Damped Oscillation Flashcards

A set of fill-in-the-blank practice flashcards covering Simple Harmonic Motion, free damped oscillations, over damping, critical damping, weak damping, logarithmic decrement, relaxation time, quality factor, and phase space diagrams.

A motion of a particle is called Simple Harmonic Motion if the force acting on it is proportional to the displacement with respect to an origin and acts along the direction __________ to the displacement.

opposite

The corresponding equation of motion for free simple harmonic motion is given by $m\ddot{x} = \_\_\_\_\_\_\_\_\_\_$.

$$-kx$$

The general solution of the differential equation $\ddot{x} = -\omega_0^2 x$ for SHM is given by $x(t) = \_\_\_\_\_\_\_\_\_\_$.

$$c_1 e^{i\omega_0 t} + c_2 e^{-i\omega_0 t}$$

In modern dynamics, a plot of displacement $x(t)$ versus velocity $\dot{x}(t)$ is known as the __________ of the dynamical system.

Phase Space

The equation $\left(\frac{\dot{x}}{\omega_0 x_0}\right)^2 + \left(\frac{x}{x_0}\right)^2 = 1$ represents an __________ in phase space, indicating periodic motion.

ellipse

In damped motion, the power dissipated due to damping force is proportional to velocity squared, where the damping force is given by $F_{\text{damping}} = \_\_\_\_\_\_\_\_\_\_$.

$$\Gamma v$$

In the constitutive differential equation of damped motion, the constant $\gamma = \frac{\Gamma}{2m}$ is called the __________.

damping factor

A damped oscillator system exhibits over damping (non-oscillatory motion) when the damping factor $\gamma$ satisfies the condition __________.

$$\gamma > \omega_0$$

A damped oscillator system exhibits critical damping when the damping factor $\gamma$ satisfies the condition __________.

$$\gamma = \omega_0$$

A damped oscillator system exhibits weak damping (oscillatory motion) when the damping factor $\gamma$ satisfies the condition __________.

$$\gamma < \omega_0$$

For a weakly damped oscillator, the modified angular frequency $\tilde{\omega}$ expressed in terms of natural frequency $\omega_0$ and damping factor $\gamma$ is given by $\tilde{\omega} = \_\_\_\_\_\_\_\_\_\_$.

$$\sqrt{\omega_0^2 - \gamma^2}$$

In a weakly damped oscillator, the time average of stored total energy decays exponentially according to $\langle E(t) \rangle = \_\_\_\_\_\_\_\_\_\_$.

$$\frac{1}{2}mA_0^2\omega_0^2 e^{-2\gamma t}$$

The average power consumed by the damping force in a weakly damped oscillator is related to the time average energy by $\langle P(t) \rangle = \_\_\_\_\_\_\_\_\_\_$.

$$2\gamma \langle E(t) \rangle$$

The logarithmic decrement $\lambda$ is defined as the natural logarithm of the ratio of two successive amplitudes separated by a half period $\frac{\tilde{T}}{2}$, giving $\lambda = \_\_\_\_\_\_\_\_\_\_$.

$$\frac{\gamma \tilde{T}}{2}$$

The time $\tau = \frac{1}{\gamma}$ required for the amplitude of a weakly damped oscillation to reduce to $\frac{1}{e}$ of its initial value is called the __________.

relaxation time

The quality factor $Q$ of a weakly damped oscillator expressed in terms of $\tilde{\omega}$ and $\gamma$ is given by $Q = \_\_\_\_\_\_\_\_\_\_$.

$$\frac{\tilde{\omega}}{2\gamma}$$

The shown phase space trajectories represent simple harmonic motion forming closed ellipses, where the difference between the red and blue curves corresponds to different __________ conditions.

initial

The displacement vs time plots shown in the figure illustrate the behavior of a __________ damped oscillator for two different initial conditions.

weakly

In the diagram shown, the sequence of peak amplitudes $A_1, A_2, A_3$ is used to define the __________ decrement.

logarithmic