MECHANICAL VIBRATIONS

Introduction Objectives Types of vibrations Simple harmonic motion (SHM) Principle of superposition applied to sim

ple harmonic motions

Beats Fourier theorem and simple problems Summary References

Introduction  

  A body is said to vibrate if it has periodic motion.

Mechanical vibration is the study of oscillatory motions of bodies.

Vibrations are harmful for engineering systems. Some times vibrations can be useful. For example, vibratory compactors are used for compacting concrete during construction work.

Excessive vibration causes discomfort to human beings, damage to machines and buildings and wear of machine parts such as bearings and gears.

Objectives

Understand the causes and effects of vibration

Know about the types of vibrations

Explain about simple harmonic motion (SHM)

Learn Principle of superposition applied to simple harmonic motions

Learn about beats

Causes of vibration

1.Bad design

2.Unbalanced inertia forces

3.Poor quality of manufacture

4.Improper bearings (Due to wear & tear or bad quality)

5.Worn out gear teeth

6.External excitation applied on the system

EFFECTS

1.Unwanted noise

2.Early failure due to cyclical stress(fatigue failure)

3.Increased wear

4.Poor quality product

5.Difficult to sell a product

6.Vibrations in machine tools can lead to improper machining of parts

BASIC TERMS

Degrees of FreedomIt is the number of coordinates required to describe the motion of the body.

Free vibration If a system, after an initial disturbance is left to vibrate on its own, the resulting vibration is known as free vibration. The frequency of free vibration is known as natural frequency of vibration, which is an important parameter in vibration analysis.

Forced vibration If a system is subjected to an external repeating type of force, the resulting vibration is known as forced vibration. The frequency of forced vibration is known as forced frequency of vibration, which is also an important parameter in vibration analysis

Undamped vibration If no energy is lost or dissipated in resistance during vibration, the resulting vibration is known as undamped vibration.

Damped vibration If energy is lost or dissipated due to resistance during vibration, it is known as damped vibration.

Linear vibration If all the basic components of a vibrating system behave linearly the resulting vibration is known as linear vibration.

Non-linear vibration If any basic component behaves non-linearly, the resulting vibration is known as non-linear vibration.

Periodic vibration If the value of excitation acting on the vibratory system is known at any given instant, the resulting vibration is known as periodic vibration.

Random vibrationIf the value of excitation acting on the vibratory system is known at any given instant, the resulting vibration is known as periodic vibration. If excitation is non-periodic, the resulting vibration is called as Random vibration.

Types of Vibrations

Based on degrees of freedom. The number of degrees of freedom for

a system is the number of kinematically independent variables necessary to completely describe the motion of every particle in the system.

Based on degrees of freedom, we can classify mechanical vibrations as follows

1.Single Degree of freedom Systems2.Two Degrees of freedom Systems3.Multidegree of freedom Systems4.Continuous Systems or systems with infinite

degrees of freedom

Another broad classification of vibrations is 1.Free and forced vibrations2.Damped and undamped vibrations

Vibration problems are classified as

1.Linear vibrations

2.Non-linear vibrations

3.Random vibrations

4.Transient vibrations

A system is linear if its motion is governed by linear differential equations.

A system is nonlinear if its motion is governed by nonlinear differential equations.

If the excitation force is known at all times, the excitation is said to be deterministic.

If the excitation force is unknown, but averages and standard derivations are known, the excitation is said to be random.In this case the resulting vibrations are also random.

Some times systems are subjected to short duration nonperiodic forces. The resulting vibrations are called transient vibrations.

One example of a nonperiodic short duration excitation is the ground motion in an earthquake

Simple Harmonic Motion(SHM)

Simple harmonic motion (SHM)     Any motion, which repeats itself after equal intervals of time, is called as periodic motion. The repetition time t is called the period of oscillation and its reciprocal 1/t is frequency of oscillation denoted by f.

The simplest form of periodic motion is harmonic motion. The harmonic motion can be represented as the projection of a straight line OP with angular speed of OP, w as shown below. Since, the motion repeats itself in 2p radians, the angular speed of OP, w can written as

Mathematically, the oscillatory motion of mass in x - direction is

The equation 4 is the equation for a SHM, where, A amplitude of the motion in mm, w circular frequency in radians / sec, t is time in seconds.

Principle of superposition applied to simple harmonic motion

Addition of two harmonic motions of similar frequency, different amplitudes and different phase angle results in a harmonic motion. Consider two harmonic motions

The resulting harmonic motion can be obtained by adding equations 5 and 6 either by analytically or by vectorially.

(a) Analytical Method

Considering

substitute equations 8 and 9 in equation 7,

The above equation is also a harmonic motion with amplitude A and phase angle q.

Resultant amplitude A To obtain amplitude of resultant motion square and add equations 8 and 9

Resultant phase angle f To obtain phase angle of resultant motion divide equation 9 by equation 8

Problem 1 :Add following two harmonics analytically x1(t) = 2 cos (wt + 0.5) and

x2(t) = 2 sin (wt + 1)

Problem 1 :Add following two harmonics analytically x1(t) = 2 cos (wt + 0.5) and

x2(t) = 2 sin (wt + 1)

Analytical solution

x(t) = x1 + x2 (t)

x(t) = 2 cos (wt + 0.5) + 5 sin (wt + 1)

x(t) = 2[ cos (wt) cos (0.5) - sin (wt) sin (0.5)] + 5[ sin (wt) cos (1) + cos (wt) sin (1)]

x(t) = sin (wt) [-2 sin (0.5) + 5 cos (1) + cos (wt) [2 cos (0.5) + 5 sin (1)]

x(t) = sin (wt) [-2 (0.4794) + 5 (0.5403) + cos (wt) [2 (0.8775) + 5(0.8414)]

x(t) = 1.742 sin (wt) + 5.962 cos (wt)

x(t) = A sin (wt + q) = A ( sin wt cos q + cos wt sin q)

x(t) = A sin (wt + q) = sin wt ( A cos q) + cos wt ( A sin q) we get from above equations.

A cos Φ = 1.742

A sin Φ = 5.962

q = tan -1 3.422 = 1.2565 radians OR 73.710

Beats

When two harmonic motions with frequencies close to one another are in the same direction, their super positioned resulting motion is like as shown in the below figure and is referred as Beats. The phenomenon of beats if often observed in machines and structures, when forcing frequency is very close to natural frequency of the machine / structure.

Fourier theorem and simple problems

With the help of Fourier series a periodic function can be analyzed in terms of sine and cosines. The application of Fourier series in vibration studies is that experimentally obtained vibration results can be represented by analytically.

If x(t) is a periodic function with time period t, then mathematically the Fourier series can be written as

Contd…

The above mathematical analysis to obtain Fourier series is referred as Harmonic analysis.