FIRST YEAR ENGINEERING Part IA Paper 1: Mechanical Engineering MECHANICAL VIBRATIONS Examples paper 3 Straightforward questions are marked with a + Tripos standard questions are marked * . Systems with two or more degrees of freedom Throughout this examples paper, assume that displacements are small and neglect the effects of damping. Free vibration +1. How many degrees of freedom have each of the dynamic systems shown in Fig. 1? (a) Planar double pendulum (b) U-tube manometer (c) Point mass in rigid frame 7 XOX ...... (d) Spring-mounted engine block (e) Planar articulated-vehicle model with pin joint (f) Tacoma Narrows bridge Fig. 1 +2. For the system shown in Fig. 2, show that the equation of free motion can be written in the form M? + Ky = 0 where M = H 0 Im1 ? 0 ? m2 ? k1+k2 -k2 , K = ? -k2 ? and ? k2 ? = May2 HH . y If the displacements in the jth normal mode of free vibration are given by yj = Yj cosajt , deduce that each natural frequency @j and its corresponding normal mode Yj must satisfy [ K - @j2 M ] Yj = 0 . Hence find the natural frequencies and normal modes for the case k1 = k2 = k and m1 = m2 =m . Sketch the two mode shapes. 1
y 1 y2 M m 1 k 1 O 1 k2 O m 2 O O Fig. 2 3. Two rigid discs with moments of inertia J and 2J are mounted on three light elastic shafts of torsional stiffness k, k , and 2k as shown in Fig. 3. The angles of rotations of the discs from their equilibrium positions are 01 and 02 respectively. Show that the equation of free torsional vibration of the system can be written in the form MÖ + K 0 = 0 where ? JO PA FRO 2J HH , K = HA-k 3k and ? ? M = 992k -k 000100 0 = 000 0L . Using the method outlined in Q2 above, find the natural frequencies and normal modes. Sketch the mode shapes. Type in the following Matlab program, or use an equivalent Python program, to check your answers: K=[2 -1 ;- 1 3]; M=[1 0;0 2]; [V,D]=eig(K,M); for n=1:2 mode=V(:,n); mode=mode/mode(1); freq2=D(n,n); disp(sprintf('Mode %i has squared frequency %g and mode [%g, %g]',n,freq2,mode)) end Do you understand the program? "eig" calculates eigenvalues and vectors: type "help eig" in the Matlab command window to see details. Save your program, to use again in question 5. 0 1 N k k 2k J Fig. 3 2J +4. Determine by inspection the normal modes of vibration of the systems shown in Fig. 4. Hence calculate the natural frequencies of each system. Describe the motion of the two masses in Fig. 4(b) for the following sets of initial conditions: (i) y? = y2 =1 and ý1 = y2 = 0; (ii) y1 =1, y2 = 0 and y =