MAE 376, Fall 2019: Lab 3
Spring mass system
A typical suspension system of a vehicle consists of a coil spring and a shock absorber (damper) at each
tire as shown in Figure 1A. This system can be modeled by a series of spring mass dampers as shown
in Figure 1B, where KT is the elastic coefficient of the tire while KS and CS represents elastic and
damping coefficients of the suspension system. MV and MW are the mass of the vehicle and the wheel
respectively. Assume that the automobile is moving with a velocity V on a rough road. In this problem,
the rough road can be modeled by a sinusoidal function with a wave length of 8 m and amplitude of
6 cm.
Spring
Damper
A)
MV
MW
KS CS
KT
8m
12 cm
YR(t): Input
YW(t)
Wheel displacement
YV(t)
Vehicle/passenger
displacement
B)
Elastic Tire
Figure 1: Suspension System on sinusoidal road
Assignment:
A1) Write the equations of motion for the suspension system shown in the figure.
A2) Rewrite the system of ODEs in matrix form and pose an eigenvalue problem, assuming the damping
coefficient CS equal to zero.
A3) Use MATLAB “eig” command to find the natural frequencies of the system for the following
settings: MW = 50 kg, MV = 1100 kg, Ks = 40 kN/m, Kt = 250 kN/m, Cs = 0 Ns/m. Display
the answer in command window.
A4) Use the model of rough road mentioned in the description to calculate the displacement of the
masses Mv and Mw from start (0 sec) to end (8 sec), for a given velocity of V = 10 m/s, as shown
in Figure 2. Plot the calculated displacements (Yv and Yw) and the input function Yr vs. time (0
to 8 sec).
Hint: Use wavelength of the given sinusoidal function and velocity of the vehicle to calculate the frequency.
Use this frequency in the equations of motion to calculate the displacement of masses.
1
8
4
12 c
sec 8/2V sec 8/V sec
Figure 2: Position of the wheel with constant velocity on a rough road at different time instants.
A5) Calculate the maximum displacement of mass Mv for V = 10 m/s and display it in the command
window.
A6) Vary the velocity of vehicle, V , from 0.1 to 100 m/s and plot the maximum displacement of mass
Mv (maximum of Yv) vs. the velocity, V .
A7) From the plot, show the corresponding velocities at which the maximum displacement of Mv is
greater than 1 m.
Bonus: Can this velocity/velocities be calculated from the natural frequency of the system given the model
of the road? If yes, show the relation.
A8) What is the relation between periodic motion of masses Mv and Mw? (Hint: Consider natural
frequencies and mode shapes.) Dispaly your answer in the command window.
Submission Instructions:
A zipped folder named as “Lab3 person#.zip”, containing:
• One pdf showing your analytical derivation of the eigenvalue problem named “Lab3 person#.pdf”.
• A single MATLAB script named “Lab3 person#.m”.
• Output should clearly state the “Natural frequency” of the system, for part A3 and “maximum
displacement” of mass Mv for part A5 and a discussion for part A8.
• All plots should be labeled properly and submitted as “Lab3 person# F1.png” and “Lab3 person# F2.png”.
The first plot should contain displacements Yv and Yw and Yr, vs. time for V = 10 m/s, and the
second plot should contain the maximum displacement, max(Yv), vs. velocity, V = [0.1, 100].
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Practice Question
A spring Mass system shown in Figure 3 can be modeled as a second order ODE equation:
x(t)
f(t)
Figure 3: Mass Spring System
Questions:
Using the free body diagram, write down the ODE equation for the spring mass system shown in Figure 1
1. Find the Non-Homogeneous solution of the displacement x(t), assuming m = 1 kg, k = 25 N/m,
b = 0 Ns/m and f(t) = 0.5sin(t).
2. Plot the displacement of the mass m for time period of t = [0, 10], from the non-homogeneous
solution found in part 1.
3. Repeat part 1 and 2 for: f(t) = 0.5sin(5t). Explain the difference.
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