This document provides an outline for a course on Mechanical Vibrations. The course will be taught in Semester 1 of 2023 by Dr. E. Shaanika. It will cover fundamental concepts of vibrations including single and multiple degree-of-freedom systems, forced and free vibrations, and damping. Students will learn vibration analysis procedures and applications such as vibration isolation and balancing of rotating machines. The goal is for students to understand vibration fundamentals, analyze vibration problems, and apply concepts to machine and structural design for vibration control. The prescribed textbook is Mechanical Vibrations by Singiresu S. Rao.
2. Mechanical Vibrations: Course Outline
Prescribed Textbook: Mechanical Vibrations, 4th edition by Singiresu S. Rao
(Available in JEDs Library)
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3. Course Content
Fundamentals of vibrations: Basic Concepts and definitions. Vibration Analysis, Harmonic
Motion. Single degree-of-freedom systems: Equation of motion, Lagrange’s equation, free
vibration of undamped and damped systems; logarithmic decrement, other forms of damping.
Forced vibration: Equation of motion, response to harmonic excitation, resonance, rotating
unbalanced, base motion excitation, response to general non-periodic excitation, impulse response
function. Design for vibration control: Vibration isolation, critical speeds of rotating shafts; practical
isolation design. Multiple degree-of-freedom systems: Equations of motion; Lagrange’s
equations, free vibration, natural frequencies and mode shapes, forced vibration, response to
harmonic excitations and normal-mode approach. Continuous systems: Introduction to continuous
systems. Vibration absorption:. Balancing of rotating machines.
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4. Learning Objectives
After completing this chapter, the reader should be able to do the following:
• Describe briefly the history of vibration
• Indicate the importance of study of vibration
• Give various classifications of vibration
• State the steps involved in vibration analysis
• Compute the values of spring constants, masses, and damping constants
• Define harmonic motion and different possible representations of harmonic
motion
• Add and subtract harmonic motions
• Conduct Fourier series expansion of given periodic function
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Chapter 1. Fundamentals of vibrations
5. 1.1 Basic Concepts of Vibration
1.1.1 Definitions
Vibrations are oscillations of a mechanical or structural system about an equilibrium position (i.e
any motion that repeats itself after an interval of time). The theory of vibration deals with the study
of oscillatory motions of bodies and the forces associated with them. A swinging pendulum is one
example of vibration.
Vibrations are initiated when an inertia element is displaced from its equilibrium position due to an
energy imparted to the system by an external source. A restoring force, or a conservative force
developed in a potential energy element, pulls the element back toward equilibrium.
Conservative forces - any transition between kinetic and potential energy conserves the total
energy of the system. This is path independent, meaning that we can start and stop at any
two points in the problem, and the total energy of the system—kinetic plus potential—at these points
are equal to each other. This is characteristic of a conservative force.
Non-conservative forces are dissipative forces such as friction or air resistance. These forces
take energy away from the system as the system progresses, energy that you can’t get back.
These forces are path dependent; therefore it matters where the object starts and stops
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6. Example 1.
When work is done on the block of Figure 1.1 to displace it from its equilibrium position,
potential energy is developed in the spring. When the block is released the spring force pulls
the block toward equilibrium with the potential energy being converted to kinetic energy. In
the absence of non-conservative forces, this transfer of energy is continual, causing the
block to oscillate about its equilibrium position.
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7. Example 2.
When the pendulum of Figure 1.2 is released from a position above its equilibrium
position the moment of the gravity force pulls the particle, the pendulum bob,
back toward equilibrium with potential energy being converted to kinetic energy.
In the absence of non-conservative forces, the pendulum will oscillate about the
vertical equilibrium position.
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8. 8
Example 3.
Non-conservative forces can dissipate or add energy to the system. The block of Figure 1.3
slides on a surface with a friction force developed between the block and the surface. The
friction force is non-conservative and dissipates energy. If the block is given a displacement
from equilibrium and released, the energy dissipated by the friction force eventually causes
the motion to cease. Motion is continued only if additional energy is added to the system as
by the externally applied force in Figure 1.4.
