Modelling a quarter car suspension system

Modelling a quarter car suspension system


This study is simulating a quarter car suspension model. Active and passive suspensions have been introduced and analysed. Example suspensions have been simulated using a theoretical and computer methods. The suspension has been built according to individual data provided and practiced on different road profile. The results of the two methods showing a close agreement. Possible differences have been discussed and a design recommendation provided.

Keywords: quarter car model, active and passive suspensions, theoretical and computer model, road profile


Sprung mass and Unsprung mass

In a vehicle with a suspension, such as an automobile, motorcycle or a tank, sprung mass (or sprung weight) is the portion of the vehicle’s total mass that is supported above the suspension. The sprung weight typically includes the body, frame, the internal components, passengers, and cargo but does not include the mass of the components suspended below the suspension components (including the wheels, wheel bearings, brake rotors, calipers).

The unsprung weight (or the unsprung mass), in a ground vehicle with a suspension, is the mass of the suspension, wheels or tracks (as applicable), and other components directly connected to them, rather than supported by the suspension. Unsprung weight includes the mass of components such as the wheel axles, wheel bearings, wheel hubs, tires, and a portion of the weight of driveshafts, springs, shock absorbers, and suspension links. A schematic of sprung and unsprung mass has been shown in Figure 1.

Figure 1. Schematic of sprung and unsprung masses

Quarter car model

A quarter car model is a simplification of a car body and suspension system within each wheel. It consists of sprung mass Ms representing a quarter mass of the vehicle body supported by spring ks and shock absorber bs. The mass of the tire with wheel and axle parts are termed as unsprung mass Mu. Tire properties can be later simplified as tire spring constant kt and tire damping constant bt. Since the damping coefficient of tire is much smaller compared to shock absorber, its value can be neglected, therefore equal to 0.

Figure 2. Simplified model of loads and displacements apply on sprung and unsprung masses

The suspension system of the vehicle designed with features stated above is usually denoted as a passive suspension system since there is no control of the system, as shown in Figure 3. If the settings of the suspension can be adjusted during the ride, then it is call semi-active or active suspension system (Figure 4) depending on the adapting method. The difference between both systems can be observed in vehicle handling performance and passengers comfort. As riding comfort goes against vehicle handling, an active suspension system tries to find a compromise between both and according to that adjust the suspension parameters. The adjustment can be done in many different ways and on the market is wide range of active suspension systems. To model simplified active suspension an extra controller with force actuator is added for generating force Fa.

In this assignment the active suspension model is designed with simple PID controlling unit. The model was created with using MATLAB Simulink software and with help of tutorials. Characteristics of all consisting parts are listed in table 1.

Table. 1 Typical suspension parameters



Ms [kg]


Mu [kg]


ks [N/m]


kt [N/m]


bs [Ns/m]


bt [Ns/m]


Analytical approach

Sprung mass and unsprung mass

At the beginning a free body diagram of sprung and unsprung mass was determined with all reactive forces as spring forces, forces due to shock absorber, gravitational forces and adjustable force from active suspension controller.


Equation of motion for sprung and unsprung mass can be obtained from Newton’s second law related to equations 1 and 3 respectively. From both EOM can be expressed the acceleration of both masses as stated in equation 2 and 4.




Free vibration

Since in steady state the gravitational forces are equal to the initial compression of the spring and tire due to mass load, Gs and Gu can be excluded from all equations. For passive suspension no Fa force is applied so the EOM become simplified. As free vibration is under consideration the undamped system is investigated for finding natural frequencies, with little ordering the final forms of EOM are:



Both equations can be rearranged to matrix form as stated below.


To find natural frequencies, the displacement of sprung and unsprung mass is assumed as harmonic motion according to following equations 8, 9. The second time derivatives of these equations represent acceleration of masses.






The displacement and acceleration equations stated above have to be inserted into equation 7 and after reordering the matrix with frequencies will be obtained as below.

