Conceptual Cab Suspension System for a Self-propelled Agricultural Machine, Part 1: Development of a Linear Mathematical Model

ARTICLE IN PRESS Biosystems Engineering (2004) 89 (4), 409–416 doi:10.1016/j.biosystemseng.2004.08.006 PM—Power and Machinery

Conceptual Cab Suspension System for a Self-propelled Agricultural Machine, Part 1: Development of a Linear Mathematical Model J. De Temmerman; K. Deprez; J. Anthonis; H. Ramon Department of Agro-Engineering and Economics, K.U. Leuven, Kasteelpark Arenberg 30, B-3001 Leuven, Belgium; e-mail of corresponding author: [email protected] (Received 13 February 2004; accepted in revised form 18 August 2004; published online 2 November 2004)

Agricultural machinery drivers are exposed to low-frequency vibrations during their work. The terrain profile, the long operating times, the forward speed of the agricultural machine and the sitting position are the key factors that lead to the suffering of the back of the agricultural machinery drivers. Suspensions systems are produced in order to reduce the health risks and the discomfort to the driver and to enable the driver to work at a faster pace. The suspension systems isolate the driver from the machine vibrations as much as possible. Modelling the motions of the suspension systems is useful to acquire insight in the dynamic characteristics of these suspensions and for design purposes. In this paper, a linear model of a cab suspension of a self-propelled agricultural machine with six degrees of freedom is developed, based on Lagrange’s equation. The model takes into account the kinetic energy of the suspension and the virtual work executed on the cab suspension. The model of the cab suspension is validated by comparing the behaviour of the model with the behaviour of an experimental test rig for different vibration signals. r 2004 Silsoe Research Institute. All rights reserved Published by Elsevier Ltd

1. Introduction Low-frequency vibrations, produced by agricultural machines, can be extremely severe, depending upon the terrain that the agricultural machine is crossing and the speed of travel. Longer operating times and constantly sitting in the same position on the agricultural machine also contribute to the injuries of the back of machinery drivers. Low-frequency vibrations between 0
5 and 10
0 Hz, caused by the roughness of the road and the unevenness of the field, are considered as the most important risk factors for back problems (Griffin, 1990; Hostens & Ramon, 2003). The frequencies to which the back is most sensitive are between 4 and 8 Hz in the vertical direction and between 1 and 2 Hz in the horizontal direction (Griffin, 1990). The main frequencies of agricultural vehicles lie in this range and without the use of adequate suspension systems, machine drivers can be exposed to heavy vibrations. It is advised to reduce the vibration exposure 1537-5110/$30.00

as low as possible, because this helps to decrease the low back pains and injuries at the back of agricultural machinery drivers (Lings & Leboeuf-Yde, 1999). Manufacturers of agricultural machines therefore produce suspensions to limit the level of vibration exposure (Choi & Han, 2003). There are four main areas which are considered in suspension design: the tyres, the frame, the cab and the seat. This paper concentrates on the development of a model of a cab suspension for a self-propelled agricultural machine. Two systems are commonly used for the cab. Either rubber blocks are used to isolate the cab from the vehicle; or alternatively, low-frequency mechanical suspension systems are used. The rubber blocks and the mechanical suspension act in each suspension system as spring-dampers that limit the vibration exposure to the agricultural machinery driver. Lowfrequency systems with a four-point suspension are the most effective type to reduce vibrations in the three linear directions (Lines et al., 1995). In this study, a four-point mechanical suspension system is modelled. 409

r 2004 Silsoe Research Institute. All rights reserved Published by Elsevier Ltd

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Notation A B B0 r 1B r 1 B1

r 1 B2 e 1 B3 r 2B

r 2 B1

r 2 B2

b1, b 2 C Ci C tij Cs D d/dt g G h 1, h 2 H1 H2 H3 I xx I yy I zz ki ktij Ks l1, l2