9. 1.1.2 Why Study Vibrations
Vibrations occur in many mechanical and structural systems. In recent times, many
investigations on vibrations have been motivated by the engineering applications such
as the design of machines, foundations, structures, engines, turbines, and control
systems. If uncontrolled, vibration can be catastrophic.
- Automobile engines vibrate, if uncontrolled, it could cause passenger discomfort.
- A vehicle traveling over rough terrain vibrates. If uncontrolled, it could cause
passenger discomfort. Vehicles are fitted with suspension systems designed to protect
passengers from discomfort.
- Vibrations of a machine tool chatter can lead to poor finish on machined parts.
- Structural failure can occur because of large dynamic stresses developed during
earthquakes or even wind-induced vibration.
- Vibrations induced by an unbalanced helicopter blade while rotating at high speeds
can lead to the blade’s failure and catastrophe for the helicopter.
- Excessive vibrations of pumps, compressors, turbomachinery, and other industrial
machines can induce vibrations of the surrounding structure, leading to inefficient
operation of the machines while the noise produced can cause human discomfort. 9
10. 1.1.3. Elementary Parts of Vibrating Systems
• In general, a vibratory system includes a means for storing potential energy
(spring or elasticity), a means for storing kinetic energy (mass or inertia) and
a means by which energy is gradually lost (damper). Let us analyse the
vibratory system of the pendulum below
• During vibration, potential energy is transferred to kinetic energy and kinetic
energy to potential energy, alternatively.
• 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.
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11. As an example of a vibratory system, consider the vibration of the simple pendulum in figure
1. Let the bob of mass m be released after being given an angular displacement At position 1
the velocity of the bob and hence its kinetic energy is zero. But it has a potential energy of
magnitude 𝑚𝑔𝑙(1 − 𝑐𝑜𝑠𝜃) with respect to the datum position 2. Since the gravitational force
mg induces a torque about the point O, the bob starts swinging to the left from position 1. This
gives the bob certain angular acceleration in the clockwise direction, and by the time it
reaches position 2, all of its potential energy will be converted into kinetic energy. Hence the
bob will not stop in position 2 but will continue to swing to position 3.
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Figure 1. A pendulum
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However, as it passes the mean position 2, a counterclockwise torque due to gravitystarts
acting on the bob and causes the bob to decelerate. The velocity of the bob reduces to zero at
the left extreme position. By this time, all the kinetic energy of the bob will be converted to
potential energy. Again due to the gravity torque, the bob continues to attain a
counterclockwise velocity. Hence the bob starts swinging back with progressively increasing
velocity and passes the mean position again. This process keeps repeating, and the pendulum
will have oscillatory motion. However, in practice, the magnitude of oscillation gradually
decreases and the pendulum ultimately stops due to the resistance (damping) offered by the
surrounding medium (air). This means that some energy is dissipated in each cycle of
vibration due to damping by the air.
13. 1.1.4. Degree of Freedom
Degree of freedom refers to the number of independent coordinates required to completely
locate all parts of a system at any given time. It is the number of kinematically independent
variables necessary to completely describe the motion of every particle in the system.
(a) One Degree of Freedom Systems
The simple pendulum shown in Fig. 1, as well as each of the systems shown in Fig. 1.11,
represents a single-degree-of-freedom system. For example, the motion of the simple pendulum
can be stated either in terms of the angle or in terms of the Cartesian coordinates x and y . If the
coordinates x and y are used to describe the motion, it must be recognized that these coordinates
are not independent. They are related to each other through the relation 𝑋2 + 𝑦2 = 𝑙2 where l is
the constant length of the pendulum. Thus any one coordinate can describe the motion of the
pendulum. In this example, we find that the choice of 𝜃 as the independent coordinate will be
more convenient than the choice of x or y.
