Damped and Forced Oscillations and Resonance

Damped Simple Harmonic Motion

In the real world, oscillations do not continue indefinitely with a constant amplitude. The amplitude of oscillations gradually decreases over time, eventually coming to rest. This reduction in amplitude is due to dissipative forces, such as friction, air resistance, or viscosity of the medium. Oscillations where the amplitude decreases over time are called damped oscillations.

Causes of Damping

Damping is caused by forces that oppose the motion and convert the mechanical energy of the oscillation into other forms of energy, primarily heat. Common damping forces include:

Viscous Damping: Friction due to the fluid medium (like air or water) the object is oscillating in. For small speeds, this force is often proportional to the velocity of the object.
Dry Friction (Coulomb Friction): Friction between solid surfaces. This force is roughly constant in magnitude and opposes the motion.
Internal Friction: Dissipation within the oscillating material itself.

Equation of Motion for Damped SHM (Linear Damping)

For many systems, the damping force is approximately proportional to the velocity of the oscillating object and acts in the opposite direction. This is called linear damping or viscous damping, and the force is $F_{damping} = -bv$, where $b$ is the damping constant (a positive value). The restoring force is $F_{restoring} = -kx$.

Applying Newton's Second Law ($F_{net} = ma$) to a mass $m$ undergoing damped linear oscillation:

$ F_{restoring} + F_{damping} = m a $

$ -kx - bv = m \frac{d^2x}{dt^2} $

$ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = 0 $

This is the differential equation for damped Simple Harmonic Motion with linear damping. The solutions to this equation describe oscillations with decreasing amplitude.

Types of Damping

The behaviour of the system depends on the relative values of the damping constant $b$, mass $m$, and force constant $k$. We define a damping ratio, often related to the natural angular frequency $\omega_0 = \sqrt{k/m}$ and a term $\gamma = b/(2m)$.

Underdamped Oscillation ($\gamma < \omega_0$, or $b^2 < 4mk$): The system oscillates with gradually decreasing amplitude. The frequency of oscillation ($\omega_d$) is slightly lower than the natural frequency ($\omega_0$) of the undamped system.
$ \omega_d = \sqrt{\omega_0^2 - \gamma^2} = \sqrt{\frac{k}{m} - \frac{b^2}{4m^2}} $
The displacement can be described by an equation like $ x(t) = A(t) \cos(\omega_d t + \phi) $, where $A(t) = A_0 e^{-\gamma t}$ is the exponentially decreasing amplitude. The time constant for damping is $1/\gamma = 2m/b$.

(Image Placeholder: A graph of displacement vs time. Show an oscillating curve whose amplitude decreases over time, tracing an exponential decay envelope.)
Critically Damped Motion ($\gamma = \omega_0$, or $b^2 = 4mk$): The system returns to the equilibrium position as quickly as possible without oscillating. This is often the desired behaviour in systems like shock absorbers or door closers.
Overdamped Motion ($\gamma > \omega_0$, or $b^2 > 4mk$): The system returns to the equilibrium position slowly and without oscillating. It takes longer to reach equilibrium than in the critically damped case.

Energy Dissipation in Damped Oscillations

In damped oscillations, the mechanical energy ($KE + PE$) of the system is not conserved. The damping force does negative work, which is dissipated as heat. The rate of energy dissipation by the damping force ($F_{damping} = -bv$) is power $P = \vec{F}_{damping} \cdot \vec{v} = (-bv)(\pm v) = -bv^2$. The mechanical energy decreases over time.

Forced Oscillations And Resonance

Damped oscillations die out over time. To maintain oscillations with a constant amplitude or to increase the amplitude, energy must be supplied to the system by an external source. When an oscillating system is subjected to a periodic external force, it undergoes forced oscillations.

Forced Oscillations

Consider a damped oscillating system (mass-spring with damping) subjected to an additional external driving force that varies periodically with time, say $F_{driving}(t) = F_0 \cos(\omega_d t)$, where $F_0$ is the amplitude and $\omega_d$ is the angular frequency of the driving force.

