CHAPTER 1
Introduction to
Mechanical Vibrations
Vibration is the motion of a particle or a
body or system of connected bodies displaced from a position
of equilibrium. Most vibrations are undesirable in machines and structures because they produce increased stresses,
energy losses, cause added wear, increase bearing loads, induce fatigue, create passenger
discomfort in vehicles, and absorb energy from the system. Rotating
machine parts need careful balancing
in order to prevent damage
from vibrations.
Vibration occurs when a system is displaced
from a position of stable equilibrium. The system
tends to return
to this equilibrium position under the action of restoring forces
(such as the elastic forces, as for a mass attached to a spring,
or gravitational forces, as for a simple
pendulum). The system
keeps moving back and forth
across its position
of equilibrium. A system is a combination of elements intended
to act together to accomplish an objective. For example, an automobile is a system whose elements
are the wheels, suspension, car body, and so
forth. A static element is one whose
output at any given time depends only on the input at that time while a dynamic element is one whose
present output depends
on past inputs. In the same way we also speak of static and dynamic systems.
A static system contains all elements while a dynamic system contains
at least one dynamic element.
A physical system
undergoing a time-varying interchange or dissipation of energy among
or within its elementary storage or dissipative devices is said to be in
a dynamic state. All of the elements in general are called passive, i.e., they are incapable
of generating net energy. A dynamic system
composed of a finite number
of storage elements
is said to be lumped or discrete, while a system containing elements, which are dense in
physical space, is called continuous. The analytical description of the dynamics of the
discrete case is a set of ordinary differential equations, while for the continuous case it is a set of partial
differential equations. The analytical formation of a dynamic system depends upon
the kinematic or geometric constraints and the physical laws governing the behaviour of
the system.
CLASSIFICATION
OF VIBRATIONS
Vibrations can be classified into three categories: free, forced, and self-excited. Free vibration of a system is vibration that occurs in the absence
of external force. An external
force that acts on the system causes
forced vibrations. In this case,
the exciting force
continuously supplies energy
to the system. Forced vibrations
may be either deterministic or random (see Fig. 1.1). Self- excited vibrations are periodic
and deterministic oscillations. Under certain conditions, the
1
equilibrium state in such a vibration system becomes
unstable, and any disturbance causes the perturbations to grow until some effect limits any further growth.
In contrast to forced vibrations, the exciting force is independent
of the vibrations and can still persist even
when the system is prevented
from vibrating.
ELEMENTARY
PARTS OF
VIBRATING SYSTEMS
In general, a vibrating system consists of a spring (a means for storing potential energy),
a mass or inertia (a means for storing kinetic energy), and a damper (a
means by which energy is gradually lost)
as shown in Fig. 1.2. An undamped
vibrating system involves
the transfer of its
potential energy to kinetic energy and kinetic energy to potential energy,
alternatively. In a damped vibrating system, some energy
is dissipated in each cycle
of vibration and should be replaced
by an external source if a steady state of vibration is to be maintained.
PERIODIC MOTION
When the motion is repeated
in equal intervals
of time, it is known as periodic motion. Simple harmonic
motion is the simplest form of periodic
motion. If x(t) represents the displacement of a
mass in a vibratory system, the motion can be expressed by the equation
x = A cos wt = A cos 2p t/t
where A is the amplitude of oscillation measured from the
equilibrium position of the mass.
1
The repetition time t
is called the period of the oscillation, and its reciprocal, f =1/
frequency. Any
periodic motion satisfies the relationship
x (t) = x (t + t)
That is Period
t = 2p/w rad/s
Frequency f =w /2p s/rad, cycles/s, or Hz
w is called the circular frequency measured in rad/sec.
The velocity and acceleration of a harmonic
displacement are also harmonic of the same frequency, but lead the displacement by p/2 and p radians, respectively. When the acceleration
˙˙
X of a
particle with rectilinear motion is always proportional to its displacement
from a fixed
point on the path and is directed towards the fixed
point, the particle is said to have simple
harmonic motion.
The motion of many vibrating
systems in general
is not harmonic. In many cases the vibrations
are periodic as in the impact force generated by a forging hammer. If x(t) is a peri- odic
function with period t, its Fourier series representation is given by
where w = 2p/t is the fundamental frequency
and a0, a1, a2, …, b1, b2, … are constant
coeffi- cients, which are
given by:
.
The harmonic functions ancos nωt or bnsinnωt are known as the harmonics of order n
of the periodic function x(t). The harmonic of order n has a period τ/n. These harmonics can be
plotted as vertical lines in a diagram of amplitude (an and bn) versus frequency (nω) and is
called frequency spectrum.
