Student ID #: ____________________ Name: _________________

SEOUL NATIONAL UNIVERSITY

SCHOOL OF MECHANICAL AND AEROSPACE ENGINEERING

SYSTEM ANALYSIS Spring 2008 Midterm Exam SOLUTION Date: April 22, 2008 (tu) Closed book, closed note 10:30~11:45AM

Student ID #: ____________________ Name: _________________

[1] (15 points) Describe followings:

(1) Linear Dynamic Systems

system: A combination of components working together to perform a specific objective

Component: single functioning unit

dynamic systems: y(t) depends on all the inputs u(tau) tau <= t, linear superposition

(2) Mathematical model Mathematical model:

-Differential equations describing (dynamic) behavior of a system -Transfer function

-state equation

dynamic systems: differential equations Static systems: algebraic equation Compromise : Accuracy versus simplicity

(3) Design and analysis

• Analysis : the investigation of the performance of a system whose mathematical model is known.

1. Derive mathematical model.

2. Parameter variations – a number of solutions.

3. Interprets and applies the result to the basic task.

• Design : (system design)

-the process of finding a system that accomplishes a given task.

Problem No (points)

Points

1(15)

2(10)

3(10)

4(15)

5(20)

6(10)

Total(80)

[2] (10 points) Solve following differential equation using Laplace Transformation.

2 4 1

(0) 0, (0) 2

y y y

y y

+ + =

= =

&& &

&

Sol )

( )

²

( )

²

1 1 1 1

( ) 4 4 1 3

1 1 7

( ) cos( 3 ) sin( 3 )

4 4 4 3

t t

Y s s

s s

y t t e⁻ t e⁻

= − + −

+ +

∴ = − +

- 과정과답이다맞으면 10점

- Y(s) 까지만정확히구했으면 5점

- 계산과정에서오류가있으면 -3점

- Cos 함수나 Sin 함수, Exponential 함수형태는맞고계수가틀리면계산오류로적용.

- 함수형태자체가틀리면오답. 즉, Y(s) 만맞았을때 5점

- 식전개가되었고 Y(s)까지구했는데 Y(s) 가틀렸으면 2점. - 틀린 Y(s)로 y(t) 까지구했어도 2 점.

[3] (10 points) Consider following mechanical system:

m=10kg, k=10 N/m.

(1) What is the natural frequency and damping ratio when b=16 Ns/m? (5

점)

0, 1 / sec, 0.8

n 2

b k k b

x x x rad

m m ω m ζ mk

+ + = = = = =

&& &

- 운동 방정식만 맞았으면 1 점.

- 고유 진동수 2 점.

- Damping ratio 3 점.

- 공식이 맞고 계산이 틀리면 -1 점 씩.

(2) “Sketch” and compare the time responses of the system when b=16 Ns/m and b=10 Ns/m. (5

점)

2

0.8 ( 16), 0.5 ( 10)

2 2

1 0.6 0.866

1.25 2

d n d

b b

for b for b

mk mk

ζ ζ

ω ω ζ ω

τ τ

= = = = = =

= − = =

= =

0 1 2 3 4 5 6 7 8 9 10

-0.2 0 0.2 0.4 0.6 0.8 1 1.2

time(s)

magnitude(m)

c=16 c=10

- 그래프 맞게 그리고 설명이 맞으면 5 점 - 그래프 맞게 그리고 설명 없으면 3 점 - 설명은 맞는 데 그래프가 틀리면, 3 점

- 그 외 시스템의 특성을 나타내는 몇 가지 계수를 맞게 구했으면 2 점 (계수를 이용해서 시스템에 대해 설명 했으면 설명이 맞은 걸로 보고 3 점)

- 설명의 일부만 맞고 그래프 틀리면 1 점.

- 설명의 일부만 맞고 그래프 맞으면 4 점

[4] (15 points) Consider a system shown in Figure below. The radius and mass of the cylinder are r and m, respectively. The stiffness of the spring is k. The displacement of the cylinder is defined as x. Assume no slip condition between the cylinder and floor.

(1) Obtain the equation of motion of the cylinder. (5점) Sol)

에너지 보존 법칙 1 ² 1 ² 1 ²

2mx +2Jθ +2kx =const x=rθ을 대입하고 시간에 대해 미분하면 3 3

0 0

2mxx+kxx= →2mx+kx=

3 0

2mx&&+kx=

- 답이 맞으면 5 점. 부분 점수 없음.

- 단, 식 전개 시 단순 계산 실수 -1 점.

(2) Obtain state equation of this system. (5점) Sol )

3 0

2mx&&+kx= 라고 놓고 풀어보면,

1 2 1

: ,

state variable x =x x = =x& &x

^{1 1}

2 2

0 1

2 0

3

x x

x k x

m

⎡ ⎤

⎡ ⎤=⎢ ⎥⎡ ⎤

⎢ ⎥ ⎢− ⎥⎢ ⎥

⎣ ⎦ ⎣ ⎦⎣ ⎦

&

&

- 부분점수없음

- 다만 (1) 에서틀린운동방정식을구하여 (2)에서맞게구했으면 3점

(3) Compute natural frequency and sketch the motion of the cylinder, i.e., sketch the time response of the displacement of the cylinder for the following initial conditions:

(0) 1, (0) 0.

x = x& =

²

2 2

2 3

3 ( ) (0) (0) ( ) 0

2

( ) , ( ) cos 2

2 3 3

n

k m

m s X s sx x kX s

s k

X s x t t

k m

s m

ω =

⎡ − − ⎤+ =

⎣ ⎦

⎛ ⎞

= ⎛ ⎞ = ⎜⎜⎝ ⎟⎟⎠ + ⎜ ⎟

⎝ ⎠

& sketch 는 시간에 따른 x(t)를 그리면 됨.