coefficient matrix of the state-space model virtual work matrix of Lagrange’s equation coefficient matrix of the state-space model stiffness matrix of the total internal virtual work stiffness matrix of the internal linear airsprings during deformation along their central axis stiffness matrix of the parasitic spring and damping forces static equilibrium matrix in the virtual work, executed by the gravity force damping matrix of the total internal virtual work damping matrix of the internal linear airsprings during deformation along their central axis damping matrix of the parasitic spring and damping forces characteristic distances of the cab suspension in the Y direction, m coefficient matrix of the state-space model linear damping of the ith air-spring, N s m1 torsional damping in the jth translational direction of the ith air-spring, N s m rad1 damping matrix of the state-space model coefficient matrix of the state-space model ordinary derivative with respect to time gravitational acceleration, m s2 transfer function matrix between the vector 0_ r1 and the vector q_ characteristic distances of the cab suspension in the Z direction, m _ and the transfer function matrix between c vector q_ transfer function matrix between y_ and the vector q_ _ and the transfer function matrix between j vector q_ moment of inertia in relation to the centre of mass around the 1X axis, kg m2 moment of inertia in relation to the centre of mass around the 1Y axis, kg m2 moment of inertia in relation to the centre of mass around the 1Z axis, kg m2 linear stiffness of the ith air-spring, N m1 torsional stiffness in the jth translational direction of the ith air-spring, N m rad1 stiffness matrix of the state-space model characteristic distances of the cab suspension in the X direction, m

l ri r l_i 0 l ri T @l ri m Ms q q_ q€ @qT 0 r1 0 _ ðTÞ r1 @ 0 r1

t T u V Wr W r1 w w xi wyi w zi W r2 Ws @W @W e @W r1 @W r1 i

@W r2

length of the ith air-spring at time t first derivative of l ri length of the ith air-spring at rest transposed partial derivative of l ri mass of the cab suspension, kg mass matrix of the state-space model vector of independent Lagrangian generalised coordinates first derivative of the vector of independent Lagrangian generalised coordinates q second derivative of the vector of independent Lagrangian generalised coordinates q transposed partial derivative of q vector from the origin of the referential axis system (0X, 0Y, 0Z) to the centre of mass of the suspension (transposed) first derivative of 0r1 partial derivative of 0r1 time, s kinetic energy of the cab suspension, J input vector of the state-space model vertical length of the air-spring, m disturbance distribution matrix of the total internal virtual work disturbance distribution matrix of the linear air-springs during deformation along their central axis vector of the generalised disturbance inputs excitation input to the cab suspension in the X direction excitation input to the cab suspension in the Y direction excitation input to the cab suspension in the Z direction disturbance distribution matrix of the parasitic spring and damping forces disturbance distribution matrix of the statespace model partial derivative of the total virtual work, J partial derivative of the external virtual work, executed by the gravity force, J partial derivative of the total internal virtual work, executed by the four air-springs during deformation along their central axis, J partial derivative of the internal virtual work, acting on the ith air-spring during deformation along the central axis of the air-spring, J partial derivative of the total internal virtual work, executed by the parasitic spring and damping forces, J

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@W r2 ij

partial derivative of the internal virtual work, executed by the parasitic spring and damping forces in the jth direction of the ith air-spring, J x translational degree of freedom along the 1X axis x state of the state-space model x_ first derivative of x 0 X, 0Y, referential axis system 0 Z 1 X, 1Y, axes with the centre of mass of the suspension 1 Z as origin y translational degree of freedom along the 1Y axis y output vector of the state-space model z translational degree of freedom along the 1Z axis bri angular vector between the initial and actual

The cab suspension is validated on the basis of an experimental test rig consisting of four air-springs, supporting steel plates that represent the cab (Fig. 1). The springs are flexible in three directions so that they can translate and rotate in each vibrational direction.

2. Procedure to derive the linearised equation of motion of a combine harvester cab The methodology to describe the equation of motion of a combine harvester cab can be based on Lagrange’s equation (Ramon & De Baerdemaeker, 1996):   d @T @T ¼B (1)  dt @_q @q where: T is the kinetic energy of the cab suspension in J; q is the vector of independent Lagrangian generalised coordinates; q_ is the first derivative of the vector of Sensors

Steel plate

Air-spring

Fig. 1. Experimental test rig, representing a cab suspension for a self-propelled agricultural machine

r b_ i T @bri r0 d j _ j

y y_ c _ c ci ki @

411

state of the ith air-spring first derivative of bri transposed partial derivative of bri vector of static deformation of the air-spring rotational degree of freedom around 1Z axis first derivative of the rotational degree of freedom j rotational degree of freedom around 1Y axis first derivative of the rotational degree of freedom y rotational degree of freedom around 1X axis first derivative of the rotational degree of freedom c angle between the air-spring and the 0Z axis angle between the projection of the air-spring on the 0X0Y plane and the 0Y axis partial derivative operator

