T3 - ME360 Winter 2011 Tutorial#3 2 Representations of...

ME360 Winter 2011 Tutorial #3
2. Representations of Dynamic Systems (Transfer Function and Block Diagram)
1) Continuing from Tutorial 2, write down the Laplace transform of the following linearized equations
and get the transfer function from the input 𝑢𝑢 ( 𝑡𝑡 ) (i.e. 𝑈𝑈 ( 𝑠𝑠 ) ) to the output 𝜃𝜃 ( 𝑡𝑡 ) (i.e. Θ ( 𝑠𝑠 ) ).
ℓ𝜃𝜃
̈
+ 𝑔𝑔𝜃𝜃 = −𝑥𝑥̈
�𝑚𝑚
𝑡𝑡
+ 𝑚𝑚
𝑝𝑝
�𝑥𝑥̈ + 𝑏𝑏𝑥𝑥̇ + 𝑚𝑚
𝑝𝑝
ℓ𝜃𝜃
̈
= 𝑢𝑢
2) The hydraulic actuator system has been linearized as follows (replacing the notations for the
deviation by the original variables, i.e. Δ𝑥𝑥̇
𝑣𝑣
= 𝑥𝑥̇
𝑣𝑣
, Δ𝑣𝑣
𝑣𝑣
= 𝑣𝑣
𝑣𝑣
, etc.)
- Valve dynamics:
𝜏𝜏
𝑣𝑣
𝑥𝑥̇
𝑣𝑣
(
𝑡𝑡
) =
−𝑥𝑥
𝑣𝑣
(
𝑡𝑡
) +
𝐾𝐾
𝑣𝑣
𝑣𝑣
𝑣𝑣
( 𝑡𝑡 )
- Flow equation:
𝑞𝑞
𝐿𝐿
( 𝑡𝑡 ) =
1
𝜌𝜌𝜌𝜌
�𝑃𝑃
𝑠𝑠
− 𝑝𝑝
𝐿𝐿 ,0
𝑥𝑥
𝑣𝑣
( 𝑡𝑡 ) −
𝑥𝑥
𝑣𝑣 ,0
2 𝜌𝜌𝜌𝜌
�
𝑃𝑃