The determinant of the matrix is calculated and expressed in equation 13 which is called characteristic equation. With a substitution of s = ω2 a discriminant of the equation is founded and later both natural frequencies obtained.






With regards to computed natural frequencies the ratios of mode shapes were determined using equations 18, 19. The initial conditions were assumed



The displacement of sprung and unsprung mass can be calculated using following equation.





From equations above the solutions were obtained and expressed in following formulas. At this position the graphs of both mass displacement was plotted.





Forced vibration

The change in road profile xr can be expressed as a harmonic excitation according to equation 29 and solution for xs and xu as harmonic function as shown above in equations 8, 9. The parameters of the quarter car model were used same as for free vibration. The external forces applied to the system are represented by actuator force Fa, gravitational forces and road profile change as seen from equations 30, 31.





An assumption with excitation frequency ω = 1 rad.s-1, ϕ = 0, amplitude of road profile Xr = 0.05 m and Fa = 0 and 1500 N was applied and also gravitational forces Gs and Gu were assumed. Equations 8, 9 were differentiated to obtain velocity and acceleration of sprung and unsprung mass and later substituted into equations 30 and 31. With ordering those equations a matrix form was created as following.


The calculated displacements were determined for ωt = π, 2π which led to canceling term with damping coefficients. The values of Xs and Xu are shown in table 2.

Tab. 2 Mass displacement with and without actuator force

Fa = 0 N

Fa = 1500 N

ωt = π

ωt = 2π

ωt = π

ωt = 2π


0.2369 m

-0.1350 m

0.1520 m

-0.0502 m


0.0695 m

0.0307 m

0.0693 m

0.0308 m

When comparing free and forced vibration the displacements of masses are higher for free vibrations, since the damping of the system is not considered. In the solutions no assumption of friction was applied so the system will oscillate without any loss.

MATLAB Simulation

In MATLAB Simulink software was created model of passive suspension (figure 6) which was later modified for design of active suspension (figure 8). For better lucidity a subsystem was created where all EOM were performed, the illustration is visible in figure 7 and 9 respectively. As seen the input variable is the road profile characterized by xr value and the outputs to be plotted are displacement of sprung/unsprung mass xs, xu, the acceleration of both masses ẍs, ẍu, the difference between sprung and unsprung mass displacement and finally also the road profile. The aim of creating these models is to compare analytical and numerical approach.

MATLAB code:

Free vibration

ms = 300;

ks = 18000;

bs = 1200;

mu = 50;

ku = 180000;

bu = 0;

b = -ks/ms -ks/mu – ku/mu;

c = (ks*ku)/(ms*mu);

D = b^2 -4*c;

sqrtD = D^(1/2);

s1 = (-b+sqrtD)/(2);

s2 = (-b-sqrtD)/(2);

w1 = (s2)^(1/2);

w2 = (s1)^(1/2);

r1 = (ks-ms*(w1)^2)/ks;

r2 = (ks-ms*(w2)^2)/ks;

t = 0:0.001:2;

Xs = 1;

xs = Xs*cos(w1*t);

xu = r1*Xs*cos(w1*t);



title(‘w1 = 7.38 rad/s’)




xs2 = Xs*cos(w2*t);

xu2 = r2*Xs*cos(w2*t);



title(‘w2 = 62.97 rad/s’)




Forced vibration

ms = 300;

ks = 18000;

bs = 1200;

mu = 50;

kt = 180000;

bt = 0;

g = 9.81;

x = pi; % phase angle (wt)

w = 1;

Xr = 0.05;

syms Xs Xu

Fa = 1500;

eqn1 = -ms*w^2*Xs*cos(x) + bs*w*Xu*sin(x) – bs*w*Xs*sin(x) – ks*Xu*cos(x) + ks*Xs*cos(x) == Fa -ms*g

eqn2 = -mu*w^2*Xu*cos(x) – bs*w*Xu*sin(x) + bs*w*Xs*sin(x) + ks*Xu*cos(x) – ks*Xs*cos(x) – kt*Xr*cos(x) + kt*Xu*cos(x) == -Fa -mu*g

