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Harmonic analysis is a branch of mathematics concerned with investigating the connections between a function and its representation in frequency.The frequency representation is found by using the Fourier transform for functions on unbounded domains such as the full real line or by Fourier series for functions on bounded domains, especially periodic functions on finite intervals.
Frequency refers to how many times per second a vibration cycle is completed. Most machinery runs at 120 Hz, although machinery is available that runs from 100–240 Hz. Frequency is dependent on the mass of the vibrating assembly, and as such can only be changed by switching out components of the assembly.
The period and frequency are determined by the size of the mass m and the force constant k, while the amplitude and phase are determined by the starting position and velocity. The velocity and acceleration of a simple harmonic oscillator oscillate with the same frequency as the position, but with shifted phases. The velocity is maximal for zero ...
where ω is the frequency of the oscillation, A is the amplitude, and δ is the phase shift of the function. These are determined by the initial conditions of the system. Because cosine oscillates between 1 and −1 infinitely, our spring-mass system would oscillate between the positive and negative amplitude forever without friction.
At a given frequency ratio, the amplitude of the vibration, X, is directly proportional to the amplitude of the force (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
In physics, complex harmonic motion is a complicated realm based on the simple harmonic motion.The word "complex" refers to different situations. Unlike simple harmonic motion, which is regardless of air resistance, friction, etc., complex harmonic motion often has additional forces to dissipate the initial energy and lessen the speed and amplitude of an oscillation until the energy of the ...
Mechanical resonance is the tendency of a mechanical system to respond at greater amplitude when the frequency of its oscillations matches the system's natural frequency of vibration (its resonance frequency or resonant frequency) closer than it does other frequencies. It may cause violent swaying motions and potentially catastrophic failure in ...
Models of the first category were presented by Laursen [8] and by Wriggers. [9] An example of the latter category is Kalker’s CONTACT model. [10] A drawback of the well-founded variational approaches is their large computation times. Therefore, many different approximate approaches were devised as well.