Sistema com Flexibilidade Mecânica

Prévia do material em texto

Problem 2.08PP
In many mechanical positioning systems there is flexibility between one part of the system and 
another. An example is shown in Fig. 1 where there is flexibility of the solar panels. Figure 2 
depicts such a situation, where a force u is applied to the mass M and another mass m is 
connected to it. The coupling between the objects is often modeled by a spring constant k with a 
damping coefficient b, although the actual situation is usually much more complicated than this, 
(a) Write the equations of motion governing this system.
(b) Find the transfer function between the control input u and the output y.
Figure 1 Communications satellite Source: Courtesy Space Systems/Loral (SSL)
Figure 2 Schematic of a system with flexibility
Step-by-step solution
step 1 of 7
Sketch the schematic o f a sjrstem with flexibility.
Step 2 of 7 ^
Sketch the free body diagram of the given figure.
iibc-y)
b ( f - y )
U
Step 3 of 7
> X
K x-y )
Step 4 of 7 ^
From the free body diagrams, we get 
mx = - J t ( x - - i ( x - j^)
= u + ^ (x - + A ( i - J>)
Simplify further.
X H— X H— X -------y -------- y = 0
m m m m
- k A . _ jt A . 1
— X -------- x + V H-------- v + — V = — u
M M M M M
Thus, the equation o f motion governing the system is obtained as
- k ^ b . k A . -
X H— X H— X -------y -------- y = 0
m m m m
- k A . . ^ A . 1
— X -------- x + V H-------- v + — V = — u
M M M M M
Step 5 of 7 ^
(b) Find ^ Limlace tra n sf^ n o f the equations
- k \ . k A . . *X H— X H— X---- y ------y = 0
m m m m
- k A . - it ^ A . 1
M M M M M
s ^ x + - X + - s x - - r - - s r = 0
— X - — sX + + — Y + — ysY = — U
M M M M M
Rewrite in matrix form 
(m ^ + b s + k - (b s + k ) ^0^
[ -(As + Jt) Ms^ + bs + k ) [ Y ; [ u )
Step 6 of 7
Solve using the Cramer’s rule, 
detl
( m ^ + b s + k 0^ 
T - ( t e + t ) u ]
+ As + it - (As + it)
det
7 =
-(A s+it) Afs^+As+it 
+ As + it
{̂ mŝ + As + itJ^Afe ̂+ As + itj - (As + it)^
Step 7 of 7
Simplify further.
y ms ̂+ As + it
^ + bs +k^ (̂ Mŝ +bs + k'j - (As + k ^
+ As + it
U mMs + (« + M')bs + (M + m) ks^