A Comparative Analysis of Linear and Nonlinear Semi-Active Suspension System

The ride quality including comfort and safety is the primary factor that is targeted in the design of an effective suspension system. The parametric behavior of suspension system is non-linear but for simplicity most of the researchers have assumed it as linear. The emphasis of this study is to analyze the non-linear behavior of basic components of the suspension system. Anon-linear mathematical model for semi-active suspension system equipped with MR (Magnetorheological) damper is developed based on a two degrees of freedom quarter vehicle model. Matlab/Simulink is used for simulation of the proposed model for different types of road disturbances. Transient response characteristics of the proposed non-linear model is compared with linear semi-active suspension model which show the difference between the responses of these models.

The ride comfort needs spring having low stiffness value while ride safety needs spring having high stiffness value. So these two conflicting requirements led to the development of semi-active and active suspension systems. Unlike passive suspension system which have fixed damping coefficient, active and semiactive suspension system consists of variable dampers.
In semi-active suspension system, the direction of the damping force depends upon the relative velocity across the damper where as its magnitude is adjustable, Carter [4]. In active suspension system, an actuator is placed parallel to the vehicle's body which moves the vehicle body in the suspension space to absorb the vibrations produced, Pekgokgoz et. al. [5].The major disadvantage of active suspension system is its high cost and huge external power. That is why automobile companies paying much attention to the development of semi-active suspension systems.
All the real world systems exhibit nonlinear behavior. The semi-active suspension system behaves linearly for low vibrations but exhibits nonlinear behavior for high vibrations. Most of the designer considered the semiactive suspension system as linear system. Very few researchers paid attention to its nonlinear behavior. Sawant et. al. [6] compared the vehicle dynamic system with nonlinear parameters subjected to actual road excitations. Non-linearities is considered only in spring, mass and damper of the passive suspension system and studied their behavior for individual and relative significance. Lajqi and Pehan [7] introduced a procedure for designing and optimizing nonlinear active and semiactive suspension systems. Hingane et. al. [8] presented the analysis of semi-active suspension system with Bingham model for MR damper for different road excitation. The comparison between passive and controlled semi-active suspension system showed the improvement of semi-active suspension system over passive.
Qazi et. al. [9] performed the modeling of passive and semi-active suspension system equipped with different control strategies. A comparison of semi-active suspension system for different control strategies is made with passive suspension system. The optimized fuzzy logic based semi-active suspension system improves the ride comfort by minimizing percentage overshoot and stabilizing time. Diala and Ezeh [10] studied the vibration transmissibility of a viscously damped isolator for a single and two degree of freedom suspension system. Non-linear viscous cubic damping is considered for the damper.
The rheological structure for an MR damper was described by the Sapinski and Filus [11]. The simplest parameter model of the MR damper used in this research is the non-linear Bingham plasticity model shown in Fig. 1. Kong et. al. [12]. f mr is the variable damping element which is placed parallel to the damper, co. The damping force for damper piston having some velocity x  , is given by Equation (1).
x is the displacement of the piston, C p0 is the hysteretic damp coefficient after MR fluid yield and f 0 is the force of the accumulator.
Researchers have developed different non-linear models of the semi-active suspension system. The non-linearity is considered in one or more of the basic components of the semi-active suspension system to study its effect on the performance of the system. Chavan et. al. [13] considered non-linearities in all passive elements of the In this study non-linearities of all the passive elements such as tire spring, suspension spring and suspension damper as well as MR damper are taken into account.

