8/19/2019 AE383_Fall2015_HW1

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Middle East Technical University Department of Aerospace Engineering

AE383 Fall 2015 Dr. Ali Türker Kutay

Dilber Derya Kaya

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Homework #1 

Due: Wednesday, 02/12/2015 16:30

1. Consider the spring-mass-damper system shown in the figure below.

a) 

(3 points) Find the response of the system when a sinusoidal force ()  2 sin()   is applied withzero initial conditions (0)   |=  0. You should obtain the () expression manually (withoutusing MATLAB) as if this was an in-class exam.

b)  (3 points) Repeat part a) using a higher frequency sinusoidal force

 ()  2 sin(5)   with the same

amplitude. Then plot the input forces for both parts a) and b) on the same figure in the range from

0 to 10 . Also plot the responses for inputs parts on a separate figure. Observe that eventhough the input forces have the same amplitudes there is a significant difference between the

amplitudes of the responses. Explain why the response amplitudes are different considering the

physics of the system.

c) 

(3 points) Find the response of the system when the system is released from an initial position of

0.2  at an initial speed of |=  0.5 ⁄ . Also find the speed of the mass ( ⁄ ) as afunction of time. You need to find the () and ′() expressions manually without using computer.Then plot () and ′() in the range from 0 to 10 .

2. Refer to the car suspension problem studied in the class.

The behavior of the system is represented by the following differential equations:

̈  (  )  (̇  ̇)  0 ̈  (  )  (̇  ̇)  (  )  (̇  ̇)  0 

where  that represents variations on the road surface is the disturbance input to the system.

a)  (2 points) Find the

2 × 1 transfer function matrix from disturbance input

 to outputs

[

]

.

  

 

 

 () 

1   2 ⁄

 

2   ( ⁄ )⁄  

  250 kg mass of the car / 4  15 kg mass of the wheel assembly  30,000 N/m spring constant of the suspension system  5,000 Ns/m damping constant of the suspension system  75,000 N/m spring constant that represents tire compressibility  1,000 Ns/m damping constant of the tire:  position of the car in m

:  position of the wheel in m

:  coordinate of the road in m

 

 

 

 

 

 

   

 

8/19/2019 AE383_Fall2015_HW1

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Middle East Technical University Department of Aerospace Engineering

AE383 Fall 2015 Dr. Ali Türker Kutay

Dilber Derya Kaya

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b)  (3 points) For the numerical values given, find the response of the car when it hits a curb with 5  height. In other words find the () expression when () is a step input with 0.05  amplitude.You may use MATLAB’s residue function in this problem.  Plot () versus .

c) 

(2 points) Repeat part b) by directly simulating the system using Matlab’s lsim function. Note that

this time you don’t get the () expression, but instead you get numerical values for  at a numberof time points. Plot () versus . Make sure that your plots for parts b) and c) are the same.

d) 

(2 points) Simulate the system one more time with the same disturbance input. This time change

the value of the tire spring constant to   150,000 N/m. This models the case where the tire ismade stiffer by pumping more air into it. Plot () versus  on the same figure with the previousresponse.

e) 

(2 points) Simulate the system one more time with the same disturbance input. Use the original

system parameters except the suspension damping coefficient. Suppose that the oil in the damper

has leaked and the damping coefficient reduced to   500 Ns/m. Plot () versus .

3. (5 points) Consider the car suspension problem in the previous question. Add a second disturbance input

to the system as a force  ()  acting on the car as shown in the figure.Modify the equations of motion given in the previous question to includethis input. Then obtain the transfer function matrix from disturbance inputs

[  ] to outputs [ ]. Finally simulate the system to find the response of  

to a step input force   with a magnitude of 200 . This simulates the casewhere a passenger sits in the car and his weight acts as a step force on the

suspension system. Use   0 for this simulation. Plot () versus .

4. (5 points) Consider the systems with the following transfer functions. Find analytical expressions for the

responses of these systems to unit step inputs (without using Matlab). Plot () versus for  for each ofthese systems. Obtain same result using MATLAB-Simulink and compare the plots ().

a)  ()    ++ b)

 

()    + c)

  ()    +√ + d)

 

()    −+ e)

 

()    −√ + 

5. (5 points) Solve the following differential equation using Laplace Transformation.

̈ 2̇ 10 −  (0)  0  ̇(0)  0 

 

 

 

 

 

 

   

 

 () 

8/19/2019 AE383_Fall2015_HW1

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Middle East Technical University Department of Aerospace Engineering

AE383 Fall 2015 Dr. Ali Türker Kutay

Dilber Derya Kaya

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6. (5 points) A function()() consists of the following zeros, poles, and gain:

Zeros at 1, and 2, poles at 0, 4, 6, and gain 5 

Find the expression for()() 

    using MATLAB. Write your MATLAB code as an m-file with explanations.

7. (5 points) Obtain the partial-fraction expansion of the following function using MATLAB:

()    10( 2)( 4)( 1)( 3)( 5) 

8.(5 points) Obtain the equations of motion for the following system:

9. (5 points) Simplify the block diagram shown in the following figure: