TheInfoList

Vibration is a mechanical phenomenon whereby
oscillation Oscillation is the repetitive variation, typically in time Time is the indefinite continued sequence, progress of existence and event (philosophy), events that occur in an apparently irreversible process, irreversible succession from the pas ...

equilibrium point In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
. The word comes from Latin ''vibrationem'' ("shaking, brandishing"). The oscillations may be , such as the motion of a pendulum—or
random In common parlance, randomness is the apparent or actual lack of pattern or predictability in events. A random sequence of events, symbols or steps often has no :wikt:order, order and does not follow an intelligible pattern or combination. I ...

, such as the movement of a tire on a gravel road. Vibration can be desirable: for example, the motion of a
tuning fork A tuning fork is an acoustic Acoustic may refer to: Music Albums * Acoustic (Bayside EP), ''Acoustic'' (Bayside EP) * Acoustic (Britt Nicole EP), ''Acoustic'' (Britt Nicole EP) * Acoustic (Joey Cape and Tony Sly album), ''Acoustic'' (Joey Cap ...

, the
reed Reed or Reeds may refer to: Science, technology, biology, and medicine * Reed bird (disambiguation) * Reed pen, writing implement in use since ancient times * Reed (plant), one of several tall, grass-like wetland plants of the order Poales * ...
in a
woodwind instrument Woodwind instruments are a family of musical instruments A musical instrument is a device created or adapted to make musical sounds. In principle, any object that produces sound can be considered a musical instrument—it is through purpose t ...
or
harmonica The harmonica, also known as a French harp or mouth organ A mouth organ is any free reed aerophone A free reed aerophone is a musical instrument that produces sound as air flows past a vibrating reed (instrument), reed in a frame. Air pressu ...

, a
mobile phone A mobile phone, cellular phone, cell phone, cellphone, handphone, or hand phone, sometimes shortened to simply mobile, cell or just phone, is a portable telephone A telephone is a telecommunications device that permits two or more use ...

, or the cone of a
loudspeaker A loudspeaker (or ''speaker driver'', or most frequently just ''speaker'') is an Acoustical engineering#Electroacoustics, electroacoustic transducer, that is, a device that converts an electrical audio signal into a corresponding sound. A ''spe ...

. In many cases, however, vibration is undesirable, wasting
energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regula ...

and creating unwanted
sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the b ...

. For example, the vibrational motions of
engine An engine or motor is a designed to convert one or more forms of into . Available energy sources include (e.g. energy of the Earth's as exploited in ), heat energy (e.g. ), , and nuclear energy (from or ). Many of these processes genera ...

s, s, or any
mechanical device A machine is a man-made Artificiality (the state of being artificial or man-made) is the state of being the product of intentional human manufacture, rather than occurring naturally through processes not involving or requiring human activity. ...

in operation are typically unwanted. Such vibrations could be caused by imbalances in the rotating parts, uneven
friction Friction is the force In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related en ...

, or the meshing of
gear Cast iron mortise wheel with wooden cogs (powered by an external water wheel) meshing with a cast iron gear wheel, connected to a pulley A pulley is a wheel on an axle or shaft (mechanical engineering), shaft that is designed to support ...

teeth. Careful designs usually minimize unwanted vibrations. The studies of sound and vibration are closely related. Sound, or pressure
wave In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular su ...

s, are generated by vibrating structures (e.g.
vocal cords In humans, vocal cords, also known as vocal chords, vocal folds or voice reeds, are folds of tissue in the throat that are key in creating sounds through vocalization. The size of vocal cords affects the pitch of voice. Open when breathing and ...
); these pressure waves can also induce the vibration of structures (e.g.
ear drum In the anatomy Anatomy (Greek ''anatomē'', 'dissection') is the branch of biology concerned with the study of the structure of organism In biology, an organism (from Ancient Greek, Greek: ὀργανισμός, ''organismos'') is a ...
). Hence, attempts to reduce noise are often related to issues of vibration.

# Types of vibration

Free vibration occurs when a mechanical system is set in motion with an initial input and allowed to vibrate freely. Examples of this type of vibration are pulling a child back on a swing and letting it go, or hitting a tuning fork and letting it ring. The mechanical system vibrates at one or more of its
natural frequencies Nature, in the broadest sense, is the natural, physical, material world or universe. "Nature" can refer to the phenomenon, phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of ...

and damps down to motionlessness. Forced vibration is when a time-varying disturbance (load, displacement, velocity, or acceleration) is applied to a mechanical system. The disturbance can be a periodic and steady-state input, a transient input, or a random input. The periodic input can be a harmonic or a non-harmonic disturbance. Examples of these types of vibration include a washing machine shaking due to an imbalance, transportation vibration caused by an engine or uneven road, or the vibration of a building during an earthquake. For linear systems, the frequency of the steady-state vibration response resulting from the application of a periodic, harmonic input is equal to the frequency of the applied force or motion, with the response magnitude being dependent on the actual mechanical system. Damped vibration: When the energy of a vibrating system is gradually dissipated by friction and other resistances, the vibrations are said to be damped. The vibrations gradually reduce or change in frequency or intensity or cease and the system rests in its equilibrium position. An example of this type of vibration is the vehicular suspension dampened by the
shock absorber A shock absorber or damper is a mechanical or hydraulic Hydraulics (from Greek#REDIRECT Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a ...
.

# Vibration testing

Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker. Alternately, a DUT (device under test) is attached to the "table" of a shaker. Vibration testing is performed to examine the response of a device under test (DUT) to a defined vibration environment. The measured response may be ability to function in the vibration environment, fatigue life, resonant frequencies or squeak and rattle sound output ( NVH). Squeak and rattle testing is performed with a special type of ''quiet shaker'' that produces very low sound levels while under operation. For relatively low frequency forcing (typically less than 100 Hz), servohydraulic (electrohydraulic) shakers are used. For higher frequencies (typically 5 Hz to 2000 Hz), electrodynamic shakers are used. Generally, one or more "input" or "control" points located on the DUT-side of a vibration fixture is kept at a specified acceleration.Tustin, Wayne.
Where to place the control accelerometer: one of the most critical decisions in developing random vibration tests also is the most neglected
', EE-Evaluation Engineering, 2006
Other "response" points may experience higher vibration levels (resonance) or lower vibration level (anti-resonance or damping) than the control point(s). It is often desirable to achieve anti-resonance to keep a system from becoming too noisy, or to reduce strain on certain parts due to vibration modes caused by specific vibration frequencies. The most common types of vibration testing services conducted by vibration test labs are sinusoidal and random. Sine (one-frequency-at-a-time) tests are performed to survey the structural response of the device under test (DUT). During the early history of vibration testing, vibration machine controllers were limited only to controlling sine motion so only sine testing was performed. Later, more sophisticated analog and then digital controllers were able to provide random control (all frequencies at once). A random (all frequencies at once) test is generally considered to more closely replicate a real world environment, such as road inputs to a moving automobile. Most vibration testing is conducted in a 'single DUT axis' at a time, even though most real-world vibration occurs in various axes simultaneously. MIL-STD-810G, released in late 2008, Test Method 527, calls for multiple exciter testing. The ''vibration test fixtureTony Araujo.
The evolution of automotive vibration fixturing
', EE-Evaluation Engineering, 2019
'' used to attach the DUT to the shaker table must be designed for the frequency range of the vibration test spectrum. It is difficult to design a vibration test fixture which duplicates the dynamic response (mechanical impedance)Blanks, H.S., "Equivalence Techniques for Vibration Testing," SVIC Notes, pp 17. of the actual in-use mounting. For this reason, to ensure repeatability between vibration tests, vibration fixtures are designed to be resonance freeBlanks, H.S., "Equivalence Techniques for Vibration Testing," SVIC Notes, pp 17. within the test frequency range. Generally for smaller fixtures and lower frequency ranges, the designer can target a fixture design that is free of resonances in the test frequency range. This becomes more difficult as the DUT gets larger and as the test frequency increases. In these cases multi-point control strategiesAraujo, T. and Yao, B., ''"Vibration Fixture Performance Qualification - A Review of Automotive Industry Best Practices," SAE Technical Paper 2020-01-1065, 2020, https://doi.org/10.4271/2020-01-1065.'' can mitigate some of the resonances that may be present in the future. Some vibration test methods limit the amount of crosstalk (movement of a response point in a mutually perpendicular direction to the axis under test) permitted to be exhibited by the vibration test fixture. Devices specifically designed to trace or record vibrations are called
vibroscope Vibroscope ( la, vibrare 'vibrate' + scope) is an instrument for observing and tracing (and sometimes recording) vibration. For example, a primitive mechanical vibroscope consists of a vibrating object with a pointy end which leaves a wave trace ...
s.

