Mathematical Methods and Mechanics

O) By applying Newton's second law to each of the particles in turn, show that the displacem€ilts xr and xc of the particles at Band C from their equilibrium positions satist/ the differential equations

trtin=-Sfua+4fuc tnic:4fua*Sbc

(6 marks)

(c) Write down the dynamic matrix of the system when the mass of each of the particles is 5 kg ana *r" stiffiress of each of the springs,4B and CD is 125 Nm-l. Find the normal modes angular frequencies for this system and the corresponding normal mode displacement ratios.

(8 marks)

(d) Describe the general initial conditions that will give normal mode motion, for each of the two normal modeg in cases where:

(i) the particles are initially stationery; (2 marks)

(ii) the particles are initially at their equilibrium positions. (2 marks)