# Dictionary Definition

vibrations n : a distinctive emotional aura
experienced instinctively; "that place gave me bad
vibrations"

# User Contributed Dictionary

## English

### Noun

vibrations- Plural of vibration

# Extensive Definition

Vibration refers to mechanical oscillations about an
equilibrium point . The oscillations may be periodic
such as the motion of a pendulum or random such as the movement of a
tire on a gravel road.

Vibration is occasionally "desirable". For
example the motion of a tuning fork,
the reed in a
woodwind
instrument or harmonica, or the cone of a
loudspeaker is
desirable vibration, necessary for the correct functioning of the
various devices.

More often, vibration is undesirable, wasting
energy and creating
unwanted sound -- noise. For example, the
vibrational motions of engines, electric
motors, or any mechanical device
in operation are typically unwanted. Such vibrations can be caused
by imbalances
in the rotating parts, uneven friction, the meshing of
gear teeth, etc. Careful
designs usually minimize unwanted vibrations.

The study of sound and vibration are closely
related. Sound, or "pressure waves", are generated by vibrating
structures (e.g. vocal cords);
these pressure waves can also induce the vibration of structures
(e.g. ear
drum). Hence, when trying to reduce noise it is often a problem
in trying to reduce vibration.

To start the investigation of the
mass-spring-damper we will 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" (we will
assume 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) F_s=- k
x \!

The force generated by the mass is proportional
to the acceleration of the mass as given by Newton’s
second law of motion.

\Sigma\ F = ma = m \ddot = m \frac

The sum of the forces on the mass then generates
this
ordinary differential equation:

- m \ddot + k x = 0.

If we assume that we start the system to vibrate
by stretching the spring by the distance of A and letting go, the
solution to the above equation that describes the motion of mass
is:

x(t) = A \cos (2 \pi f_n t) \!

This solution says that it will oscillate with
simple
harmonic motion that has an amplitude of A and a frequency
of f_n, but what is f_n? f_n is one of the most important
quantities in vibration analysis and is called the undamped natural
frequency.

f_n is defined for the simple mass-spring system
as:

f_n = \sqrt \!

Note: Angular
frequency \omega (\omega=2 \pi f) with the units of radians per
second is often used in equations because it simplifies the
equations, but is normally converted to “standard” frequency (units
of Hz or equivalently
cycles per second) when stating the frequency of a system.

If you know the mass and stiffness of the system
you can determine the frequency at which the system will vibrate
once it is set in motion by an initial disturbance using the above
stated formula. Every vibrating system has one or more natural
frequencies that it will vibrate at once it is disturbed. This
simple relation can be used to understand in general what will
happen to a more complex system once we add mass or stiffness. For
example, the above formula explains why when a car or truck is
fully loaded the suspension will feel “softer” than unloaded
because the mass has increased and therefore reduced the natural
frequency of the system.

#### What causes the system to vibrate under no force?

These formulas describe the resulting motion, but
they do not explain why the system oscillates. The reason for the
oscillation is due to the conservation
of energy. In the above example we have extended the spring by
a value of A and therefore have stored potential
energy (\tfrac k x^2) in the spring. Once we let go of the
spring, the spring tries to return to its un-stretched state and in
the process accelerates the mass. At the point where the spring has
reached its un-stretched state it no longer has energy stored, but
the mass has reached its maximum speed and hence all the energy has
been transformed into kinetic
energy (\tfrac m v^2). The mass then begins to decelerate
because it is now compressing the spring and in the process
transferring the kinetic energy back to its potential. This
transferring back and forth of the kinetic energy in the mass and
the potential energy in the spring causes the mass to
oscillate.

In our simple model the mass will continue to
oscillate forever at the same magnitude, but in a real system there
is always something called damping that dissipates the energy and
therefore the system eventually bringing it to rest.

### Free vibration with damping

We now add a "viscous" damper to the model that
outputs a force that is proportional to the velocity of the mass.
The damping is called viscous because it models the effects of an
object within a fluid. The proportionality constant c is called the
damping coefficient and has units of Force over velocity (lbf s/ in
or N s/m).

