Lecture Set - 1

Motivation to study vibration
• Vibration involves in our day-to-day life in following manner

• Vibration in machine components. Example – turbines, engine balancing, centrifugal machine etc.
• Our eardrums vibrate, vibration of lungs
• light waves undergo vibration
• Mechanical engineering motivation of vibration study
• In general, machine component subjected to vibration can fail because of material fatigue resulting from the cyclic variation of the
induced stress. Furthermore, the vibration causes more rapid wear of machine parts such as bearings and gears and also creates
excessive noise.
• Whenever the natural frequency of vibration of a machine or civil structure (like bridges etc.) coincides with the frequency of the
external excitation, there occurs a phenomenon known as resonance, which leads to excessive deflections and failure - An example
is Tacoma Narrows bridge destruction due to during wind-induced vibration.
• Nonlinearity is inevitable.
Vibration - introduction
• Any motion that repeats itself after an interval of time is called vibration or oscillation. The swinging of a
pendulum and the motion of a plucked string are typical examples of vibration. The theory of vibration deals
with the study of oscillatory motions of bodies and the forces associated with them.
• Components of a vibratory system – component for storing potential energy (spring or elasticity), a means for
storing kinetic energy (mass or inertia), damper, and external forcing.
Degree of freedom
• The minimum number of independent coordinates required to determine completely the positions of all parts
of a system at any instant of time defines the number of degrees of freedom of the system.
Classification of vibration
• Based on presence of externa forcing
• Free vibration
• Forced vibration
• Based on presence of damping
• Undamped vibration
• Damped vibration
• Based on physical nature of system
• Discrete or lumped parameter system
• Single degree of freedom
• Multi degree of freedom
• Continuous or flexible system – infinite degree of freedom
• Based on linearity
• Linear vibration
• Nonlinear vibration
Classification of vibration
• Based on randomness
• Deterministic vibration - If the value or magnitude of the excitation (force or motion) acting on a vibratory system is known at any
given time, the excitation is called deterministic. The resulting vibration is known as deterministic vibration.
• Random vibration - In some cases, the excitation is nondeterministic or random; the value of the excitation at a given time cannot
be predicted. Random vibration is alaysed and solved using statistical method.
• The scope of present course – Deterministic, nonliner vibration

Linear and nonlinear vibration
• Linearity: In a system, output is proportional to input. In mathematical sense, the relationship of output and
input or the function can be graphically represented as a straight line.
• Example:
• Definition: In mathematics, a linear function f(x) is a function that satisfies the two properties:
Additivity: f(x + y) = f(x) + f(y).
Homogeneity: f(αx) = α f(x) for all α.
This is also known as principle of superposition.
• Nonlinearity: A system or function which is not linear is known as nonlinear.
• Example:
Free Vibration of an linear undamped System
• Solution of the equation of motion
Free Vibration of an linear undamped System
• Solution of the equation of motion
Assume solution in form
Characteristics equation
Where, Natural frequency od the system

Free Vibration of an Undamped System
The general solution
By using the identity
Two unknowns of the solution can be determined by providing initial condition
The final form of solution

The solution can be expressed in harmonic form by taking form as
In another way, we can write

Further
The final harmonic form of solution

• Graphical representation of harmonic oscillator
Forced Vibration of an linear forced damped System
• Viscous damped forced vibration
Total solution
Equation for homogeneous solution

Forced Vibration of an damped System (single degree of freedom)
• Total solution
Solution of equation
• Particular solution of equation
• Non-dimensional form
Amplitude ratio or Magnification factor, M =

Graphical representation of solution
• A graph of amplitude ratio vs frequency ratio
Graphical representation of solution
Resonance
• Maximum value of magnification factor or amplitude ratio
As zeta value decreases, the difference in these two

amplitude ratio decreases
Quality factor and experimentally determining damping ratio
• We define quality factor as
We know that, loss of energy in one complete cycle

Nonlinear vibration – an example
• Example: simple pendulum
• Equation of motion
• Equation of motion – a nonlinear system
• Energy equation
• The above equation we can write as

•
• Phase plane of a simple pendulum

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Motivation to study vibration

• Vibration involves in our day-to-day life in following manner

• The scope of present course – Deterministic, nonliner vibration

Assume solution in form

Where, Natural frequency od the system

By using the identity

Two unknowns of the solution can be determined by providing initial condition

The final form of solution

In another way, we can write

The final harmonic form of solution

Equation for homogeneous solution

Amplitude ratio or Magnification factor, M =

As zeta value decreases, the difference in these two

We know that, loss of energy in one complete cycle

• The above equation we can write as

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