PROBLEM 1: (30%) The schematic shown below represents a pulley-driven machine with a flexible support. The three coordinates shown are absolute coordinates drawn with respect to the static equilibrium position of the system. Assume there is no slip between the pulley and cable. The cables are also inextensible and are always in tension. θ x 2 x 1 (a) Derive the algebraic constraints between the three coordinates given. How many independent coordinates are there in this system? (b) Draw the free body diagrams of the pulley and the mass M 2. (c) Derive the differential equation of motion using θ as the independent coordinate. Given: We are given the drawing above and the assumptions associated with the schematic diagram. Find: (a) We begin by deriving the constraint equations relating the three coordinates defined in the schematic diagram. x 1 θ R Constraints (2): x 1 = Rθ x 2 = 2x 1 x 2 M 2
2 (b) The free body diagrams are given below. We need not include gravity in the free body diagrams because gravity has been taken into account already in the definition of the coordinates, which are all drawn with respect to the equilibrium position of the system. x 1 T L R θ CM K 1 x 1 O T R T L M 2 K 2 x 2 x 2 (c) Euler s equation can then be applied about point O, which is inside the pulley at the point of contact tangency with the cable, and Newton s second law can be applied to M 2 in the positive vertical direction to minimize the number of equations needed. M = I θ O O 2 ( ICM MR ) = + ( KxR 2RT ) = 1 1 + + FM = M 2 2x2 = T K x We must then eliminate two of the coordinates in favor of the selected independent coordinate, which we selected to be θ: L 2 2 θ 2 2 ( ICM + MR ) θ = ( K1R θ + 2RTL ) 2MR θ = T+ 2KRθ 2 L 2 Then we substitute the second (force) equation above into the first (moment) equation as follows: 2 2 2 ( ICM MR R M2) θ R ( K1 K2) + + 4 + + 4 θ = 0 L At this point, we check our final equation of motion to make sure it is reasonable. The inertia coefficient is positive; the stiffness coefficient is also positive. The units are also correct.
3 PROBLEM 2: (40%) The transfer function between the Laplace transform of the input U(s) and the Laplace transform of the output Y(s) of a system is given by the expression: H(s) = ( )( s + 9) Answer the following questions. (a) For zero initial conditions, calculate the response displacement y(t) to an impulse force u(t)=δ(t) using Laplace transform techniques. (b) Show that the differential equation relating u(t) to y(t) is given by, y + 10y + 9y = u + 3u (c) Calculate the poles and zeros of the transfer function H(s) and explain how the poles relate to the response you obtained in part (a). (d) For a step input u(t)=10u s (t), calculate the steady-steady response y ss. (e) Derive the transfer function between the Laplace transform of the input U(s) and the Laplace transform of the acceleration A(s). (a) For zero initial conditions, calculate the response y(t) to an impulse u(t)=δ(t). The Laplace transform of δ(t) is 1 and the relationship between the output and the input is given by Y(s)=H(s)U(s) for zero initial conditions. The response Y(s) can then be written as follows: Y(s) = H(s)U(s) Y(s) = ( )( s + 9) 1 Then we must apply partial fraction expansion techniques to expand Y(s) as follows,
4 s+ 3 A B = + s+ 1 s+ 9 s+ 1 s+ 9 ( )( ) ( ) ( ) s+ 3= A s+ 9 + B s+ 1 1 = A+ B A= 1 B 3= 9A+ B 3= 9 9B+ B= 9 8B B= 6/8= 3/4, A=+ 1/4 1/4 3/4 Y() s = + s+ 1 s+ 9 Then we can apply the inverse Laplace transform tables to this expression to calculate y(t): 1 t 3 yt () = e + e 4 4 (b) Show that the differential equation relating u(t) to y(t) is given by, y+ 10y + 9y = u + 3u 9t We can start with the differential equation and apply the Laplace transform to both sides, or we can start with the transfer function, clear denominators on both sides, and then apply the inverse Laplace transforms. We choose to use the latter method. The solution process is given below: Y() s s+ 3 = U() s s 1 s 9 ( + )( + ) 2 ( + + ) = ( + ) s 10s 9 Y( s) s 3 U( s) y+ 10y + 9y = u + 3u (c) Calculate the poles and zeros of the transfer function H(s) and explain how the poles relate to the response you obtained in part (a). The poles of the transfer function are found by setting the denominator equal to zero and the zeros are found by setting the numerator equal to zero: ( ) ( s + 9)= 0 s 1 = 1 Two poles s 2 = 9
5 = 0 s = 3 One zero The poles determine the form of the response in part (a) in that there are two first-order exponentially decaying terms in y(t) corresponding to the two real negative poles. (d) For a step input u(t)=10u s (t), calculate the steady-steady response displacement y ss. The steady-state response to a step input can be calculated using the final value theorem. The Laplace transform of the response to this step input is given by: Y(s) = H(s)U(s) Y(s) = ( )( s + 9) Then the final value theorem can be applied because there is only one pole at the origin and both of the other poles are in the left plane: = 10 s yss lims 0 s s s s s 0 10 ( + 1)( + 9) = lim 10 30 10 = = 9 3 ( s+ 1)( s+ 9) (e) Derive the transfer function between the Laplace transform of the input force U(s) and the Laplace transform of the response acceleration A(s). Acceleration is the second derivative of displacement; therefore, the Laplace transform of A(s) is given by s 2 Y(s). With this formula for A(s), the transfer function between U(s) and A(s) is given by: ( ) ( )( s + 9) A(s) U(s) = s2 Y(s) U(s) = s2
6 PROBLEM 3: (30%) Consider the schematic diagram shown below that represents a model for an oil-drilling rig. The input rotation θ in (t) is used to turn a drill shaft of length L with a drill head of rotary inertia I h. The drill head turns against soil that produces a viscous torque characterized by the coefficient B. The polar area moment of the shaft is J, and the shear modulus is G. The output rotation of the drill head is given by θ out (t). Answer the following questions: (a) Calculate the rotary stiffness K of the drill shaft given that the rotational deflection of a shaft of length L, polar area moment J, and shear modulus G due to a torque T is TL/JG. (b) Draw the free body diagram of the drill head. (c) Derive the differential equation of motion relating the input rotation to the output rotation. (a) Calculate the rotary stiffness K of the drill shaft given that the rotational deflection of a shaft of length L, polar area moment of inertia J, and shear modulus G due to a torque T is TL/JG. By rearranging the formula that was provided, the rotary stiffness can be calculated: θ = TL JG T θ = JG L = K (b) Draw the free body diagram of the drill head. The free body diagram for the drill head is shown below.
7 (c) Derive the differential equation of motion relating the input rotation to the output rotation. By writing down Euler s equation for the drill head, the equation of motion can be derived: M = I =+ K( ) B Head JG JG I hθout + B θout + θout = θin L L θ θ θ θ h out in out out