F_d = - c v = - c \dot = - c \frac \!

By summing the forces on the mass we get the
following ordinary differential equation:

- m \ddot + \dot + x = 0.

The solution to this equation depends on the
amount of damping. If the damping is small enough the system will
still vibrate, but eventually, over time, will stop vibrating. This
case is called underdamping--this case is of most interest in
vibration analysis. If we increase the damping just to the point
where the system no longer oscillates we reach the point of
critical damping (if the damping is increased past critical damping
the system is called overdamped). The value that the damping
coefficient needs to reach for critical damping in the mass spring
damper model is:

- c_c= 2 \sqrt

To characterize the amount of damping in a system
a ratio called the damping ratio (also known as damping factor and
% critical damping) is used. This damping ratio is just a ratio of
the actual damping over the amount of damping required to reach
critical damping. The formula for the damping ratio (\zeta ) of the
mass spring damper model is:

- \zeta = .

For example, metal structures (e.g. airplane
fuselage, engine crankshaft) will have damping factors less than
0.05 while automotive suspensions in the range of 0.2-0.3.

The solution to the underdamped system for the
mass spring damper model is the following:

- x(t)=X e^ \cos() , \ \ \omega_n= 2\pi f_n

The value of X, the initial magnitude, and \phi ,
the phase
shift, are determined by the amount the spring is stretched.
The formulas for these values can be found in the references.

The major points to note from the solution are
the exponential term and the cosine function. The exponential term
defines how quickly the system “damps” down – the larger the
damping ratio, the quicker it damps to zero. The cosine function is
the oscillating portion of the solution, but the frequency of the
oscillations is different from the undamped case.

The frequency in this case is called the "damped
natural frequency", f_d , and is related to the undamped natural
frequency by the following formula:

- f_d= \sqrt f_n

The damped natural frequency is less than the
undamped natural frequency, but for many practical cases the
damping ratio is relatively small and hence the difference is
negligible. Therefore the damped and undamped description are often
dropped when stating the natural frequency (e.g. with 0.1 damping
ratio, the damped natural frequency is only 1% less than the
undamped).

The plots to the side present how 0.1 and 0.3
damping ratios effect how the system will “ring” down over time.
What is often done in practice is to experimentally measure the
free vibration after an impact (for example by a hammer) and then
determine the natural frequency of the system by measuring the rate
of oscillation as well as the damping ratio by measuring the rate
of decay. The natural frequency and damping ratio are not only
important in free vibration, but also characterize how a system
will behave under forced vibration.

### Forced vibration with damping

In this section we will look at the behavior of
the spring mass damper model when we add a harmonic force in the
form below. A force of this type could, for example, be generated
by a rotating imbalance.

- F= F_0 \cos \!

If we again sum the forces on the mass we get the
following ordinary differential equation:

- m \ddot + \dot + x = F_0 \cos

The steady state
solution of this problem can be written as:

- x(t)= X \cos \!

The result states that the mass will oscillate at
the same frequency, f, of the applied force, but with a phase shift
\phi .

The amplitude of the vibration “X” is defined by
the following formula.

- X=

Where “r” is defined as the ratio of the harmonic
force frequency over the undamped natural frequency of the
mass-spring-damper model.

- r=\frac

The phase shift , \phi, is defined by following
formula.

- \phi= \arctan

The plot of these functions, called "the
frequency response of the system", presents one of the most
important features in forced vibration. In a lightly damped system
when the forcing frequency nears the natural frequency (r \approx 1
) the amplitude of the vibration can get extremely high. This
phenomenon is called resonance
(subsequently the natural frequency of a system is often referred
to as the resonant frequency). In rotor bearing systems any
rotational speed that excites a resonant frequency is referred to
as a critical
speed.

If resonance occurs in a mechanical system it can
be very harmful-- leading to eventual failure of the system.
Consequently, one of the major reasons for vibration analysis is to
predict when this type of resonance may occur and then to determine
what steps to take to prevent it from occurring. As the amplitude
plot shows, adding damping can significantly reduce the magnitude
of the vibration. Also, the magnitude can be reduced if the natural
frequency can be shifted away from the forcing frequency by
changing the stiffness or mass of the system. If the system cannot
be changed, perhaps the forcing frequency can be shifted (for
example, changing the speed of the machine generating the
force).

