Vibration in a system can be a source of problems. For example, the deck on a ship could vibrate due to the engine which represents a forcing function. This system may be simply modelled by a mass, representing the deck, a spring representing the stiffness of the deck and a forcing function,representing the engine, on the other end of the spring. A vibration absorber is a mechanism which can be attached to the deck in order to absorb the energy in the system by vibrating itself with the deck remaining static. The vibration absorber can be modelled by a mass and spring and these are attached to the first mass spring system, as shown in the diagram. The whole system is modelled by a two spring, two mass system with forcing on one end. This question is concerned with modelling the steady state oscillation of the system. Forcing due to engine Spring representing stiffness of deck K Mass of deck Let the deck be represented by the mass m1 with stiffiness k and displacement from equilibrium at time as 1 (t) Let the vibration absorber be represented by the spring of stiffness k2 and mass m2 with displacement from equilibrium at time t given as z2(t) The displacements are given by the second order inhomogeneous simultaneous differential equations: Vibrationabsorbe 2 dt2 2 where p is the angular frequency of the forcing oscillation For, a laboratory model the parameter values are m1-0.47 m2 0.12 k1-12.2 k2 -2.7 The steady state solutions (ie. solutions after the system has settled down) of the equations of motion will be of the form 1 (t)- a(p)sin(pt) and z2(t)- b(p)sin(pt) Use these two solutions as trial solutions and determine the amplitudes of the displacements a(p) and b(p) by substituting these trial functions into the equations of motion. Note that these amplitudes are functions of p. We are interested in the behaviour of the amplitudes ap)and b(p) . Plot these functions in Mathcad and examine the behaviour.