# Compilation for old exam, need advise for solutions.

1. May 15, 2007

### laplaces

1. The problem statement, all variables and given/known data
Information:
The gravity defines the point of equilibrium for the system. The mathematical models you are to find are all movements around the point of equilibrium. The force of gravity is NOT to be included in these models.

Numbers used throughout the exercise:

m1= 300kg m2= 15kg
k2= $$3.0\times 10^5\frac N m$$

Schematical figure for the suspension of the carl:

m1 [kg]: 1/4 of the chassy's(car) mass
k1 [N/m]: springconstant
d [Ns/m]: suspensionconstant
m2 [kg]: mass of the wheel
k2 [N/m]: springconstant of the wheel
u(t) [m] : vertical position of the road
x(t) [m]: the wheels vertical position related to the position of equilibrium
y(t) [m]: chassy's vertical postition related to the position of equilibrium

Simplified model for the car suspension:

In the first part of the exercise we are to focus on a simplified mathematical model of the dynamical system. Assume the wheel has no mass and that it is completly rigid.

1. Use newtons law for the balance of forces, and create a differential equation for the chassy's dynamical movement through the point of equilibrium.

2. The transferfunction (laplace transformation of the differential equation)H1(s) between the signal in u(s) and the signal out y(s) can be written in the form:
$$H_1(s)=\frac {Y(s)} {U(s)}=\frac{b_1\cdot s+1}{a_2\cdot s^2 +a_1\cdot s + 1}$$

Determin the coefficients: $$a_2, a_1 og b_1$$

3. The suspension is to be dimensioned sot that$$\omega_0=10 rad/s og \zeta=0.5$$ determine $$k_1 og d$$. What are the eigenvalues for the system? Determine the bandwith of the system?

4. Insert the transferfunction $$H_1$$(s) in simulink, and simulate the chassy's movement with a leap in u(t) of 1,0 m.

5. Draw the bodediagram for the transferfunction $$H_1$$(s) in the frequencyarea 1-100 rad/s.

6. Assume the suspension is tested with a sinusoidal formed signal with an amplitude of 0,05 m. The contribution on the suspension is then given by u(t)=0.05$$\cdot sin(\omega\cdot t)$$.

7. Insert the transferfunction $$H_1(s)$$ in simulink. and simulate the chassy's movement when the contribution is a sinusoidal function with an amplitude of 0,05m.og . Simulate for the angular frequencies 1, 5, 10 og 20 rad/s. Are the simulation results corresponding with the analysisresults you got in 6.?

8. Now assume the suspension is tested with a periodical square signal with an amplitude of 0,05m. Use a periodical square signal with an equal length and width up and down and find it's fouriertransform.

9.Analyze, using the results from the bodediagram of $$H_1(s)$$, how large proportion of the different harmonics of the squarepulse get through to the chassy's movement at different angular frequencies.

10. Add the transferfunction $$H_1(s)$$ to simulink, and simulate the chassy's movement when the contribution is a square periodical signal with an amplitude of 0,05m. Simulate for the angular frequencies 1, 5 , 10 og 20 rad/s. is this the same results as you got from the analysis you did ini 9?

Part 2

Now you are to include the wheel in the model. It has mass, suspension and it is represented in the figure below with the mass$$m_2$$ and the constant for the suspension $$k_2$$

1. Use Newtons law for balance of force, and model the differential equations for the masses $$m_1 and m_2$$ dynamical movement through the point of equilibrium.

Use the following conditions, contributions and measurement:

$$x_1=y, x_2=\frac{dy}{dt} , x_3=x , x_4=\frac{dx}{dt}$$ , u, and measures $$y=x_1$$ and write the differentialfunction in the matriceform:
$$\frac {d\vec{x}}{dt}= A\cdot \vec{x} + B\cdot \vec{u}$$
$$y=C\cdot \vec{x}$$

Write the content in matrices A, B and C.

make sure that the mathematical model is correct.

2. the transferfunction$$H_2(s)$$ between the signal in U(s) and the signal outY(s) can be written like:

$$H_2(s)=\frac{Y(s)}{U(s)}=\frac{b_1\cdot s +1}{a_4\cdot s^4 + a_3\cdot s^3 + a_2\cdot s^2 + a_1\cdot s + 1}$$

Determin $$a_4, a_3, a_2, a_1 og b_1.$$

To find the transferfunction use the laplacetransformation on the differential equation, and then eliminate X(s). In matlab= H=$$C\cdot (sI-A)^{-1} \cdot B$$

3. Determine the eigenvalues of the system with the values on$$k_1$$ and d that you found in 3. part 1. write the eigenvalues like $$r\cdot e^{j\phi}$$. what is the bandwith of this model?

(In matlab: eig(A) or roots([$$a_4 a_3 a_2 a_1 1$$] or pole(sys)

4. Add the transferfunction $$H_2(s)$$ to simulink, and simulate the chassy's movement with a leap in u(t) of 1,0m. Use the same values on $$k_1$$ and d that you found in 3. part 1. add the step response to the simplified model in the same diagram so that it is easier to compare the stepresponse.

5. Try and change the rigidity of the springstiffness (halve and double it), and simulate the step responses(leave d as you found it in 3. part 1)

6. Try and change the dampingcoefficient (halve and double it),and simulate the step responses ($$k_1$$ should be what you found in 3. part 1)

7. Draw the bodediagram of the transferfunction$$H_2$$(s) in the frequency area 1-10000 rad/s. Use the same values for k1 and d that you found in 3. part 1. Put the bodediagram from the simplified model in the same diagram so it is easier to compare the bodediagrams of the models. Are the bodediagrams correct with focus on the models bandwith?

8. An even more precise model of how the driver experience comfort while driving, is achieved by including that the driver is seated in a carseat with springs. Determine the dimesion of Matrices A, B and C that you get if you model this system on matriceform.

$$\frac {d\vec{x}}{dt}= A\cdot\vec{x} + B\cdot\vec{u}$$
$$y=C\cdot\vec{x}$$

any help will be greatly appreciated.

2. Relevant equations

3. The attempt at a solution
I have only come up with something that could be a solution for 1. in part 1 but I need to make sure that it is correct before I continue further. Now the gravity is not to be included. When they say newtons law of forcebalance I suggest $$\sum{F}=ma$$ and that the start values for y(t), x(t) and y'(t), x'(t) all equal 0. My set up (assuming it acts like a low pass filter with signal in x(t) and signal out y(t)):

$$m\cdot \frac{d^2y}{dt^2}+d\cdot\frac{dy}{dt}+k=m\cdot\frac{d^2x}{dt^t}$$

I need confirmation that this is right, and further help beyond that is much appreciated.

Last edited: May 16, 2007
2. May 16, 2007

### steven10137

you have provided alot of information and many problems, perhaps you would care to ask a more specific question from the piece of work that you are having a lot of trouble understanding ...

steve
- many people may neglect this type of thread otherwise

3. May 16, 2007

### laplaces

You are right.

I should be more specific but the fact is I know I will have a lot of trouble with this assignment, which is why I posted the whole thing To be more specific:

In part 1.1 You see I am supposed to create a model of a differential equation, in the bottom you see my attempt to do so. What I need now is confirmation that my model is the right one, or if it is wrong I could use some advise on how to put it up. you just need to read the top questions for that one

Thanks to whoever helps me out here, it will be greatly appreciated.