MATHEMATICAL MODELING
A two degrees of freedom semi-active quarter car model is shown in Fig. 2. It consists of sprung mass, M s which is connected to the unsprung mass, M u through semi-active suspension system. The semi-active suspension system consists of spring stiffness K s ,suspension damper C s and a MR damper placed between sprung mass and unsprung mass. The tire stiffness is represented by K t while tire damping is neglected because of its minimum effect on final results.x s , x u and x t are the sprung mass, unsprung mass and tire vertical displacement respectively. This quarter car model revels the basic properties of full vehicle model.
f mr is the controllable damping force generated by MR damper. The non-linear behavior of MR damper has been described by the Bingham plasticity model Kong et. al. [12].
The Equations (2-3) of motion for the designed quarter car model with two degrees of freedom are given as: k s1 , k s2 and k s3 are suspension stiffness coefficients, k t1 , k t2 and k t3 are tire stiffness coefficients and C s1 and C s2 are suspension damping coefficients and f d is the nonlinear damping force described in Equation (1). k s2 and k s3 , k t2 and k t3 and C s3 are the stiffness and damper nonlinear responses at higher loads.

SIMULATION
The simulation results for the nonlinear semi-active The values of various parameters for the nonlinear semiactive suspension system are given in Table 1 as described by Sawant and Tamboli [14] and Muresan [15].

VALIDATION
Validation of the linear model is performed by analytical method. First of all, the transfer function is determined by taking Laplace transform of the dynamic equations of the system with zero initial conditions. Then the solution of the system is determined for a unit step response by using Matlab. The Laplace transforms equations are given in Equations (4-5).
(m s S 2 + C s S + k s )X s + F mr = (C s S + k s )X u  (C s S + k s )X s + F mr = (m u S 2 + C s S + k s + k t )X u (5) The suspension displacement obtained through Laplace transform is given by Equation (6).

FIG. 4. SIMULINK MODEL FOR NON-LINEAR SEMI-ACTIVE SUSPENSION SYSTEM
Equation (6) is obtained from the Equations (4-5) by solving it for suspension displacement, x s.
The suspension displacement response obtained through analytical method is shown in Fig. 5 which is very similar to the Simulink model suspension displacement shown in Fig. 6.

SIMULATION RESULTS
For analysis of the nonlinear system, the Simulink model of Fig. 3 is used. A step and a sinusoidal road disturbance is used for system excitation to study its response for a variety of road profiles.  Fig. 7 and a double cosine road bumpgivenis shown in Fig. 8.
where a is the amplitude of the sine wave. The comparison is based on the sprung mass and unsprung mass displacement of the semi-active suspension system. Sprung mass displacements for both linear and non-linear system for sinusoidal road disturbance are shown in Fig. 9. Unsprung mass displacement for both linear and non-linear system for sinusoidal road disturbance are shown in Fig. 10. Sprung mass displacements for both linear and non-linear system for step input are shown in Fig. 11. Unsprung mass displacements for linear and non-linear system for step input are shown in Fig. 12.

RESULTS AND DISCUSSION
It is obvious from the transient response of the suspension system that a difference between linear and non-linear system does exists. This difference is based on the settling time, maximum overshoot and steady state error. The reference displacement is steady state response of the linear system. The performance of both types of the systems were studied on 20 seconds scale.
For sinusoidal road input of Figs. 9-10, the difference between the linear and non-linear system for the sprung and unsprung mass displacement based on different parameters are tabulated in Table 3. Table 3 shows that maximum displacements for the linear system are greater than the non-linear system. The settling time for the sprung mass displacement of the linear system is 12.5% whereas the settling time for the sprung mass displacement of the non-linear system is 25%. The settling time for the unsprung mass displacement of the linear system is 11 % whereas the settling time for the unsprung mass displacement of the non-linear system is 26 %.
For step input of Figs. 11-12, the difference between the linear and non-linear system for the sprung and unsprung mass displacement based on different parameters are tabulated in Table 4. Table 4 shows that maximum displacement of sprung mass and unsprung mass for the linear system is smaller than the non-linear system. Settling time for the sprung mass of the linear system is 15% while for non-linear system it is 50.1%. Similarly settling time for the sprung mass of the linear system is 16% while for non-linear system it is 50%. Therefore,it is necessary to include non-linearity in modeling the dynamic response of the vehicle suspension system.