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15. (b) Two-Degree of freedom system
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X1 and X2 𝜃1 𝑎𝑛𝑑 𝜃2 X and 𝜃 or ( x or y and X)
Figure 3
16. (c)Three degree of freedom system
A single particle free to move in space has three degrees of freedom, and a suitable choice of
generalized coordinates is the Cartesian coordinates (x, y, z) of the particle with respect to a
fixed reference frame. As the particle moves in space, its position is a function of time.
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Figure 4
17. An unrestrained rigid body has six degrees of freedom, three
coordinates for the displacement of its centre of mass, and angular
rotation about three coordinate axes, as shown below. However
constraints may reduce that number.
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Figure 5. Six Degrees of freedom
18. 1.2. Discrete and Continuous Systems
The systems shown in section 1.1 above can be described using a finite number of degrees of
freedom. Some systems, especially those involving continuous elastic members, have an
infinite number of degrees of freedom. An example is a cantilever beam shown in Fig. 6.
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Figure 6. Continuous System
19. 1.3. Classification of Vibration
Vibrations are classified by the number of degrees of freedom necessary for their modelling,
the type of forcing they are subject to, and the assumptions used in the modelling.
1.3.1 By DOF
Vibrations of systems that have a finite number of degrees of freedom are called discrete
systems. A system with one degree of freedom is called a single degree-of-freedom (SDOF)
system. A system with two or more degrees of freedom is called a multiple degree-of-
freedom (MDOF) system. A system with an infinite number of degrees of freedom is called a
continuous system or distributed parameter system.
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20. 1.3.2 Free and Forced Vibration
If the vibrations are a result of an initial disturbance, with the system is left to vibrate on its
own and no other source is present, the resulting vibrations are called free vibrations. No
external force acts on the system e.g. a simple pendulum. If the vibrations are caused by an
external force or motion, the vibrations are called forced vibrations. A good example is the
oscillations occurring in a diesel engine. If the external input is periodic, the vibrations are
harmonic. Otherwise, the vibrations are said to be transient. If the input is stochastic, the
vibrations are said to be random.
Figure 7. A random excitation (force)
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21. 1.3.3 Undamped and Damped Vibration
If the vibrations are assumed to have no source of energy dissipation, they are called
undamped. If a dissipation source is present, the vibrations are called damped and are further
characterized by the form of damping. For example, if viscous damping is present, they are
called viscously damped.
1.3.4 Linear and Nonlinear Vibration
If the mass, springs and dampers behave in a linear manner, the vibrations are termed as linear.
Solving linear vibrations is easy as the mathematical techniques for solving linear DEs are well
developed. If, components behave non-linearly, the resulting vibration is called nonlinear.
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22. 1.4. Vibration Analysis Procedure
A vibratory system is a dynamic system for which variables such as the excitations (inputs)
and responses (outputs) are time dependent. The response of a vibrating system generally
depends on the initial conditions as well as the external excitations. The analysis of a
vibrating system usually involves mathematical modelling, derivation of the governing
equations, solution of the equations, and interpretation of the results.
1.4.1 Mathematical Modelling
The purpose of mathematical modelling is to represent all the important features of the
system for the purpose of deriving the mathematical (or analytical) equations governing the
system s behaviour. E.g. for a bob of a pendulum, we take its mass and ignore the size, and
for a spring, we take the spring constant and ignore mass and damping.
1.4.2 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.
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23. 1.4.3 Solution of the Governing Equations
The equations of motion must be solved to find the response of the vibrating system.
1.4.4 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.
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Figure 8. Physical and mathematical
model of a motorcycle with a rider
Figure 8 (a) shows a motorcycle with a rider.
Develop a sequence of three mathematical models
of the system for investigating vibration in the
vertical direction. Consider the elasticity of the
tires, elasticity and damping of the struts (in the
vertical direction), masses of the wheels, and
elasticity, damping, and mass of the rider.
25. 1.5. Spring Elements
The force developed by a spring is proportional to the amount of deformation and is given by
F = 𝑘x
Where k is the spring constant and x is the deformation. Actual springs are nonlinear. We will assume
deformations are small so that springs behave in a linear manner.