# Vibration analysis

Vibration analysis (VA), applied in an industrial or maintenance environment aims to reduce maintenance costs and equipment downtime by detecting equipment faults. VA is a key component of a
condition monitoring Condition monitoring (colloquially, CM) is the process of monitoring a parameter of condition in machinery (vibration, temperature etc.), in order to identify a significant change which is indicative of a developing fault. It is a major component of ...
(CM) program, and is often referred to as
predictive maintenance Predictive maintenance techniques are designed to help determine the condition of in-service equipment in order to estimate when maintenance should be performed. This approach promises cost savings over routine or time-based preventive maintenance, ...
(PdM). Most commonly VA is used to detect faults in rotating equipment (Fans, Motors, Pumps, and Gearboxes etc.) such as imbalance, misalignment, rolling element bearing faults and resonance conditions. VA can use the units of Displacement, Velocity and Acceleration displayed as a time waveform (TWF), but most commonly the spectrum is used, derived from a
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in t ...
of the TWF. The vibration spectrum provides important frequency information that can pinpoint the faulty component. 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 In mechanics Mechanics (Ancient Greek, Greek: ) is the area of physics concerned with the motions of physical objects, more specifically the relationships among force, matter, and motion. Forces applied to objects result in Displacement (vec ...

. The mathematics used to describe its behavior is identical to other simple harmonic oscillators such as the
RLC circuit An RLC circuit is an electrical circuit An electrical network is an interconnection of electronic component, electrical components (e.g., battery (electricity), batteries, resistors, inductors, capacitors, switches, transistors) or a model of ...

. Note: This article does not include the step-by-step mathematical derivations, but focuses on major vibration analysis equations and concepts. 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 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: :$F_s=- k x. \!$ The force generated by the mass is proportional to the acceleration of the mass as given by
Newton's second law of motion Newton's laws of motion are three law Law is a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its ...
: :$\Sigma\ F = ma = m \ddot = m \frac.$ The sum of the forces on the mass then generates this
ordinary differential equation In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and the derivatives of those functions. The term ''ordinary'' is used in contrast with the term partial ...
: $\ m \ddot + k x = 0.$ 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 mass is: :$x\left(t\right) = A \cos \left(2 \pi f_n t\right). \!$ This solution says that it will oscillate with
simple harmonic motion In mechanics and physics, simple harmonic motion (sometimes abbreviated ) is a special type of Periodic function, periodic motion where the restoring force on the moving object is directly Proportionality (mathematics), proportional to the magni ...

that has an
amplitude The amplitude of a period Period may refer to: Common uses * Era, a length or span of time * Full stop (or period), a punctuation mark Arts, entertainment, and media * Period (music), a concept in musical composition * Period, a descriptor for ...

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: :$f_n = \sqrt. \!$ Note:
angular frequency In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...
ω (ω=2 π ''f'') with the units of radians per second is often used in equations because it simplifies the equations, but is normally converted to ordinary frequency (units of or equivalently cycles per second) when stating the frequency of a system. If the mass and stiffness of the system is known, the formula above can determine the frequency at which the system vibrates once set in motion by an initial disturbance. Every vibrating system has one or more natural frequencies that it vibrates at once disturbed. This simple relation can be used to understand in general what happens 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 feels ″softer″ than unloaded—the mass has increased, reducing the natural frequency of the system.

### What causes the system to vibrate: from conservation of energy point of view

Vibrational motion could be understood in terms of
conservation of energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...
. In the above example the spring has been extended by a value of x and therefore some
potential energy In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors. Common types of potential energy include the gravitational potentia ...

($\tfrac k x^2$) is stored in the spring. Once released, the spring tends to return to its un-stretched state (which is the minimum potential energy state) and in the process accelerates the mass. At the point where the spring has reached its un-stretched state all the potential energy that we supplied by stretching it has been transformed into
kinetic energy In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular ...
($\tfrac m v^2$). 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. Thus oscillation of the spring amounts to the transferring back and forth of the kinetic energy into potential energy. In this simple model the mass continues to oscillate forever at the same magnitude—but in a real system, ''damping'' always dissipates the energy, eventually bringing the spring to rest.