independent Lagrangian generalised coordinates q; B is the virtual work matrix; and t is the time in s. Operator d is used for ordinary derivatives whereas operator @ is applied for partial derivatives. For linear time-invariant systems, the kinetic energy T is not a function of the vector of Lagrangian generalised coordinates q; only of its derivative q_ (Ramon & De Baerdemaeker, 1996). By this, Eqn (1) simplifies to   d @T ¼B (2) dt @_q The procedure to derive the equation of motion starts by determining the mass m in kg, the centre of mass, the moments of inertia I xx ; I yy and I zz in kg m2 in relation to the centre of mass and the degrees of freedom of the cab suspension (Haug, 1989; Ramon & De Baerdemaeker, 1996; Shabana, 1989; Wittenburg, 1977). These degrees of freedom are translated into the vector of independent Lagrangian generalised coordinates q: The model test rig consists of four air-springs, modelled with a linear stiffness ki in N m1 and damping Ci in N s m1, orientated as illustrated in Fig. 2. The damping Ci can be seen as an inherent damping, typical for an air-spring. The subscript i refers to the properties of the ith air-spring. The characteristic distances of the suspension are named l1, l2, b1, b2, h1 and h2 in m, the excitation inputs to the ith air-spring of the suspension are indicated by wxi ; wyi and wzi and the symbol V represents the vertical length in m of the air-spring. The suspension contains six degrees of freedom: three translational degrees of freedom along the three axes 1 X, 1Y and 1Z, represented by x, y and z and three rotational degrees of freedom around these axes, given by c, y and j.

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Z

l2

l1


b2 b1

x

y 0

h1

z

ψ

Z

1

X

h2

k1, C1

V

k4, C4

θ

γi 1

wy4

k2, C2

Y

wx4

wz4

wy1

wx1

κi

wz 1

wx2 0

0

X

wy2

wz2

Y

Fig. 2. Schematic illustration of the cab suspension: 0X, 0Y and 0Z, referential axis system; c, y and j, rotational degrees of freedom and l1 and l2, b1 and b2, h1 and h2, characteristic distances in the axis system 1X, 1Y and 1Z with the origin as the centre of mass of the suspension system; x, y and z, translational degrees of freedom along the axes; V, vertical length of the air-spring; ki and Ci, linear stiffness and damping of the ith air-spring; ci ; angle between the air-spring and the 0Z axis; ki ; angle between the projection of the airspring on the 0X0Y plane and the 0Y axis; and wxi, wxi and wxi, excitation input to the cab suspension in respectively the X, Y and Z direction

2.1. The kinetic energy of the combine cab In case the suspension translates along the 1X axis, the Y axis and the 1Z axis and rotates around those three axes and when 0r1 represents the vector from the origin of the axis system (0X, 0Y, 0Z) to the centre of mass of the suspension (origin of (1X, 1Y, 1Z)), the kinetic energy T of the cab with suspension can be expressed as

where: H 1 ; H 2 ; H 3 and G are the transfer function matrices. By this, the kinetic energy can be written as a quadratic function in q_ for linear time-invariant systems

1

_ 2 þ 1 I yy y_ 2 þ 1 I zz j_ 2 T ¼ 12 m0 r_T1 0 r_1 þ 12 I xx c 2 2

(3)

_ y_ and j _ are the first derivatives of the angular where: c; rotations c; y and j; also called the angular velocities around the 1X axis, the 1Y axis and the 1Z axis, respectively; and 0 r_ðTÞ 1 is the (transposed) first derivative of the vector 0r1. For linear systems, c; y; j and 0r1 can be expressed as a linear function of the vector of independent Lagrangian generalised coordinates q with components x, y, z, c; y and j (Anthonis et al., 2003). The first derivatives of these functions are then c_ ¼ H 1 q_

(4)

y_ ¼ H 2 q_

(5)

j_ ¼ H 3 q_

(6)

0

(7)

r_1 ¼ G q_

T ¼ 12 m_qT G T G q_ þ 12 I xx q_ T H T1 H 1 q_ þ

1 2

I yy q_ T H T2 H 2 q_ þ 12 I zz q_ T H T3 H 3 q_

ð8Þ

2.2. The virtual work, performed by different forces A mechanism only shows a dynamic behaviour if it is subjected to one or more forces. These forces are divided into internal and external forces. To calculate the virtual work, performed by the suspension on the cab body, the forces generated by the suspension must be determined. These forces are the internal air-spring forces along the central axis of the air-springs, the parasitic spring and damping forces of the air-springs and the external gravity force. 2.2.1. The virtual work, executed by the air-springs during deformation along its central axis The virtual work, executed by the linear air-springs, is expressed by Ramon and De Baerdemaeker (1996) as rT r r0 r0 _r @W r1 i ¼ @l i fki ðl i  l i þ di Þ þ C i l i g

