This document provides an introduction to vibration, including: 1) A brief history of vibration studies from ancient times to modern finite element analysis. 2) Examples of unwanted vibration that can cause failures and discomfort, and examples of desirable vibration in musical instruments and medical/industrial applications. 3) Key concepts in vibration including degrees of freedom, discrete/continuous systems, free/forced vibration, damped/undamped vibration, linear/nonlinear vibration, and deterministic/random vibration. 4) An overview of the typical vibration analysis procedure including mathematical modeling, derivation of governing equations, solution of equations, and interpretation of results.

1. KIG3003 Mechanics of Machines
and Vibration
Lecture 1: Introduction to vibration
(Part 1)
Prepared by
mfsoong

2. Where we are
Week Topic Week Topic
1 Fundamentals of vibration 8 Vibration under general force
Response under general periodic force
2 Free vibration of 1-DOF systems
Undamped translational & rotational
systems
9 Vibration under general force
Response under non-periodic force
3 Free vibration of 1-DOF systems
Damped translational & rotational
systems
10 Laplace transform
4 Characteristic roots and
corresponding solutions
11 2-DOF systems
Free & forced vibration of undamped
systems
5 Harmonic excitation
Response of undamped & damped
systems
12 2-DOF systems
Coordinate coupling & principal
coordinates
6 Harmonic excitation
Base excitation & rotating unbalance
13 Multi-DOF systems
Equations of motion in matrix form
7 Frequency transfer functions 14 Multi-DOF systems
Modal analysis of forced vibration

3. What is vibration?
• Any motion that repeats itself after an interval of time =
vibration, oscillation
• Mechanical vibration:
• Consists of 3 elements
Mass / Inertia
Stiffness
Damping “mass-spring-damper” system
• Involves transfer of kinetic energy  potential energy
• study of oscillatory motions of bodies and the forces
associated with them

4. History of vibration
• People became aware of vibration – musical instruments,
but did not study it scientifically
• 582 – 507 B.C: Pythagoras – conducted experiments on a
vibrating string – monochord – developed concept of pitch
• Observed that if two like strings of different lengths are
subject to the same tension, the shorter one emits a higher
note

5. History of vibration
• 350 B.C. – Aristotle wrote
treatises on music and
sound
• 320 B.C. – Aristoxenus
wrote a three-volume work
entitled ‘Elements of
Harmony’
• 300 B.C. – Euclid wrote a
treatises ‘Introduction to
Harmonics’
• A.D. 132 – Zhang Heng
invented the world’s first
seismograph to measure
earthquakes:

6. History of vibration
• More scientific studies
started from Galileo Galilei
(1564 – 1642):
• Experimenting simple
pendulum
• published ‘Discourses
Concerning Two New
Sciences’ (1638); described
resonance, frequency,
length, tension & density of
a vibrating stretched string
Lagrange, Poisson, Rayleigh,
……,  until modern times
• Robert Hooke (1635 – 1703)
– relation btw pitch and
frequency of string
vibration
• Joseph Sauveur (1653 –
1716) – acoustic
• Newton’s (1642 – 1727)
three laws of motion

7. History of vibration
• 1950s
• Beginning of high-speed digital computers - possible to treat
moderately complex systems & to generate approximate solutions
• Turner, Clough, Martin and Topp presented the finite element
method as known today
• Finite element method enabled engineers to use digital
computers to conduct numerically detailed vibration
analysis of complex mechanical, vehicular, and structural
systems displaying thousands of degrees of freedom

8. Importance of studying vibration
• Early times
• To understand natural
phenomena – develop math
theories to describe system
• Recent times
• Focus on engineering
applications – design of
systems
• There are also good uses of
vibration
• Vibrations lead to excessive
deflections and failure on
machines and structures:
• To reduce vibration through
proper design of machines
and their mountings

9. Examples of vibration (unwanted)
• Failures in machine and
structures, e.g. fatigue
• Discomfort due to noise &
vibration – road vehicles
• Motion sickness – various
transportations
• Earthquakes

10. Example of vibration (desirable)
• Musical instruments: guitar – string; drum – membrane; etc.
• Medical – e.g. high frequency vibration probe for heart
disease treatment (www.flowcardia.com)
• Vibration test rig / shakers – scientific studies, product R&D
• Industrial / consumer products:

11. Basic concepts:
What is vibration?
• Any motion that repeats itself after an interval of time =
vibration, oscillation
• Mechanical vibration:
• Consists of 3 elements
Mass / Inertia
Stiffness
Damping
• Transfer of kinetic energy  potential energy
• There are other kinds of system with oscillatory response:
electrical RLC circuit – elec-mech analogy
Must it always be
single-object?

12. Basic concepts:
Degrees of freedom
• Minimum number of independent coordinates required to
determine completely the positions of all parts of a system
at any instant of time
1-DOF systems
2-DOF systems
More DOFs?

13. Basic concepts:
Discrete & continuous systems
• Many practical systems can be described using a finite
number of DOFs – discrete or lumped parameter systems
• E.g. representation of passenger car with suspension system
• Rigid body vibration
• Some systems with elastic members have infinite number of
DOFs – continuous or distributed systems
• E.g. structural or machine systems that have deformable members
• Elastic body vibration
• Approximate as discrete: ↑ DOFs ↑ accuracy

14. Basic concepts:
Free & forced vibrations
• 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
• If a system is subjected to
an external force, the
resulting vibration is known
as forced vibration
• Force can be any type,
usually repeating (periodic)

15. Basic concepts:
Undamped & damped vibrations
• Undamped vibration:
• No energy is lost or dissipated in friction or other resistance during
oscillation
• Damped vibration:
• Energy is dissipated from the system, motion gradually decreases
• Practically, damping is present in vibrating systems;
becomes important in vibration near resonance

16. Basic concepts:
Linear & non-linear vibrations
• If all basic components of a
vibratory system behave
linearly, the resulting
vibration is known as linear
vibration
• If any of the basic
components behave
nonlinearly, the vibration is
called nonlinear vibration

17. Basic concepts:
Deterministic & random vibration
• Deterministic vibration:
• If the value or magnitude of the excitation (force or motion) is
known at any given time, the vibration is called deterministic
• Nondeterministic or random vibration:
• If the excitation is random, this results in nondeterministic vibration

18. Vibration analysis procedure
(1) Mathematical modeling
Represent all the important
features of a system to get
math equations governing the
system’s behavior
(3) Solution of the governing
equations
Use math techniques (e.g.
Laplace) to solve differential
equations to get response of
vibrating system; complex
systems – numerical solutions
(2) Derivation of governing
equations
Use principles of dynamics
with the aid of free-body
diagram; usually obtain
differential equations: e.g.
Newton’s laws, Principle of
conservation of energy,
D’Alembert’s principle
(4) Interpretation of the
results

19. Example of modeling
• Forging hammer:
Extent of modeling depends
on desired accuracy &
computational simplicity