Mathematical Methods and Mechanics

Qustion 5

(a) Find the eigenvalues and the corresponding eigenvectors for the matrix

lz -r -21lo r -41. [-, -r ,.]

O) The eigenvalues of a certain 2 x 2mafrix A me -2 and 3. Let

(17 marks)

B=A2 -2A - 3I.

What are the values of the trace and the determinant of B? Show how you obtained your answers.

(8 marks)

Question 6

Three model springs AB, BC and CD each have stiffiress ft and natural length /o are arrange horizontally as shown in Figure Q6. Two particles of equal firass ra are attached to the springs at B and C and the ends A and D are fixed to tvro points a horizontal distance 3lsapxt. The system is free to move in the horizontal line AD.

Fieufe O6

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

(a) Let xr and xz be the displacements of the particles B and C from their equilibrium positions. Show that the equations of motion of the particles are

ffiir= -2b1 + fuz ffi*z: fur - 2*l2 .

(8 marks)

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

(l l marks)