MMAN2300 Problem Solving Exercise #10

In the system below, as cylinder of mass m and radius r rolls without slipping on the floor and the

block with mass m rolls on a pair of massless wheels. The centre of the cylinder is connected to the

wall at left by a spring with constant k1 and to the block by a spring with constant k2. The position of

the centre of the cylinder is given by x1 and the position of the block is given by x2. The positions are

measured from equilibrium.

(a) Write an expression for the potential energy of the system as a function of x1 and x2

(b) Write an expression for the kinetic energy of the system as a function of and

(c) Write the Lagrangian of the system.

(d) Find the partial derivatives:

,

,

and

(e) Use the results from (d) to write Lagranges equations of motion for the system

(f) Write the results from (e) in matrix form

(g) Let m = 1.5 kg, k1 = 12 N/m and k2 = 5 N/m. Find the natural frequencies of the system

(h) Find the modes of the system

Optional (not for marks):

(i) Find the mass-normalised modes of the system

k1 k2

x1 x2

m m