Exam Solutions 31-40

Exam Solutions 31-40 - Question 3 The input of the system...

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Unformatted text preview: Question 3 The input of the system illustrated below is the force F applied to the mass m 1 . F k x 1 x 2 x 3 m 1 m 2 m 3 b 2 b 1 The system is composed of three masses m 1 , m 2 and m 3 , two dampers b 1 , b 2 and a spring k. The energy stored in the spring k is equal to E(Δx) = 1 2 · k 1 · Δx 2 + 1 4 · k 2 · sin 4 (Δx) where Δx = x 1- x 2 . The damping effects can be modeled by the linear relation F b = b · v where for each damper the the accompanying damping coefficient b and velocity v needs to be substituted. 1. Draw a bond-graph of the system considering F as the input force. Answer The Bond Graph of the system is as follows: F F : S e F b 1 C :: E (Δx) F c Δv 1 Δv 1 Δv 1 F 1 F 3 v 1 v 1 v 1 ΔF 1 ΔF 1 ΔF 1 1 1 1 I : m 1 I : m 2 I : m 3 R : b 1 R : b 2 F 2 F b 2 Δv 2 ΔF 2 v 3 v 2 v 2 v 2 2. Annotate the bond-graph and calculate from it the state space differential equation of the form ˙x = f(x , F) describing the dynamics of the system. Answer Inertia 1 F 1 = ˙p 1 v 1 = p 1 m 1 Inertia 2 F 2 = ˙p 2 v 2 = p 2 m 2 Inertia 3 F 3 = ˙p 3 v 3 = p 3 m 3 Resistor 1 F b 1 = b 1 · Δv 1 Resistor 2 F b 2 = b 2 · Δv 2 Spring Δ˙x = Δv 1 F C = k 1 · Δx + k 2 · sin 3 (Δx) · cos(Δx) 1-junctions F = ΔF 1 + F 1 ΔF 1 = F C + F b 1 ΔF 1 = F 2 + ΔF 2 0-junctions v 1 = v 2 + Δv 1 v 2 = v 3 + Δv 2 Solving the equations F 1 = F- ΔF 1 = F- F C- F b 1 ˙p 1 = F- k 1 · Δx- k 2 · sin 3 (Δx) · cos(Δx)- b 1 · Δv 1 F 2 = ΔF 1- ΔF 2 = F C + F b 1- F b 2 ˙p 2 = k 1 · Δx + k 2 · sin 3 (Δx) · cos(Δx) + b 1 · Δv 1- b 2 · Δv 2 F 3 = ΔF 2 = F b 2 ˙p 3 = b 2 · Δv 2 Δ˙x = Δv 1 Δv 1 = v 1- v 2 = p 1 m 1- p 2 m 2 Δv 2 = v 2- v 3 = p 2 m 2- p 3 m 3 Express dynamic equations as a function of the state variables ˙p 1 = F- k 1 · Δx- k 2 · sin 3 (Δx) · cos(Δx)- b 1 · ( p 1 m 1- p 2 m 2 ) ˙p 2 = k 1 · Δx + k 2 · sin 3 (Δx) · cos(Δx) + b 1 · ( p 1 m 1- p 2 m 2 )- b 2 · ( p 2 m 2- p 3 m 3 ) ˙p 3 = b 2 · ( p 2 m 2- p 3 m 3 ) Δ˙x = p 1 m 1- p 2 m 2 Question 4 The following Bode plot of magnitude and phase of a system G(s) 10-4 10-2 10...
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This note was uploaded on 08/13/2010 for the course EE EE 302 taught by Professor Taufik during the Spring '10 term at Cal Poly.

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Exam Solutions 31-40 - Question 3 The input of the system...

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