Dynamic model for an independent suspension

The paper presents the dynamic model with 4 degrees of freedom of an independent suspension with MacPherson type coil springs, taking into account the anti-roll bar. Starting from the constructive scheme, a kinematic analysis necessary to obtain the dynamic model is carried out. The differential displacement equations are obtained starting from the expressions of kinetic and potential energy, by applying Lagrange’s equations. Next, the expressions of the inertia, stiffness and damping matrices and, finally, the matrix expression of pitch-roll vibrations are presented. At the end of the paper, in the numerical application carried out with a calculation program in MATLAB, the influence of the values of the damping constants on the dynamic forces at the wheels is studied. The numerical results are presented in the form of diagrams.


Introduction
The dynamic and passing performance of a vehicle moving on unimproved roads or in an advanced state of wear is characterized, first of all, by the qualities of the suspension and, secondly, by the energy performance of the engine [1], [2].
An incorrect operation of the suspension together with the wheel guidance system will influence the handling, maneuverability and stability of the vehicle [3], [4].
The suspension analyzed in this paper is that of an independent suspension with wheels guided by a crank mechanism -oscillating slide, also known as the MacPherson suspension.This type of suspension, in which the shock absorber also plays the role of a pivot, has the advantage of an insignificant change in the gauge and the inclination of the wheel during cornering, the reduction of the unsuspended mass and a simple design [1], [4].Such a suspension system, when heavily damped, increases vehicle handling but decreases comfort.When the suspension system is slightly damped, passenger comfort is high, but during sharp turns the vehicle may lose stability [5], [6].
A semi-active suspension system can achieve a compromise between comfort and stability, choosing appropriate values for the elasticity constant of the spring and the damping constant of the shock absorber [6], [7].
To establish the limits between which the values of the elastic and damping constants can vary, a dynamic model with 4 degrees of freedom will be presented.This model can be used for a control strategy of semi-active suspensions [8], [9].

The kinematic scheme of the suspension and the model used for the study of vibrations
Figure 1 shows the model of a front drive axle with suspension with helical springs, type MacPherson, in a certain position.
The used notations were: 1 -suspended mass, 2 -suspension spring, 3 -shock absorber, 4 -rim, 5knuckle, 6 -side shafts, 7 -lower control arm, 8 -anti-roll bar.To simplify the calculations, the projection of the suspension mechanism in the plane ZOY is considered.The adopted model is a model with 4 degrees of freedom (left wheel displacement, right wheel displacement, bounce and roll angle of suspended mass).
The used notations are: OZ -the symmetry axis of the suspension in the resting position; C -the center of gravity of the suspended mass; O -the point of intersection between the axis OZ and the line that joins the centers of the wheels, with the vehicle being in the resting position; 0 O -the point of intersection between the axis OZ and the plane tangent to the inside of the rim, with the suspension being in the resting position; -points on the anti-roll bar; G , ' G -the points of articulation of the anti-roll bar on the body;  -specific constructive angle.

The dynamic model
For the dynamic model, represented in figure 1, the next notations where used: z -the displacement of the body relative to the equilibrium position on an horizontal road,  -the roll angle, To obtain the expression of the angle 1  , the projection on the axis OZ of the vector equation is used: (1) To simplify the expressions, it is taken into account that the displacements  , z and 1  are small.The expression is thus obtained: (2) from which it results: (3) where: (4) To obtain the expression of the angle 2  , the projection on the OZ axis of the vector equation is used: (5) It is also taken into account that the displacements  , z and 2  are small and the following expression is obtained: from which results:

Differential equations
The following notations are used: Lagrange's equations are used to establish the differential equations of displacement.We will first need to perform a kinematic analysis.The reference system ′ is chosen.
The coordinates of the points C , F , ' F , D and ′ are given by the expressions:  (12) The expression for kinetic energy will be rewritten as: To obtain the speeds, relations ( 8) -( 12) are derived in relation to time.
Neglecting the non-linear terms, the next expressions are obtained: which, replaced in the kinetic energy expression, lead to the expression: The potential energy has the expression: In the previous relationship 01 s , 02 s , 03 s and 04 s indicate the static deflections.The static deflections check the system of equations in the case where: To obtain the system of differential equations, we apply Lagrange's equations: where the generalized coordinates k q are: The static deflections resulting from equations (17) are: After performing the calculations, the expression of the inertia matrix is obtained: and the expression of the stiffness matrix: where the term 44 k has the expression: To obtain the damping matrix, the constants 5 k and g are replaced by zero in the stiffness matrix and  . (25) The excitation vector   F is deduced from the equations ( 19): According to relations (20), the displacement vector   Z has the expression: The system of differential equations of displacement, written in matrix form is: .(28) Equation (28) obtained is the matrix differential equation of bounce-roll vibrations.

