hw4_sol - MEEN 364 Fall 2004 Homework Set 4 Due September...

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MEEN 364 Homework Set 4 Fall 2004 September 30, 2004 Homework Set 4 – Due September 30, 2004 @ 5:00 PM 1. a) There are two rigid bodies in the system. The system has two degrees of freedom, represented by the linear displacements of mass m 1 , x ( t ), and mass m 2 , y ( t ). Kinetics Stage The free body diagram of the masses m 1 and m 2 is shown bellow. If we consider the coordinate system fixed to the inclined plane, we will see the all the forces in y-direction, perpendicular to x , balance each other. Applying Newton’s Law to both masses, and shortening x(t) and y(t) as as x, y we get: F(t) k 1 (x - y) b 1 (x ˙ - y ˙ ) = m 1 x ¨ m 1 x ¨ + b 1 x ˙ + k 1 x = F(t) + b 1 y ˙ + k 1 y ( 1 ) k 1 (x - y) + b 1 (x ˙ - y ˙ ) k 2 y b 2 y ˙ = m 2 y m 2 y + ( k 1 + k 2 ) y + (b 1 + b 2 ) y ˙ = k 1 x + b 1 x ˙ ( 2 ) b) State-space Representation Let the states of the system be defined as 12 34 x xx y x = = = y = && 13 24 x = = x ( 3 ) Taking those values into (1) and (2), we get: m 1 x ˙ 3 + b 1 x 3 + k 1 x 1 = F(t) + b 1 x 4 + k 1 x 2 x ˙ 3 = 1 m 1 F(t) k 1 m 1 x 1 + k 1 m 1 x 2 b 1 m 1 x 3 + b 1 m 1 x 4 ( 4 ) x(t) y(t) m 1 k 1 (x(t) - y(t)) m 2 b 1 (x ˙ (t) - y ˙ (t)) F(t) k 2 y(t) k 1 (x(t) - y(t)) b 1 (x ˙ (t) - y ˙ (t)) b 2 y ˙ (t) 1
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MEEN 364 Homework Set 4 Fall 2004 September 30, 2004 m 2 x ˙ 4 + ( k 1 + k 2 ) x 2 + (b 1 + b 2 ) x 4 = k 1 x 1 + b 1 x 3 x ˙ 4 = k 1 m 2 x 1 k 1 + k 2 m 2 x 2 + b 1 m 2 x 3 b 1 + b 2 m 2 x 4 ( 5 ) Now we express (3), (4) and (5) in matrix form: 11 22 33 1 44 2 2 0010 0 0001 0 () 1 0 xx kk bb Ft mm m k b mmmm ⎡⎤ ⎢⎥ −− =⋅ ++ ⎣⎦ & & & & + And the output is: [] 1 2 3 4 0100 x x y x x 2
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MEEN 364 Homework Set 4 Fall 2004 September 30, 2004 2. Kinematics stage The given system has two degrees of freedom, represented by the angular displacement of the pulley, ‘ θ ’, and the translational displacement of the block, ‘x’. The angular velocity and the angular acceleration will be and , and translational velocity and translational acceleration will be and . This completes the kinematics stage. Assume that x is larger than r θ . θ & & & x & x & & I, r Kinetics stage Free body diagram of the pulley is shown in the fig.2 (a) Writing the Newton’s law of motion, we get.
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hw4_sol - MEEN 364 Fall 2004 Homework Set 4 Due September...

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