## Free vibration with damping

When a "viscous" damper is added to the model this outputs a force that is proportional to the velocity of the mass. The damping is called viscous because it models the effects of a fluid within an object. The proportionality constant ''c'' is called the damping coefficient and has units of Force over velocity (lbf⋅s/in or N⋅s/m). :$F_\text = - c v = - c \dot = - c \frac.$ Summing the forces on the mass results in the following ordinary differential equation: :$m \ddot + c \dot + kx = 0.$ The solution to this equation depends on the amount of damping. If the damping is small enough, the system still vibrates—but eventually, over time, stops vibrating. This case is called underdamping, which is important in vibration analysis. If damping is increased just to the point where the system no longer oscillates, the system has reached the point of critical damping. If the damping is increased past critical damping, the system is
overdamped Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples inclu ...

. The value that the damping coefficient must reach for critical damping in the mass-spring-damper model is: :$c_\text = 2 \sqrt.$ To characterize the amount of damping in a system a ratio called the
damping ratio Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
(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 fuselages, engine crankshafts) have damping factors less than 0.05, while automotive suspensions are in the range of 0.2–0.3. The solution to the underdamped system for the mass-spring-damper model is the following: :$x\left(t\right)=X e^ \cos\left\left( \sqrt \omega_n t - \phi \right\right) , \qquad \omega_n = 2\pi f_n.$ The value of ''X'', the initial magnitude, and $\phi,$ the
phase shift In physics Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular succ ...

, are determined by the amount the spring is stretched. The formulas for these values can be found in the references.

### Damped and undamped natural frequencies

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", $f_\text,$ and is related to the undamped natural frequency by the following formula: :$f_\text= f_n\sqrt.$ 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 “rings” 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 behaves under forced vibration.

## Forced vibration with damping

The behavior of the spring mass damper model varies with the addition of a harmonic force. A force of this type could, for example, be generated by a rotating imbalance. :$F= F_0 \sin\left(2 \pi f t\right). \!$ Summing the forces on the mass results in the following ordinary differential equation: :$m \ddot + c\dot + k x = F_0 \sin\left(2 \pi f t\right).$ The
steady state In systems theory Systems theory is the interdisciplinary study of system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influen ...

solution of this problem can be written as: :$x\left(t\right)= X \sin\left(2 \pi f t +\phi\right). \!$ 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 the following formula. :$\phi= \arctan\left \left(\frac \right\right).$ 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 Resonance describes the phenomenon of increased amplitude The amplitude of a ic is a measure of its change in a single (such as or ). There are various definitions of amplitude (see below), which are all s of the magnitude of the differ ...
(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 In solid mechanics Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of force In physics ...

. 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 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 $F_0.$ This deflection is called the static deflection $\delta_.$ 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 $\delta_.$ 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 isolationVibration isolation is the process of isolating an object, such as a piece of equipment, from the source of vibrations. Vibration is undesirable in many domains, primarily engineered systems and habitable spaces, and methods have been developed to ...
. More damping actually reduces the effects of vibration isolation when ''r'' ≫ 1 because the damping force (''F'' = ''cv'') is also transmitted to the base. * Whatever the damping is, the vibration is 90 degrees out of phase with the forcing frequency when the frequency ratio ''r'' = 1, which is very helpful when it comes to determining the natural frequency of the system. * Whatever the damping is, when ''r'' ≫ 1, the vibration is 180 degrees out of phase with the forcing frequency * Whatever the damping is, when ''r'' ≪ 1, the vibration is in phase with the forcing frequency

### Resonance causes

Resonance is simple to understand if the spring and mass are viewed 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 external force acting on them they transfer energy back and forth at a rate equal to the natural frequency. In other words, to efficiently pump energy into both mass and spring requires that the energy source 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, a push is needed at the correct moment to make the swing get higher and higher. As in the case of the swing, the force applied need not be high to get large motions, but must just add energy to 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, there is a point when the energy dissipated by the damper equals the energy added 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, theoretically, the motion will continue to grow 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#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
that takes a signal as a function of time (
time domain Time domain refers to the analysis of mathematical functions Mathematics (from Greek: ) includes the study of such topics as quantity Quantity is a property that can exist as a multitude or magnitude, which illustrate discontinuity ...
) and breaks it down into its harmonic components as a function of frequency (
frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or Signal (information theory), signals with respect to frequency, rather than time. Put simply, a time-dom ...
). For example, by applying a force to the mass–spring–damper model that repeats the following cycle – a force equal to 1
newton Newton most commonly refers to: * Isaac Newton (1642–1726/1727), English scientist * Newton (unit), SI unit of force named after Isaac Newton Newton may also refer to: Arts and entertainment * Newton (film), ''Newton'' (film), a 2017 Indian fil ...
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 A square wave is a non-sinusoidal periodic waveform in which the amplitude alternates at a steady frequency Frequency is the number of occurrences of a repeating event per unit of time A unit of time is any particular time Tim ...

