Analysis of an Automobile Suspension

by

Derek MaximHieu NguyenRyan ParentEric Twiest

School of EngineeringGrand Valley State University

EGR 350 – VibrationsSection A

Instructor: Dr. Ali Mohammazadeh

August 4, 2006

Introduction

Modeling the suspension of an automobile is of interest for many automotive and

vibrations engineers. Of importance for these engineers are the ride quality of the vehicle

traversing over broken roads and control of body motion. When traveling over rough terrain, the

vehicle exhibits bounce (up and down), pitch (rotation about the center of gravity along the

vehicle's length) and roll (rotation about the center of gravity along the vehicle's width) motions.

For this project, the bounce and pitch motion of the car over rough roads are of interest and will

be analyzed in this report.

Assumptions

For the analysis, it will be assumed that the vehicle is a rigid body with a suspension that

will be modeled as a two-degree-of-freedom (DOF) system. The setup of the suspension will

consist of equivalent springs in which the stiffness of the tire and the spring are combined, and

equivalent dampers that account for the shock absorber and the damping of the tire.

Theory

Figure 1 shows the two DOF system schematic that was used to determine the equations

of motion of the vehicle.

Figure 1: Spring-mass-damper model of the vehicle

To determine the equations of motion, Lagrange's equations, also known as the energy

method, were utilized. Equation (1) shows the general form of Lagrange's equations

(1)

where L is the sum of the kinetic and potential energies, or

(2)

where T is the kinetic energy, U is the potential energy of the system. The terms qi and Qi from

Eq. (1) represents a degree of freedom and the non-conservative work for each DOF (subscript i

denoting the first and second degrees of freedom); represents the derivative of qi.

To derive the equations of motion using Lagrange, the degrees of freedom i needs to be

defined. This is shown in Eqns. (3) and (4).

(3)

(4)

Next, the kinetic energy of the system is shown in Eq. (5),

(5)

where M is the mass of the body, is the bounce velocity of the body about its center of

gravity, J is the polar moment of inertia, and is the angular acceleration of the body.

The potential energy of the system is shown in Eq. (6)

(6)

where k1 and k2 are the equivalent spring rates of the front and rear suspension, xCG is the

displacement of the body's center of gravity, l1 and l2 are the distances from the center of gravity

to the front suspension and rear suspensions, and y1 and y2 are the input functions of the road for

the front and rear of the system.

Combining Eqs. (5) & (6) produces the energy equation, Eq. (7)

(7)

The equations for non-conservative work for both degrees of freedom are shown in Eqs.

(8) & ( 9)

(8)

(9)

where Q1 and Q2 are non-conservative work for q1 and q2, c1 and c2 are the damping coefficients

of the system and and are the time derivatives of the road input function.

Finally, taking the derivatives of the q terms and combining all of the equations into the

form of Eq. (1), the equations of motion for the system are

(10)

(11)

The parameters of the system are as follows: k1 = k2 = 30000 N/m, c1 = c2 = 3000 N*s/m,

M = 2000 kg, J = 2500 kg*m2, l1 = 1 m, and l2 = 1.5 m. Substituting these values and expanding

Eqs. (10) & (11) yields Eqs. (12) & (13)

(12)

(13)

The car is traveling at 13.88 m/s over road that is assumed to be sinusoidal in cross-

section with an amplitude of 10 millimeters (0.01 meters) and having a wavelength of 5 meters.

With this information, the input functions y1 and y2 are defined in Eqs. (14) & (15)

(14)

(15)

Where, t is the time traveled and π is the time shift that accounts for the time that it takes for the

rear suspension to negotiate the "bump" that the front suspension had negotiated.

Results

SIMULINK

The system was simulated using MATLAB's SIMULINK program. Figure 2 shows the

schematic that was used for analysis.

Figure 2: SIMULINK model of the two-degree-of-freedom system

The schematic shown in Figure 2 was used to determine the natural frequencies ω1 and ω2 of the

system. Using MATLAB, the modes of vibration, which are due to the system possessing two

different natural frequencies, were calculated to determine ω1 and ω2 in SIMULINK. From

MATLAB, the first and second modes of vibration were 0.477 and -0.596 (see MATLAB

results). Figures 3 and 4 show the plots produced by SIMULINK, which contains the natural

frequencies, and verify the MATLAB results. From Figures 3 and 4, the natural frequencies

were determined from the "Power Spectral Density" plots (middle graphs) and were ω1 = 5.1

rad/s and ω2 = 6.5 rad/s.