Elastic elements like beams like the cantilever shown in Figure 9 also behave like springs. We know that the
static deflection of a cantilever beam at the free end is
𝛿 =
𝑊𝑙3
3𝐸𝐼
Where W=mg, m = mass, E=Young’s modulus, and I is the moment of inertia of the cross section.
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Figure 9
26. 1.5.1. Springs in parallel
In General if we have n springs with spring constants 𝑘1, 𝑘2, … 𝑘𝑛 in parallel, then the
equivalent spring constant 𝑘𝑒𝑞 is calculated as
𝑘𝑒𝑞 = 𝑘1 + 𝑘2 + ⋯ + 𝑘𝑛
1.5.2 Springs in Series
1
𝑘𝑒𝑞
=
1
𝑘1
+
1
𝑘2
+ … +
1
𝑘𝑛
Example1: Equivalent k of a truck suspension
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Figure 10. shows the suspension system of a freight truck with a parallel-spring arrangement. Find
the equivalent spring constant of the suspension if each of the three helical springs is made of steel
with a shear modulus of 𝐺 = 80 x 109𝑁/𝑚2 and has five effective turns, mean coil diameter D =
20 cm, and wire diameter d =2
Figure 10
𝑘 =
𝐺𝑑4
8𝑛𝐷3
27. Example 2: Torsional Spring constant of a Propeller Shaft.
Determine the torsional spring constant of the steel propeller shaft shown in Fig. 11
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Figure 11
𝑘 =
𝐺𝐽
𝑙
28. 1.6 Mass or Inertia Elements
The mass or inertia element is assumed to be a rigid body. It is common to disregard a mass if it is negligible
compared to another mass, the same way we neglected the mass of the beam in Fig. 9 and only considered the
weight (W) of the mass. Mass gains or loses kinetic energy when its velocity changes
❑Combination of masses
In practical applications, masses appear in combination. In order to simplify our analysis, we can replace these
several masses by a single equivalent mass.
Case 1. Translational Masses. Masses be attached to a rigid bar that is pivoted at one end (Fig. 12). We replace
the three masses with 𝑚𝑒𝑞 located at A
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Figure 12
29. Case 2: Coupled Translational and Rotational Masses
Figure 13 represents a rack and pinion.
The rack has:
▪ a mass m,
▪ Translational velocity ሶ
𝑥
▪ Pinion coupled
The pinion has:
▪ Mass moment of inertia 𝐽𝑜
▪ Rotational velocity ሶ
𝜃
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Figure 13
These two masses can be
combined into 1 translational
or rotational mass
30. Example. Find the equivalent mass of the system shown in Fig. 14, where the rigid link 1 is
attached to the pulley and rotates with it.
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Figure 14
31. 1.6. Damping Elements
During vibration, the vibrational energy is gradually converted to heat or sound. The
mechanism by which the vibrational energy is gradually converted onto heat or sound is
called damping. Damping can be modelled as one or more of the following.
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1.6.1. Viscous Damping
This is the most common damping used in
vibration analysis. It occurs when a mechanical
system vibrates in a fluid medium. The
resistance offered by the fluid to the moving
body causes energy to be lost. In viscous
damping, the damping force is proportional to
the velocity of the moving body.
32. 1.6.2 Coulomb or Dry friction Damping
The damping (friction) force is a constant and acts opposite to the direction of motion of the
vibrating body. It occurs due to the rubbing of surfaces that are either dry or have insufficient
lubrication. 𝐹 = 𝜇𝑁
1.6.3 Material or Solid or Hysteretic Damping
Ideally, if the material is stressed below its yield point and then unloaded, the stress-strain
curve for the unloading follows the same curve for the loading. However, in a real engineering
material, internal planes slide relative to one another and molecular bonds are broken, causing
conversion of strain energy into thermal energy and causing the process to be irreversible. i.e.
not all the energy is recovered. In a vibrating mechanical system an elastic member undergoes
a cyclic load-displacement relationship.
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33. The loading is repeated over each cycle. The existence the hysteresis leads to
energy dissipation from the system during each cycle, which causes natural
damping, called hysteretic damping.
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