Wize University Physics Textbook (Master) > Periodic Motion: Oscillations

Underdamped Oscillations

Damped Oscillation

In real systems, there are always resistive forces that oppose oscillation. These resistive forces cause the total energy of the system to decay. A type of these resistive forces is proportional to the speed of the oscillator and is show by:

F_{resistive}=-bv

where $b$ is the damping constant and shows the strength of the resistive force.

Image illustrating Damped Oscillation Theory

The net force acting on the oscillator in this case is equal to:

F=-kx-bv=ma

The solution of the above equation for $x\left(t\right)$ and the behaviour of the physical system depend on the magnitude of the damping constant.

Underdamped Condition

For small values of damping constants $\left(\frac{b}{2m}<\sqrt{\frac{k}{m}}\right)$ , the object undergoes an underdamped oscillation which is described by:

\boxed{x\left(t\right)=A_oe^{-\frac{bt}{2m}}\cos\left(\omega't+\varphi\right)}

The Time Constant $\tau$ is defined as $\tau=\dfrac{2m}{b}$ and it is a measure of damping strength. It is a time it takes for the amplitude of the oscillation to drop to $1/e$ of its initial value.

Underdamped oscillation happens with a different angular velocity than the one for SHM. This angular frequency is known as damped angular frequency and is usually shown by $\omega_{damped}$ ,or $\omega'$ :

\boxed{\ \ \omega'=\sqrt{\omega_o^2-\left(\frac{b}{2m}\right)^2}=\sqrt{\frac{k}{m}-\left(\frac{b}{2m}\right)^2}\ \ }

$\omega_o$ is the angular frequency of the undamped system (SHM) and is also known as natural frequency.

Energy in a Damped Oscillation

Energy is no longer conserved in a damped oscillation. The total energy of the system could still be obtained by $\frac{1}{2}kA^2$ but here $A$ is a function of time:

\boxed{E\left(t\right)=\frac{1}{2}kA^2\left(t\right)=\frac{1}{2}kA_o^2e^{-\frac{bt}{m}}}

Example: Decaying Amplitude vs Decaying Energy

A $0.3\ kg$ mass undergoes a damped oscillation with the damping constant $5\ \frac{kg}{s}$ . How long does it take for the amplitude and total energy of the oscillation to be $\frac{1}{4}$ of their initial values?

Solution:

Let's set $t_1$ to be the time at which amplitude is dropped to $\frac{1}{4}$ of its initial value.

$A\left(t_1\right)=A_{0\ }e^{-\dfrac{bt_1}{2m}}=\dfrac{1}{4}A_0$ $\to e^{-\dfrac{bt_1}{2m}}=\dfrac{1}{4}\to\ln\left(\dfrac{1}{4}\right)=-\dfrac{bt_1}{2m}=-\ln\left(4\right)$

$\to\ln\left(4\right)=\dfrac{bt_1}{2m}\to t_1=\dfrac{2m\ln\left(4\right)\ }{b}=\dfrac{2\times0.3\times\ln\left(4\right)}{5}=0.1663\ s$

Now let's do the same calculations for energy and set $t_2$ as the time at which energy is dropped to $\frac{1}{4}$ of its initial value.

$E\left(t_2\right)=\dfrac{1}{2}kA_0^2e^{-\dfrac{bt_2}{m}}=\dfrac{1}{4}E_0=\dfrac{1}{4}\times\dfrac{1}{2}kA_0^2$ $\to e^{-\dfrac{bt_2}{m}}=\dfrac{1}{4}\to\ln\left(4\right)=\dfrac{bt_2}{m}$

$\to t_2=\dfrac{m\ln\left(4\right)}{b}=\dfrac{0.3\ln\left(4\right)}{5}=0.0832\ s$

As you can see it takes shorter for energy to decay to one quarter of its initial value compared to amplitude.

Extra Practice

Underdamped Oscillation

A mass-spring (m=1kg, k=10N/m, b=1 Kg/s) starts oscillation at time 0;

a) What is the frequency of oscillation?

b) By how much does the amplitude decrease after 3s?

Simple Harmonic Motion: Pendulum

A large pendulum has 10 m length and it has a mass of 50 kg attached to that. The pendulum is oscillating in an under-damped condition. The initial amplitude of the oscillation is 1.2 m and it drops to 0.90 m after 5 days. Calculate the time it takes for the total energy of this pendulum to drop by 1/10 factor.

Simple harmonic motion: Pendulum

A large pendulum has $10\ m$ length and it has a mass of $50\ kg$ attached to that. The pendulum is oscillating in an under-damped condition. The initial amplitude of the oscillation is $1.2\ m$ and it drops to $0.90\ m$ after 5 days. Calculate the time it takes for the total energy of this pendulum to drop by $\frac{1}{10}$ factor (in days) . Include 3 decimal places in your answer.

Underdamped Oscillations

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Damped Oscillation

Underdamped Condition

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Energy in a Damped Oscillation

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Example: Decaying Amplitude vs Decaying Energy

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Extra Practice