Question 2 A pendulum consisting of a bar “B” and a mass “C” is suspended from a fixed pivot at “O” as shown in figure 2– a. A light spring is attached to the pendulum at point “P” and is anchored at a fixed point “Q”. When the pendulum is in equilibrium the line OC is at 450 from the vertical and the angle OPQ is 900. Calculate the natural frequency of the pendulum for small oscillations about the equilibrium position. The following data is available: Bar “B” Mass “C” Spring PQ mass 1 kg 6 kg centre of gravity G1 G2 radius of gyration about the centre of gravity 0,1 m 0,025 m spring stiffness 700 N/m

Q1: The system shown has two masses. Beam of mass (Jo#m L² kg.m²) rotates about fixed point (O) and its free end is connected to disk rotates about fixed point (O₂). Consider all connecting links are massless and rigid. Find 1- The displacements of points A, B, and C in addition to the rotations of masses, all in terms of 0. 2- Find the equation of motion (EOM) in terms of 0. 3- What is the natural frequency of the system? L/2 A Jo=m L² k L/2 Joz-m R² full 0₂ R U B C

A pendulum is made up of a spring with a mass attached to the end. The spring is arranged to lie in a straight line, by wrapping the spring around a massless bar. The equilibrium length of the bar is “l”. Using lagrange’s equations, find the equations of motion.

QUESTION 2. A smooth rod with a length of 21 is bent as in Figure 2 and small oscillations are made around the frictionless pin in B. If the period of minor oscillations is denoted by To when B = 0, find the beta angle where the period of minor oscillations is 2To. hand written plz B 7 Figure 2.

Vibration

+ b 3. A tuning fork is an example of a "resonant system", that is, one that has a low damping ratio. A model of a (half) tuning fork is rod of mass m with air drag modeled as a damper connected approximately at its middle, and a torsional spring at the base of the cantilever. Assume the cantilever rotates back and forth about its pivot by a small angle o. Because other forces are much larger, you can reasonably neglect the effect of gravity in this model. The moment of inertia of a cantilevered beam pinned at one end is equal to ml2, where I is its length. (a) What is the natural frequency of this system if m= 0.1 kg and k = 2548 Nm/rad, and 1 = 0.1 m. Please give the value in rad/s and Hz. (b) Find an upper bound on the damping coefficient b that insures that the damping ratio of the tuning fork is no greater than = 0.01. (c) For this damping ratio, what is the damped natural frequency of the system? (d) For this damping ratio, what is the decay rate (the size of the exponent…

Figure below shows a rod with mass M = 100 kg, cross section area A = 10 cm². The rod is supported by 4 springs having stiffnesses of k = 1 MN/m. A tip mass with m = 120 kg is attached to the rod. Assume x is the degree of freedom of the equivalent system. The Young's modulus, E, is not known. If the natural frequency of the equivalent mass-spring system is 40 Hz, determine the Young's modulus, E. Assume the length of the rod is L = 5 m. Find the nearest answer. k ww E, A, M k F @www.ww wint L 195 GPa 179 GPa 116 GPa 45 GPa 236 GPa 000 00

A body weighing 5kg is attached to a very light bar as depicted below. The set-up is free to rotate in the vertical plane about pin support O. The force applied at point C makes the bar oscillate(via small rotation) about the position of equilibrium illustrated below a. Derive the differential equation of motion for the steady state angular displacement (theta) of the bar

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 .A moveable 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.

Problem (1): Derive the equation of motion of the system shown in Figure and find: • The natural frequency of vibration • The damping ratio • The free damped frequency of vibration If m= 10 kg, c = 50 N-s/m, k = 1000 N/m, L= 1 m 44 3k Uniform rigid bar, mass m 2 2