. The Fourier transform of the square wave generates a
frequency spectrum The power spectrum S_(f) of a x(t) describes the distribution of into frequency components composing that signal. According to , any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over ...
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- functions such as transients (e.g. impulses) and random functions. The Fourier transform is almost always computed using the
fast Fourier transform A fast Fourier transform (FFT) is an algorithm that computes the discrete Fourier transform (DFT) of a sequence, or its inverse (IDFT). Fourier analysis converts a signal from its original domain (often time or space) to a representation in t ...
(FFT) computer algorithm in combination with a
window function In and , a window function (also known as an apodization function or tapering function) is a that is zero-valued outside of some chosen , normally symmetric around the middle of the interval, usually near a maximum in the middle, and usually t ...
. 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 in general gives multiple harmonic forces. The second mathematical tool, the
superposition principle The superposition principle, also known as superposition property, states that, for all linear system In systems theory Systems theory is the interdisciplinary study of systems, i.e. cohesive groups of interrelated, interdependent parts that ...
, allows the summation of the solutions from multiple forces if the system is
linear Linearity is the property of a mathematical relationship (''function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out se ...
. 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

The solution of a vibration problem can be viewed as an input/output relation – where the force is the input and the output is the vibration. Representing the force and vibration in the frequency domain (magnitude and phase) allows the following relation: :$X\left(i\omega\right)=H\left(i\omega\right)\cdot F\left(i\omega\right) \text H\left(i\omega\right)= .$ $H\left(i\omega\right)$ is called the
frequency response In electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons The electron is a subatomic particle In physical sciences, subatomic particles ...
function (also referred to as the
transfer function In engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad rang ...

, but not technically as accurate) and has both a magnitude and phase component (if represented as a
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

, a real and imaginary component). The magnitude of the frequency response function (FRF) was presented earlier for the mass–spring–damper system. :$, H\left(i\omega\right), =\left , \= , \text r=\frac=\frac.$ The phase of the FRF was also presented earlier as: :$\angle H\left(i\omega\right)= -\arctan\left \left(\frac \right\right).$ For example, calculating 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. Applying the 1 Hz square wave from earlier allows the calculation of 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 to. 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 a known force over a range of frequencies is applied, and if the associated vibrations are measured, the frequency response function can be calculated, thereby characterizing the system. This technique is used in the field of experimental
modal analysis Modal analysis is the study of the dynamic properties of systems in the frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or Signal (inform ...
to determine the vibration characteristics of a structure.

# Multiple degrees of freedom systems and mode shapes

The simple mass–spring–damper model is the foundation of vibration analysis, but what about more complex systems? The mass–spring–damper model described above is called a single
degree of freedom Degree may refer to: As a unit of measurement * Degree symbol (°), a notation used in science, engineering, and mathematics * Degree (angle), a unit of angle measurement * Degree (temperature), any of various units of temperature measurement ...
(SDOF) model since the mass is assumed to only move up and down. In more complex systems, the system must be discretized into more masses that move in more than one direction, adding degrees of freedom. The major concepts of multiple degrees of freedom (MDOF) can be understood by looking at just a 2 degree of freedom model as shown in the figure. The equations of motion of the 2DOF system are found to be: :$m_1 \ddot + \left(c_1+c_2\right) \dot - c_2 \dot+ \left(k_1+k_2\right) x_1 - k_2 x_2= f_1,$ :$m_2 \ddot - c_2 \dot+ \left(c_2+c_3\right) \dot - k_2 x_1+ \left(k_2+k_3\right) x_2 = f_2. \!$ This can be rewritten in
matrix Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols, or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the material in between a eukaryoti ...
format: :$\beginm_1 & 0\\ 0 & m_2\end\begin\ddot\\ \ddot \end + \begin c_1+c_2 & -c_2\\ -c_2 & c_2+c_3\end\begin\dot\\ \dot\end+\begink_1+k_2 & -k_2\\ -k_2 & k_2+k_3\end\begin x_1\\ x_2\end=\begin f_1\\ f_2\end.$ A more compact form of this matrix equation can be written as: :$\beginM\end\begin\ddot\end+\beginC\end\begin\dot\end+\beginK\end\begin x\end=\begin f \end$ where $\beginM\end,$ $\beginC\end,$ and $\beginK\end$ are symmetric matrices referred respectively as the mass, damping, and stiffness matrices. The matrices are NxN square matrices where N is the number of degrees of freedom of the system. The following analysis involves the case where there is no damping and no applied forces (i.e. free vibration). The solution of a viscously damped system is somewhat more complicated.Maia, Silva. ''Theoretical And Experimental Modal Analysis'', Research Studies Press Ltd., 1997, :$\beginM\end\begin\ddot\end+\beginK\end\begin x\end=0.$ This differential equation can be solved by assuming the following type of solution: :$\begin x\end=\begin X\ende^.$ Note: Using the exponential solution of $\begin X\ende^$ is a mathematical trick used to solve linear differential equations. Using
Euler's formula Euler's formula, named after Leonhard Euler, is a mathematics, mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex number, complex exponential function. Euler's ...