(9)

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where: l ri is the length of the ith air-spring at time t; l ri is r the length of the ith air-spring at rest; l_i is the first r rT derivative of l i ; @l i is the transposed partial derivative 0 of l ri ; dri is the vector of static deformation of the ith airspring; and @W r1 i is the partial derivative of the virtual work that acts on the ith air-spring in J. The superscript r refers to the considered internal air-spring force operating on the mechanism, in this case the machine cab and the subscript i represents the ith linear airT r spring. The vectors l ri ; @l ri and l_i can be written as a function of the vector of independent Lagrangian generalised coordinates q and its derivative q_ and the excitation input vector w; which contains the input excitations in each direction, such that the total virtual work by deformation of the air-springs along the central axis becomes  @W r1 ¼ @qT 1 B r1 qþ2 B r1 q_ þ W r1 w (10) where: 1 Br1 represents the stiffness matrix of the linear air-springs; 2 Br1 is the damping matrix of the linear airsprings; W r1 is the disturbance distribution matrix of the air-springs; @qT is the transposed partial derivative of q; and @W r1 is the partial derivative of the total virtual work, executed by the four linear air-springs. 2.2.2. The virtual work, executed by the parasitic spring and damping forces of the air-spring Apart from the deformation of the air-springs along the central axis, the air-springs also contain a parasitic torsional and bending stiffness around their central axis respectively around their axis lying in the plane perpendicular to the central axis. By non-ideal behaviour of the air-springs, some damping is present which can be approximated by linear damping characteristics. These parasitic effects are introduced by directly connecting the air-springs to the cab and the base of the vehicle. To avoid these parasitic effects, the airsprings should be connected through spherical joints to the base of the vehicle and the cab. However, the test rig as depicted in Fig. 2, would then be in an unstable equilibrium. Even a small disturbance would force the cab in a ‘hanging’ position instead of in a ‘standing’ position on the air-springs. The virtual work, done by parasitic spring and damping effects, is calculated by considering the angle between the initial and actual state of the air-springs and given by T

r

t r r t _ @W r2 ij ¼ @bi fkij bi þ C ij bi g

(11)

where: bri represents the angular vector between the r initial and actual state of the ith air-spring; b_ i is the first T derivative of bri ; @bri is the transposed partial derivative of bri ; ktij is the torsional stiffness in the jth direction of the ith spring-damper; C tij is the torsional damping in

413

the jth direction of the ith spring-damper; and @W r2 ij is the partial derivative of the virtual work, executed by the parasitic spring and damping forces in the jth direction of the ith air-spring. These angular displacer ments bri ; @bri and b_ i can be written as a linear function of the vector of independent Lagrangian generalised coordinates q and its derivative q_ and the input vector w such that the total virtual work, executed by the parasitic spring and damping forces, becomes  @W r2 ¼ @qT 1 B r2 qþ2 B r2 q_ þ W r2 w (12) where: 1 B r2 represents the stiffness matrix of the parasitic spring and damping forces; 2 Br2 is the damping matrix of the parasitic spring and damping forces; W r2 is the disturbance distribution matrix of the parasitic spring and damping forces; and @W r2 is the partial derivative of the total virtual work, executed by the parasitic spring and damping forces. 2.2.3. The virtual work, executed by the gravity force The gravity force that works on the combine cab is only a function of the mass of the suspension m and the gravitational acceleration g. The virtual work, executed by the gravity force is expressed as @W e ¼ @0 r1 ðmgÞ ¼ @qT1 B e3

