Dynamics of Machinery

Module 3
Free Undamped Single Degree of
Freedom System of Vibration
By Prof. Parag S. Sarode
Assistant Professor,
VCET, Vasai
 Basics of Vibration
⚫ What is Vibration?
Any motion that repeats itself after an
interval of time is called vibration or
oscillation.
⚫ Examples of Vibration
The motion of a plucked string, Motion of
Swinging arm of a pendulum, Ringing bell
 Elements of Vibrating system
⚫ Spring or elastic member: Which stores
potential energy
⚫ Mass: Which stores kinetic energy
⚫ Damper: By which energy is gradually lost.

⚫ The vibration of a system involves the transfer

of its potential energy to kinetic energy and of
kinetic energy to potential energy, alternately.
⚫ If the system is damped, some energy is
dissipated in each cycle of vibration and must be
replaced by an external source if a state of
steady vibration is to be maintained.
Basic terms in Vibration
⚫ Number of Degrees 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.

Fig 3.1 : Single degree of freedom system

Basic terms in Vibration

Fig. 3.2 : Two degree of freedom

system
Basic terms in Vibration

Fig. 3.3 : Three degree of freedom system

Basic terms in Vibration

Fig. 3.4 : Cantilever beam (Infinite degrees of freedom

system)
Vibrating System types
⚫ Discrete and continuous Systems:

⚫ Systems with a finite number of degrees of

freedom are called discrete or lumped
parameter systems.

⚫ Systems with an infinite number of degrees

of freedom are called continuous or
distributed systems.
 Classification of Vibration
⚫ Free Vibration:
⚫ If a system, after an initial disturbance, is left
to vibrate on its own, the ensuing vibration is
known as free vibration. No external force
acts on the system.
⚫ The oscillation of a simple pendulum is an
example of free vibration.
 Classification of Vibration
⚫ Forced Vibration:
⚫ If a system is subjected to an external force
(often, a repeating type of force), the
resulting vibration is known as forced
vibration.
⚫ The oscillation that arises in machines such
as diesel engines is an example of forced
vibration.
Classification of Vibration
⚫ Undamped Vibration:
⚫ If no energy is lost or dissipated in friction or
other resistance during oscillation, the
vibration is known as undamped vibration.

⚫ Damped Vibration:
⚫ If any energy is lost or dissipated in friction
or other resistance during oscillation, it is
called as damped vibration.
Classification of Vibration
⚫ Linear Vibration:
⚫ If all the basic components of a vibratory system
the spring, the mass, and the damper behave
linearly, the resulting vibration is known as linear
vibration.
⚫ Non linear Vibration:
⚫ If, however, any of the basic components behave
nonlinearly, the vibration is called nonlinear
vibration
Types of Free Vibrations

Fig.3.5 Types of free vibration

⚫ Longitudinal Vibration:
⚫ When the particles of a bar or disc move parallel to
the axis of the shaft, then the vibrations are known as
longitudinal vibrations as shown in fig. (a).
⚫ The bar is elongated and shortened alternately and
thus the tensile and compressive stresses are
inducted in the bar.
⚫ The motion of spring mass system is longitudinal
vibrations.
⚫ Transverse Vibrations :
⚫ When the particles of the bar or disc move
approximately perpendicular to the axis of the bar,
then the vibrations are known as transverse
vibrations as shown in fig.(b).
⚫ In this case, bar is straight and bent alternately.
Bending stresses are induced in the bar.
⚫ Torsional Vibrations :
⚫ When the particles of the bar or disc get alternately
twisted and untwisted on account of vibratory
motion of suspended body, it is said to be undergoing
torsional vibrations as shown in fig. (c).
⚫ In this case, torsional shear stresses are induced in
the bar.
Classification of Vibration
⚫ 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
⚫ Random Vibration
⚫ The value of the excitation at a given time
cannot be predicted.
⚫ Examples of random excitations are wind
velocity, road roughness, and ground motion
Spring Elements
⚫ A spring is said to be linear if the elongation or
reduction in length x is related to the applied force F
as
F = k.x (3.1)
⚫ where k is a constant, known as the spring constant
or spring stiffness or spring rate.
⚫ The spring constant k is always positive and denotes
the force (positive or negative) required to cause a
unit deflection (elongation or reduction in length) in
the spring.
⚫ The work done (U) in deforming a spring is stored as strain
or potential energy in the spring, and it is given by
Spring Constant of a rod
⚫ Find the equivalent spring constant of a uniform rod
of length l, cross-sectional area A , and Young’s
modulus E subjected to an axial tensile or
compressive force F as shown in fig.
The elongation or
shortening δ of the
road under the axial
tensile (or
compressive) force F,
 Spring Constant of a Cantilever beam
⚫ Find the equivalent spring constant of a cantilever beam
subjected to an concentrated load F at its end as shown
in fig. The deflection of the
beam δ due to
concentrated load F=W
is given by,

