FIRST YEAR ENGINEERING Part IA Paper 1: Mechanical Engineering MECHANICAL VIBRATIONS Examples paper 2 Straightforward questions are marked with a + Tripos-standard questions are marked * . Second-order systems: Steady Harmonic Vibrations You are advised wherever possible to arrange differential equations into one of the standard forms [cases (a), (b) and (c)] given on pages 8-13 of the 2000 Mechanics Data Book. Sufficient accuracy is usually obtained by reading from the corresponding graphs, interpolating as necessary. +1. Figure 1 shows a mass m attached to a vertical wall by a spring of stiffness k and a dashpot of rate . If the mass is excited by a force f (t) , and the displacement of the mass from its equilibrium position is y, show that the displacement y is related to the force f(t) by the differential equation m dt2 d2y dy + ky = f(t) . + 2 dt Show that the equation of motion may be written in the form for case (a) given on page 8 of the Mechanics Data Book for harmonic response, namely n ÿ + 25 y + y = x . Write down expressions for the undamped natural frequency On and the damping factor § . How is the input x related to the exciting force f (t) ? State the units of x . When f(t) = F cosot where F is a constant, it is observed that y = Y cos(ot - 0). Use the Data Book to determine the response amplitude Y and the phase lag ø when @ = 0 and o . How do your answers depend on the damping factor § ? 2 y H - m f(t) k Fig. 1 0 C +2. A machine of mass m = 100 kg is anchored to a wall by a spring of stiffness k = 400 N/m and by a dashpot of rate 2 = 80 Ns/m as shown in Fig. 1. Use your results from Q1 to calculate the undamped natural frequency @) and the damping factor & for the machine. The machine is vibrating at an angular frequency of @= 1.8 rad/s under the action of a harmonic force of amplitude F = 10 N. Find the amplitude Y of steady-state vibration and its phase lag relative to the applied force f (t) = F cos@t . If 2 is increased to 400 Ns/m , what is the new value of the damping factor & ? Find the new value of the response amplitude Y under the action of the same force. 1
+3. In a simple AC ammeter, the recorded current ir is related to the input current ij by the differential equation 16 df2 1 d2ir + 4 dt 1 dir + ir = ij . Find the undamped natural frequency On and the damping factor & for oscillations of the instrument. When the input current is given by ij = Ij cos @t , the steady-state recorded current is ir = Ir cos(@t