[A,B] = equationsToMatrix([eqn1, eqn2], [Xs, Xu])

X = linsolve(A,B);

Xs = X(1)

Xu = X(2)

Simulink model of pothole road profile for passive and active suspension

Fig 6. Typical pothole road

Figure 7. Simulink model of pothole road profile for passive (top) and active suspension (bottom)

Simulink model of random road profile for passive and active suspension

Figure 8. Simulink model of random road profile for passive (top) and active suspension (bottom)

At first a model with passive suspension was subjected to analysis on road with same assumptions as above: harmonic sinusoidal excitation with frequency 1 rad/s and amplitude 0.05 m. The used marking for mass displacement is – xs, xu, road profile – xr and acceleration of mass – xsddot, xuddot.

The same conditions were applied for force vibration with applied PID controller and maximum actuator force Fa = 1500 N.

When a comparison of calculated values from analytical approach and Simulink model for free and force vibration was made the obtained results were equal for particular ωt. From graphs above can be seen that better performance is reached without PID controller for sinusoidal road type.

Figure 13 Pothole road profile

The second road type under consideration was simulating potholes on the road. The characteristic were assumed as in previous tasks Xr = 0.05 m and ω = 1rad/s.

The damping of the sprung and unsprung mass after hitting the pothole is very clear from figures above. Since the spring stiffness is lower compared to tire stiffness and Ms is higher than Mu, the sprung mass oscillate for longer time. The displacements are similar but for controlling suspension with actuator force the magnitude is lower.

The third road type was set to random profile as used in tutorial example with variance of 0.0005 m again for passive and active models.

On this road profile both systems have similar behavior. The one with PID controller shows slightly lower displacement for sprung mass.

The last simulation of active suspension was performed with another sinusoidal road profile and different driving schemes. The amplitude of the road profile Xr = 0.1 m and frequency range 1, 10 and 100 rad/s was applied which is connected with the velocity of the vehicle.

From figures 15, 16 and 17 can be seen the effect of frequency on mass displacement and acceleration. For lower frequencies the displacement is higher for sprung mass and for higher frequencies the displacement decreases. For lower frequencies the acceleration is lower for unsprung mass and for higher values of ω the tendency goes in opposite direction.


Using the input data (e.g. Xr=?, m=?, …..), the result shows, a lower frequency generate higher displacement for sprung mass and for higher frequencies the displacement decreases. Therefore, a change on parameters (e.g. Ku=? to Ku=?) suggested to improve the performance of suspension system and reduce the displacement and noise(?). The data has been selected according the individual data given to each student.

Please create at least two bullet point recommendations.


In this assignment an active suspension system was investigated by analytical approach and MATLAB Simulink modelling. Free vibration analysis was performed from which natural frequencies and modes of the system were obtained. In force vibration loads from road change, active suspension system and gravitational forces were considered and the displacement values of both masses compared for both solutions. Different road profiles were simulated for both systems and compared. In general active suspension provides better performance than passive suspension but the limitation of the controlling and adjusting is the cost of the system. For this reason the passive suspension still represents majority of vehicle production.


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[2] Chahyadi, H. D. (2019). Simulation and Analysis of Two-Mass Suspension Modification Using MATLAB Programming. ACMIT Proceedings. 3. 160-165. 10.33555/acmit.v3i1.39.

[4] Hassaan, G. (2014). Car Dynamics using Quarter Model and Passive Suspension, Part I: Effect of Suspension Damping and Car Speed. International Journal of Computer Techniques, 1(2).

[5] Tran, D. (2014). Quarter Car [online] Available at: [Accessed 13 March. 2020].

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[7] (n.d.).Introduction: Simulink Modeling. [online] Available at: [Accessed 13 March. 2020]