Question

3. In the diagram shown. The mass M is free to move on frictionless rollers in the x direction and is attached to springs k1 and k2. The shaft and disc are free to rotate on frictionless bearings, without translation. The moment of inertia of the disc is I and it is connected to spring k2 at position r. The disc is subject to an external torque T(t) = Tsin?t k1 = 5 N/m, k2 = 10 N/m, M = 2 kg, r = .050 m. The mass of disc is 4 kg and its diameter is 200 mm. T = 5 Nm, ? = 2.89 rad/s. (a) Derive the equations of motion for the system, you may use Newtonian or LaGrange method. (8 marks) (b) Collect the coefficients of x and ? and write the equation in matrix form. (3 marks) (c) If the system was allowed to oscillate freely what would be the natural frequencies. (5 marks) (d) If the spring k2 and disc were to be used as a vibration absorber for resonant oscillations of mass M and spring k1, what value would the spring k2 need to be. (4 marks) Hint: You will need to determine the effective mass of disc at position r. Assume: r? > x, forces to the right positive and clockwise rotation and moments positive. No damping. I = mr^2 / 2 for disc about polar axis. La Grange Equation. d/dt( dKE/dq_i? ) - dKE/dq_i + dPE/dq_i + dDE/dq_i? = Q_i Where q_i is a generalised co-ordinate. Dot notation signifies differentiation with respect to time.

          3.
In the diagram shown. The mass M is free to move on frictionless rollers in the x direction and is attached to springs k1 and k2. The shaft and disc are free to rotate on frictionless bearings, without translation. The moment of inertia of the disc is I and it is connected to spring k2 at position r.
The disc is subject to an external torque T(t) = Tsin?t
k1 = 5 N/m, k2 = 10 N/m, M = 2 kg, r = .050 m. The mass of disc is 4 kg and its diameter is 200 mm.
T = 5 Nm, ? = 2.89 rad/s.
(a) Derive the equations of motion for the system, you may use Newtonian or LaGrange method. (8 marks)
(b) Collect the coefficients of x and ? and write the equation in matrix form. (3 marks)
(c) If the system was allowed to oscillate freely what would be the natural frequencies. (5 marks)
(d) If the spring k2 and disc were to be used as a vibration absorber for resonant oscillations of mass M and spring k1, what value would the spring k2 need to be. (4 marks)
Hint: You will need to determine the effective mass of disc at position r.
Assume: r? > x, forces to the right positive and clockwise rotation and moments positive. No damping.
I = mr^2 / 2 for disc about polar axis.
La Grange Equation. d/dt( dKE/dq_i? ) - dKE/dq_i + dPE/dq_i + dDE/dq_i? = Q_i
Where q_i is a generalised co-ordinate. Dot notation signifies differentiation with respect to time.

3.
In the diagram shown. The mass M is free to move on frictionless rollers in the x direction and is attached to springs k1 and k2. The shaft and disc are free to rotate on frictionless bearings, without translation. The moment of inertia of the disc is I and it is connected to spring k2 at position r.
The disc is subject to an external torque T(t) = Tsin?t
k1 = 5 N/m, k2 = 10 N/m, M = 2 kg, r = .050 m. The mass of disc is 4 kg and its diameter is 200 mm.
T = 5 Nm, ? = 2.89 rad/s.
(a) Derive the equations of motion for the system, you may use Newtonian or LaGrange method. (8 marks)
(b) Collect the coefficients of x and ? and write the equation in matrix form. (3 marks)
(c) If the system was allowed to oscillate freely what would be the natural frequencies. (5 marks)
(d) If the spring k2 and disc were to be used as a vibration absorber for resonant oscillations of mass M and spring k1, what value would the spring k2 need to be. (4 marks)
Hint: You will need to determine the effective mass of disc at position r.
Assume: r? > x, forces to the right positive and clockwise rotation and moments positive. No damping.
I = mr^2 / 2 for disc about polar axis.
La Grange Equation. d/dt( dKE/dqi? ) - dKE/dqi + dPE/dqi + dDE/dqi? = Qi
Where qi is a generalised co-ordinate. Dot notation signifies differentiation with respect to time.

Added by Brian G.

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