EVALUATION OF THE FREQUENCY RESPONSE OF TRACTOR CAB ANGULAR MOVEMENTS
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1 EVALUATION OF THE FREQUENCY RESPONSE OF TRACTOR CAB ANGULAR MOVEMENTS M. Mattetti, G. Molari, M. Pesce, M. Grillo, M. Forte, E. Sedoni ABSTRACT. The mission profiles of tractors have been modified recently due to a rise in their use as transport vehicles. This use of the tractor together with an increasing maximum velocity on the road emphasizes the containment problem of the vibration transmitted to the driver. In particular, a significant contribution to the increasing global vibration level transmitted to the driver is given by low-frequency movements. Recent work has demonstrated the capability of the ISO standards used for tests in the automotive sector to characterize the dynamic behavior of tractors. In this work, one test, derived from the ISO standards mentioned above, was used to evaluate the capability of the suspension system to isolate the cab from the angular oscillations coming from the driveline. In particular, the frequency response of the cab angular movements was measured on two tractors, with and without an implement connected to the three-point linkage, and was compared with that measured on the tractor s driveline. The results showed a considerable amplification of the roll from the driveline to the cab, particularly with an implement, and a negligible amplification of the yaw. Keywords. Cab, Random steer input, Roll frequency, Tractor, Vibration. The number of fatal accidents in agriculture has decreased in recent years (Eurostat, 2), due, in particular, to an increased safety level of agricultural machinery and updating of equipment. However, the agricultural sector remains one of the professions most affected by injuries. The exposure of the body to vibrations poses serious health risks. Whole-body vibrations can cause low-back pain, spinal disorders, abdominal pain, digestive disorders, and vision problems, as has been demonstrated by a large number of studies (Griffin, 99; Hulshof and Van Zanten, 987). These effects cause a reduction of comfort and productivity (Fairley, 99). With the goal of reducing these risks, limits to vibration exposure have been imposed by European directive 22/44/EC (EC, 22). The tractor driver is subject to vibration, particularly at low frequency and high intensity, caused by the contact between the tires and implements with the soil (Rakheja and Sankar, 984). The use of the tractor as a transport vehicle together with an increasing maximum velocity on the road emphasizes the containment problem of the vibration transmitted to the driver. Recent studies have clarified that Submitted for review in August 2 as manuscript number PM 9349 approved for publication by the Power & Machinery Division of ASABE in March 22. The authors are Michele Mattetti, Doctoral Student, and Giovanni Molari, Associate Professor, Department of Agricultural Economics and Engineering, University of Bologna, Bologna, Italy; Marco Pesce, Vehicle Dynamics Expert, and Massimo Grillo, Test Engineer, Centro Ricerche Fiat, Vehicle Dynamics Department, Orbassano, Italy; and Michele Forte, Tractor CVT and Transmission Projects, and Enrico Sedoni, Agricultural Product Evaluation, CNH Tractor Engineering, Modena, Italy. Corresponding author: Giovanni Molari, Department of Agricultural Economics and Engineering, University of Bologna, via G. Fanin, 427 Bologna, Italy; phone: ; giovanni. molari@unibo.it. the vibration level on the seat, caused by the tractor s angular movements, is higher than that on the cab floor, and is increased by soft tires (Scarlett et al., 2). This effect is highlighted in modern tractors, in which the position of the driver is moved from the rolling axle. Even though the vibration level is influenced more by transverse and longitudinal accelerations than by vertical acceleration (ISO, 997), manufacturers have concentrated on the control of vertical acceleration. Despite the difficulties of isolating the transverse and longitudinal accelerations (Griffin, 99), some attempts have been made to develop seat suspension systems with negative stiffness (Lee et al., 27) that allow a low natural frequency (up to. Hz). However, these systems are not installed on tractors because of their high cost. In addition, very low spring stiffness causes an excessive static and dynamic motion of the operator with respect to the vehicle controls, and driving the tractor becomes, as a consequence, uncomfortable and unsafe. Presently, the horizontal suspension systems adopted on some seats provide only a reduction of the discomfort, moving the operator in phase with the seat when it is subject to transverse and longitudinal vibrations (Donati, 22; Joseph and Griffin 27), and the suspension systems adopted in tractor cabs primarily absorb the pitch of the frame (Scarlett et al., 2). In this context, it is fundamental to understand the cab s angular movements with the goal of improving the present solutions for isolating the tractor cab from the frame, particularly during transport on roads (Lines et al., 99). Recent studies have clarified the capability of some standards used in the automotive sector to analyze the handling behavior of tractors, after the introduction of the necessary modifications for the different characteristics of tractors with respect to cars (Molari et al., 2). In this study, the tractor configuration with an implement mounted to the Transactions of the ASABE Vol. (2): American Society of Agricultural and Biological Engineers ISSN
2 three-point hitch, which is becoming more and more frequent due to increasing farm sizes (Clay and Hemingway, 2), has been demonstrated as critical for the understanding of tractor dynamics. The goal of this study is to evaluate the capability of the methodology mentioned above to measure the roll and yaw behavior of the tractor cab and to analyze the capability of the tractor s suspension system to isolate the cab from the angular vibration of the frame in configurations with and without an attached implement. MATERIALS AND METHODS TRACTOR TESTING Two tractors were used, both with kw engine power and 4 km h - maximum speed, with and without an implement. The implement on each tractor was designed by taking into account the tractor s wheelbase and mass. In particular, the distance between the center of gravity of the implement and the pins of the three-point linkage (instead of the center of the back wheels, due to the reduced differences between the two tractors with respect to the distance between the pins of the three-point linkage and the center of the back wheels) was proportional to the wheelbase, and the mass of the implement was proportional to the mass of the tractor. The implement was mounted on the three-point linkage and set to the maximum height. The configurations of the tractors used in the tests are reported in table. Both tractors had cabs connected to the front frame with rubber silencer blocks and with viscous dampers and coil springs in the rear. Tractor TA used less stiff elastic elements in the front. Roll and yaw movements were allowed within the silencer block compliance. Both tractors had conventional front axle suspension in which the axle freely pivoted with respect to the frame on the tractor s longitudinal axis. The axle was linked to the frame through hydro-pneumatic elements that provided springing and damping functions. Tractor TA had only one hydro-pneumatic element, while tractor TB had two. DESCRIPTION OF THE TESTS The tests were made using ISO 74 (ISO, 23) and ISO 8726 (ISO, 988). These standards, frequently used for characterizing the transient response behavior of vehicles in the frequency domain, were used in this study to measure [a] Table. Tested tractor configurations. [a] Tractor TA Tractor TB Width 249 mm 26 mm Wheelbase 288 mm 292 mm Cab mass 8 kg 9 kg Total mass 8 kg (TA), 9 kg (TA-I) 89 kg (TB), kg (TB-I) Load on front axle 39% (TA), 32% (TA-I) 4% (TB), 3% (TB-I) Front tire type and pressure Goodyear Optitrac DT88 4/6 R3, Michelin Mach X BIB 6/7 R3, Rear tire type and pressure Goodyear Optitrac DT88 6/6 R42, Michelin XM 28 7/7 R42, TA-I and TB-I indicate tractors TA and TB with an implement. the roll behavior of the cab and frame. The capability of the suspension systems installed on tractors to isolate the cab from the rolling vibration of the tractor frame was also evaluated at a frequency range lower than 2 Hz. The tests were carried out on an oval track with 4 m radius turns at each end, two straight stretches of 8 m, and a width of 6.2 m. The tractors covered a straight stretch with a constant speed of 2 and 4 km h - with six repetitions for each test. The tests were performed with and without an implement, applying an oscillation to the steering wheel with a frequency range between.2 and. Hz in 4 s and an amplitude necessary to obtain a maximum lateral acceleration of.2g (Mattetti et al., 29). DEFINITION OF THE VARIABLES The roll velocity ( ϕ f, c and yaw velocity ( f ϕ ), pitch velocity ( ϑ f, ϑ c ), ψ, ψ c ) of the frame (subscript f) and cab (subscript c) were measured correspondingly in the longitudinal, lateral, and vertical directions, referring to the vehicle axis system defined by ISO 88 (ISO, 99). These quantities and the lateral acceleration of the frame ( a* yϕ, f ) were measured with two inertial platforms (Land- Mark VG LN, Gladiator Technologies, Snoqualmie, Wash.), one fixed on the cab access steps and the other on the cab floor near the seat base. The longitudinal ( v * x ) and transverse ( v * y ) velocities were measured in the same position as the first inertial platform, using an optical transducer (Corrsys HS-CE, Corrsys-Datron, Wetzlar, Germany). The steering wheel angle (δ H ) was measured from the straightahead position with a measuring steering wheel (MSW, Corrsys-Datron, Wetzlar, Germany). All signals were sampled at 2 Hz and filtered with a low-pass filter with a cutoff frequency of 3 Hz. The instrumentation used respects ISO s prerequisites (ISO, 26). In all repetitions, the steering wheel angle records had to have an adequate frequency content in accordance with ISO 8726 (ISO, 988). To analyze the variables measured, a reference point independent of the point of measurement was necessary. To compare the values obtained from the two different tractors, a point on the longitudinal symmetry plane m above the soil and fixed at the longitudinal position of the tractor s center of gravity was chosen as the reference. Therefore, the longitudinal (v x ) and transverse (v y ) velocities were measured at the tractor reference point and, as a consequence, the sideslip angle (β) was calculated using the following equations: v = v d +ϑ d x * x ψ f y f z v = v ψ d +ϕ d * y y f x f z v β = v where d x, d y, and d z are the distances from the measuring point to the reference point in the longitudinal, transverse, y x () (2) (3) 364 TRANSACTIONS OF THE ASABE
3 and vertical directions, respectively, and v * x and v * y are the measured longitudinal and transverse velocities, respectively. The lateral frame acceleration of the tractor reference point (a yf ) was derived from the measured lateral acceleration of the frame ( a* yϕ, f ), modified by the influence of the roll angle at the frame (ϕ f ) and by the different position of the transducer with respect to the tractor reference point, using the following equations: a * y, f = ayϕ, fcos ϕ f gsin ϕ f a*, gϕ (4) yϕ f f * 2 ay, f = ay, f ψ fdx+ψ fdy () where ϕ f was calculated from the numerical integration of the ϕ f measurement, the yaw acceleration of the frame ( ψ f ) was calculated by the numerical derivation of ψ f, and g is the acceleration of gravity. The frequency response functions of the frame roll gain ( ϕ f / ayf, ), the cab roll gain ( ϕ c / ay, f ), the frame yaw gain ( ψ f / δh ), and the cab yaw gain ( ψ c / δh) were obtained as means of six acquisitions, and the magnitude, phase, and coherence function were calculated (Mattetti et al., 2), where ϕ c is the roll angle at the cab. The frequency responses of ( ϕ f / ayf, ) and ( ϕ c / ay, f ) were used to evaluate the dynamic response of the roll as a function of the lateral acceleration of the vehicle. The frequency responses of ( ψ f / δh ) and ( ψ c / δh) were used to evaluate the yaw as a function of the steering wheel angle. The magnitude of the frequency response of ( ϕ f / ayf, ) and ( ψ f / δh ) would be more suitable if constant in a frequency range lower than Hz (Salaani et al., 24; Metz, 24). This implies a constant response of the tractor, independent of the steering wheel speeds. The amount of roll is strongly dependent on the lateral acceleration, and the amount of yaw is strongly dependent on the steering wheel angle. On the other hand, ( ϕ c / ay, f ) and ( ψ f / δh ) are better if they are low and constant, without resonance effects, in a frequency range lower than 2 Hz, to not considerably increase the seat s lateral acceleration. The magnitude of the frequency response of ( ϕ c / ay, f ) and ( ψ c / δh) should be lower than the magnitude of the frequency response of ( ϕ f / ayf, ) and ( ψ f / δh ), respectively, indicating that the cab suspension isolates the driver with respect to the frame s angular movements. As a consequence, the frequency response of the roll transmissibility, ϕc / ϕ f, calculated as the ratio between the magnitude of ( ϕ c / ay, f ) and ( ϕ f / ayf, ), and the frequency response of the yaw transmissibility, ψ c / ψ f, calculated as the ratio between the magnitude of ( ψ c / δh) and ( ψ f / δh ), should be constant