(A simple pendulum). Consider the dynamics of the pendulum depicted below, where u denotes an input torque provided by a DC motor. Mg

Consider the damped mass-spring system for mass of 1.5 kg, spring constant 10 N/m, damping 3 kg/s and an oscillating force 4.7 cos(@t) Newtons. That is, 1.5x" + 3x' + 10x = 4.7 cos(@t). What positive angular frequency o leads to maximum practical resonance? W = help (numbers) What is the maximum displacement of the mass in the steady state solution when the we are at practical resonance: C(@) = help

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

III) A relatively well-praised model for a car suspension is depicted in Figure 5: 4 4 Body Mass MW Suspension Mass b₂ ↑" Figure 5 - Model of the suspension of a car. Considering it is a system with two degrees of freedom, WRITE the equations that model the free vibrations of this system.

a) A two-degree-of-freedom coupled pendulum model is shown in Figure Q1a. The parameters of the model are: k=100 N/m; m=10 kg and L=1 m. The equations of motion of the system are given below. J₁ä + (mgL + kĽ² ) 8₁ - kľ² 0₁₂ = 0 J₁₂+(mgL+kĽ²)₂-kĽ² 0₁ = 0 i) Calculate the natural frequencies. ii) Draw the mode shapes. m Figure Q1a. b) In the above system discuss the implications of the following changes in coupling stiffness. i) k is reduced considerably to 0.1 N/m. ii) k is increased considerably to 1x105 N/m. c) Draw a schematic of modal test arrangement of a beam using instrumented impact hammer. d) Why is it necessary to measure force in a modal test?

1. A thin disk attached to the end of an elastic beam has three uncoupled modes of vibration. The longitudinal motion, transverse motion and the torsional oscillations are all kinematically independent. Calculate the following for the system shown. a. Longitudinal stiffness b. Transverse stiffness c. Torsional stiffness mumm 65 cm r= 10 mm E-200 x 10° N/m² G=80 x 10º N/m²

Derive the governing equation of motion of the mass-spring system on an inclined plane as shown in the figure below. Then determine the natural frequency of the given system. Please remember to state all assumptions and include a free-body diagram. ki ww k: m www A

PROBLEM1: Determine the equation of motion of the following systems: 1080 al*k www 0.51 ghiv Vin & Masa Less, Rigid L J Figure 2-Problem 1-2 a3*m

4. A student followed strictly correct procedure in the lab during the experiment of moment of inertia. He measured the mass and diameter of plastic cylinder with value of m = 0.36 kg and D = 0.1 m. The period of system without plastic cylinder is To = 0.667 s. The period of system with plastic cylinder T₁ = 0.92 s. Find: The spring constant K of the system. If a metallic cylinder and a plastic cylinder have same diameter and same thickness, which one has larger moment of inertia? Explain.

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