Mathematical Methods and Mechanics

Question 6

Two model springs OA and AB have stiffiress 4k and 3& respectively, and equal natural lengttr.s /e. A particle ofmass 2n is attached to the springs atA and another, of mass 3m at 8. The end O of the first spring is fixed and the system is free to oscillate along a horizontal line as shown in the figure.

OAB Figure 06

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

(a)

o)

Let rr and xz be the displacements of the particles ,{ and 3 from their respective equilibrium positions, away from the fixed point O. Show that the equations of motion of the particles are

Zmlir-*7f;,1 +35s, friz= fut- kz.

(8 marks)

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

(12 marks)

The idtial positions of the particles are such that the spnngs OA and,4.B each have length f /e. The particles are released from rest tom these positions. Show that the resultant motion is one of the normal modes, and write down expressions for the displacements of the two particles at time t for this motion.

(5 marks)

{c)