with the frequency and lower than to isolate the cab from the frame s angular movements. The phase of the frequency response functions was evaluated to show the delay in response with respect to the roll and yaw of the frame and cab. A similar phase delay between the roll and yaw gains of the cab and frame is advisable to avoid operator disorientation due to a relative movement of the cab to the frame. This is particularly important on tractors compared to other vehicles with cab suspensions because the driver is experienced with frame movements. Finally, the coherence functions were shown, exposing the capability of the frequency response function to interpolate the original data. Values higher than 9% show a good correlation. RESULTS AND DISCUSSION The trend of the magnitude, phase, and coherence function of the frequency response of ( ϕ f / ayf, ) and ( ϕ c / ay, f ) for the tractors, with and without an implement, are reported in figure for 2 km h - and in figure 2 for 4 km h -. In accordance with ISO standards, all angles are reported in degrees. For all conditions tested, the magnitude is positive in all frequency ranges analyzed and decreases with increasing frequency due to the inertia moment with respect to the longitudinal axis of the vehicle (figs. a, b, 2a, and 2. The implement increases the roll gain magnitude at low frequencies but reduces the natural roll frequency, from.2 to.8 Hz, due to the increasing longitudinal inertia. Increasing the speed from 2 to 4 km h - does not influence the natural roll frequency, which remains constant. Indeed, the roll is caused by radial deformation of the tires, and the tire radial stiffness is constant with increasing speed (Lines and Murphy, 99). The trend of the roll gain of the cab is similar to the trend of the roll gain of the frame, but an amplification of the first value is relevant around the vehicle s natural frequency, particularly for tractor TA with an implement at 2 and 4 km h -. This is probably caused by an increase of the unsuspended mass and by the small oscillation of the lower link of the lift due to implement linkage clearances that increase the lateral load transfer. The implement effect is stronger at 4 km h - than at 2 km h - due to an increase of the implement yaw accelerations. Tractor TA is more sensitive to frame roll movements than tractor TB because of its less rigid elastic elements. The frequency response trends of the roll gain phase (figs. c, d, 2c, and 2d) show a phase delay from the cab to the frame, which is more evident around the resonance frequency, for both tractors, especially with an implement. The values of the coherence function (figs. e, f, 2e, and 2f) are close to up to the natural roll frequency for both tractors without an implement. This represents the goodness of the frequency response in that range. For the tractors with an implement, a considerable reduction of the coherence function is shown with respect to the roll frequency. (2):
4 Roll gain phase [ ] c) Roll gain phase [ ] d) e) f) Figure. Roll gain frequency responses at 2 km h - : ( magnitude, (c) phase, and (e) coherence function without implement; and ( magnitude, (d) phase, and (f) coherence function with implement ( = frame roll gain of tractor TA, = cab roll gain of tractor TA, = frame roll gain of tractor TB, and = cab roll gain of tractor TB). R oll gain phase [ ] c) Roll gain phase [ ] d) e) f) Figure 2. Roll gain frequency responses at 4 km h - : ( magnitude, (c) phase, and (e) coherence function without implement; and ( magnitude, (d) phase, and (f) coherence function with implement ( = frame roll gain of tractor TA, = cab roll gain of tractor TA, = frame roll gain of tractor TB, and = cab roll gain of tractor TB). 366 TRANSACTIONS OF THE ASABE
5 Yaw gain magnitude [s - ] Yaw gain phase [ ] yaw gain coherence c) e) Yaw gain magnitude [s - ] yaw gain phase [ ] Yaw gain coherence d) f) Figure 3. Yaw gain frequency responses at 2 km h - : ( magnitude, (c) phase, and (e) coherence function without implement; and ( magnitude, (d) phase, and (f) coherence function with implement ( = frame yaw gain of tractor TA, = cab yaw gain of tractor TA, = frame yaw gain of tractor TB, and = cab yaw gain of tractor TB). Yaw gain magnitude [s - ] Yaw gain phase [ ] Yaw gain coherence c) e) Yaw gain magnitude [s - ] Yaw gain phase [ ] Yaw gain coherence d) f) Figure 4. Yaw gain frequency responses at 4 km h - : ( magnitude, (c) phase, and (e) coherence function without implement; and ( magnitude, (d) phase, and (f) coherence function