Numerical application
The numerical application is made on the data of a hypothetical medium class vehicle for which it was adopted: 515 0  m kg suspended mass (unsuspended mass is subtracted from the mass of the front axle), -the moment of inertia of the lower arm,  Hz.Next, the behavior of the structure, in the case of harmonic and random excitations, is studied [6].In the case of harmonic excitations, sinusoidal unevenness of the roadway was considered.The car is moving at a constant speed of , with constant step The displacement amplification factors are: Due to the symmetry of the curves of the amplification factors,  1 ,  2 are identical, and the amplification factor  4 is practically zero.In figure 2   Figure 4 and figure 5 present the amplification factors of the dynamic normal forces at the wheels, given by the relations: The graphics have the same allure due to symmetry.The allure of the amplification factors of the dynamic normal forces at the wheels is influenced by the value of the damping constant of the damper, along with the increase of the frequency value.
A first maximum is at the frequency As the static force has the value of There is a small difference between the displacements of the two wheels, a difference due to the anti-roll bar.All these values were obtained for a vehicle speed of  = 30 m/s. m/s), the wheels do not come off the roadway.

Conclusions
The presented model has 4 degrees of freedom and is closer to the case of a real vehicle because it takes into account the anti-roll bar.Compared to the models in which the differential equations of motion are obtained by applying the momentum theorem and the theorem of kinetic moment in relation to the center of gravity, in this paper the differential equations of motion were obtained using Lagrange's equations.
Considering the 4 generalized coordinates (left wheel displacement, right wheel displacement, bounce and roll angle of suspended mass) and applying Lagrange's equations, the 4 differential equations of motion were obtained.To solve the system of differential equations of motion, the differential equations were put in matrix form.
In the presented example, equal and in-phase excitations at the two wheels were considered.Thus, the bouncing vibrations of the center of gravity were highlighted.In this way it was possible to check the accuracy of the presented model in comparison with the model with 4 degrees of freedom obtained with the theorem of impulse and kinetic moment in relation to the center of gravity The model can be extended for 7 degrees of freedom, so taking also into account the movements of the rear axle.In this way, the roll movement and the influence of the anti-roll bar can be correctly appreciated.
Analyzing the form of the matrices of inertia, stiffness and damping obtained for this model, a strong similarity is observed with the forms of the matrices obtained for the model with 4 degrees of freedom with independent suspensions, model without anti-roll bar [6].
The dynamic model obtained can be used for a control of a semi-active suspension and is useful when the excitations of the wheels on the same axle are not equal as in the analyzed example.

Figure 1 .
Figure 1.Suspension model in a certain position.

1  , 2  2 h
-the rotation angles of the arms AB and ' ' B A , relative to the body; -the unevenness of the rolling path, 1 k , k -the elastic constant of the anti-roll bar.

0m 1 m 2 m
-the suspended mass of the car, C J -the moment of inertia of the suspended mass in relation to the longitudinal axis passing through the point C , B J -the moment of inertia of the suspension arm in relation to the axis of rotation of the body, -the equivalent mass for a wheel, hub and 2/3 of the side shaft[6], -two thirds of the mass of the suspension arm[6], static deformations of the springs.

1  and 2 
and figure 3 the curves of the amplification factors  1 ,  2 and respectively  4 are represented, for the 3 values of the damping constants The maxima of the amplification factors are around their natural frequencies.If the first maximum is not influenced by the value of the damping constant of the damper, the second maximum decreases with the increase of the value of the damping constant of the damper: from 1.81 at the frequency of 9.55 Hz for 1000  c Ns/m to the value of 1.01 at the frequency of 7.95 Hz for 2000  c Ns/m.Also, the damping factor  3 is slightly influenced by the values of the damping constants.

Figure 4 .
Figure 4.The dynamic force amplification factor at the right wheel.

Figure 5 .
Figure 5.The dynamic force amplification factor at the left wheel.
a maximum that is very little influenced by the value of the damping constant of the damper.The amplification factor curve continues to have increasing values towards the frequencies 3

Table 1 .
The effective values of the displacements 1 z , 2 z and of the effective forces at the wheels.