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7 Ιουν 2024 · The damping equation provides a mathematical representation of the damping force acting on a system. This force opposes the motion and helps dissipate energy, reducing the amplitude as time progresses.
Describe the motion of driven, or forced, damped harmonic motion. Write the equations of motion for forced, damped harmonic motion. In the real world, oscillations seldom follow true SHM. Friction of some sort usually acts to dampen the motion so it dies away, or needs more force to continue.
The general equation for an exponentially damped sinusoid may be represented as: = where: y ( t ) {\displaystyle y(t)} is the instantaneous amplitude at time t ; A {\displaystyle A} is the initial amplitude of the envelope;
In the absence of a damping term, the ratio k/m would be the square of the circular frequency of a solution, so we will write k/m = n 2 with n > 0, and call n the natural circular frequency of the system. Divide the equation through by m: x ̈ + (b/m) ̇x + 2 n x = 0.
This kind of motion is called critically-damped. The easiest way to get a handle on this is to simply plug the condition into the solution, Equation 8.3.4 : \[x\left(t\right) = \left[A\sin\phi\right]e^{-\sqrt\frac{k}{m} t}\]
Eq.(4) is the desired equation of motion for harmonic motion with air drag. It models what is known as damped harmonic oscillations, and is more realistic than the case where b is assumed to be zero. It can thus be readily applied to most every-day oscillating systems provided they can be defined one-dimensionally.
In the presence of damping, the equation of motion looks like \[M \frac{d^{2}}{d t^{2}} \psi(t)=-M \Gamma \frac{d}{d t} \psi(t)-K \psi(t) ,\] where \(M \Gamma\) is the matrix that describes the velocity dependent damping.