- 고유진동수 맞으면 1 점, 틀리면 부분 점수 없음.

- X(s) 맞게 구했으면 1 점

- x(t) 맞게 구했으면 2 점 ( 라플라스 변환을 이용하지 않고 구했으면 3 점으로 인정) - 그래프 맞게 그렸으면 1 점

- X(s) 나 x(t) 를 잘못 구했으면 부분점수 없음

- (1)에서 잘못 구한 운동 방정식으로 맞게 풀었으면 3 점 (고유 진동수 포함)

[5] (20 points) Consider a quarter car vehicle suspension model shown in the Figure.

(1) Assuming that z_s, z_u, and z_r are displacements of sprung mass, unsprung mass, and road from a reference, respectively, obtain a state equation with following definitions:

[ _{s u}, _s, _{u r}, _u]^T x= z −z z z& −z z&

[ ,_{s s u}, _{u r}]^T

y= z z&& −z z −z

운동방정식

( ) ( ) ( )

( ) ( )

u u s s u s s u t r u

s s s s u s s u

m z k z z b z z k z z

m z k z z b z z

= − + − + −

= − − − −

1 1

2 2

3 3

4 4

0 1 0 1

0 0

0

0 0 0 1 1

0

s s s

s s s

r

s s t s

u u u u

x k b b x

m m m

x x

x x z

x k b k b x

m m m m

⎡ − ⎤

⎢ ⎥

⎢ ⎥

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥

⎢ ⎥ ⎢ − − ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥

⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ + ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ − ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦

⎣ ⎦ ⎢ ⎢ ⎢ ⎣ − − ⎥ ⎥ ⎥ ⎦ ⎣ ⎦

1 1

2 2

3 3

4

0

1 0 0 0

0 0 1 0

s s s

s s s

k b b

m m m x

y x

y x

y x

⎡ ⎤

⎡ ⎤

⎢ − − ⎥

⎢ ⎥

⎢ ⎥

⎡ ⎤ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ = ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦

⎣ ⎦

-

^정답 10점

-

^{운동방정식까지만맞게풀고답이안나왔으면}3점

-

^{운동방정식맞고중간계산틀린상태에서} state equation 까지다구했으면 5점

-

^{틀린운동방정식으로} state equation까지구했으면 3점

-

^{운동방정식틀리고} state equation 까지구하지못했으면 1점

bs

kt

ks

P ms

mu

z

^u

z

s

z

r

bs

kt

k

s

P ms

m

u

z

^u

z

s

z

r

u

(2) Assume that an actuator is placed between sprung and unsprung masses. In this case, the road displacement is considered as a disturbance, d , to the system and the actuator force is considered as a control input, u , to the system. Obtain a state equation of following form:

u d

x&=Ax+B u+B d

u d

y=Cx+D u+D d

운동방정식

( ) ( ) ( )

( ) ( )

u u s s u s s u t r u

s s s s u s s u

m z k z z b z z k z z u

m z k z z b z z u

= − + − + − −

= − − − − +

1 1

2 2

3 3

4 4

0 1 0 1 0

1 0

0 0

0 0 0 1 0 1

1 0

s s s

s s s s

r

s s t s

u

u u u u

x k b b x

m m m m

x x

u z

x x

x k b k b x

m

m m m m

⎡ − ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ − − ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ + ⎢ ⎥ + ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ − ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦

⎣ ⎦ ⎢ ⎢ ⎢ ⎣ − − ⎥ ⎥ ⎥ ⎦ ⎣ ⎦ ⎢ ⎢ ⎣ − ⎥ ⎥ ⎦

1 1

2 2

3 3

4

0 1

1 0 0 0 0

0 0 1 0 0

s s s

s s s s

k b b

m m m x m

y x

y u

y x

x

⎡ ⎤ ⎡ ⎤

⎡ ⎤

⎢ − − ⎥ ⎢ ⎥

⎢ ⎥

⎢ ⎥ ⎢ ⎥

⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ + ⎢ ⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥

⎣ ⎦

⎢ ⎥ ⎢ ⎥

⎣ ⎦

⎣ ⎦

-

^정답 10점

-

^{운동방정식까지만맞게풀고답이안나왔으면}3점

-

^{운동방정식맞고중간계산틀린상태에서} state equation 까지다구했으면 5점

-

^{틀린운동방정식으로} state equation까지구했으면 3점

-

^{운동방정식틀리고} state equation 까지구하지못했으면 1점

[6] (10 points) For the matlab m-file given below, sketch the result.

clear;

clc;

num=[7];

den=[10 15 60];

t=0 : 0.01 : 10;;

step(num,den,t)

grid on;

title(‘ step response’ ,’fontsize’,15);

xlabel(‘ _{t sec}’ ,’fontsize’,15); ylabel(‘ _y(t)’ ,’fontsize’,15);

Î

2

1 7

( ) 10 15 60

X s = s s s

+ +

이를 Laplace 역변환하면

3 3

4 4

7 7 1 7 1

( ) cos 87 87 sin 87

60 60 4 1740 4

t t

x t = − ⋅e⁻ ⎛⎜⎝ t⎞⎟⎠− ⋅e⁻ ⎛⎜⎝ t⎞⎟⎠ 이를그리면

0 2 4 6 8 10

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

0.16 step response

t sec (sec)

y(t)

- 정답 10점(7/60으로수렴하는지확인) - X(s)는맞으나 x(t)가틀리면 3점

- x(t)는맞지만그래프가틀리면 5점

- x(t) 는맞고그래프는맞지만라벨이나타이틀, grid 없거나틀린경우 7점