A mass attached to the end of a cantilever beam is subjected to an impulse of 5 Ns, in a direction normal to the axis of the beam and is set in oscillations. If the mass m = 700 kg, the damping ratio, length, and Young's modulus E are respectively 0.3, 3 m, and 7x 1010 N/m2, find the approximate peak response of the mass. The moment of inertia of the beam is 0.6 m4.

For the mechanical system shown, a- Find the equivalent mass, equivalent spring stiffness and equivalent damping coefficient when x (the displacement of the 2 kg block) is used as the generalized coordinate. b- Derive the equation of motion of the system in terms of equivalent system parameters (equivalent mass, equivalent spring stiffness and equivalent damping coefficient). 3000 N/m 2 kg 200 N.s/m /=0.04 kg-m² r = 10 cm 1000 N/m 1 kg I' 400 N.s/m ial

4. Derive the equation of motion of the system depicted below. does the gravity affect the damping ratio and the natural frequency? k T m www g

For the system below, the mass, m is attached to the end of a cantilever beam of length, l, elasticity, E, and moment of inertia, I. Derive the equation of motions and find the natural frequencies in two cases by lagrange energy method    a) k1=k2=0 b) k1≠k2≠0

3.5. A spring-mass is excited by a force F, sin ut. At resonance, the amplitude is measured to be 0.58 cm. At 0.80 resonant frequency, the amplitude is measured to be 0.46 cm. Determine the damping factor of the system.

For the system shown below, the disc of mass M rolls without slip. Assume that the mass M = 11 kg, the damping constant C - 209 Ns/kg and the stiffnesses K, = 9,036 N/m, K2 = 4,855 N/m, the radius of the disc R = 0.4 m and the moment of inertia of the disc about its centre I = MR² What is the damping ratio of the free vibration of this system? www K2 K1 M *No Slip

Pravinbhai

k 0.5m 1m The figure shows a slender rod with a mass of 6 kg. Attached to the rod is a spring of stiffness k = 250 N/m.Amoveable point mass m1 = 1 kg is position at a distance X from the pivot point. You can assume small angle oscillations from the equilibrium position shown in the figure. Calculate the required position, X, of the moveable point mass m1 needed to give the system a natural frequency of 5.3 rad/s. Give your answer to two decimal places.

A concrete part with a mass of 800 kg is hung on a 6 m long steel cable. You determine the spring constant to be 3.924 \times 10 ^ 6 N/m. What is the oscillation period when the steel beam oscillates in the direction of the cable? Provide the result in seconds.

A fixed-fixed steel beam, of length 5 m, width 0.5 m, and thickness 0.1 m, carries an electric motor of mass 75 kg and speed 1200 rpm at its mid-span, as shown in Fig.. A rotating force of magnitude 5000N is developed due to the unbalance in the rotor of the motor. A. Find the amplitude of steady-state vibrations by disregarding the mass of the beam, assuming negligible damping. B. What will be the amplitude if the mass of the beam is considered? C. Discuss that how the amplitude of vibration can be reduced in both cases Fo wt 2.