The following are some other points in regards to
the forced vibration shown in the frequency response plots.

- At a given frequency ratio, the amplitude of the vibration, X, is directly proportional to the amplitude of the force F_0 (e.g. If you double the force, the vibration doubles)
- With little or no damping, the vibration is in phase with the forcing frequency when the frequency ratio r 1
- When rF_0 . This deflection is called the static deflection \delta_. Hence, when r>1 the amplitude of the vibration is actually less than the static deflection \delta_. In this region the force generated by the mass (F=ma) is dominating because the acceleration seen by the mass increases with the frequency. Since the deflection seen in the spring, X, is reduced in this region, the force transmitted by the spring (F=kx) to the base is reduced. Therefore the mass-spring-damper system is isolating the harmonic force from the mounting base—referred to as vibration isolation. Interestingly, more damping actually reduces the effects of vibration isolation when r>>1 because the damping force (F=cv) is also transmitted to the base.

#### What causes resonance?

Resonance is simple to understand if you view the
spring and mass as energy storage elements--with the mass storing
kinetic energy and the spring storing potential energy. As
discussed earlier, when the mass and spring have no force acting on
them they transfer energy back forth at a rate equal to the natural
frequency. In other words, if energy is to be efficiently pumped
into both the mass and spring the energy source needs to feed the
energy in at a rate equal to the natural frequency. Applying a
force to the mass and spring is similar to pushing a child on
swing, you need to push at the correct moment if you want the swing
to get higher and higher. As in the case of the swing, the force
applied does not necessarily have to be high to get large motions;
the pushes just need to keep adding energy into the system.

The damper, instead of storing energy, dissipates
energy. Since the damping force is proportional to the velocity,
the more the motion the more the damper dissipates the energy.
Therefore a point will come when the energy dissipated by the
damper will equal the energy being fed in by the force. At this
point, the system has reached its maximum amplitude and will
continue to vibrate at this level as long as the force applied
stays the same. If no damping exists, there is nothing to dissipate
the energy and therefore theoretically the motion will continue to
grow on into infinity.

#### Applying "complex" forces to the mass-spring-damper model

In a previous section only a simple harmonic
force was applied to the model, but this can be extended
considerably using two powerful mathematical tools. The first is
the Fourier
transform that takes a signal as a function of time (time domain)
and breaks it down into its harmonic components as a function of
frequency (frequency
domain). For example, let us apply a force to the
mass-spring-damper model that repeats the following cycle--a force
equal to 1 newton for 0.5
second and then no force for 0.5 second. This type of force has the
shape of a 1 Hz square
wave.

The Fourier transform of the square wave
generates a frequency
spectrum that presents the magnitude of the harmonics that make
up the square wave (the phase is also generated, but is typically
of less concern and therefore is often not plotted). The Fourier
transform can also be used to analyze non-periodic
functions such as transients (e.g. impulses) and random functions.
With the advent of the modern computer the Fourier transform is
almost always computed using the Fast
Fourier Transform (FFT) computer algorithm in combination with
a window
function.

In the case of our square wave force, the first
component is actually a constant force of 0.5 newton and is
represented by a value at "0" Hz in the frequency spectrum. The
next component is a 1 Hz sine wave with an amplitude of 0.64. This
is shown by the line at 1 Hz. The remaining components are at odd
frequencies and it takes an infinite amount of sine waves to
generate the perfect square wave. Hence, the Fourier transform
allows you to interpret the force as a sum of sinusoidal forces
being applied instead of a more "complex" force (e.g. a square
wave). In the previous section, the vibration solution was given
for a single harmonic force, but the Fourier transform will in
general give multiple harmonic forces. The second mathematical
tool, "the principle of superposition",
allows you to sum the solutions from multiple forces if the system
is linear. In
the case of the spring-mass-damper model, the system is linear if
the spring force is proportional to the displacement and the
damping is proportional to the velocity over the range of motion of
interest. Hence, the solution to the problem with a square wave is
summing the predicted vibration from each one of the harmonic
forces found in the frequency spectrum of the square wave.