(13)

where: 1 B e3 is the static equilibrium matrix; @0 r1 is the partial derivative of 0r1; and @W e is the partial derivative of the total virtual work, executed by the gravity force. The superscript e refers to the considered external gravity force. As the 1 B e3 matrix cannot be written as a linear function of the vector of Lagrangian generalised coordinates q (De Temmerman, 2003), it does not contribute to the dynamic model but only influences the static equilibrium. However, the impact of the gravitaty force on the overall virtual work, which is the sum of the separate virtual works, is negligible in comparison with the impact of the linear and torsional stiffnesses. 2.3. The equation of motion of the combine cab The expression for the kinetic energy and the virtual work of the combine cab contain all factors, necessary for Lagrange’s equation [Eqn (1)]. The left part of Lagrange’s equation is found by differentiating the kinetic energy to q_ and afterwards deriving it to the time t:   d @T dt @_q ¼ ðmG T G þ I xx H T1 H 1 þ I yy H T2 H 2 þ I zz H T3 H 3 Þ€q ð14Þ where: q€ is the second derivative of q: The right part of Lagrange’s equation is found by calculating the total

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virtual work of the suspension. The total virtual work is the sum of the three virtual works, described above. The total virtual work can then be written as a linear function in the variation of the transposed vector of generalised coordinates @qT and the matrix B [Eqn (1)].  (15) @W ¼ @qT 1 B r qþ2 B r q_ þ W r w where: W r represents the disturbance distribution matrix of the total internal virtual work; and @W is the partial derivative of the total virtual work. The factor between brackets represents the right part of Lagrange’s equation. The equation of motion can be rewritten in the form M s q€ þ C s q_ þ K s q ¼ W s w

(16)

where: M s represents the mass matrix; C s is the damping matrix; K s is the stiffness matrix of the state-space model; and W s describes the way each component of w acts on the mechanism.

2.4. The state-space model of the combine cab The equation of motion [Eqn (16)] is used as a basis for the state-space model, represented by the following equations: x_ ¼ Ax þ B 0 u

4. The validation procedure The validation procedure compares the behaviour of the linear model with the behaviour of the experimental test rig. The test rig is excited in the frequency range between 0
7 and 7
0 Hz with swept sines as input signals. The eigenfrequencies of all degrees of freedom of the test rig are determined and compared with the eigenfrequencies of the model. 4.1. Linear character of the experimental test rig The calculated model is based on linear equations. A system is linear when its frequency response function (FRF) is independent of the input signal, i.e. the ratio of the Fourier transform of the output over the Fourier transform of the input is constant. By this, comparison of the FRF of two different vibration measurements (swept sine and random signal) on the experimental test rig examines the linear character of the test rig. The amplification of both signals for the vertical degree of freedom is illustrated in Fig. 3. Both signals are almost similar such that the experimental test rig shows a linear behaviour below 5 Hz. Beyond 5 Hz, the behaviour of the test rig is considered as non-linear. This is also visible in the figure where the FRF becomes more dependent on the input signal.

(17) 4.2. Comparison of the eigenfrequencies

y ¼ Cx þ Du

(18)

where: x is the state; x_ is the first derivative of x; u is the input vector, which is equal to w; y is the output vector; and A, B 0 ; C and D are the coefficient matrices of the state-space model. Matrix D shows up because the displacements under the spring-dampers of the suspension are taken as an input.

The comparison of the eigenfrequencies of the model with the eigenfrequencies of the experimental test rig, deduced on the basis of the swept sines as input signals, shows to what extent the model is able to describe the behaviour of the test rig. Table 2 gives an overview of Table 1 Overview of the numerical values of the mechanical parameters Mechanical parameter Characteristic distances

3. Determination of the mechanical parameters The model, deduced above, is a function of the distances between the centre of mass of the test rig and the connections of the test rig with the air-springs, the mass of the system, the moments of inertia, the linear stiffness, the damping and the torsional constants (torsional stiffness and torsional damping) in each translational direction of the air-springs. All these parameters are measured or determined by calculations on the experimental test rig. Table 1 gives an overview of the numerical values of all these mechanical parameters.

Mass of the suspension Moments of inertia Linear stiffness Linear damping Torsional stiffness Torsional damping

Symbol

Value

l1 l2 b1 b2 h1 h2 m Ixx Iyy Izz ki Ci ktij C tij

0
400 m 0
400 m 0
400 m 0
400 m 0
014 m 0
016 m 111
88 kg 7
16 kg m2 4
78 kg m2 11
92 kg m2 2
64 103 N m1 60 N s m1 80 N m rad1 15 N m s rad1

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15 10

This validation can also be analysed in the time domain. The displacement signals of the model and the measurement have a quasi identical course. Figure 6 represents the measured and simulated displacement signals for the vertical degree of freedom. This validation indicates that the choice of the linear model was adequate.

