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65-460: Vibration can be desirable: for example, the motion of a tuning fork , the reed in a woodwind instrument or harmonica , a mobile phone , or the cone of a loudspeaker . In many cases, however, vibration is undesirable, wasting energy and creating unwanted sound . For example, the vibrational motions of engines , electric motors , or any mechanical device in operation are typically unwanted. Such vibrations could be caused by imbalances in

130-452: A /12 if the prongs have rectangular cross-section of width a along the direction of motion. Tuning forks have traditionally been used to tune musical instruments , though electronic tuners have largely replaced them. Forks can be driven electrically by placing electronic oscillator -driven electromagnets close to the prongs. A number of keyboard musical instruments use principles similar to tuning forks. The most popular of these

195-460: 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). Summing the forces on the mass results in the following ordinary differential equation: The solution to this equation depends on

260-430: 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. The Fourier transform

325-654: A 360- hertz steel tuning fork as its timekeeper, powered by electromagnets attached to a battery-powered transistor oscillator circuit. The fork provided greater accuracy than conventional balance wheel watches. The humming sound of the tuning fork was audible when the watch was held to the ear. Alternatives to the common A=440 standard include philosophical or scientific pitch with standard pitch of C=512. According to Rayleigh , physicists and acoustic instrument makers used this pitch. The tuning fork John Shore gave to George Frideric Handel produces C=512. Tuning forks, usually C512, are used by medical practitioners to assess

390-763: 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

455-446: A fixed tone. The main reason for using the fork shape is that, unlike many other types of resonators, it produces a very pure tone , with most of the vibrational energy at the fundamental frequency . The reason for this is that the frequency of the first overtone is about ⁠ 5 / 2 ⁠ = ⁠ 25 / 4 ⁠ = 6 + 1 ⁄ 4 times the fundamental (about 2 + 1 ⁄ 2 octaves above it). By comparison,

520-463: A frequency of 32,768 Hz in the ultrasonic range (above the range of human hearing). It is made to vibrate by small oscillating voltages applied by an electronic oscillator circuit to metal electrodes plated on the surface of the crystal. Quartz is piezoelectric , so the voltage causes the tines to bend rapidly back and forth. The Accutron , an electromechanical watch developed by Max Hetzel and manufactured by Bulova beginning in 1960, used

585-539: 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. Vibrational motion could be understood in terms of conservation of energy . In the above example the spring has been extended by a value of x and therefore some potential energy ( 1 2 k x 2 {\displaystyle {\tfrac {1}{2}}kx^{2}} )

650-453: A patient's hearing. This is most commonly done with two exams called the Weber test and Rinne test , respectively. Lower-pitched ones, usually at C128, are also used to check vibration sense as part of the examination of the peripheral nervous system. Orthopedic surgeons have explored using a tuning fork (lowest frequency C=128) to assess injuries where bone fracture is suspected. They hold

715-399: A piezoelectric device. Upon coming in contact with solids, amplitude of oscillation goes down, the same is used as a switching parameter for detecting point level for solids. For liquids, the resonant frequency of tuning fork changes upon coming in contact with the liquids, change in frequency is used to detect level. Vibroscope Vibroscope ( Latin : vibrare 'vibrate' + scope )

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780-470: A real fracture while wondering if a response means a sprain. A systematic review published in 2014 in BMJ Open suggests that this technique is not reliable or accurate enough for clinical use. Tuning forks also play a role in several alternative therapy practices, such as sonopuncture and polarity therapy . A radar gun that measures the speed of cars or a ball in sports is usually calibrated with

845-451: 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 fixture used to attach the DUT to the shaker table must be designed for

910-406: A solid sheet is slid in between the prongs of a vibrating fork, the apparent volume actually increases , as this cancellation is reduced, just as a loudspeaker requires a baffle in order to radiate efficiently. Commercial tuning forks are tuned to the correct pitch at the factory, and the pitch and frequency in hertz is stamped on them. They can be retuned by filing material off the prongs. Filing

975-436: A standard temperature. The standard temperature is now 20 °C (68 °F), but 15 °C (59 °F) is an older standard. The pitch of other instruments is also subject to variation with temperature change. The frequency of a tuning fork depends on its dimensions and what it is made from: where The ratio I / A in the equation above can be rewritten as r /4 if the prongs are cylindrical with radius r , and

