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Theory of Vibrations with Applications

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ISBN 13: 9780136510680

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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 Hz , servohydraulic electrohydraulic shakers are used. For higher frequencies typically 5 Hz to 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.

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. The vibration test fixture 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 [3] of the actual in-use mounting.

For this reason, to ensure repeatability between vibration tests, vibration fixtures are designed to be resonance free [3] 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 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. 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. The mathematics used to describe its behavior is identical to other simple harmonic oscillators such as 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. 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 negative sign indicates that the force is always opposing the motion of the mass attached to it:.

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

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.

Theory of Vibration with Applications 5th Edition

For the simple mass—spring system, f n is defined as:. 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. Vibrational motion could be understood in terms of conservation of energy. 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. 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. 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 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.

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If the damping is increased past critical damping, the system is overdamped. The value that the damping coefficient must reach for critical damping in the mass-spring-damper model is:. This damping ratio is just a ratio of the actual damping over the amount of damping required to reach critical damping. For example, metal structures e. The solution to the underdamped system for the mass-spring-damper model is the following:. 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 cosine function is the oscillating portion of the solution, but the frequency of the oscillations is different from the undamped case.

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.


Theory of Vibration with application 5th Solution

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