Figure 3: SIMULINK plot results for the first mode of vibration showing the bounce (left graph) and pitch response (right graph); Power Spectral Density graph used to determine natural

frequency ω1

Figure 4: SIMULINK plot results for the second mode of vibration showing bounce (left plot) and pitch (right plot) response where natural frequency ω2 can be determined from the Power

Spectral Density graph

In addition, SIMULINK was used to model the response of the system to the road

conditions. Once road conditions were modeled, the SIMULINK model was modified using a

slider gain to reduce the pitch motion of the vehicle. Figures 5 and 6 show the response of the

system under the given car parameters and Figures 7 and 8 show the response when the gains on

the dampers in the system were modified to achieve the most desirable results. Comparing

Figures 5 and 6 to Figures 7 and 8 the figures, it was easy to see that by increasing damping gain

by a factor of 10, pitching motion decreases from 5x10-4 meters to less than 1x10-4 meter.

Bounce motion also decreases from 3x10-3 meters to 1x10-3 meters.

Figure 5: Bounce (left) and pitch motion (right) plot results for the unadjusted modeling of the system under original conditions

Figure 6: SIMULINK model used to determine the response of the system

Figure 7: Bounce (left) and pitch motion (right) response plot results for the system with higher viscosity (increased gain) dampers

Figure 8: SIMULINK model with slider gain block included to reduce the pitching motion of the system

MATLAB

MATLAB, a mathematical processing software, was used to compare and verify the

model analyzed in SIMULINK. The program was also used to compare the responses of the

system using a function known as "lsim" and modal analysis. Attached at the end of this report

are the codes used to run lsim and the modal analysis.

Before the analysis of the system was performed using the lsim function, the modes and

natural frequencies of the system were determined. Figure 9 shows the plot of the modes

produced in MATLAB. From Figure 9, mode 1 is seen to have an oscillation of lower amplitude

than mode 2, which has an oscillation of higher amplitude. Using modal analysis, the

displacement degrees of freedom due to natural frequencies ω1 and ω2 were u1 = [-0.0197,

0.0094] meters and u2 = [0.0105, 0.0176] meters. From these results, it can be concluded that

mode 2 has a greater effect on the system than mode 1.

Figure 9: Plot of the modes of the system; mode 1 is shown to have an oscillation with a smaller frequency than mode 2

To use the lsim function in MATLAB. To convert the equations into transfer functions,

the equations themselves must undergo a Laplace transformation. The generic equation for the

transfer function is shown in Eq. (16), whereas the specific transfer functions of the system, after

undergoing the Laplace transformation, are shown in Eqs. (17)-(20) (see Appendix A for

derivation of these equations).

(16)

(17)

(18)

(19)

(20)

With these transfer functions entered into MATLAB, the frequency response plots of the bounce

and pitching motion were created and are shown in Figures 10 and 11.

Figure 10: Bounce motion plot resulting from the analysis of the system using the lsim function

Figure 11: Pitching motion plot of the system resulting for the use of the lsim function

Comparing Figures 10 and 11 to Figure 5 (SIMULINK plot of the system), it can be seen that

both models show similar behavior to the road input, with small differences in amplitude. The

response of the front and rear suspensions to the road using lsim analysis are shown in Figures

12-13 and Figures 14-15. Figure 12 shows the front suspension response to bounce, Figure 13

shows the pitching response of the same suspension, Figure 14 shows the rear suspension

response to bounce, and Figure 15 shows the pitching motion response.

Figure 12: Front suspension response to bounce using the lsim function

Figure 13: Pitching response of the front suspension to the road input

Figure 14: Bounce response of the rear suspension to road input

Figure 15: Pitching response of the rear suspension to road input

Modal analysis was performed using MATLAB to compare the response of the system to

the lsim analysis and the matrices needed to perform the analysis can be seen in Appendix A.

However, it was not completed at the time of writing, so it cannot be proved in this report that

the response from the use of the lsim function is similar to the response resulting from modal

analysis. It is expected that the results would be similar, assuming that the matrices included in

this report from modal analysis were correct and the parameters and input functions were

transformed correctly.

Conclusions

Using MATLAB to model the suspension system (albeit simplified two-degree-of

freedom compared to a system that can be modeled with as much as ten degrees of freedom), it

was found that the suspension with front and rear spring rates of 30,000 Newton per meter, front

and rear dampers of a rate of 3,000 Newton-second per meter for a 2,000-kg vehicle quells the

excitation produced by the road in approximately 1.5 seconds. The second mode of vibration

was found to contribute the bounce and pitch motion of the vehicle more than the first mode of

vibration. The response of the system using modal analysis was also performed to verify the

response of the system. Though the eigenvalues and eigenvectors were determined using

MATLAB, unfortunately, the response of the system from the analysis was incomplete at the

time of writing.

SIMULINK was also used to model the suspension system and it was found to be within

agreement with the MATLAB model. Using the slider gain to increase or decrease the damping

rate on the SIMULINK model, it was found that by increasing the damping gain (and therefore

damping rate), the bounce and pitch motions of the vehicle decreased by a factor of

approximately 5 and 3, respectively.