Question 1 - Consider the mass m sliding without friction on the circular wire of radius R. The mass is connected by a spring of constant k to the point O, and a force F(t) with constant direction is acting on it. Gravity is included and the spring is at its natural length (un-stretched) when the mass is at point C and θ = 0.
(i) Using Lagrange's equations derive the equation of motion of the mass using θ as the independent variable. Do not assume small angles.
(ii) Prove that θe = 0 is an equilibrium position.
(iii) Neglecting the applied force, linearize the equation of motion close to the equilibrium position θe = 0. Solve analytically the linearized equation for initial conditions θ(0) = 0.1, θ·(0) = -0.05.
Question 2 - In the mechanism below the only mass is m and gravity is included. All springs and dampers are at their natural lengths when θ = 0. Assume that (approximately) the mass moves vertically.
(i) Derive the equation of motion without assuming small angles.
(ii) Linearize the equation of motion for small angles and find the natural frequency and the viscous damping ratio of the linearized system.
Question 3 - Consider the cam - lifter - rocker arm - valve assembly shown below. The cam provides a small motion r(t) to the roller at the end of the lifter. The lifter is modeled by stiffness k2 in parallel with the viscous damping c, and is assumed to be mass-less. The rocker arm is rigid and has mass moment of inertial about the pivot point P. The valve has mass m and is assumed to be rigid. The response of interest is the position y(t) of the valve. The system is at rest when y = x = θ = 0 and r(t) is at its minimum value (where θ is the angle of rotation of the rocker arm); then the two springs and the damper are un-stretched. Assume that the lifter roller is always in contact with the cam and that the valve never comes in contact with the valve seat. Find the equation(s) of motion for this system considering y(t) as one of the dependent variables.
