Physical damping

The damping is the effect generated by the entry of a Système, which tends to be opposed to the variations of the exit of the system.

Explanation

In any real system, part of total energy is dissipated, generally in heat, which creates a force damping.

In mechanics, this one depends on the Speed of the body. In many cases one can suppose that the system is linear, damping being then proportional to the speed (see oscillating Systèmes with a degree of freedom).

In electricity, damping indicates the resistive effect of a circuit RLC.

One defines the damping coefficient C by:

$\backslash \; bold\; \{F\}\; =\; -\; C\; \backslash \; bold\; \{v\}$

Example: Mass-Arise-shock absorber

Study system ideal Mass-Arise-Shock absorber, with Mass m fixed (in the direction where the body keeps the same mass throughout the study), a constant of stiffness K , and a damping coefficient C :

$\backslash \; bold\; \{F\_r\}\; =\; -\; K\; \backslash \; bold\; \{X\}$

$\backslash \; bold\; \{F\_a\}\; =\; -\; C\; \backslash \; frac\; \{D\; \backslash \; bold\; \{X\}\}\; \{dt\}$

The mass is a free body. The inertial reference mark is supposed, therefore the first Vecteur is parallel to the spring and the shock absorber. According to the conservation of the momentum:

$\backslash \; bold\; \{F\_r\}\; +\; \backslash \; bold\; \{F\_a\}\; =\; m\; \backslash \; frac\; \{d^2\; \backslash \; bold\; \{X\}\}\; \{dt\}$

$-\; K\; X\; -\; C\; \backslash \; frac\; \{dx\}\; \{dt\}\; =\; m\; \backslash \; frac\; \{d^2x\}\; \{dt\}$

Ordinary differential equation

It is a ordinary differential equation of the second order. It is linear, homogeneous and with constant coefficients: $m\; \backslash \; frac\; \{d^2x\}\; \{dt^2\}\; +\; C\; \backslash \; frac\; \{dx\}\; \{dt\}\; +\; K\; X\; =\; 0$

In order to simplify the equation, we define two parameters:

The natural pulsation of the system: $\backslash \; omega\_0\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{K\}\; \{m\}\}$

The Rate of depreciation: $\backslash \; zeta\; =\; \backslash \; frac\; \{C\}\; \{2\; \backslash \; sqrt\; \{K\; m\}\}$

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

Transitory mode of the system

The behavior of the system depends on the natural pulsation, and the rate of depreciation. In particular, it strongly depends on the nature of $\backslash \; Omega$.

Pseudoperiodic mode

, the roots $\backslash \; Omega$ is complex and combined. The solution is the sum of two exponential complexes:

$X\; (T)\; =\; Ae^\; \{\backslash \; omega\_1\; T\}\; +\; Be^\; \{\backslash \; omega\_2\; T\}$

$X\; (T)\; =\; Ae^\; \{-\; \backslash \; omega\_0\; (\backslash \; zeta\; -\; J\; \backslash \; sqrt\; \{1\; -\; \backslash \; zeta^2\})\; T\}\; +\; Be^\; \{-\; \backslash \; omega\_0\; (\backslash \; zeta\; +\; J\; \backslash \; sqrt\; \{1\; -\; \backslash \; zeta^2\})\; T\}$

One can rewrite the solution in a trigonometrical form:

$X\; (T)\; =\; e^\; \{-\; \backslash \; frac\; \{T\}\; \{\backslash \; tau\}\}\; T)\; +\; Bsin\; (\backslash \; omega\_d\; T)$

where $\backslash \; tau\; =\; \backslash \; frac\; \{2m\}\; \{C\}$ is the time-constant of the system, and $\backslash \; omega\_d\; =\; \backslash \; omega\_0\; \backslash \; sqrt\; \{1\; -\; \backslash \; zeta^2\}$ is the own pseudo-pulsation of the system. It is noticed that it is always strictly higher than the natural pulsation.

One determines most of the time the constant ones has and B thanks to the initial conditions $x\_0$ and $\backslash \; dowry\; \{x\_0\}$:

$\backslash \; frac\; \{dx\}\; \{dt\}\; =\; e^\; \{-\; \backslash \; frac\; \{T\}\; \{\backslash \; tau\}\}\; (B\; -\; \backslash \; frac\; \{has\}\; \{\backslash \; tau\}\; cos\; (\backslash \; omega\_d\; T)\; -\; has\; +\; \backslash \; frac\; \{B\}\; \{\backslash \; tau\}\; sin\; (\backslash \; omega\_d\; T))$

One solves the system of linear equations:

$\backslash \; begin\; \{boxes\}\; has\; =\; x\_0\; \backslash \; \backslash \; B\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; omega\_d\}\; (\backslash \; dowry\; \{x\_0\}\; +\; \backslash \; frac\; \{x\_0\}\; \{\backslash \; tau\})\; \backslash \; end\; \{boxes\}$

The general homogeneous solution is obtained:

$X\; (T)\; =\; e^\; \{-\; \backslash \; frac\; \{T\}\; \{\backslash \; tau\}\}\; cos\; (\backslash \; omega\_d\; T)\; +\; \backslash \; beta\; sin\; (\backslash \; omega\_d\; T)$

Critical aperiodic mode

$\backslash \; zeta\; =\; 1$, the root $\backslash \; Omega$ is real and double. The solution is the product of a polynomial of order 1 and exponential real:

