Mathematical Methods and Mechanics

Question I

Consider the following simultaneous linear algebraic equations,

x1 *6x2*fu3:5 h*3xz*bt=4 2r1+3x2+lut=l

where ft is an unspecified constant.

(a) By using the method of Gauss elimination withpartial pivoting only,rcducethe augmented matrix to upper triangular form.

(8 marks)

O) Hence, identifu the value(s) of ft, for which the equations have

- aunique solution, - no solution.

(4 marks) You are not required tofind the soluti.on of the equatiansfor ony case.

Question 2

A particle l, of mass 2 kg is connected to the left-hand wall at O by amodel spring of stiffrress 20 Nm-' and natur.al length 1.5 m, and on the right by another model damper of damping constant 4 Ns rn-'. The position of the particle at time / is x, as measured from o.

/o = l.5m m=Zkg

Fisure 02

The vibrating system is oscillating horizontally as shown in Figure Q2. The forcing functionflr) is on the displacement at B.

(a) Draw a diagram clearly expressing the forces acting on the particle, the positive rdirection, and define these forces.

(5 marks)