with implement ( = frame yaw gain of tractor TA, = cab yaw gain of tractor TA, = frame yaw gain of tractor TB, and = cab yaw gain of tractor TB). The trends of the magnitude, phase, and coherence function of the frequency response of ( ψ f / δh ) and ( ψ c / δh) for the tractors, with and without an implement, are reported in figure 3 for 2 km h - and in figure 4 for 4 km h -. For the yaw gain (figs. 3a, 3b, 4a, and 4, the magnitude decreases with increasing frequency due to the inertia moment around the vertical axis of the vehicle, which reduces the capability of the vehicle to respond to the oscillations applied to the steering wheel. The cab yaw gains are similar to the frame yaw gains for both tractors, as (2):
6 7 2 kmh kmh Roll trasmissibility 4 3 Roll trasmissibility Figure. Trend of the roll transmissibility of the cab suspension at ( 2 km h - and ( 4 km h - ( = tractor TA, = tractor TA-I, = tractor TB, and = tractor TB-I). 2 kmh kmh Yaw trasmissibility.6 Yaw trasmissibility Figure 6. Trend of the yaw transmissibility of the cab suspension at ( 2 km h - and ( 4 km h - ( = tractor TA, = tractor TA-I, = tractor TB, and = tractor TB-I). a consequence of the elastic mount yaw compliances, which are stiffer than the longitudinal compliances. The cab and frame phases are similar (figs. 3c, 3d, and 4c), with the consequence that the cab follows the frame. Finally, the coherence function has high values in all frequency ranges (figs. 3e, 3f, 4e, and 4f). The trends of the roll and yaw transmissibility of the cab suspension at 2 and 4 km h -, with and without an implement, are shown in figures and 6. The cab introduces additional roll movements with respect to those measured on the frame for both tractors. This is more evident for tractor TA with an implement. Consequently, the seat transverse acceleration increases, as has been shown by other researchers (Scarlett et al., 27). A possible solution to this problem could be a connection of the hydraulic cylinders between the lower link of the lift to the frame (Ferhadbegović et al., 27), decoupling of the vehicle from the implement (Bogala and Ulrich, 29), or a fully suspended tractor with semi-active control (Hammes and Meyer, 2). Moreover, the cab introduces reduced yaw movements with a frequency higher than.8 Hz, particularly in tractor A without an implement. In any case, the implement reduces the yaw acceleration with an increase of the inertia. CONCLUSIONS In the present study, the roll and yaw transmissibility of the cab suspension were evaluated in two tractors. The two tractors, similar with regard to their mass and dimensions, have different geometries in their front axle suspension and similar cab suspension systems, with the only differences being the stiffness of the elastic elements. The methodology 368 TRANSACTIONS OF THE ASABE
7 used to characterize the tractors in transient motion was used successfully to obtain an analysis of the roll and yaw behavior of the cab with respect to the frame. The tests showed an amplification of the roll from the frame to the cab, in particular at.2 Hz for the tractors without an implement and at.8 Hz for the tractors with an implement. The decrease of the resonance frequency was caused by an increase of the longitudinal inertia moment of the vehicle by adding the implement. The implement strongly influenced the cab roll response. This was due to an increase of the unsuspended mass, which makes the cab more sensitive to the frame s rolling resonant mode. For this reason, it is advisable to use a stiff cab suspension system. The cab also introduced reduced yaw movements with a frequency higher than.8 Hz. In any case, the implement reduced the yaw acceleration with an increase of the inertia around the vertical axis of the vehicle. The cab s natural roll frequency is included in the critical range for human perception of transverse acceleration that increases the vibration level and reduces the comfort of the driver. Solutions to reduce the cab rolling oscillations are necessary to reduce the vibration level transmitted to the driver and to increase the driver s comfort, especially for tractors with a power rating higher than kw. This objective can be reached with the introduction of a rear axle suspension that can increase frame roll damping, with cabs designed to absorb the frame roll movement, or, as an alternative, with implement mounting systems that can separate the vibration transmitted from the implement to the frame. REFERENCES Bogala, A., and A. Ulrich. 