1040-468: A surface or with an object, and emits a pure musical tone once the high overtones fade out. A tuning fork's pitch depends on the length and mass of the two prongs. They are traditional sources of standard pitch for tuning musical instruments. The tuning fork was invented in 1711 by British musician John Shore , sergeant trumpeter and lutenist to the royal court. A tuning fork is a fork-shaped acoustic resonator used in many applications to produce

1105-422: A system behaves under forced vibration. 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. Summing the forces on the mass results in the following ordinary differential equation: The steady state solution of this problem can be written as: The result states that the mass will oscillate at

1170-420: A tuning fork is specifically used in the Weber and Rinne tests for hearing in order to bypass the middle ear . If just held in open air, the sound of a tuning fork is very faint due to the acoustic impedance mismatch between the steel and air. Moreover, since the feeble sound waves emanating from each prong are 180° out of phase , those two opposite waves interfere , largely cancelling each other. Thus when

1235-445: A tuning fork. Instead of the frequency, these forks are labeled with the calibration speed and radar band (e.g., X-band or K-band) they are calibrated for. Doubled and H-type tuning forks are used for tactical-grade Vibrating Structure Gyroscopes and various types of microelectromechanical systems . Tuning fork forms the sensing part of vibrating point level sensors . The tuning fork is kept vibrating at its resonant frequency by

1300-464: A very high acoustic impedance ), is partly converted into audible sound in air which involves a much greater motion ( particle velocity ) at a relatively low pressure (thus low acoustic impedance). The pitch of a tuning fork can also be heard directly through bone conduction , by pressing the tuning fork against the bone just behind the ear, or even by holding the stem of the fork in one's teeth, conveniently leaving both hands free. Bone conduction using

1365-418: Is overdamped . The value that the damping coefficient must reach for critical damping in the mass-spring-damper model is: 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

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1430-415: Is almost always computed using the fast Fourier transform (FFT) computer algorithm in combination with a window function . Tuning fork A tuning fork is an acoustic resonator in the form of a two-pronged fork with the prongs ( tines ) formed from a U-shaped bar of elastic metal (usually steel ). It resonates at a specific constant pitch when set vibrating by striking it against

1495-417: 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 on a smoked surface of a rotating cylinder . Vibroscopes are used to study properties of substances. For examples, polymers ' torsional modulus and Young's modulus may be determined by vibrating

1560-414: 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

1625-428: Is defined by the following formula. [REDACTED] 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 ≈ 1 {\displaystyle r\approx 1} ) the amplitude of the vibration can get extremely high. This phenomenon

1690-450: Is easier to tune other instruments with this pure tone. Another reason for using the fork shape is that it can then be held at the base without damping the oscillation. That is because its principal mode of vibration is symmetric, with the two prongs always moving in opposite directions, so that at the base where the two prongs meet there is a node (point of no vibratory motion) which can therefore be handled without removing energy from

1755-420: 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

1820-414: 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. When

1885-501: 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 ( 1 2 m v 2 {\displaystyle {\tfrac {1}{2}}mv^{2}} ). The mass then begins to decelerate because it

1950-463: 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, by applying 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

2015-473: Is the Rhodes piano , in which hammers hit metal tines that vibrate in the magnetic field of a pickup , creating a signal that drives electric amplification. The earlier, un-amplified dulcitone , which used tuning forks directly, suffered from low volume. The quartz crystal that serves as the timekeeping element in modern quartz clocks and watches is in the form of a tiny tuning fork. It usually vibrates at

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2080-421: 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

2145-505: 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 of the TWF. The vibration spectrum provides important frequency information that can pinpoint

2210-437: 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

2275-484: The RLC circuit . 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. 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

2340-650: The DUT-side of a vibration fixture is kept at a specified acceleration. 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

2405-401: 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

2470-470: The axis under test) permitted to be exhibited by the vibration test fixture. Devices specifically designed to trace or record vibrations are called vibroscopes . 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 (CM) program, and is often referred to as predictive maintenance (PdM). Most commonly VA

2535-400: The damping ratio ( ζ {\displaystyle \zeta } ) of the mass-spring-damper model is: 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: The value of X ,

2600-441: 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 , {\displaystyle f_{\text{d}},} and is related to the undamped natural frequency by the following formula: The damped natural frequency

2665-432: 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 strategies 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

