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Encyclopedia > Vibration
Look up vibration in
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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. Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... Random redirects here. ...

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. A tuning fork is a simple metal two-pronged fork with the tines formed from a U-shaped bar of elastic material (usually steel). ... A reed is a thin strip of material which vibrates to make music. ... A woodwind instrument is an instrument in which sound is produced by blowing against an edge or by a vibrating with air a thin piece of wood known as a reed. ... A harmonica is a free reed wind instrument. ... For the Marty Friedman album, see Loudspeaker (album) An inexpensive low fidelity 3. ...

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 minimise unwanted vibrations. Sound is a disturbance of mechanical energy that propagates through matter as a wave. ... This article is about noise as in sound. ... For other uses, see Engine (disambiguation). ... For other kinds of motors, see motor. ... This article is about devices that perform tasks. ... Engine balance is the design, construction and tuning of an engine to run smoothly. ... For other uses, see Friction (disambiguation). ... For other uses, see Gear (disambiguation). ...

The study of sound and vibration are closely related. Sound, pressure waves, are generated by vibrating structures (e.g. vocal cords) and pressure waves can generate vibration of structures (e.g. ear drum). Hence, when trying to reduce noise it is often a problem in trying to reduce vibration. A WAVES Photographer 3rd Class The WAVES were a World War II era division of the U.S. Navy that consisted entirely of women. ... Laryngoscopic view of the vocal folds. ... The tympanum or tympanic membrane, colloquially known as eardrum, is a thin membrane that separates the outer ear from the middle ear. ...


Types of vibration

Free vibration occurs when a mechanical system is set off with an initial input and then allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and then letting go or hitting a tuning fork and letting it ring. The mechanical system will then vibrate at one or more of its natural frequencies and damp down to zero. For other uses, see Frequency (disambiguation). ...

Forced vibration is when an alternating force or motion is applied to a mechanical system. Examples of this type of vibration include a shaking washing machining due to an imbalance, transportation vibration (caused by truck engine, springs, road, etc), or the vibration of a building during an earthquake. In forced vibration the frequency of the vibration is the frequency of the force or motion applied, but the magnitude of the vibration is strongly dependent on the mechanical system itself. This article is about the natural seismic phenomenon. ...

Vibration testing

Vibration testing is accomplished by introducing an forcing function into a structure usually with some type of shaker. Generally, one or more points on the structure are controlled to a specified vibration level. Two typical types of vibration test performed are random and sine test. Sine test are performed to survey the structural response of the device under test (DUT). A random test a generally conducted to replicate a real world environment.

Vibration analysis

The fundamentals of vibration analysis can be understood by studying the simple mass-spring-damper model. Indeed, even a complex structure such as an automobile body can be modeled as a summation of simple mass-spring-damper models. The mass-spring-damper model is an example of a simple harmonic oscillator and hence the mathematics used to describe its behavior is identical to other simple harmonic oscillators such as the RLC circuit. A harmonic oscillator is either a mechanical system in which there exists a returning force F directly proportional to the displacement x, i. ... An RLC circuit (also known as a resonant circuit or a tuned circuit) is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. ...

Note: In this article the step by step mathematical derivations will not be included, but will focus on the major equations and concepts in vibration analysis. Please refer to the references at the end of the article for detailed derivations.

Free vibration without damping

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). Image File history File links No higher resolution available. ...

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. Newtons First and Second laws, in Latin, from the original 1687 edition of the Principia Mathematica. ...

 Sigma F = ma = m ddot{x} = m frac{d^2x}{dt^2}

The sum of the forces on the mass then generates this ordinary differential equation: In mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of only one independent variable, and one or more of its derivatives with respect to that variable. ...

m ddot{x} + 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 the 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 fn, but what is fn? fn is one of the most important quantities in vibration analysis and is called the undamped natural frequency. Simple harmonic motion is the motion of a simple harmonic oscillator, a motion that is neither driven nor damped. ... It has been suggested that pulse amplitude be merged into this article or section. ...

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

 f_n = {1over {2 pi}} sqrt{k over m} !

Note: Angular frequency ω (ω = 2πf) 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. It has been suggested that this article or section be merged into Angular velocity. ...

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 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 hence 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 {1}{2} 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 any energy stored, but the mass has reached its maximum speed and hence all the energy has been transformed into kinetic energy (tfrac {1}{2} 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 into 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. Look up conservation of energy in Wiktionary, the free dictionary. ... Potential energy can be thought of as energy stored within a physical system. ... The cars of a roller coaster reach their maximum kinetic energy when at the bottom of their path. ...

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 comes to rest.

Free vibration with damping

Mass Spring Damper Model

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 in 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). Image File history File links Mass-Spring-Damper. ...

 F_d = - c v = - c dot{x} = - c frac{dx}{dt} !

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

m ddot{x} + { c } dot{x} + {k } 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 will stop vibrating over time. This case is called underdamping--the case 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{k m}

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 (ζ) of the mass spring damper model is:

zeta = { c over 2 sqrt{k m} }.

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^{-zeta omega_n t} cos({sqrt{1-zeta^2} omega_n t - phi}) ,   omega_n= 2pi f_n

The value of X, the initial magnitude, and φ, the phase shift, are determined by the amount the spring is stretched. The formulas for these values can be found in the references. Image File history File links Size of this preview: 438 × 599 pixel Image in higher resolution (733 × 1002 pixel, file size: 65 KB, MIME type: image/png) Lzyvz For vibration article I, the creator of this work, hereby release it into the public domain. ... This article is about a portion of a periodic process. ...

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, fd, and is related to the undamped natural frequency by the following formula:

f_d= sqrt{1-zeta^2} 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 and 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 would, for example, be generated by a rotating imbalance.

F= F_0 cos {(2 pi f t)} !

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

m ddot{x} + { c } dot{x} + {k } x = F_0 cos {(2 pi f t)}

The steady state solution of this problem can be written as: HELLO EVERYONE!! Steady state is a more general situation than Dynamic equilibrium. ...

x(t)= X cos {(2 pi f t -phi)} !

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

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

X= {F_0 over k} {1 over sqrt{(1-r^2)^2 + (2 zeta r)^2}}

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


The phase shift , φ, is defined by following formula.

phi= arctan {left (frac{2 zeta r}{1-r^2} right)}

Forced Vibration Response Image File history File links Size of this preview: 800 × 285 pixel Image in higher resolution (2455 × 875 pixel, file size: 209 KB, MIME type: image/png) lzyvzl Created for use with Vibration Article I, the creator of this work, hereby release it into the public domain. ...

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 the resonant frequency is referred to as the critical speed. Mechanical Resonance is the debut album by the American rock band Tesla. ... There are very few or no other articles that link to this one. ...

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 resonance may occur and 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 F0 (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 and 180 degrees out of phase when the frequency ratio r >1
  • When r<<1 the amplitude is just the deflection of the spring under the static force F0. This deflection is called the static deflection δst. Hence, when r<<1 the effects of the damper and the mass are minimal.
  • When r>>1 the amplitude of the vibration is actually less than the static deflection δst. 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.

Vibration isolation is the process of isolating an object, such as a piece of equipment, from the source of vibrations. ...

What causes resonance?

Resonance is simple to understand if you view the spring and mass as energy storage elements--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 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 amplitude 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 to 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 break 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. In mathematics, the Fourier transform is a certain linear operator that maps functions to other functions. ... Time-domain is a term used to describe the analysis of mathematical functions, or real-life signals, with respect to time. ... Frequency domain is a term used to describe the analysis of mathematical functions with respect to frequency. ... For other uses, see Newton (disambiguation). ... A square wave is a kind of basic waveform. ...

How a 1 Hz square wave can be represented as a summation of sine waves(harmonics) and the corresponding frequency spectrum

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. Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (841 × 841 pixel, file size: 170 KB, MIME type: image/gif) LZYVZL Vibration Article I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the... Image File history File links Size of this preview: 600 × 600 pixel Image in higher resolution (841 × 841 pixel, file size: 170 KB, MIME type: image/gif) LZYVZL Vibration Article I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this document under the... Familiar concepts associated with a frequency are colors, musical notes, radio/TV channels, and even the regular rotation of the earth. ... In mathematics, a periodic function is a function that repeats its values after some definite period has been added to its independent variable. ... The Fast Fourier Transform (FFT) is an efficient algorithm to compute the discrete Fourier transform (DFT) and its inverse. ... In signal processing, a window function (or apodization function) is a function that is zero-valued outside of some chosen interval. ...

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 the 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 the 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. In linear algebra, the principle of superposition states that, for a linear system, a linear combination of solutions to the system is also a solution to the same linear system. ... A linear system is a model of a system based on some kind of linear operator. ...

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)= {X(omega) over F(omega)}

H(ω) 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. In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = −1. ...

|H(omega)|=left |{X(omega) over F(omega)} right|= {1 over k} {1 over sqrt{(1-r^2)^2 + (2 zeta r)^2}},   where  r=frac{f}{f_n}=frac{omega}{omega_n}

The phase of the FRF was also presented earlier as:

angle H(omega)= arctan {left (frac{2 zeta r}{1-r^2} right)}

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.

Frequency Response Model

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 then calculate the frequency response function, and hence characterize the system. This technique is used in the field of experimental modal analysis to determine the vibration characteristics of a structure.
Image File history File links Size of this preview: 800 × 436 pixel Image in higher resolution (2471 × 1346 pixel, file size: 161 KB, MIME type: image/png) lzyvzl For use in Vibration article I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this... Image File history File links Size of this preview: 800 × 436 pixel Image in higher resolution (2471 × 1346 pixel, file size: 161 KB, MIME type: image/png) lzyvzl For use in Vibration article I, the creator of this work, hereby grant the permission to copy, distribute and/or modify this... Modal analysis is the study of the dynamic properties of structures under vibrational excitation. ...


  • 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

Testing Labs

NSWCDD, Dahlgren, VA at Pumpkin Neck

Other Resources

See also

External links

  • http://mywebsite.bigpond.com/npajkic/vibration/undamped_free/index.html

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