Rate of depreciation

Number without dimension. It is about the factor which takes into account the damping, and which governs with him only the mode of the système.
The rate of depreciation of a material depends on the temperature and the frequency.

For a deadened harmonic oscillator, with a mass m, a damping coefficient C, and a constant of stiffness K, the rate of depreciation is: $\backslash \; zeta\; =\; \backslash \; frac\; \{C\}\; \{2\; \backslash \; sqrt\; \{K\; m\}\}$.

Mathematical definition of the rate of depreciation

The ordinary differential equation modelling a deadened harmonic oscillator is: $m\; \backslash \; frac\; \{d^2x\}\; \{dt^2\}\; +\; C\; \backslash \; frac\; \{dx\}\; \{dt\}\; +\; K\; X\; =\; 0$

By using the natural pulsation of the Oscillating harmonic $\backslash \; omega\_0\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{K\}\; \{m\}\}$ and the definition of the rate of depreciation, the differential equation becomes: $\backslash \; frac\; \{d^2x\}\; \{dt^2\}\; +\; 2\; \backslash \; zeta\; \backslash \; omega\_0\; \backslash \; frac\; \{dx\}\; \{dt\}\; +\; \backslash \; omega\_0^2\; X\; =\; 0$

One solves the characteristic Polynôme:

$\backslash \; omega^2\; +\; 2\; \backslash \; zeta\; \backslash \; omega\_0\; \backslash \; Omega\; +\; \backslash \; omega\_0^2\; =\; 0$

From where $\backslash \; Omega\; =\; \backslash \; omega\_0\; (-\; \backslash \; zeta\; \backslash \; pm\; \backslash \; sqrt\; \{\backslash \; zeta^2\; -\; 1\})$

• Periodic : if ω then the solution is purely imaginary is a sinusoid the form $e^\; \{\backslash \; pm\; J\; \backslash \; omega\_0\; T\}$. This corresponds to the case of a harmonic oscillator. It appears for the borderline case $\backslash \; zeta\; =\; 0$.

• Pseudoperiodic : if ω complex is , then the solution is the product of exponential decreasing and a sinusoid of the form $e^\; \{\backslash \; omega\_0\; (-\; \backslash \; zeta\; \backslash \; pm\; J\; \backslash \; sqrt\; \{1\; -\; \backslash \; zeta^2\})\; T\}$. This phenomenon appears for .

• Aperiodic criticism : it is the border between the pseudoperiodic mode and the aperiodic mode. It is often the optimal solution with a problem of damped oscillations. It appears for the borderline case $\backslash \; zeta\; =\; 1$.

• Aperiodic : if ω is real, then the solution is simply exponential decreasing without oscillation. It appears for the case $\backslash \; zeta\; >\; 1$.

See Too

Internal bonds

• physical Damping

• Measurement by Dynamic Mechanical Analysis (DMA)