#### Frequency response model

We can view the solution of a vibration problem
as an input/output relation--where the force is the input and the
output is the vibration. If we represent the force and vibration in
the frequency domain (magnitude and phase) we can write the
following relation:

- X(\omega)=H(\omega)* F(\omega) \ \ or \ \ H(\omega)= .

H(\omega) is called the frequency response
function (also referred to the transfer function, but not
technically as accurate) and has both a magnitude and phase
component (if represented as a complex
number, a real and imaginary component). The magnitude of the
frequency response function (FRF) was presented earlier for the
mass-spring-damper system.

- |H(\omega)|=\left | \right|= , \ \ where\ \ r=\frac=\frac

The phase of the FRF was also presented earlier
as:

- \angle H(\omega)= \arctan .

For example, let us calculate the FRF for a
mass-spring-damper system with a mass of 1 kg, spring stiffness of
1.93 N/mm and a damping ratio of 0.1. The values of the spring and
mass give a natural frequency of 7 Hz for this specific system. If
we apply the 1 Hz square wave from earlier we can calculate the
predicted vibration of the mass. The figure illustrates the
resulting vibration. It happens in this example that the fourth
harmonic of the square wave falls at 7 Hz. The frequency response
of the mass-spring-damper therefore outputs a high 7 Hz vibration
even though the input force had a relatively low 7 Hz harmonic.
This example highlights that the resulting vibration is dependent
on both the forcing function and the system that the force is
applied.

The figure also shows the time domain
representation of the resulting vibration. This is done by
performing an inverse Fourier Transform that converts frequency
domain data to time domain. In practice, this is rarely done
because the frequency spectrum provides all the necessary
information.

The frequency response function (FRF) does not
necessarily have to be calculated from the knowledge of the mass,
damping, and stiffness of the system, but can be measured
experimentally. For example, if you apply a known force and sweep
the frequency and then measure the resulting vibration you can
calculate the frequency response function and then characterize the
system. This technique is used in the field of experimental
modal
analysis to determine the vibration characteristics of a
structure.

## See also

- Cushioning
- Critical speed
- Damping
- Mechanical resonance
- Modal analysis
- Mode shape
- Noise, Vibration, and Harshness
- Quantum vibration
- Random vibration
- Shock
- Simple harmonic oscillator
- Sound
- Structural dynamics
- Torsional vibration
- Vibration analysis of machines
- Vibration isolation
- Wave
- Whole body vibration
- FFT

## References

- Inman, Daniel J., Engineering Vibration, Prentice Hall, 2001, ISBN 0-13-726142
- Rao, Singiresu, Mechanical Vibrations, Addison Wesley, 1990, ISBN 0-201-50156-2
- Thompson, W.T., Theory of Vibrations, Nelson Thornes Ltd, 1996, ISBN 0-412-783908
- Hartog, Den, Mechanical Vibrations, Dover Publications, 1985, ISBN 0-486-647854

## External links

- Hyperphysics Educational Website, Concepts
- Thermotron Industries, of Electrodynamic Vibration Testing Handbook
- Nelson Publishing, Evaluation Engineering Magazine
- Essay Undamped free vibration
- Essay Viscously damped free vibration
- Structural Dynamics and Vibration Laboratory of McGill University
- Normal vibration modes of a circular membrane

vibrations in German: Vibration

vibrations in Spanish: Vibración

vibrations in Persian: ارتعاش

vibrations in French: Vibration

vibrations in Hungarian: Rezgés

vibrations in Icelandic: Titringur

vibrations in Italian: Vibrazione

vibrations in Dutch: trilling

vibrations in Japanese: 振動運動

vibrations in Portuguese: Vibração

vibrations in Russian: Виброизоляция

vibrations in Russian: Вибрация

vibrations in Swedish: Vibration

vibrations in Turkish:
Titreşim