and taking only the real part of the solution it is the same cosine solution for the 1 DOF system. The exponential solution is only used because it is easier to manipulate mathematically. The equation then becomes: :$\begin-\omega^2 \begin M \end + \begin K \end \end \beginX\ende^=0.$ Since $e^$ cannot equal zero the equation reduces to the following. :$\begin\beginK\end-\omega^2 \begin M \end \end \begin X \end=0.$

## Eigenvalue problem

This is referred to an
eigenvalue In linear algebra, an eigenvector () or characteristic vector of a Linear map, linear transformation is a nonzero Vector space, vector that changes at most by a Scalar (mathematics), scalar factor when that linear transformation is applied to i ...

problem in mathematics and can be put in the standard format by pre-multiplying the equation by $\beginM\end^$ :$\begin\beginM\end^\beginK\end-\omega^2 \begin M \end^ \beginM\end\end\beginX\end=0$ and if: $\beginM\end^\beginK\end=\beginA\end$ and $\lambda=\omega^2 \,$ :$\begin\beginA\end-\lambda\beginI\end\end\beginX\end=0.$ The solution to the problem results in N eigenvalues (i.e. $\omega_1^2,\omega_2^2,\cdots\omega_N^2$), where N corresponds to the number of degrees of freedom. The eigenvalues provide the natural frequencies of the system. When these eigenvalues are substituted back into the original set of equations, the values of $\beginX\end$ that correspond to each eigenvalue are called the eigenvectors. These eigenvectors represent the
mode shape A normal mode of an oscillation, oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixe ...
s of the system. The solution of an eigenvalue problem can be quite cumbersome (especially for problems with many degrees of freedom), but fortunately most math analysis programs have eigenvalue routines. The eigenvalues and eigenvectors are often written in the following matrix format and describe the modal model of the system: :$\begin^\diagdown \omega_^2 \end=\begin \omega_1^2 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & \omega_N^2 \end \text \begin \Psi \end=\begin \begin \psi_1 \end \begin \psi_2 \end \cdots \begin \psi_N \end \end.$ A simple example using the 2 DOF model can help illustrate the concepts. Let both masses have a mass of 1 kg and the stiffness of all three springs equal 1000 N/m. The mass and stiffness matrix for this problem are then: :$\beginM\end=\begin1 & 0\\ 0 & 1\end$ and $\beginK\end=\begin2000 & -1000\\ -1000 & 2000\end.$ Then $\beginA\end=\begin2000 & -1000\\ -1000 & 2000\end.$ The eigenvalues for this problem given by an eigenvalue routine is: :$\begin ^\diagdown \omega_^2 \end=\begin 1000 & 0 \\ 0 & 3000 \end.$ The natural frequencies in the units of hertz are then (remembering $\scriptstyle \omega=2 \pi f$) $\scriptstyle f_1=5.033 \mathrm$ and $\scriptstyle f_2=8.717 \text.$ The two mode shapes for the respective natural frequencies are given as: :$\begin \Psi \end=\begin \begin \psi_1 \end \begin \psi_2 \end \end= \begin \begin -0.707 \\ -0.707 \end_1 \begin 0.707 \\ -0.707 \end_2 \end.$ Since the system is a 2 DOF system, there are two modes with their respective natural frequencies and shapes. The mode shape vectors are not the absolute motion, but just describe relative motion of the degrees of freedom. In our case the first mode shape vector is saying that the masses are moving together in phase since they have the same value and sign. In the case of the second mode shape vector, each mass is moving in opposite direction at the same rate.

