Review of Mechanical Vibrations

Contents

Basic concepts

Free vibration of single DOF with and without damping

Forced vibration of single DOF

Force and motion isolation

2 DOF system

Natural frequency

Basic Concepts

Degree of freedom

Contd..

Simple Harmonic Motion

Q.The response of a system is given by:-

Determine (a) Amplitude (b) Period (c) Frequency(d) Phase angle (e) Response in the abovestated form

Contd..

Types of Vibration:-

SDOF(Single degree of freedom) & MDOF(Multiple DOF)

Undamped, damped and forced

Harmonic and transient

Linear and non-linear

Deterministic and random

Contd..

Some important definitions:-Periodic motion

Time period

Frequency

SHM

Amplitude

Free vibrations

Forced vibrations

Natural frequency

Resonance

Damping

Degree of Freedom

Contd..

Q. A body is subjected to two harmonic motions as given :-

What extra harmonic motion should be given to the body to bring it to static equillibrium?

Free vibration of single DOF without damping

Characteristic equation:-

Q. The bell crank arm is pivoted at O and has mass moment of inertia I.Find natural frequencyof the system.

Contd..

Soln:-Find K.E and P.EE=K.E.+P.E.Apply dE/dt=0

Q. Find natural frequency of the system

Contd..

Equivalent springs:-In parallelIn series

Q. Find equivalent spring stiffness for the case:-

Free vibration of single DOF with damping

Characteristic equation:-

Contd..

Logarithmic decrement:-

Contd..

Q. A damped system has following elements:-mass=4kg; k=1 kN/m; C=40 N-s/mDetermine:-a.) Damping factorb.) Natural frequecy of damped oscillationsc.) Logarithmic decrementd.) No of cycles after which the original amplitude is reduced to 20%

Contd..

Sol:-

Contd..

Q. A machine of mass 75 kg is mounted on springs and is fitted with a dashpot to damp out vibrations. There are three springs each of stiffness 10 N/mm and it is found that the amplitude of vibration diminishes from 38.4 mm to 6.4 mm in two complete oscillations. Assuming that the damping force varies as the velocity, determine : 1. the resistance of the dashpot at unit velocity 2. the ratio of the frequency of the damped vibration to the frequency of the undamped vibration 3. the periodic time of the damped vibration

Forced Vibrations of single DOF

Characteristic Equation:-

k

k

Contd..

In general xc dies out very fast, so solution is given by xp. So,

Graphical Solution:-

Contd..

Magnification Factor:-The ratio of the amplitude of the steady-state response to the static deflection under the action of force F0 is known as magnification factor (MF).

Contd..

Salient point about graph of MFvs ratio of frequencyAt

MF cannot reach infinity owing to friction which dampens the vibrations

Contd..

Q. A machine part having a mass of 2.5 kg vibrates in a viscous medium. A harmonic exciting force of 30 N acts on the part and causes a resonant amplitude of 14 mm with a period of 0.22 second. Find the damping coefficient. If the frequency of the exciting force is changed to 4 Hz, determine the increase in the amplitude of the forced vibrations upon the removal of the damper.

Contd..

Q. A single-cylinder vertical diesel engine has a mass of 400 kg and is mounted on a steel chassis frame. The static deflection owing to the weight of the chassis is 2.4 mm. The reciprocating masses of the engine amounts to 18 kg and the stroke of the engine is 160 mm. A dashpot with a damping coefficient of 2 N/mm/s is also used to dampen the vibrations. In the steady-state of the vibrations, determine:(i) the amplitude of the vibrations if the driving shaft rotates at 500 rpm(ii) the speed of the driving shaft when the resonance occurs.

Force and Motion Isolation

A major part of 4th Module, so will be discussed in detail over there.

2 Degrees of Freedom

Natural Frequency

Already studied in this mobule

Natural Frequency

Already studied in this mobule

Practice Problems

Q.Find the natural frequency for the given systems:-

Q. A 3 kg object is attached to spring and will stretch the spring 392 mm by itself. There is no damping in the system and a forcing function of the form, is attached to the object and the system will experience resonance. If the object is initially displaced 20 cm downward from its equilibrium position and given a velocity of 10 cm/sec upward find the displacement at any time t.

THANK-YOU