Fundamentals of Vibration
Natural frequency and damping ratio. There is a standard, and useful, normalization of the second order homogeneous linear constant coefficient ODE mx + bx˙. The practice amplitude of vibrations becomes progressively smaller as energy is lost due to friction between the oscillating body and the particles in the air. lower half-power point (lower frequency at the half-power level) There are many ways to show or derive the damping ratio relationships described in table 1 .
Advanced Search Summary The theory of linear viscoelasticity is the simplest constitutive model that can be adopted to accurately predict the small-strain mechanical response of materials exhibiting the ability to both store and dissipate strain energy.
An important result implied by this theory is the relationship existing between material attenuation and the velocity of propagation of a mechanical disturbance. The functional dependence of these important parameters is represented by the Kramers—Kronig KK equations, also known as dispersion equations, which are nothing but a statement of the necessary and sufficient conditions to satisfy physical causality.
This paper illustrates the derivation of exact solutions of the KK equations to provide explicit relations between frequency-dependent phase velocity and material damping ratio or equivalently, quality factor.
The assumptions that form the basis of the derivation are not beyond those established by the standard theory of viscoelasticity for a viscoelastic solid. The explicit expression for phase velocity as a function of damping ratio was derived by means of the theory of linear singular integral equations, and in particular by the solution of the associated Homogeneous Riemann Boundary Value Problem.
It is shown that the same solution may be obtained also by using the implications of physical causality on the Fourier Transform.
On the other hand, the explicit solution for damping ratio as a function of phase velocity was found through the components of the complex wavenumber. The exact solutions make it possible to obtain frequency-dependent material damping ratio solely from phase velocity measurements, and conversely. Hence, these relations provide an innovative and inexpensive tool to determine the small-strain dynamic properties of geomaterials.
It is shown that the obtained rigorous solutions are in good agreement with well-known solutions based on simplifying assumptions that have been developed in the fields of seismology and geotechnical engineering. In addition, the theoretical results have been validated using recently published experimental material functions for fine-grained soils and for polymethyl methacrylate.
Damped Harmonic Oscillator
A distinctive feature of linear viscoelasticity theory is that, in contrast to linear elasticity where material constants are prescribed, the description of a viscoelastic constitutive model requires the specification of material functions of either frequency or time. The more damping present in a mechanical system, the shorter the time to stop moving.
Note that if the value of one of these damping forms is known, the other forms can be mathematically derived. It is just a matter of using the equations to transform the value to a different form. Why have different ways of expressing damping? Mostly, this is to make it easier to discuss differences in damping. This means that any vibration set in motion in the structure would decay faster due to the increased damping.
Red would be considered to have more damping than green. Depending on the form being used to express damping, the value may be higher or lower. Cautions When calculating damping from a FRF, there are several items which can influence the final results: It is broken into discrete data points at a fixed frequency interval or resolution.
For example, this could be a 1. This will influence how the frequency values f0, f1 and f2 are determined.
Bode plot of the same system response acquired with four different frequency resolutions: