Mechanical Vibrations Ss Rao

[Andreas Mbili]

Vibrationfrom Latin vibrāre 'to shake') is a mechanical phenomenon whereby oscillations occur about an equilibrium point. Vibration may be deterministic if the oscillations can be characterised precisely (e.g. the periodic motion of a pendulum), or random if the oscillations can only be analysed statistically (e.g. the movement of a tire on a gravel road).

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 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.[1]

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 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.

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.[1] 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.[3]

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 fixture[4] 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)[5] of the actual in-use mounting. For this reason, to ensure repeatability between vibration tests, vibration fixtures are designed to be resonance free[5] 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 strategies[6] 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 vibroscopes.

Vibration analysis (VA), applied in an industrial or maintenance environment aims to reduce maintenance costs and equipment downtime by detecting equipment faults.[7][8] VA is a key component of a condition monitoring (CM) program, and is often referred to as predictive maintenance (PdM).[9] 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.[10]

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 the faulty component.

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 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 \displaystyle \zeta ) of the mass-spring-damper model is:

The value of X, 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 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 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.

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 is almost always computed using the fast Fourier transform (FFT) computer algorithm in combination with a window function.

In the case of our square wave force, the first component is actually a constant force of 0.5 newton and is represented by a value at 0 Hz in the frequency spectrum. The next component is a 1 Hz sine wave with an amplitude of 0.64. This is shown by the line at 1 Hz. The remaining components are at odd frequencies and it takes an infinite amount of sine waves to generate the perfect square wave. Hence, the Fourier transform allows you to interpret the force as a sum of sinusoidal forces being applied instead of a more "complex" force (e.g. a square wave).

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.

where [ M ] , \displaystyle \beginbmatrixM\endbmatrix, [ C ] , \displaystyle \beginbmatrixC\endbmatrix, and [ K ] \displaystyle \beginbmatrixK\endbmatrix 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.

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