Mechanical Vibrations v i b r a t
Mechanical Vibrations v i b r a t i o n s Dr. Adnan Dawood Mohammed. (Professor of Mechanical Engineering)
Free vibration of single degree of freedom system v q Introduction i b r a t i o n s . 1
Free vibration of single degree of freedom system v i b r a t i o n s . q Examples
Free vibration of single degree of freedom system v i b r a t i o n s . q Examples
Free vibration of single degree of freedom system v q Equation of motion i Equation of motion b r a t i o Newton's 2 nd law of motion Other methods n s D’Alembert’s principle. Virtual displacement Energy conservation
Free vibration of single degree of freedom system v q Newton's 2 nd law of motion i b r a t Draw the free-body diagram i o n s . Select a suitable coordinate Determine the static equilibrium configuration of the system 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 v q Newton's 2 nd law of motion i b r a t i o n s . � Energy conservation Kinetic energy Potential energy
Free vibration of single degree of freedom system v i b r a t i o n s . q Vertical system
Free vibration of single degree of freedom system v i b r a t i o n s . q. Solution to the equation of motion:
Free vibration of single degree of freedom system v i b r a t i o n s .
Free vibration of single degree of freedom system v i b r a t i o n s .
Free vibration of single degree of freedom system v i b r a t i o n s .
Free vibration of single degree of freedom system v i b r a t i o n s . Graphical representation x(t ) = A cos(ωnt – ϕ )
Free vibration of single degree of freedom system v i b r a t i o n s . Torsional vibration
Free vibration of single degree of freedom system v � Natural Frequency (ωn) : i � It is a system property. It depends, mainly, on the stiffness and the mass of vibrating system. a � It has the units rad. /sec, or cycles/sec. (Hz) t � It is related to the natural period of oscillation (τn) such that, τn = 2π/ωn b r i o n s . and ωn = 2 π fn where fn is the natural frequency in Hz.
Free vibration of single degree of freedom system q. Example 2. 1 v The column of the water tank shown in Fig. i is 90 m high and is made of reinforced b concrete with a tubular cross section of inner diameter 2. 4 m and outer diameter 3 m. r The tank mass equal 3 x 105 kg when filled a with water. By neglecting the mass of the t column and assuming the Young’s modulus of reinforced concrete as 30 Gpa. determine i the following: o � the natural frequency and the natural n period of transverse vibration of the water tank s � 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 v i b Example 2. 1 solution: Initial assumptions: r a t i o n s . 1. the water tank is a point mass 2. the column has a uniform cross section 3. the mass of the column is negligible
Free vibration of single degree of freedom system v i b r a t i o n s . Example 2. 1 solution: a. Calculation of natural frequency: 1. Stiffness: , But: So: 2. Natural frequency :
Free vibration of single degree of freedom system v i b r a Example 2. 1 solution: b. Finding the response: 1. x(t ) = A cos (ωn t - ϕ ) t i o n s . So, x(t ) = 0. 3 cos (0. 9829 t )
Free vibration of single degree of freedom system v i b Example 2. 1 solution: c. Finding the max. velocity: r a t i o n s . Finding the max. acceleration :
Free vibration of single degree of freedom system v i b r a t i o n s . Example 2. 2 :
Free vibration of single degree of freedom system Example 2. 2 : solution
Free vibration of single degree of freedom system v i q. Simple pendulum Governing equation: b r a t i Assume θ is very small o n s Natural frequency (ωn).
Free vibration of single degree of freedom system v i b r a t i o n s . � Solution
Free vibration of single degree of freedom system v i b r a t i o n s . q. 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 v i q. Solution the governing equation is found as: b r a Assume small angle of vibration: t i o So: n s .
Free vibration of single degree of freedom system v i b r a t i o n s . q. Example 2. 5: Q 2. 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
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