Mathematical Methods and Mechanics

(d) In each case calculate the eigenvalues for the matrix M. Hence classiS each equilibrium point and sketch the vector fields in the neighbourhood of that point.

(9 marks)

Question 7

Three model springs AB, BC and CD, each of natural length /0, have stiffrress 2k,3k and I respectively. A particle of mass 2m is attached to the springs at B and another particle of mass m, is attached at C as shown in Figure Q7. The end A and D of the springs are fixed to two points which are a horizontal distance316apaft and the system is free to oscillate along a horizontalline AD.

Fisurp O7

You may assume that the only forces acting on the particles in a horizontal direction are those due to the springs.

(a)

(b)

(c)

Write down and describe in vector form, the changes in the spring forces acting on the particles, when the particles at .B and C arc displaced from their equilibrium positions by distances xr and.r2 respectively in the direction away from the fxed poirtt l, for the caSe.r2 > rr > 0'

(3 marks)

Write down the equations of motion of the two particles. (6 marks)

Find and relate the normal mode angular frequencies of the system to the corresponding normal mode displacement ratios.

(12 marks)

The initial positions of the particles are such that .B is f lo to the left of its equilibrium position while C is ] /o to the right of its equilibrium position. The particles are released from rest from these positions. Describe the resultant motion is one of the normal mode, and write down the angular frequency of the motion'

(4 marks)

(d)