ME 375 EXAM #1 Tueday February 1, 006 Diviion Adam 11:30 / Savran :30 (circle one) Name Intruction (1) Thi i a cloed book examination, but you are allowed one 8.5x11 crib heet. () You have one hour to work all three problem on the exam. (3) Ue the olution procedure we have dicued: what i given, what are you aked to find, what are your aumption, what i your olution, doe your olution make ene. You mut how all of your work to receive any credit. (4) You mut write neatly and hould ue a logical format to olve the problem. You are encouraged to really think about the problem before you tart to olve them. Pleae write your name in the top right-hand corner of each page. Problem No. 1 (30%) Problem No. (35%) Problem No. 3 (35%) TOTAL (***/100%)
February 1, 006 PROBLEM NO. 1 (30%) You are given the following chematic of a mechanical ytem. A dic of ma M 1, radiu R, and moment of inertia I 1 roll without lipping on a ma M. The pring K and damper C are attached a hown in the figure and the force f(t) i applied to M. Gravity i alo acting downward in the figure. The tranlational motion of M i denoted by x and the angular motion of I 1 i θ. g θ x M 1,I 1 C R f(t) M K No friction (a) Derive the expreion relating the tranlational motion of the dic to x to θ. Motion of dic = Motion of body M = x + Rθ + Motion of relative to M
3 February 1, 006 (b) Draw the free body diagram for the dic M 1 howing all of the appropriate force. M 1 g K(x +Rθ) f 1 f (c) Draw the free body diagram for the ma M howing all appropriate force. Cx M g f 1 f N f(t)
4 February 1, 006 PROBLEM NO. (35%) You are given the following tranfer function: Y() + 1 = F + + () (a) Find the equation of motion relating the force f(t) to the repone yt (). Firt, we clear denominator on both ide of the equation above: ( + + ) = ( + ) Y( ) 1 F( ) then we take the invere Laplace tranform y+ y + y = f + f (b) Find the teady tate repone y (t) for a unit tep input f(t)=1, t>0 ec. We can ue the final value theorem to find the teady tate repone to a unit tep becaue the root of the ytem are: + + = 0 1, = 1± j The final value i then calculated a follow: y t = lim Y( ) ( ) 0 + 1 = lim F( ) 0 + + + 1 1 = = lim 0 + + 1
5 February 1, 006 (c) Find the tranfer function relating the force f(t) to the acceleration yt (). Acceleration i the econd derivative of diplacement o we multiply by ( + 1) A F F () Y() = = () () + + (d) What force f(t) would reult in a teady tate acceleration yt () = 1? The final value theorem can be applied again to determine what F() produce a unit teady tate acceleration: a t = lim A( ) ( ) 0 ( + 1) = lim F( ) 0 + + = 1 The only way thi equation can be true i if F ( ) = 3 which ha the invere Laplace tranform of f( t) = t
6 February 1, 006 PROBLEM NO. 3 (35%) The rigid body below of ma M and rotational ma moment of inertia I i upported by two pring, each with tiffne K, at the point hown on the body. Thi chematic diagram i often ued to model the bounce and pitch motion of an automobile on it upenion. A vertical force, P, act on the body at the center of ma. Aume mall angle of rotation, θ, and ignore the effect of gravity. Anwer the following quetion. M, I x θ a b K P K (a) Draw the free-body diagram for the chematic hown above including all of the neceary force expreed in term of the given parameter and coordinate. The motion of point a and b are needed to compute the deflection in the pring. Thee motion are calculated by adding the motion of the center of ma to the motion of point a and b relative to the center of ma a follow: x b =x +bθ x a =x -aθ Then the free body diagram can be completed: b θ a x K(x -aθ) P K(x +bθ)
7 February 1, 006 (b) Show that the tranfer function between the force, P, and the repone, x and θ, are given by the following expreion when a=b, ( ) P( ) X = 1 Θ( ) and = 0 M + K P( ) Give a phyical explanation for why the econd of thee tranfer function i zero. By the Newton and Euler law, the equation of motion are given by: ( ) Mx + Kx K a b θ = P ( ) θ ( ) I θ + K a + b K a b x = 0 When a = b, thee equation reduce to Mx + Kx = P I θ + a Kθ = 0 o the tranfer function are given by X() 1 Θ() = and = 0 P () M + K P () The econd tranfer function i zero becaue no moment i produced by P around the center of ma when a = b; input force, P, do not produce rotational motion.
8 February 1, 006 (c) Find the TOTAL repone, x and θ, to a unit tep input, P(t)=1 for t>0 with initial condition, x (0) = 0 m and x (0) = 0 m/ and θ ( 0) = 0 rad and θ (0) = 0.5 rad/. Expre the repone in term of the unknown parameter M, I, K, and a. What i different about the natural frequencie of ocillation in the two repone and why? Give a phyical explanation. The equation of motion when a = b were given by, Mx + Kx = P I θ + a Kθ = 0 o the repone are given by the following expreion in the Laplace domain: 1 Mx(0) Mx (0) X() = P() + + M + K M + K M + K Iθ(0) I θ(0) Θ() = + I + ak I + ak which yield the following repone to the given input and initial condition: X M 1 1 D1 D+ D3 1/K () = = + = K M + K M + K M + K 1 1 K x() t = co t K K M 0.5I 1 Θ( ) = = 0.5 I ak + a K + I I a K θ () t = 0.5 in t ak I xt ( ) ha a frequency determined by the two parallel pring and ma M θ (t) ha a frequency determined by the two pring and moment of inertia I