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)). ℓ𝜃𝜃̈+𝑔𝑔𝜃𝜃=−𝑥𝑥̈�𝑚𝑚𝑡𝑡+𝑚𝑚𝑝𝑝�𝑥𝑥̈+𝑏𝑏𝑥𝑥̇+𝑚𝑚𝑝𝑝ℓ𝜃𝜃̈=𝑢𝑢2) The hydraulic actuator system has been linearized as follows (replacing the notations for the deviation by the original variables, i.e. Δ𝑥𝑥̇𝑣𝑣=𝑥𝑥̇𝑣𝑣, Δ𝑣𝑣𝑣𝑣=𝑣𝑣𝑣𝑣, etc.) -Valve dynamics: 𝜏𝜏𝑣𝑣𝑥𝑥̇𝑣𝑣(𝑡𝑡) =−𝑥𝑥𝑣𝑣(𝑡𝑡) +𝐾𝐾𝑣𝑣𝑣𝑣𝑣𝑣(𝑡𝑡)-Flow equation: 𝑞𝑞𝐿𝐿(𝑡𝑡) =1𝜌𝜌𝜌𝜌�𝑃𝑃𝑠𝑠− 𝑝𝑝𝐿𝐿,0𝑥𝑥𝑣𝑣(𝑡𝑡)−𝑥𝑥𝑣𝑣,0 2𝜌𝜌𝜌𝜌�𝑃𝑃
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