## Illustration of a multiple DOF problem

When there are many degrees of freedom, one method of visualizing the mode shapes is by animating them using structural analysis software such as
Femap Femap (''Finite Element Modeling And Postprocessing'') is an engineering analysis program sold by Siemens PLM Software that is used to build finite element models of complex engineering problems ("pre-processing") and view solution results ("post- ...
,
ANSYS Ansys, Inc. is an American company based in Canonsburg, Pennsylvania. It develops and markets multiphysics engineering simulation software for product design, testing and operation and offers its products and services to customers worldwide. Ans ...
or VA One by ESI Group. An example of animating mode shapes is shown in the figure below for a
cantilever A cantilever is a rigid structural elementStructural elements are used in structural analysis Structural analysis is the determination of the effects of load Load or LOAD may refer to: Aeronautics and transportation *Load factor (aeronaut ...

ed -beam as demonstrated using
modal analysis Modal analysis is the study of the dynamic properties of systems in the frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or Signal (inform ...
on ANSYS. In this case, the
finite element method The finite element method (FEM) is a widely used method for numerically solving differential equations In mathematics, a differential equation is an equation In mathematics, an equation is a statement that asserts the equality (mathematics) ...
was used to generate an approximation of the mass and stiffness matrices by meshing the object of interest in order to solve a discrete eigenvalue problem. Note that, in this case, the finite element method provides an approximation of the meshed surface (for which there exists an infinite number of vibration modes and frequencies). Therefore, this relatively simple model that has over 100 degrees of freedom and hence as many natural frequencies and mode shapes, provides a good approximation for the first natural frequencies and modes. Generally, only the first few modes are important for practical applications.
Note that when performing a numerical approximation of any mathematical model, convergence of the parameters of interest must be ascertained.

## Multiple DOF problem converted to a single DOF problem

The eigenvectors have very important properties called orthogonality properties. These properties can be used to greatly simplify the solution of multi-degree of freedom models. It can be shown that the eigenvectors have the following properties: :$\begin\Psi\end^\beginM\end\begin\Psi\end=\begin ^\diagdown m_ \end,$ :$\begin\Psi\end^\beginK\end\begin\Psi\end=\begin ^\diagdown k_ \end.$ $\begin ^\diagdown m_ \end$ and $\begin ^\diagdown k_ \end$ are
diagonal matrices In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. An example of a 2×2 diagonal matrix is \left begin 3 & 0 \\ 0 & 2 \end\right/math>, while ...
that contain the modal mass and stiffness values for each one of the modes. (Note: Since the eigenvectors (mode shapes) can be arbitrarily scaled, the orthogonality properties are often used to scale the eigenvectors so the modal mass value for each mode is equal to 1. The modal mass matrix is therefore an
identity matrix In linear algebra, the identity matrix of size ''n'' is the ''n'' × ''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structu ...

) These properties can be used to greatly simplify the solution of multi-degree of freedom models by making the following coordinate transformation. :$\begin x \end= \begin \Psi \end \begin q \end.$ Using this coordinate transformation in the original free vibration differential equation results in the following equation. : $\beginM\end\begin \Psi \end \begin \ddot \end + \begin K \end \begin \Psi \end \begin q\end=0.$ Taking advantage of the orthogonality properties by premultiplying this equation by $\begin\Psi\end^$ :$\begin\Psi\end^\beginM\end\begin \Psi \end\begin\ddot\end+\begin\Psi\end^\beginK\end \begin \Psi \end \begin q\end=0.$ The orthogonality properties then simplify this equation to: :$\begin ^\diagdown m_ \end \begin\ddot\end+\begin^\diagdown k_\end \begin q\end=0.$ This equation is the foundation of vibration analysis for multiple degree of freedom systems. A similar type of result can be derived for damped systems. The key is that the modal mass and stiffness matrices are diagonal matrices and therefore the equations have been "decoupled". In other words, the problem has been transformed from a large unwieldy multiple degree of freedom problem into many single degree of freedom problems that can be solved using the same methods outlined above. Solving for ''x'' is replaced by solving for ''q'', referred to as the modal coordinates or modal participation factors. It may be clearer to understand if $\begin x \end= \begin \Psi \end \begin q \end$ is written as: :$\begin x_n \end= q_1\begin \psi \end_1 +q_2\begin \psi \end_2 +q_3\begin \psi \end_3 +\cdots + q_N\begin \psi \end_N.$ Written in this form it can be seen that the vibration at each of the degrees of freedom is just a linear sum of the mode shapes. Furthermore, how much each mode "participates" in the final vibration is defined by q, its modal participation factor.

