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# Mechanical Systems

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```									MECH 529 – Laboratory 5

“Mechanical Systems Simulation”

Introduction:
This week we got started talking about the equations of mechanical systems. This
lab and homework you will investigate the effect of the selection of damping and spring
constants on a vehicle suspension system. The model for the vehicle suspension is a
“bicycle” type model as shown and derived on p325-327 of Palm.

Learning Objectives:
1) Techniques in larger scale and nonlinear modeling of mechanical systems in
2) Use 2nd order systems analogies to understand system response

The first step to modeling the suspension system is to model the shape of the road.
We would like to model a “washboard” road to test the performance of the suspension
system under off-road conditions. Because we are using a bicycle model of the suspension,
a speed bump will hit the second wheel a few moments after it hits the first wheel. In
Simulink we can model the road with the upper half of a sine wave and we can model the
time delay between the first and last axle with a transport delay. An example of this model
is shown in Figure 1 and its response is shown in Figure 2.

Sine Wave
max                          3
y2
MinMax
0

Constant
du/dt             4
y2dot
Derivative

1
y1
Transport
Delay

du/dt                    2
y1dot
Derivative1   Transport
Delay1

Figure 1. Model a half-sinusoid road. Outputs of the model include
the wheel displacements and velocities at each axle

1
1.2       Road Displacement at Wheel 1, y1
Road Displacement at Wheel 2, y2
1

0.8
Distance, m
0.6

0.4

0.2

0
0   2              4        6             8   10
Time, sec
Figure 2. Displacement outputs of the road model, derivatives
of road displacement are not shown here
For the next step of tuning the suspension model, we will use a step function for the
road shape. With a step height of 0.1 m, this simulates driving up a 10 cm curb. To make
the simulation more modular, place the road model into a subsystem by selecting the
blocks of the road model, right clicking and selecting “Create Subsystem.”

The equations of motion of the vehicle suspension are derived on p325-327 of Palm.
The technique for modeling this system is the same as we used in previous weeks. First, set
up the integrators that relate the state variables (   , and  ) to their derivatives. Label
the input and output of the integrators, as shown in Figure 3

1                      1
s                      s
out theta dot           out theta

1                      1
s                      s
in xddot                in xdot
out xdot                 out x

Figure 3. Simulink™ diagram relating state variables to their derivatives

The next step is to put the ODEs in explicit form so that
and                               . Then construct the equation for                          and   and connect the

2
signal that represents and back to the input of the integrators above. In previous
exercises, we did that by physically constructing the equations for and out of the low
level math blocks, such as add, subtract and multiply. This time we will use Simulink’s™
multiplexer block and function block to avoid making spaghetti.
The multiplexer block allows us to make a vector out of a number of time signals.
For this problem, the equations 4.8-1 and 4.8-2 in the book are a function of 8 variables.
Putting the time signals of those 8 variables into the Mux block saves them as a vector. The
output vector of the Mux block is a vector u. We can get to the signal number 4 in the
multiplexed signal by referring to u(4) in the function block. This allows our Simulink™
diagram to be much cleaner. In my example solution, I set the inputs of the multiplexer in
order so that                            .
To construct equations 4.8-1 and 4.8-2 from the book, set up the constants that you
will need in the MATLAB™ workspace.

>> c1 = 100; c2 = 100; L1 = 2; L2 = 2.5; k1 = 10000; k2 = 10000;
>> m=2000; IG = 2000;

To simplify the construction of the equations, I broke them up into terms. For
example term 1 of equation 4.8-1 is              .

>> (c2*L2^2 + c1*L1^2)*u(5)

This equation is entered into the Fcn block, which can be found under the Functions
and Tables Library. The output of this block is summed with the other terms of equation
4.8-1 to get an equation for . Place this equation into a subsystem by selecting a number
of blocks, right clicking and selecting “Create Subsystem.”
1                 f(u)
Mux of
Equation 4.8.1
Variables
Term 1

f(u)

Equation 4.8.1
Term 2

f(u)                   -K-        1
Theta DDot
Equation 4.8.1              1/IG
Term 3

f(u)

Equation 4.8.1
Term 4

f(u)

Equation 4.8.1
Term 5

Figure 4. Subsystem for Equation 4.8-1
The subsystem for equation 4.8-1 and 4.8-2 are similar enough that you can copy the first
subsystem to get a good start on writing the terms of equation 4.8-2. Attach Scopes and the To
Workspace blocks to output some relevant signals.

3
y1

y 1dot

y2

y 2dot
Mux of Variables   Theta DDot

Equation 4.8.1
1                  1
s                  s
out theta dot        out theta
Mux of Variables      x DDot

Equation 4.8.2

1                  1
s                  s
in xddot           in xdot
out xdot            out x

Figure 5. Simulink block diagram of the vehicle suspension model

Task 3 – Choosing damping coefficients to achieve ride quality
Although this system is a multi-degree of freedom system (x and ) we will assume
that all axles have the same damping constants. Choose a damping constant so that when
you go over a small step you get approximately a second order response with an equivalent
damping ratio of 0.4. To plot a second order response with damping ratio of 0.4, run the
following piece of code in the MATLAB command window.

>>   s = tf('s');
>>   zeta = 0.4
>>   wn = 1
>>   sys = 1/(s^2+2*zeta*wn*s + wn^2)
>>   step(sys)

4
Step Response
1.4

1.2

1

0.8

Amplitude
0.6

0.4

0.2

0
0   5                   10   15
Time (sec)

Figure 6. Example 2nd order response with damping ratio of 0.4

1) Complete this model by adding the weight of the vehicle to the equations in
Palm, and determine the approximate value of the suspension damping
constants that provides an approximate damping ratio of 0.4. Plot and
label the vehicle suspension response to a repeating washboard road.
2) Use Simulink™ to calculate the maximum force on each of the suspension
components as it goes over the washboard road. Plot and compare the
force exerted by the dampers to the force in the springs.
3) What is the physical meaning of when the force in the spring/damper
systems changes signs? Find a road condition that makes the force in the
spring/damper systems go negative. Think “Dukes of Hazzard”.

Writeup due 2/26 at the beginning of Laboratory.

Fully document your homework solutions using this handout as a template. All
problem statements must be copied to the solutions. All diagrams and plots must be
labeled with units and symbols, and must be captioned.

5

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