Mathematical Methods and Mechanics

Question 5

Consider the non-linear system of differential equations

f+x+t *+u+y+l

(a) Find the two equilibrium points of the system. (8 marks)

(b) For each of the equilibrium points (Xo, Yo), define the excesses a and v by x = u * Xo and y = v + 16 and use these to find a linear approximation u : M u for the system of equations near the equilibrium point where u: (zr, v)r.

(9 marks)

(c) In each case find the eigenvalues for the matrix M. Hence classify each equilibrium point.

(8 marks)

Question 6

Three model springs AB, BC and CD have stiffness k, 4k and k respectively, and equal natural lengths /p. Particles of equal mass tn are attached to the springs at B and C, and the ends A and D are fixed to two points a horizontal distance 6ls apart as shown in Figure Q6. The question is concerned with the longitudinal vibrations of the system.

6Io

(a)

Fieure 06

Suppose that when the particles are in equilibrium, the length of the spring AB is x.o. The length of the spring CD when the particles are in equilibrium will also be xro.

(i) Draw a force diagram for the forces acting on each particle. (2 marks)

(ii) Write down the spring forces H1 and H2 acting on the mass at B, in terms of x., and /0, and use a unit vector to speciff its direction.

(2 marks)

(iiD By considering the spring forces acting on the mass at B, find in terms of /6 alone, the lengths of the three springs when the particle is in equilibrium.

(3 marks)

x= i=