak at 2.15 m/s. Subharmonic frequency was observed clearly at 4.32 m/s indicating chaotic dynamics of the tractor Continuous wavelet transform analysis A continuous wavelet transform analysis was conducted and presented in a graphical figure called coefficient plots for all selected time series data shown in Figs. 4(a) and 4(b). The choice of wavelet was the Coiflets level 4 with a scale setting ranging from 1 to 32 [Wavelet Toolbox User s Guide, 2002]. From the results of the wavelet analysis, a uniform feature throughout time and scale in the coefficient plot indicates periodic vibration. At forward velocities 1.95 m/s and 2.15 m/s, the coefficient plot showed approximately three features. One feature is uniformly distributed throughout time and scale, which is the forcing frequency of the vibration. Secondly, the feature in the scale range from 5 to 9 showed a lesser peak at higher frequency of vibration. Thirdly, in the scale of 1 to 4, the pattern distribution indicated the noise of the signal. Figure 4(b) at velocity 2.88 m/s showed approximately four dominant features observed from the coefficient plot. The dominant feature was observed throughout time and scale; the other four features have scales that are not so different from each other, and they exist periodically over time. Noise can be observed in the scale range of 1 to 5. This scenario probably indicates period-doubling vibration of the tractor. From the coefficient plots at velocities 3.52 m/s, 4.02 m/s and 4.32 m/s, the features of these coefficients indicate some degree of transient characteristics (v = 3.52 m/s) observed over time, and existence of varying features at different scale range and time at 4.02 m/s and 4.32 m/s. These are the characteristic features of the chaotic dynamics of the tractor. At forward velocities 0.95 m/s and 1.42 m/s, the existence of features that have scale values of almost equal and half of the dominant feature was due to the gravel road surface and not due to the artificial test track profile [Garciano et al., 2002]. Therefore at this condition, there is a quasi-periodic vibration with a low signal to noise ratio. 4. Conclusions Experimental investigation of the dynamic characteristics of a farm tractor was conducted with the forward velocity of 0.63 m/s to 4.50 m/s using an artificial test track installed on a gravel farm road. The analysis was conducted using nonlinear time series, frequency spectrum and continuous wavelet transform. The nonlinear time series analysis detected the nonlinear resonance at 2.15 m/s. The subharmonic frequency at 4.52 m/s indicated the chaotic dynamics and was confirmed by the trend of the largest Lyapunov exponents analysis. The continuous wavelet transform analysis estimated the quasi-periodic vibration at forward velocities of 1.95 m/s and 2.15 m/s. The coefficient plots during chaotic vibration at 3.52 m/s, 4.02 m/s and 4.32 m/s showed varying feature distribution. At forward velocities of 0.95 m/s and 1.42 m/s, the existence of features that have scale values of almost equal and half of the dominant feature was due to the influence of the gravel road surface and not because of the artificial test track profile. Experimentally obtained bifurcation was clearly observed from the three different patterns of the coefficient plots from quasi-periodic vibration at 2.15 m/s, period-doubling at 2.88 m/s and chaos at 3.52 m/s. Acknowledgment This research is supported by the Japan Society for the Promotion of Science grant-in-aid for scientific research. References Crolla, A. A. & Horton, D. N. [1984] Factors affecting the dynamic behavior of higher speed agricultural vehicles, J. Agricul. Engin. Res. 30, Garciano, L. O., Torisu, R., Takeda, J. & Yoshida, J. [2001] Resonance identification and mode shape analysis of farm tractors, J. Japanese Soc. Agricul. Mach. 63,

7 Experimentally Obtained Bifurcation Phenomenon in Chaotic Tractor 231 Garciano, L. O., Torisu, R., Takeda, J. & Sakai, K. [2002] Random, quasi-periodic and chaotic vibrations of farm tractors, J. Japanese Soc. Agricul. Mach. 64, Garciano, L. O., Torisu, R., Takeda, J. & Yoshida, J. [2003] Wavelet analysis of tractor nonlinear dynamics, ASAE Int l Presentation Meeting, Paper No , American Society of Agricultural Engineers, St. Joseph, MI. Goodyear Tire and Rubber Company [2001] Goodyear Farm Handbook, available at ca/tires/farm/pdf/goodyearfarmhandbook.pdf. Kim, K. U. & Rehkugler, G. E. [1987] A review of tractor dynamics and stability, Trans. Amer. Soc. Agricul. Engin. 30, Mitchell, B. W., Zachariah, G. L. & Liljedahl, J. B. [1972] Prediction and control of tractor stability to prevent rearward overturning, Trans. Amer. Soc. Agricul. Engin. 15, Newland, D. E. [1994a] Wavelet analysis of vibration, Part 1: Theory, J. Vibr. Acoust. 116, Newland, D. E. [1994b] Wavelet analysis of vibration, Part 2: Wavelet Maps, J. Vibr. Acoust. 116, Rehkugler, G. E. [1982] Tractor steering dynamics Simulated and measured, Trans. Amer. Soc. Agricul. Engin. 25, Sakai, K. [2000] Experimental analysis of nonlinear dynamics and chaos in bouncing tractors, J. Japanese Soc. Agricul. Mach. 62, Smith, D. W., Perumpral, J. V. & Liljedahl, J. B. [1974] The kinematics of tractor sideways overturning, Trans.Amer.Soc.Agricul.Engin.17, 1 3. Wavelet Toolbox User s Guide [2002] The MathWorks, Inc. 3, Apple Hill Drive, Natick, MA. Wolf, A., Swift, J. B., Swinney, H. L. & Vastano, J. A. [1985] Determining Lyapunov exponents from a time series, Physica D16,

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