5. Conclusion This paper describes the procedure for deriving an analytical, three-dimensional model of a combine cab,

10 0

A m plification, dB

the eigenfrequencies of the model and the test rig for each degree of freedom. The difference between the measured and simulated eigenfrequencies is caused by the values of the mechanical parameters. The torsional stiffness is a largely fluctuating parameter. Fitting the model to the measurements through the adaptation of the torsional stiffnesses of the air-springs allows to achieve the same eigenfrequency for each degree of freedom. This can be seen in Fig. 4 for the vertical degree of freedom of the combine cab. Finally, when a random signal is used as input to the experimental test rig and the model, the FRFs of the experimental test rig and the model still show identical natural eigenfrequencies for each degree of freedom. The course of the FRFs for the vertical degree of freedom is represented in Fig. 5. The similarity between both signals is most obvious in the region below 5 Hz. The torsional parameter values in this simulation remain the modified torsional values.

−10 −20 −30

0

−40

−5

1

2

3

−10

4 5 Frequency, Hz

6

Fig. 4. The frequency response function for the vertical degree of freedom of the combine harvester cab with modified torsional stiffness parameter of the suspension: —, measurement;





, model

−15 −20 −25 1

2

3 4 Frequency, Hz

5

15

6

Fig. 3. The frequency response function of the swept sine (—) and the random signal (





) for the vertical degree of freedom of the laboratory experiment Table 2 Comparison between the eigenfrequencies of the model and the experimental test rig Degree of freedom

x translation y translation z translation x rotation y rotation z rotation

Eigenfrequencies, Hz Model

Test rig

2
1 2
3 2
2 3
6 3
6 5
0

2
3 2
2 2
5 3
3 3
1 5
2

10 Amplification, dB

Amplification, dB

5

5 0 −5

−10 1

1.5

2

2.5

3 3.5 4 4.5 Frequency, Hz

5

5.5

6

Fig. 5. The frequency response function of the model and the measurement for the vertical degree of freedom of the combine harvester cab: —, measurement;





, model

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References 3

Displacement, mm

2 1 0 −1 −2 −3 107

108

109

110 111 Time, s

112

113

114

Fig. 6. The measured and simulated vertical displacements of the combine harvester cab: —, measurement;





, model

based on Lagrange’s equation. This general methodology can be used for any kind of cab suspension of a selfpropelled agricultural machine. The linear model is a function of several mechanical parameters that are determined by measurements or calculations. Linearity of the model of the combine cab seems to be valid up to 5 Hz. The adaptation of the torsional values, which are difficult to measure, increases the accuracy of the model. The analogy between the signals in the frequency domain and the time domain shows that the model is able to describe the motions of the cab suspension for each excitation.

Anthonis J; Hostens I; Mouazen A M; Moshou D; Ramon H (2003). A generalized modeling technique for linearized motions of mechanisms with flexible parts. Journal of Sound and Vibration, 266(3), 553–572 Choi S B; Han Y M (2003). MR seat suspension for vibration control of commercial vehicle. International Journal of Vehicle Design, 31(2), 202–215 De Temmerman J (2003). Comfortevaluatie en design van een cabineophanging. Comfort evaluation and design of a cab suspension. MCSc Thesis, KULeuven, Belgium Griffin M J (1990). Handbook of Human Vibration. Academic Press, London, UK Haug E J (1989). Computer Aided Kinematics and Dynamics of Mechanical Systems, Vol. I, Basic Methods. Allyn and Bacon, Massachusetts, USA Hostens I; Ramon H (2003). Descriptive analysis of combine cabin vibrations and their effect on the human body. Journal of Sound and Vibration, 266(3), 453–464 Lines J; Stiles M; Whyte R (1995). Whole body vibration during tractor driving. Journal of Low Frequency Noise and Vibration, 14, 87–95 Lings S; Leboeuf-Yde C (1999). Whole-body vibration and low back pain: a systematic, critical review of the epidemiological literature 1992–1999. International Archives of Occupational and Environmental Health, 73, 290–297 Ramon H; De Baerdemaeker J (1996). A modelling procedure for linearized motions of tree structured multibodies, part 1: derivation of the equations of motion. Computers and Structures, 59(2), 347–360 Shabana A A (1989). Dynamics of Multibody Systems. John Wiley & Sons, New York, USA Wittenburg J (1977). Dynamics of Systems of Rigid Bodies. B.G. Teubner, Stuttgart, Germany