DISCRETE AND
CONTINUOUS SYSTEMS
Most of the mechanical and structural systems
can be described using a finite number of degrees of freedom. However, there are some
systems, especially those include continuous
elastic members, have an
infinite number of degree of freedom. Most mechanical and structural systems have elastic
(deformable) elements or components as members and hence have an infinite number of degrees of freedom.
Systems which have a finite number of degrees of free
dom are known
as discrete or lumped parameter systems, and those systems with an infinite number of
degrees of freedom are called continuous or distributed systems.
VIBRATION ANALYSIS
The outputs of a vibrating system, in general,
depend upon the initial conditions, and external excitations. The vibration analysis
of a physical system may be summarised by the four steps:
1.
Mathematical Modelling of a Physical System
2.
Formulation of Governing Equations
3.
Mathematical Solution of the Governing Equations
1. Mathematical modelling of a physical system
The purpose of the mathematical modelling is to determine the existence and nature of the system,
its features and aspects, and the physical
elements or components involved in the physical system.
Necessary assumptions are made to simplify the modelling. Implicit
assump- tions are used that include:
(a)
A physical system can be treated as a continuous
piece of matter
(b) Newton’s laws of motion can be applied by assuming that the earth is an internal frame
(c)
Ignore or neglect the relativistic effects
All components or elements of the physical
system are linear.
The resulting mathemati- cal model may be linear
or non-linear, depending on the given
physical system. Generally speaking, all physical systems
exhibit non-linear behaviour. Accurate mathematical modelling of any physical system will lead to non-linear differential equations governing
the behav- iour of the system. Often, these non-linear
differential equations have either no solution or difficult
to find a solution. Assumptions are made to linearise a system, which permits quick solutions
for practical purposes. The advantages of linear models are the following:
(1) their
response is proportional to input
(2) superposition
is applicable
(3) they
closely approximate the behaviour of many dynamic systems
(4) their response
characteristics can be obtained from the form of system equations
without a detailed solution
(5) a
closed-form solution is often possible
(6) numerical
analysis techniques are well developed, and
(7) they serve as a basis
for understanding more complex non-linear system behaviours. It should, however,
be noted that in most non-linear problems
it is not possible to obtain closed-form analytic solutions for the
equations of motion. Therefore, a computer simulation
is often used for the response analysis.
When analysing the results obtained from
the mathematical model, one should realise that the mathematical model is only an approximation to the true or real physical system
and therefore the actual
behaviour of the system may be different.
2. Formulation of governing equations
Once the mathematical model is
developed, we can apply the basic laws of nature and the principles of
dynamics and obtain the differential equations that govern the behaviour of the
system. A basic law of nature is a physical law that is applicable to all
physical systems irrespective of the material from which
the system is constructed. Different materials
behave differently under
different operating conditions. Constitutive equations provide information about the materials of which a system is made. Application of
geometric constraints such as the kinematic relationship between displacement, velocity, and acceleration is often necessary
to complete the mathematical modelling of the physical system. The
application of geometric constraints is necessary in order to formulate the required boundary
and/or initial conditions.
The resulting mathematical model may be
linear or non-linear, depending upon the
behaviour of the elements or components of the dynamic system.
3. Mathematical solution of the governing equations
The mathematical modelling of a physical
vibrating system results
in the formulation of the governing equations of motion.
Mathematical modelling of typical systems leads to a sys- tem of differential equations
of motion. The governing equations
of motion of a system are solved
to find the response of the system.
There are many techniques available
for finding the solution, namely,
the standard methods
for the solution
of ordinary differential equations, Laplace transformation methods, matrix methods, and
numerical methods. In general, exact analytical solutions are available
for many linear dynamic systems,
but for only a few non- linear
systems. Of course,
exact analytical solutions are always preferable to numerical or approximate solutions.
4. Physical interpretation of the
results
The solution of the governing equations of motion
for the physical
system generally gives the
performance. To verify the validity of the model, the predicted performance is com- pared with the experimental results. The model may have to be refined or a new model is developed
and a new prediction compared with the experimental results. Physical interpretation of the results is an important
and final step in the analysis procedure. In some situations, this may involve (a) drawing general
inferences from the mathematical solution, (b) develop-
ment of design
curves, (c) arrive at a simple
arithmetic to arrive
at a conclusion (for a typical or specific
problem), and (d) recommendations
regarding the significance of the results and
any changes (if any) required
or desirable in the system involved.
1.5.1 COMPONENTS OF VIBRATING SYSTEMS
(a) Stiffness elements
Some times it requires finding out the equivalent spring stiffness values when a continuous
system is attached to a discrete system or when there are a number of spring elements
in the system. Stiffness of continuous elastic elements such as rods, beams, and shafts, which produce restoring elastic forces, is obtained from deflection considerations
When there are several springs arranged in parallel as shown in Fig. 1.6, the equivalent
spring constant is given by algebraic sum of the stiffness of individual springs. Mathematically,
keq =