$X\; (T)\; =\; (+\; LT\; has)\; e^\; \{\backslash \; omega\_0\; T\}$

As $\backslash \; omega\_0$ is real, it translates either a pulsation but a time-constant, therefore one notes $\backslash \; tau\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; omega\_0\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{m\}\; \{K\}\}$

$X\; (T)\; =\; (+\; LT\; has)\; e^\; \{-\; \backslash \; frac\; \{T\}\; \{\backslash \; tau\}\}$

One determines most of the time the constant ones has and B thanks to the initial conditions $x\_0$ and $\backslash \; dowry\; \{x\_0\}$:

$\backslash \; frac\; \{dx\}\; \{dt\}\; =\; (B\; -\; \backslash \; frac\; \{has\}\; \{\backslash \; tau\}\; -\; B\; \backslash \; frac\; \{T\}\; \{\backslash \; tau\})\; e^\; \{-\; \backslash \; frac\; \{T\}\; \{\backslash \; tau\}\}$

One solves the system of linear equations:

$\backslash \; begin\; \{boxes\}\; has\; =\; x\_0\; \backslash \; \backslash \; B\; =\; \backslash \; dowry\; \{x\_0\}\; +\; \backslash \; frac\; \{x\_0\}\; \{\backslash \; tau\}\; \backslash \; end\; \{boxes\}$

The general homogeneous solution is obtained:

$X\; (T)\; =\; (x\_0\; +\; \backslash \; beta\; T)\; e^\; \{-\; \backslash \; frac\; \{T\}\; \{\backslash \; tau\}\}$

Aperiodic mode

$\backslash \; zeta\; >\; 1$, the roots $\backslash \; Omega$ is real and distinct. The solution is the sum of two exponential real:

$X\; (T)\; =\; Ae^\; \{\backslash \; omega\_1\; T\}\; +\; Be^\; \{\backslash \; omega\_2\; T\}$

As $\backslash \; omega\_1$ and $\backslash \; omega\_2$ is real, it translate either a pulsation but a time-constant, therefore one notes $\backslash \; tau\_1\; =\; -\; \backslash \; frac\; \{1\}\; \{\backslash \; omega\_1\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; omega\_0\; (\backslash \; zeta\; -\; \backslash \; sqrt\; \{\backslash \; zeta^2\; -\; 1\})\}$ and $\backslash \; tau\_2\; =\; -\; \backslash \; frac\; \{1\}\; \{\backslash \; omega\_2\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; omega\_0\; (\backslash \; zeta\; +\; \backslash \; sqrt\; \{\backslash \; zeta^2\; -\; 1\})\}$.

$X\; (T)\; =\; has\; e^\; \{-\; \backslash \; frac\; \{T\}\; \{\backslash \; tau\_1\}\}\; +\; B\; e^\; \{-\; \backslash \; frac\; \{T\}\; \{\backslash \; tau\_2\}\}$

One determines most of the time the constant ones has and B thanks to the initial conditions $x\_0$ and $\backslash \; dowry\; \{x\_0\}$:

$\backslash \; frac\; \{dx\}\; \{dt\}\; =\; -\; \backslash \; frac\; \{has\}\; \{\backslash \; tau\_1\}\; e^\; \{-\; \backslash \; frac\; \{T\}\; \{\backslash \; tau\_1\}\}\; -\; \backslash \; frac\; \{B\}\; \{\backslash \; tau\_2\}\; e^\; \{-\; \backslash \; frac\; \{T\}\; \{\backslash \; tau\_2\}\}$

One solves the system of linear equations:

$\backslash \; begin\; \{boxes\}\; has\; =\; \backslash \; frac\; \{\backslash \; tau\_1\; \backslash \; tau\_2\}\; \{\backslash \; tau\_1\; -\; \backslash \; tau\_2\}\; (\backslash \; dowry\; \{x\_0\}\; +\; \backslash \; frac\; \{x\_0\}\; \{\backslash \; tau\_2\})\; \backslash \; \backslash \; B\; =\; \backslash \; frac\; \{\backslash \; tau\_1\; \backslash \; tau\_2\}\; \{\backslash \; tau\_2\; -\; \backslash \; tau\_1\}\; (\backslash \; dowry\; \{x\_0\}\; +\; \backslash \; frac\; \{x\_0\}\; \{\backslash \; tau\_1\})\; \backslash \; end\; \{boxes\}$

Lexicon

• Coefficient damping: Expression in kilograms by second. It is observed that there exists a whole of forces external with the body, which is proportional to the speed of the body. One indicates by damping coefficient the relationship between these forces and speed.

• Constant of stiffness: Expression in Newtons by Meter. It is observed that there exists a whole of forces external with the body, which is proportional to the displacement of the body. One indicates by constant of stiffness the relationship between these forces and displacement.

• Time-constant: Expression in second. Generally, one notes the power of exponential negative only utilizing time like the relationship between this one and a coefficient, homogeneous him also at a time, which takes the name of time-constant. It translates a scale time for the balance of the modelled phenomenon.

• own Pseudo-pulsation: Expression in radians a second. It is about the pulsation of the pseudoperiodic mode, related to the frequency of the phenomenon deadened.

See too

Internal bonds

• Circuit RLC
• Oscillation
• Oscillating harmonic
• Resonance
• Rate of depreciation

Random links:

Mond lady | Blued Ukrainian | Robert Anderson (officer) | Hoya (Germany) | Sui-ren | Garryowen

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org