29. Development of a control system for improving handling performance of tractor. In Proc. 67th Intl. Conf. on Agricultural Engineering (LAND.TECHNIK - AgEng 29). Dusseldorf, Germany: VDI-Verlag. Clay, R., and P. Hemingway. 2. Engineering tractors for higher speed. ASAE Distinguished Lecture Series, No. 2. St. Joseph, Mich.: ASAE. Donati, P. 22. Survey of technical preventative measures to reduce whole-body vibration effects when designing mobile machinery. J. Sound and Vibration 23(): EC. 22. Directive 22/44/EC of the European Parliament and of the Council. Strasbourg, France: European Community. Eurostat. 2. Health and safety at work. Brussels, Belgium: European Commission. Available at: ec.europa.eu/portal/page/portal/health/health_safety_work. Fairley, T. E. 99. Predicting the discomfort caused by tractor vibration. J. Ergonomics 38(): Ferhadbegović, B., S. Böttinger, and H. D. Kutzbach. 27. Handling analysis of agricultural tractor using multi-body simulation. In Proc. 6th Intl. Conf. on Agricultural Engineering (LAND.TECHNIK - AgEng 27). Dusseldorf, Germany: VDI-Verlag. Griffin, M. J. 99. Handbook of Human Vibration. st ed. London, U.K.: Academic Press. Hammes, S., and H. Meyer H. 2. Development and investigation of completely semi-active suspension system for full spring mounted tractors. In Proc. Intl. Conf. on Agricultural Engineering (AgEng 2). Wallingford, U.K.: CABI. Available at: 2/ pdf. Hulshof, C., and B. V. Van Zanten Whole-body vibration and low-back pain: A review of epidemiologic studies. Intl. Archives Occup. and Environ. Health 9(3): ISO Road vehicles: Transient open-loop response test method with pseudo-random steering input. ISO Geneva, Switzerland: International Organization for Standardization. ISO. 99. Road vehicles: Vehicle dynamics and road-holding ability: Vocabulary. ISO 88. Geneva, Switzerland: International Organization for Standardization. ISO Mechanical vibration and shock: Evaluation of human exposure to whole-body vibration: Part. General requirements. ISO Geneva, Switzerland: International Organization for Standardization. ISO. 23. Road vehicles: Lateral transient response test methods: Open-loop test methods. ISO 74. Geneva, Switzerland: International Organization for Standardization. ISO. 26. Road vehicles: Vehicle dynamics test methods: Part. General conditions for passenger cars. ISO 37-. Geneva, Switzerland: International Organization for Standardization. Joseph, J. A., and M. J. Griffin. 27. Motion sickness from combined lateral and roll oscillation: Effect of varying phase relationships. Aviation, Space, and Environ. Med. 78(): Lee, C. M., V. N. Goverdovskiy, and A. I. Temnikov. 27. Design of springs with negative stiffness to improve vehicle driver vibration isolation. J. Sound and Vibration 32(4-): Lines, J. A., and K. Murphy. 99. The stiffness of agricultural tractor tyres. J. Terramechanics 28(): Lines, J. A., M. Stiles, and R. T. Whyte. 99. Whole-body vibration during tractor driving. J. Low-Frequency Noise and Vibration 4(2): Mattetti, M., G. Molari, M. Forte, E. Sedoni, M. Pesce, and M. Grillo. M. 29. Characterisation of the behaviour of a highpower tractor on the road. In Proc. 33rd CIOSTA and CIGR Section V Intl. Conf., Reggio Calabria, Italy: Artemis. Mattetti, M., G. Molari, M. Pesce, M. Grillo, M. Forte, and E. Sedoni. 2. Frequency roll response of farm tractor s cabs. In Proc. Intl. Conf. on Agricultural Engineering (AgEng 2). Wallingford, U.K.: CABI. Available at: Metz, L. D. 24. What constitutes good handling? SAE Paper No Warrendale, Pa. Society of Automotive Engineers. Molari, G., M. Mattetti, M. Pesce, M. Grillo, M. Forte, and E. Sedoni E. 2. Evaluation of a tractor s driving performance on the road. Trans. ASABE 4(): Rakheja, S., and S. Sankar Suspension design to improve tractor ride: II. Passive cab suspension. SAE Paper No Warrendale, Pa. Society of Automotive Engineers. Salaani, M. K., G. J. Heydinger, and P. A. Grygier. 24. Closedloop steering system model for the national advanced driving simulator. SAE Paper No Warrendale, Pa. Society of Automotive Engineers. Scarlett, A. J., J. S. Price, D. A. Semple, and R. M. Stayner. 2. Whole-body vibration on agricultural vehicles: Evaluation of emission and estimated exposure levels. Research Report 32. Bedford, U.K.: Silsoe Research Institute. Available at: Scarlett, A. J., J. S. Price, and R. M. Stayner. 27. Whole-body vibration: Evaluation of emission and exposure levels arising from agricultural tractors. J. Terramechanics 44(): (2):
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