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2730-411: The distance of A and releasing, the solution to the above equation that describes the motion of mass is: This solution says that it will oscillate with simple harmonic motion that has an amplitude of A and a frequency of f n . The number f n is called the undamped natural frequency . For the simple mass–spring system, f n is defined as: Note: angular frequency ω (ω=2 π f ) with

2795-486: The end of the vibrating fork on the skin above the suspected fracture, progressively closer to the suspected fracture. If there is a fracture, the periosteum of the bone vibrates and fires nociceptors (pain receptors), causing a local sharp pain. This can indicate a fracture, which the practitioner refers for medical X-ray. The sharp pain of a local sprain can give a false positive. Established practice, however, requires an X-ray regardless, because it's better than missing

2860-577: The ends of the prongs raises the pitch, while filing the inside of the base of the prongs lowers it. Currently, the most common tuning fork sounds the note of A = 440 Hz , the standard concert pitch that many orchestras use. That A is the pitch of the violin's second-highest string, the highest string of the viola, and an octave above the highest string of the cello. Orchestras between 1750 and 1820 mostly used A = 423.5 Hz, though there were many forks and many slightly different pitches. Standard tuning forks are available that vibrate at all

2925-475: 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. 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

2990-444: 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 . The mathematics used to describe its behavior is identical to other simple harmonic oscillators such as

3055-402: The first overtone of a vibrating string or metal bar is one octave above (twice) the fundamental, so when the string is plucked or the bar is struck, its vibrations tend to mix the fundamental and overtone frequencies. When the tuning fork is struck, little of the energy goes into the overtone modes; they also die out correspondingly faster, leaving a pure sine wave at the fundamental frequency. It

3120-423: 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. 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

3185-405: The frequency range of the vibration test spectrum. It is difficult to design a vibration test fixture which duplicates the dynamic response (mechanical impedance) of the actual in-use mounting. For this reason, to ensure repeatability between vibration tests, vibration fixtures are designed to be resonance free within the test frequency range. Generally for smaller fixtures and lower frequency ranges,

3250-400: The initial magnitude, and ϕ , {\displaystyle \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

3315-424: 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

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3380-399: 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: The force generated by

3445-402: The mass is proportional to the acceleration of the mass as given by Newton's second law of motion : The sum of the forces on the mass then generates this ordinary differential equation :   m x ¨ + k x = 0. {\displaystyle \ m{\ddot {x}}+kx=0.} Assuming that the initiation of vibration begins by stretching the spring by

3510-429: The oscillation (damping). However, there is still a tiny motion induced in the handle in its longitudinal direction (thus at right angles to the oscillation of the prongs) which can be made audible using any sort of sound board . Thus by pressing the tuning fork's base against a sound board such as a wooden box, table top, or bridge of a musical instrument, this small motion, but which is at a high acoustic pressure (thus

3575-487: The pitches within the central octave of the piano, and also other pitches. Tuning fork pitch varies slightly with temperature, due mainly to a slight decrease in the modulus of elasticity of steel with increasing temperature. A change in frequency of 48 parts per million per °F (86 ppm per °C) is typical for a steel tuning fork. The frequency decreases (becomes flat ) with increasing temperature. Tuning forks are manufactured to have their correct pitch at

3640-440: The polymers and measuring their frequency of vibration under certain external forces. Similar approach works to determine linear density of thread-shaped objects, such as fibers , filaments , and yarn . Vibroscopes are also used to study sound in different areas of the mouth during speech. Jean-Marie Duhamel published about an early recording device he called a vibroscope in 1843. This physics -related article

3705-455: The process of subtractive manufacturing . Free vibration or natural 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 and damps down to motionlessness. Forced vibration

3770-498: The rotating parts, uneven friction , or the meshing of gear teeth. Careful designs usually minimize unwanted vibrations. The studies of sound and vibration are closely related (both fall under acoustics ). 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, attempts to reduce noise are often related to issues of vibration. Machining vibrations are common in

3835-419: The same frequency, f , of the applied force, but with a phase shift ϕ . {\displaystyle \phi .} The amplitude of the vibration “X” is defined by the following formula. 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 , {\displaystyle \phi ,}

3900-428: 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

3965-415: 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

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4030-417: 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

4095-575: 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 Hz 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

4160-437: 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,

4225-400: 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 . Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker. Alternately,

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