Introduction
Mechanical vibration is the
motion of a particle or body which oscillates about a position of equilibrium.
Most vibrations in machines and structures are undesirable due to increased
stresses and energy losses. It deals with the relationship between forces
acting on the mechanical system and the oscillatory motion of the mechanical
system about a point within the system.
The main reasons for vibration are as follows :
- Unbalanced
centrifugal force in the system. This is caused because of nonuniform
material distribution in a rotating machine element.
- Elastic
nature of the system.
- External
excitation is applied to the system.
- Winds may
cause vibrations of certain systems such as electricity lines, telephone
lines, etc
Classification of Vibrations
Free and forced 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. For example oscillation of the simple pendulum
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. For example oscillation
that arises in machines such as diesel engines.
Damped and Undamped vibration
Damped vibration - If any energy is lost or dissipated in friction or other resistance during oscillation, it is called damped vibration.
Undamped vibration -
If no energy is lost, the vibration is known as undamped vibration.
In many physical systems, the amount of damping is so
small that it can be disregarded for most engineering purposes. However, consideration
of damping becomes extremely important in analyzing vibratory systems near
resonance.
Linear and nonlinear 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.
If, however, any of the basic components behave nonlinearly, the vibration is
called nonlinear vibration.
If the vibration is linear, the principle of superposition
holds, and the mathematical techniques of analysis are well developed. For
nonlinear vibration, the superposition principle is not valid, and techniques
of analysis are less well known.
Types of Free Vibration:
1. Torsional Vibration:
Torsional vibration is the
angular vibration of an object, commonly a shaft along its axis of
rotation. It is often a concern in power transmission systems using rotating
shafts or couplings where it can cause failures if not controlled. The second
effect of torsional vibrations applies to passenger cars.
Torsional vibrations can lead to
seat vibrations or noise at certain speeds. Both reduce the comfort.
2. Axial or Longitudinal Vibrations:
Axial vibration is a kind
of longitudinal shafting vibration which occurs in the crankshaft
because of the radial as well as tangential forces.
3. Transverse Vibrations:
A vibration in which the element
moves to and fro in a direction perpendicular to the direction of the advance
of the wave.
Vibration Measurement
When faced with a vibration
problem, engineers generally start by making some measurements to try to
isolate the cause of the problem. There are two common ways to
measure vibrations:
1. An accelerometer
is a small electro-mechanical device that gives an electrical signal
proportional to its acceleration. The picture shows a typical 3
axis accelerometer.
2. A
displacement transducer is similar to an accelerometer but gives an electrical
signal proportional to its displacement.
Displacement transducers are generally preferable
if you need to measure low-frequency vibrations; accelerometers behave better
at high frequencies.
The most common procedure is to mount three accelerometers
at a point on the vibrating structure, so as to measure accelerations in three
mutually perpendicular directions. The velocity and displacement are then
deduced by integrating the accelerations.
Applications of Mechanical Vibrations:
The applications of Mechanical Vibrations are as
follows.
- Identification
of the system: If
you want to calculate the mass, stiffness, and damping of a vibratory
system then you need to do the vibration analysis which is used in
structural health monitoring.
- Design of
components: When
you are designing the components of an automobile, you need to look at the
consideration that the vibrations excited by the engine should not match
with the other components and if it does, there may be a chance of failure
like loosening of bolts, etc.
- Vibrations
may cause loosening of parts from the machine
- If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is reached, and dangerously large oscillations may occur which may result in the mechanical failure of the system.
Stages of Vibration Analysis :
Free vibration without damping
To start the investigation of the mass-spring–damper assume the damping is negligible and that there is no external force applied to the mass (i.e. free vibration). The force applied to the mass by the spring is proportional to the amount the spring is stretched "x" (assuming the spring is already compressed due to the weight of the mass). The proportionality constant, k, is the stiffness of the spring and has units of force/distance (e.g. lbf/in or N/m). The negative sign indicates that the force is always opposing the motion of the mass attached to it:
πs = ‒ππ.
