Mechanical Vibrations Dr. Adnan Dawood Mohammed (Professor of Mechanical Engineering)
Free vibration of single degree of freedom system vibrations . Introduction 1
Free vibration of single degree of freedom system vibrations . Examples
Free vibration of single degree of freedom system vibrations . Examples
Free vibration of single degree of freedom system vibrations . Equation of motion Equation of motion Newton's 2nd law of motion Other methods D’Alembert’s principle Virtual displacement Energy conservation
Free vibration of single degree of freedom system vibrations . Newton's 2nd law of motion Select a suitable coordinate Determine the static equilibrium configuration of the system Draw the free-body diagram Apply Newton s second law of motion The rate of change of momentum of a mass is equal to the force acting on it.
Free vibration of single degree of freedom system vibrations . Newton's 2nd law of motion Energy conservation Potential energy Kinetic energy
Free vibration of single degree of freedom system vibrations . Vertical system
Free vibration of single degree of freedom system vibrations . Solution to the equation of motion:
Free vibration of single degree of freedom system vibrations .
Free vibration of single degree of freedom system vibrations .
Free vibration of single degree of freedom system vibrations .
Free vibration of single degree of freedom system vibrations . Graphical representation x(t ) = A cos(ωnt – ϕ )
Free vibration of single degree of freedom system vibrations . Torsional vibration
Free vibration of single degree of freedom system vibrations . Natural Frequency (ωn) : It is a system property. It depends, mainly, on the stiffness and the mass of vibrating system. It has the units rad./sec, or cycles/sec. (Hz) It is related to the natural period of oscillation (τn) such that, τn = 2π/ωn and ωn = 2 π fn where fn is the natural frequency in Hz.
Free vibration of single degree of freedom system vibrations . Example 2.1 The column of the water tank shown in Fig. is 90m high and is made of reinforced concrete with a tubular cross section of inner diameter 2.4m and outer diameter 3m. The tank mass equal 3 x 105 kg when filled with water. By neglecting the mass of the column and assuming the Young’s modulus of reinforced concrete as 30 Gpa. determine the following: the natural frequency and the natural period of transverse vibration of the water tank the vibration response of the water tank due to an initial transverse displacement of tank of 0.3 m and zero initial velocity. the maximum values of the velocity and acceleration experienced by the tank.
Free vibration of single degree of freedom system vibrations . Example 2.1 solution: Initial assumptions: the water tank is a point mass the column has a uniform cross section the mass of the column is negligible
Free vibration of single degree of freedom system vibrations . Example 2.1 solution: a. Calculation of natural frequency: 1. Stiffness: , But: So: 2. Natural frequency :
Free vibration of single degree of freedom system vibrations . Example 2.1 solution: b. Finding the response: 1. x(t ) = A cos (ωn t - ϕ ) So, x(t ) = 0.3 cos (0.9829 t )
Free vibration of single degree of freedom system vibrations . Example 2.1 solution: c. Finding the max. velocity: Finding the max. acceleration :
Free vibration of single degree of freedom system vibrations . Example 2.2 :
Free vibration of single degree of freedom system Example 2.2 : solution
Free vibration of single degree of freedom system vibrations . Simple pendulum Governing equation: Assume θ is very small Natural frequency (ωn)
Free vibration of single degree of freedom system vibrations . Solution
Free vibration of single degree of freedom system vibrations . Example 2.3 Any rigid body pivoted at a point other than its center of mass will oscillate about the pivot point under its own gravitational force. Such a system is known as a compound pendulum (as shown). Find the natural frequency of such a system.
Free vibration of single degree of freedom system vibrations . Solution the governing equation is found as: Assume small angle of vibration: So:
Free vibration of single degree of freedom system vibrations . Example 2.5: Q2.45 Draw the free-body diagram and derive the equation of motion using Newton s second law of motion for each of the systems shown in Fig