The equation of motion for such a system is:

$ m \frac{d^2x}{dt^2} + b \frac{dx}{dt} + kx = F_0 \cos(\omega_d t) $

The resulting motion of the system has two parts:

Transient Response: Initially, the system oscillates at its natural damped frequency, and the amplitude of this oscillation decays over time due to damping.
Steady-State Response: After the transient part dies out, the system oscillates with a constant amplitude and the same frequency as the driving force ($\omega_d$). This is the forced oscillation.

The amplitude of the steady-state forced oscillation depends on the amplitude of the driving force ($F_0$), the properties of the oscillating system ($m, k, b$), and the frequency of the driving force ($\omega_d$).

Resonance

Resonance is a phenomenon that occurs when the frequency of the driving force is close to the natural frequency of the oscillating system. When this happens, the amplitude of the forced oscillations becomes very large.

The natural frequency of an undamped system is $\omega_0 = \sqrt{k/m}$. For a damped system, the frequency of free oscillations is $\omega_d = \sqrt{\omega_0^2 - \gamma^2}$. The frequency at which the amplitude of forced oscillations is maximum is called the resonance frequency. For small damping, the resonance frequency is very close to the natural frequency $\omega_0$.

The amplitude of forced oscillations is given by (for driving frequency $\omega_{drive}$):

$ A(\omega_{drive}) = \frac{F_0}{\sqrt{(k - m\omega_{drive}^2)^2 + (b\omega_{drive})^2}} $

Resonance occurs when the denominator is minimum, which happens when $(k - m\omega_{drive}^2)^2$ is minimum, i.e., when $k - m\omega_{drive}^2 = 0$, or $\omega_{drive}^2 = k/m = \omega_0^2$. So, the resonance frequency is $\omega_{res} = \omega_0 = \sqrt{k/m}$ in the case of zero damping ($b=0$). When damping is present ($b>0$), the amplitude is maximum at a frequency slightly lower than $\omega_0$, but often $\omega_{res} \approx \omega_0$ if damping is small.

At resonance ($\omega_{drive} \approx \omega_0$), the amplitude becomes large, inversely proportional to the damping constant $b$. If there were no damping ($b=0$), the amplitude at resonance would theoretically become infinite, though this doesn't happen in reality due to unavoidable damping and the fact that Hooke's Law breaks down at very large displacements.

Graph of amplitude vs driving frequency for forced oscillations, showing resonance peak.

(Image Placeholder: A graph with Driving Frequency (omega_drive) on the x-axis and Amplitude (A) on the y-axis. Show curves starting from non-zero amplitude at omega_drive=0, rising to a peak near omega_0, and then falling for higher frequencies. Show multiple curves for different damping constants b: lower damping results in a sharper, higher peak at resonance.)

Applications and Consequences of Resonance

Musical Instruments: Resonance is crucial for producing sound in many instruments. Soundboards of string instruments resonate with the vibrating strings, amplifying the sound. Air columns in wind instruments resonate at specific frequencies.
Tuning Radios/TVs: Tuning into a specific station involves adjusting the resonant frequency of an electrical circuit to match the frequency of the incoming radio waves.
MRI: Magnetic Resonance Imaging uses the principle of resonance of atomic nuclei in a magnetic field.
Destructive Resonance: Resonance can be destructive if large amplitudes are induced in structures. Examples include:
- Bridges collapsing due to wind or marching soldiers (if the driving frequency matches the natural frequency of the bridge). The famous Tacoma Narrows Bridge collapse is often cited as an example of aeroelastic flutter, which is a form of resonance.
- Buildings vibrating during earthquakes (if the earthquake frequency components match the building's natural frequency).
Microwave Ovens: Microwave ovens work by generating microwaves that resonate with the water molecules in food, heating them efficiently.

Resonance is a powerful phenomenon that can be harnessed for useful purposes or needs to be carefully avoided in engineering design to prevent catastrophic failure.