系統動態分析與模擬 Quiz 1
April 19, 2013 1:10-3:00
Close Books / Close Notes
Total 98 Points
1. Please explain or answer the following problems. (32 Points)
You may like to answer them as detailed as possible. However, please carefully arrange
your time.
(a) (12 Pts) In class, we have addressed the state-space representation of a
dynamic system. Please try your best and use your own words to answer the
following questions:
1. What are state variables? For mechanical systems, we usually choose
displacements and velocities as state variables, why?
2. What is a state matrix? What is its important physical meaning for
explaining system dynamics?
3. Please make a brief comparison between state-space representation and
input-output representation of system dynamics.
(b) (5 Pts) Please tell us the definition of analogous system. What is the
advantage to use the concept of analogous system in system dynamics? (4
Points)
(c) (5 Pts) In chapter 2, we have mentioned “the law of displacement”. I must
acknowledge that our introduction is not very clear. Here, please try your
best to write your own explanation as much as possible.
(d) (10 Pts) In order to test your ability of using Matlab/Simulink for solving
system responses (which is important for homework and term projects), please
draw SIMULINK block diagrams for the system shown below.
M 1 
x1  K1 x1  Bx2  ( K 2  K 3 ) x2  M 1 g  f a (t )
with all initial conditions =0.
M 2 
x1  M 2 
x2  Bx2  ( K 2  K 3 ) x2  M 2 g
SOL:
1
系統動態分析與模擬 Quiz 1
April 19, 2013 1:10-3:00
2.
(15 Points) Consider the spring-loaded pendulum system shown in the following figure.
Assume that the spring force acting on the pendulum is zero when the pendulum is
vertical ( = 0). Assume also that the friction involved is negligible and the angle of
oscillation, , is small. Obtain a mathematical model of the system.
2
系統動態分析與模擬 Quiz 1
April 19, 2013 1:10-3:00
3.
(16 Pts) Consider the system shown below, a spring-mass system is driven by a
displacement input u, which is connect with the mass through a damper b. The
displacements y and u are measured from their respective equilibrium positions.
(a) (8 Pts) Please write the equation of motion of the system.
(b) (8 Pts) Please express the equation of motion into a state-space form. (請注意你的直覺答
案是否違反 state-space model 之規定)
Sol:
3
系統動態分析與模擬 Quiz 1
April 19, 2013 1:10-3:00
However, the above expression violates the basic definition of state space form.
one possible choice is shown below:
4.
As a result,
(15 Pts) Consider the system shown below, please
(a) (10 Pts) Draw the free body diagram and find the equation of motions.
(b) (5 Pts) Please find the corresponding state-space representation. The output
is the elongation of each spring.
4
系統動態分析與模擬 Quiz 1
April 19, 2013 1:10-3:00
SOL:
5.
(20 Pts) In the mechanical system shown below, the cable is wrapped around the disk
and does not slip or stretch. The input is the foce fa(t), and the output is the
displacement x.
The springs are deflected when =x=0.
(a) (10 Pts) Please draw free body diagrams and write the equation of motions.
(b) (10 Pts) Please find the input-output differential equation relating output: x and
input: fa(t). Please using K1=K2=K3=B1=B2=M=J=R=1 for performing
computation.
5
系統動態分析與模擬 Quiz 1
April 19, 2013 1:10-3:00
SOL:
6