Where E is Young’s modulus, I moment of inertia

of the cross section of beam about bending axis
ZZ, Spring constant k can be obtained as,
Combination of Springs
⚫ Case I : springs in ⚫ Case II : springs in
parallel series
Equivalent k of a rigid bar connected
by springs
⚫ For small angular
displacement of rod θ
Spring constant of Associated with
restoring force due to gravity
⚫ The restoring
moment or torque (T)
created by the weight
of the mass (mg)
about the pivot point
O is given by
⚫ T = mg (l sin θ)
(1)
⚫ For small angular
displ. Sin θ ≈ θ,
⚫ So, T = mgl θ
(2)
⚫ T= k tθ
Equivalent stiffness of Inclined Springs
⚫ A vertical displacement
x of point B will cause
the spring (boom) to
deform by an amount
and the spring (cable)
to deform by an
amount The length of
the cable FB, l1
⚫ l12 = 3 2 + 10 2 - 2(3)(10)
cos 135° = 151.426,
 Definitions
⚫ Cycle.
⚫ The movement of a vibrating body from its
undisturbed or equilibrium position to its extreme
position in one direction, then to the equilibrium
position, then to its extreme position in the other
direction, and back to equilibrium position is called
a cycle of vibration.
⚫ Amplitude.
⚫ The maximum displacement of a vibrating body
from its equilibrium position is called the amplitude
of vibration.
⚫ Period of oscillation.
⚫ The time taken to complete one cycle of motion is
known as the period of oscillation or time period and
 Definitions
⚫ Cycle.
⚫ The movement of a vibrating body from its
undisturbed or equilibrium position to its extreme
position in one direction, then to the equilibrium
position, then to its extreme position in the other
direction, and back to equilibrium position is called
a cycle of vibration.
⚫ Amplitude.
⚫ The maximum displacement of a vibrating body
from its equilibrium position is called the amplitude
of vibration.
⚫ Period of oscillation.
⚫ The time taken to complete one cycle of motion is
known as the period of oscillation or time period and
 Definitions
⚫ Frequency of oscillation.
⚫ The number of cycles per unit time is called the
frequency of oscillation or simply the frequency and is
denoted by f. Thus
⚫ f=1/τ=ω/2π
⚫ Phase angle.
⚫ Consider two vibratory motions denoted by
 Definitions
⚫ Natural frequency.
⚫ If a system, after an initial disturbance, is left to vibrate on
its own, the frequency with which it oscillates without
external forces is known as its natural frequency.
⚫ A vibratory system having n degrees of freedom will have,
in general, n distinct natural frequencies of vibration.
⚫ Beats.
⚫ When two harmonic motions, with frequencies close to
one another, are added, the resulting motion exhibits a
phenomenon known as beats.
 Definitions
⚫ Octave.
⚫ When the maximum value of a range of frequency is
twice its minimum value, it is known as an octave
band. For example, each of the ranges 75 – 50 Hz,
150 – 300 Hz, and 300 – 600 Hz can be called an
octave band. In each case, the maximum and
minimum values of frequency, which have a ratio of
2:1, are said to differ by an octave.
⚫ Decibel.
⚫ The various quantities encountered in the field of
vibration and sound (such as displacement, velocity,
acceleration, pressure, and power) are often
represented using the notation of decibel.
 Harmonic Motion
⚫ Oscillatory motion may
repeat itself regularly, as
in the case of a simple
pendulum, or it may
display considerable
irregularity, as in the
case of ground motion
during an earthquake.
⚫ If the motion is repeated
after equal intervals of
time, it is called periodic
motion.
⚫ The simplest type of
periodic motion is
harmonic motion.
⚫ The motion imparted to
the mass m due to the
Harmonic Motion
 Harmonic Motion

Fig. Displacement, velocity and acceleration as rotating vectors

 Harmonic Motion

Fig. Vectorial addition of harmonic functions

 Vibration Analysis Procedure
⚫ Step 1: Mathematical Modeling

Fig.3.5 Modeling of forging hammer

Vibration Analysis Procedure
⚫ The purpose of mathematical modeling is to
represent all the important features of the system
for the purpose of deriving the mathematical (or
analytical) equations governing the systems
behaviour.
⚫ The mathematical model should include enough
details to allow describing the system in terms of
equations without making it too complex.
⚫ The mathematical model may be linear or nonlinear,
depending on the behaviour of the systems
components.
⚫ Sometimes the mathematical model is gradually
improved to obtain more accurate results.
Vibration Analysis Procedure
⚫ Step 2: Derivation of Governing Equations
⚫ Once the mathematical model is available, we use the
principles of dynamics and derive the equations that
describe the vibration of the system. The equations of
motion can be derived conveniently by drawing the
free-body diagrams of all the masses involved.
⚫ The free-body diagram of a mass can be obtained by
isolating the mass and indicating all externally
applied forces, the reactive forces, and the inertia
forces.
⚫ The equations of motion of a vibrating system are
usually in the form of a set of ordinary differential
Vibration Analysis Procedure
⚫ Step 2: Derivation of Governing Equations..
.
⚫ The equations may be linear or nonlinear,
depending on the behaviour of the components of
the system.
⚫ Several approaches are commonly used to derive the
governing equations.
⚫ Among them are Newton’s second law of motion,
D Alembert’s principle, and the principle of
conservation of energy.
Vibration Analysis Procedure
⚫ Step 3: Solution of the Governing
Equations
⚫ The equations of motion must be solved to find the
response of the vibrating system.
⚫ Depending on the nature of the problem, we can use
one of the following techniques for finding the
solution: standard methods of solving differential
equations, Laplace transform methods, matrix
methods, and numerical methods.
Vibration Analysis Procedure
⚫ Step 4: Interpretation of the Results
⚫ The solution of the governing equations gives the
displacements, velocities, and accelerations of the
various masses of the system.
⚫ These results must be interpreted with a clear view
of the purpose of the analysis and the possible
design implications of the results.
Mathematical Model of a Motorcycle
 References
⚫ Mechanical Vibrations, S. S. Rao, 5 th edition, Prentice
Hall