Undamped free vibration pdf
For mechanical oscillations in the form of machining context, see Machining vibrations. Vibration is a mechanical phenomenon whereby oscillations occur about an equilibrium point. Undamped free vibration pdf 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.
The studies of sound and vibration are closely related. Hence, attempts to reduce noise are often related to issues of vibration. Car Suspension: designing vibration control is undertaken as part of acoustic, automotive or mechanical engineering. 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 and damps down to motionlessness.
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. 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.
Vibration testing is accomplished by introducing a forcing function into a structure, usually with some type of shaker. For higher frequencies, electrodynamic shakers are used. Generally, one or more “input” or “control” points located on the DUT-side of a fixture is kept at a specified acceleration. The most common types of vibration testing services conducted by vibration test labs are Sinusoidal and Random. 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 the frequency range of the vibration test spectrum. This section does not cite any sources. Fast Fourier Transform 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. 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. This solution says that it will oscillate with simple harmonic motion that has an amplitude of A and a frequency of fn. The number fn is called the undamped natural frequency. Vibrational motion could be understood in terms of conservation of energy.
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. This damping ratio is just a ratio of the actual damping over the amount of damping required to reach critical damping. 05, while automotive suspensions are in the range of 0.
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.
The plots to the side present how 0. 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. The plot of these functions, called “the frequency response of the system”, presents one of the most important features in forced vibration. In rotor bearing systems any rotational speed that excites a resonant frequency is referred to as a critical speed. 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. The following are some other points in regards to the forced vibration shown in the frequency response plots. 1 the effects of the damper and the mass are minimal. 1, which is very helpful when it comes to determining the natural frequency of the system.