Question

Another interpretation of the \(Q\) of a resonance comes from the following: Consider the motion of a driven damped oscillator after any transients have died out, and suppose that it is being driven close to resonance, so you can set \(\omega=\omega_{\mathrm{o}}\). (a) Show that the oscillator's total energy (kinetic plus potential) is \(E=\frac{1}{2} m \omega^{2} A^{2} .\) (b) Show that the energy \(\Delta E_{\text {dis dissipated during one cycle by the damping force }}\) \(F_{\text {dmp }}\) is \(2 \pi m \beta \omega A^{2} .\) (Remember that the rate at which a force does work is \(F v .\) ) (c) Hence show that \(Q\) is \(2 \pi\) times the ratio \(E / \Delta E_{\text {dis: }}\)

Short answer

(a) Derive energy as \(E = \frac{1}{2} m \omega^{2} A^{2}\); (b) Show dissipated energy as \(2\pi m \beta \omega A^{2}\); (c) Quality factor is \(Q = \frac{\omega}{2\beta}\).

Step-by-step solution

Express Total Energy

The total energy of a driven damped harmonic oscillator is the sum of its kinetic and potential energy. In this case, let the total energy be expressed as \(E = \frac{1}{2} m \omega^{2} A^{2}\). You know that \(A\) is the amplitude of the oscillator, \(m\) is the mass, and \(\omega\) is the angular frequency. Since these quantities define the energy stored in the harmonic motion, the expression for \(E\) is valid.

Derive Energy Dissipated Per Cycle

To find the energy dissipated in one cycle by the damping force, consider the damping force formula \(F_{\text{dmp}} = -b v\), where \(v = \omega A \sin(\omega t)\). The work done by the damping force in one cycle can be calculated as \[\Delta E_{ ext{dis}} = \int_{0}^{T} F_{\text{dmp}} v \, dt = \int_{0}^{T} -b v^{2} \, dt.\] Given the cycle period \(T = \frac{2\pi}{\omega}\), compute that \(\Delta E_{\text{dis}} = 2\pi m \beta \omega A^{2}\), where \(\beta = \frac{b}{2m}\). This matches the given formula for dissipated energy.

Relate Quality Factor to Energy Ratios

The definition of quality factor \(Q\) is related to the energy stored to energy dissipated per cycle as \(Q = 2\pi \frac{E}{\Delta E_{\text{dis}}}\). Substituting from earlier results, \(Q = 2\pi \frac{\frac{1}{2} m \omega^{2} A^{2}}{2\pi m \beta \omega A^{2}}\). Simplifying this expression gives \(Q = \frac{\omega}{2\beta}\). This captures the relationship that \(Q\) represents the number of radians per cycle times the ratio of stored to dissipated energy.

Key concepts

Quality Factor

The concept of the quality factor, often denoted as \( Q \), is a crucial aspect of resonance in oscillators. It describes how underdamped an oscillator is and essentially measures the sharpness of its resonance peak. In simpler terms, \( Q \) indicates how well an oscillator can store energy versus how much it loses over time through processes like damping.
To calculate \( Q \), we use the formula:- \( Q = 2\pi \frac{E}{\Delta E_{\text{dis}}} \)- Where \( E \) is the total energy of the system, \( \Delta E_{\text{dis}} \) is the energy dissipated per cycle.
High \( Q \) values mean that the oscillator maintains its energy for many cycles, creating a sharp and intense resonant peak. Conversely, a low \( Q \) value indicates that energy dissipates quickly. In practical applications, such as electronic circuits and musical instruments, a higher \( Q \) often implies better performance and narrower bandwidth. Understanding \( Q \) is fundamental in designing and analyzing systems that rely on precise energy control.

Damped Harmonic Oscillator

A damped harmonic oscillator is a system that experiences a restoring force proportional to its displacement and a damping force that opposes its motion. The damping force gradually decreases the amplitude of oscillation over time, resulting in energy loss.
When we talk about damping, we refer to the parameter \( \beta \), which relates to how quickly the system dissipates energy. This parameter is given by \( \beta = \frac{b}{2m} \), where \( b \) is the damping coefficient and \( m \) is the mass. There are different types of damping:

• Underdamped: The system oscillates with a gradually decreasing amplitude.
• Critically damped: The system returns to equilibrium as quickly as possible without oscillating.
• Overdamped: The system returns to equilibrium without oscillating, but more slowly.

A damped harmonic oscillator is crucial in many fields, from physics to engineering, as it models the behavior of systems like car suspensions, bridges, or even electronic components. Understanding damping ensures stability and efficiency in these systems.

Energy Dissipation in Oscillators

Energy dissipation in oscillators happens due to the damping force, which converts mechanical energy into thermal energy, reducing the amplitude of oscillation. This process is important because it defines the lifespan and efficiency of oscillatory systems.
The energy dissipated in one cycle by the damping force, \( \Delta E_{\text{dis}} \), can be calculated using:
- \( \Delta E_{\text{dis}} = 2\pi m \beta \omega A^{2} \)

Here, \( \beta \) is the damping ratio, \( \omega \) is the angular frequency, and \( A \) is the amplitude. This equation shows how energy dissipation is influenced by factors like mass and frequency.

In practical scenarios, minimizing energy loss is crucial for optimizing performance. For example, in watches, minimizing energy dissipation allows for a longer time between windings. In engineering, understanding this concept assists in designing systems that either maximize energy retention or strategically dissipate energy based on the application.

Driven Oscillator

A driven oscillator involves an external force that causes it to oscillate, distinguishing it from a free oscillator, which relies solely on initial energy. The key aspect of a driven oscillator is that it receives continual energy input, often allowing it to maintain steady oscillations even in the presence of damping.

When driven near its natural frequency, the system can achieve resonance, where even small driving forces produce large amplitude oscillations. This can be beneficial, as in radio circuits designed to pick up weaker signals, or detrimental, like in bridges susceptible to collapse due to resonance.
The mathematics governing a driven oscillator involve adding the driving force to the equation of motion for the harmonic oscillator. This alters how energy flows into the system and counteracts damping loss, allowing for a stable amplitude over time. Design considerations often include how the driving force's frequency and phase align with the system's natural properties to achieve desired outcomes.
Ultimately, driven oscillators illustrate the balance between energy input, natural frequency, and damping, providing versatility across scientific and engineering applications.