The force generated by the mass is proportional to the acceleration of the mass as given by Newton's second law of motion:
∑π = πΠ° = παΊ = (d²π/dπ©²).
The sum of the forces on the mass then generates this ordinary differential equation:
παΊ + ππ = 0.
Assuming that the initiation of vibration begins by stretching the spring by the distance of A and releasing, the solution to the above equation that describes the motion of the mass is:
(π©) = A cos(2πΉπβπ©).
This solution says that it will oscillate with a simple harmonic motion that has an amplitude of A and a frequency of fn. The number fn is called the undamped natural frequency. For the simple mass-spring system, fn is defined as:
πβ = (1/2πΉ)√(π/π) .
Parts of a Vibrating System :
A vibratory system basically consists of three
elements, namely the mass, the spring, and the damper. In a vibrating body,
there is an exchange of energy from one form to another. Energy is stored by
mass in the form of kinetic energy (1/2 mv2), in the spring in the form of
potential energy (1/2 kx ), and dissipated in the damper in the form of heat
energy which opposes the motion of the system.
Natural Frequency of Free Longitudinal Vibrations
The natural Frequency of the
Free Longitudinal Vibrations can be determined by the following methods.
1. Equilibrium Method
2. Energy method
3. Rayleigh’s method
Energy methods for analysis
For undamped free vibration, the total energy in the
vibrating system is constant throughout the cycle. Therefore the maximum
potential energy V,, is equal to the maximum kinetic energy T,, although these
maxima occur at different times during the cycle of vibration. Furthermore,
since the total energy is constant,
T + V = constant,
and thus d/dπ© (T + V)
= 0.
Applying this method to the case, already considered, of a body of mass
m fastened to a spring of stiffness π, when the body is displaced a distance x from its
equilibrium position,
strain energy (SE) in spring = ½ππ²
kinetic energy (KE) of body = ½ππ²
Hence V = ½ππ²
and T = ½ππ²
Thus d/dπ© (½ππ² + ½ππ² ) = 0
that is
αΊ + (π/π)= 0, as before in equation
This is a very useful method for certain types of problems
in which it is difficult to apply Newton's laws of motion.
Alternatively, assuming SHM, if π = πβ cos ππ©
the maximum SE, Vmax = ½ππ²β,
and
the maximum KE, Tmax = ½π(πβπ)².
Thus, since Tmax = Vmax,
½ππ²β = ½(πβπ)²,
or π = √(π/π) rad/s.
Energy methods can also be used in the analysis of
the vibration of continuous systems such as beams.
The frequency of vibration is found by considering the conservation of energy in the system; the natural frequency is determined by dividing the expression for potential energy in the system by the expression for kinetic energy.
Conclusion
- Mechanical
Vibration is a measurement of a periodic process of oscillations with
respect to an equilibrium point.
- We experience
these mechanical vibrations in everyday life
- There are
useful as well as harmful vibrations.
- There can
be Free/Forced, Damped/Undamped, Linear/Nonlinear OR
Deterministic/Random Vibration OR combinations of these mechanical
vibrations.
- Modeling
of systems can be Single Degree of freedom, 2 DOF, Multi DOF, and
Continuous Systems.
References
1) Research on ‘Impacts of mechanical vibrations on the production machine to its rate of change of technical state’; by Ε tefΓ‘nia SalokyovΓ‘, Radoslav Krehel’, Martin PollΓ‘k; July 4, 2016.
2) ‘Mechanical Vibration Sound waves are mechanical vibrations in solid, liquid or gas’; by Techniques and instrumentation in Analytical Chemistry, 2002.
3) Research on ‘Mechanical vibration monitoring based on wireless sensor network and sparse’; Bayes. Xinjun Lei & Yunxin Wu ; EURASIP; Journal on Wireless Communications and Networking volume 2020
4)
Research on ‘Impacts of mechanical vibrations on
the production machine to its rate of change of technical state’;July, 2016 ,Advances in Mechanical Engineering
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