## Rigid-body mode

An unrestrained multi-degree of freedom system experiences both rigid-body translation and/or rotation and vibration. The existence of a rigid-body mode results in a zero natural frequency. The corresponding mode shape is called the rigid-body mode.

*
Acoustic engineering Acoustical engineering (also known as acoustic engineering) is the branch of engineering dealing with sound and oscillation, vibration. It includes the application of acoustics, the science of sound and vibration, in technology. Acoustical engineers ...
*
Anti-vibration compound An anti-vibration compound is a temperature-resistant mixture of a liquid with fine particles, which is used to reduce oscillations Oscillation is the repetitive variation, typically in time, of some measure about a central value (often a point ...
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Balancing machine A balancing machine is a measuring tool used for balancing rotating machine parts such as rotors for electric motors, fans, turbine A turbine ( or ) (from the Greek , ''tyrbē'', or Latin Latin (, or , ) is a classical language belonging to ...
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Base isolation Seismic base isolation, also known as base isolation, or base isolation system, is one of the most popular means of protecting a structure against earthquake forces. It is a collection of structural elements which should substantially decoupl ...
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Cushioning Package cushioning is used to protect items during shipment. Vibration and impact shock during shipment and loading/unloading are controlled by cushioning to reduce the chance of product damage. Cushioning is usually inside a shipping container ...
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Critical speed In solid mechanics Solid mechanics, also known as mechanics of solids, is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of force In physics ...

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Damping ratio Damping is an influence within or upon an oscillatory system that has the effect of reducing or preventing its oscillation. In physical systems, damping is produced by processes that dissipate the energy stored in the oscillation. Examples incl ...
* Dunkerley's method *
Earthquake engineering Earthquake engineering is an interdisciplinary Interdisciplinarity or interdisciplinary studies involves the combination of two or more academic disciplines into one activity (e.g., a research project). It draws knowledge from several other ...
* Elastic pendulum *
Fast Fourier transform A fast Fourier transform (FFT) is an algorithm In and , an algorithm () is a finite sequence of , computer-implementable instructions, typically to solve a class of problems or to perform a computation. Algorithms are always and are u ...
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Mechanical engineering Mechanical engineering is an engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineerin ...

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Mechanical resonance in a mechanical oscillatory system Mechanical resonance is the tendency of a mechanics, mechanical system to respond at greater amplitude when the frequency Frequency is the number of occurrences of a repeating event per unit of time. It is ...
*
Modal analysis Modal analysis is the study of the dynamic properties of systems in the frequency domain In physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or Signal (infor ...
*
Mode shape A normal mode of an oscillation, oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixe ...
*
Noise and vibration on maritime vesselsNoise and vibration on maritime vessels are not the same but they have the same origin and come in many forms. The methods to handle the related problems are similar, to a certain level, where most shipboard noise Noise is unwanted sound consider ...
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Noise, vibration, and harshness Noise, vibration, and harshness (NVH), also known as noise and vibration (N&V), is the study and modification of the noise and vibration characteristics of vehicles, particularly cars and trucks. While noise and vibration can be readily measured, ha ...
* Pallesthesia *
Passive heave compensationPassive Degrees of freedom (mechanics), heave compensation is a technique used to reduce the influence of Wind wave, waves upon lifting and drilling operations. A simple passive heave compensator (PHC) is a soft spring which utilizes Vibration isolat ...
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Pendulum A pendulum is a weight suspended from a pivot Pivot may refer to: *Pivot, the point of rotation in a lever A lever ( or ) is a simple machine consisting of a beam or rigid rod pivoted at a fixed hinge, or '':wikt:fulcrum, fulcrum''. A lever ...

* Quantum vibration * Random vibration *
Ride quality Ride quality refers to a vehicle's effectiveness in insulating the occupants from undulations in the road surface (e.g., bumps or corrugations). A vehicle with good ride quality provides a comfort for the driver and passengers. Importance Good ri ...
* Rayleigh's quotient in vibrations analysis *Shaker (testing device) *Shock (mechanics), Shock *Shock and vibration data logger *Simple harmonic oscillator *Sound *Structural acoustics *Structural dynamics *Tire balance *Torsional vibration *Tuned mass damper *Vibration Calibrator, Vibration calibrator *Vibration control *Vibration isolation *Wave *Whole body vibration