Question

Consider a damped oscillator, with natural frequency \(\omega_{\mathrm{o}}\) and damping constant \(\beta\) both fixed, that is driven by a force \(F(t)=F_{0} \cos (\omega t) .\) (a) Find the rate \(P(t)\) at which \(F(t)\) does work and show that the average rate \(\langle P\rangle\) over any number of complete cycles is \(m \beta \omega^{2} A^{2} .\) (b) Verify that this is the same as the average rate at which energy is lost to the resistive force. (c) Show that as \(\omega\) is varied \(\langle P\rangle\) is maximum when \(\omega=\omega_{\mathrm{o}} ;\) that is, the resonance of the power occurs at \(\omega=\omega_{\mathrm{o}}\) (exactly).

Short answer

The average power over a number of cycles is \(m \beta \omega^2 A^2\). It equals the energy loss rate, and peaks at \(\omega = \omega_o\).

Step-by-step solution

Expression for Power

The instantaneous power delivered by the force is given by:\[ P(t) = F(t) \cdot v(t) \]where \(v(t)\) is the velocity of the oscillator. For \[ F(t) = F_{0} \cos(\omega t) \]the equation for motion of the damped driven oscillator is:\[ m \frac{d^2 x}{dt^2} + m \beta \frac{dx}{dt} + m \omega_{o}^2 x = F_{0} \cos(\omega t) \]

Solution to the Differential Equation

Solving the equation of motion, we assume a steady-state solution of the form:\[ x(t) = A \cos(\omega t - \delta) \]Differentiating, we find the velocity:\[ v(t) = \frac{dx}{dt} = -A\omega \sin(\omega t - \delta) \]

Expression for Instantaneous Power

Instantaneous power becomes:\[ P(t) = F_{0} \cos(\omega t) \times (-A\omega \sin(\omega t - \delta)) \]Which simplifies using the trigonometric identity: \[ \cos(a) \sin(b) = \frac{1}{2} [\sin(a+b) + \sin(a-b)] \]

Average Power Calculation

We integrate over one complete cycle to find the average power:\[ \langle P \rangle = \frac{1}{T} \int_{0}^{T} P(t) \, dt \]This uses the orthogonality of sine and cosine functions, resulting in terms involving sine and cosine over a period cancelling out, leading to:\[ \langle P \rangle = F_{0} \omega \beta A^{2} \]

Equivalent Energy Loss Rate

For a damped oscillator, the power loss to the resistive force (friction) is:\[ P_{ ext{loss}} = m \beta \left(\frac{dx}{dt}\right)^2 \]The time-averaged rate of energy loss can also be calculated and shown to be:\[ \langle P_{\text{loss}} \rangle = m \beta \omega^2 A^2 \] which matches the average power input.

Resonance Condition

The amplitude \(A\) is related to the driving frequency \(\omega\) as:\[ A = \frac{F_{0}}{m\sqrt{(\omega_{o}^2 - \omega^2)^2 + 4\beta^2 \omega^2}} \]The power is a maximum when the denominator of \(A\) is minimized. Differentiating and calculating, this occurs when:\[ \omega = \omega_{o} \]

Key concepts

Natural Frequency

Natural frequency refers to the rate at which a system oscillates in the absence of any driving or damping forces. In simple terms, it is the frequency at which a system tends to vibrate naturally. For a damped oscillator, this natural frequency is denoted by \( \omega_{\mathrm{o}} \). This means that if the system is displaced from its equilibrium position and no external forces act on it other than the damping, it will oscillate at this frequency.
Natural frequency is a fundamental property of oscillatory systems and is dependent on the system's internal characteristics such as mass and stiffness. For instance, a simple mass-spring system in the absence of damping will have a natural frequency \( \omega_{\mathrm{o}} \) given by \( \omega_{\mathrm{o}} = \sqrt{\frac{k}{m}} \), where \( k \) is the spring constant and \( m \) is the mass.

• The higher the mass, the lower the natural frequency as it takes more time to complete a cycle.
• The higher the stiffness, the higher the natural frequency as the system can return more quickly to equilibrium.

Understanding natural frequency is crucial in designing systems to avoid unwanted oscillations, especially near their natural frequency where systems become more sensitive to external forces.

Damping Constant

The damping constant, often represented as \( \beta \), is a measure of how quickly oscillations in a system dissipate or decrease over time. In a damped oscillator, this damping constant determines how the amplitude of oscillations decays and how fast the energy is lost to friction or resistive forces.
The damping effect is critical in determining the nature of oscillatory motion. There are three primary types of damping based on the value of the damping constant \( \beta \):

• Under-damping: Occurs when \( \beta \) is small, causing the system to oscillate with gradually decreasing amplitude.
• Critical damping: The exact amount of damping \( \beta = \beta_c \) needed to return to equilibrium as quickly as possible without oscillating.
• Over-damping: Occurs when \( \beta \) is large, causing the system to return to equilibrium without oscillating, but slower than in the critically damped case.

The damping constant is essential in many applications to control oscillations and stabilize systems, such as in automotive shock absorbers or building structures subject to dynamic forces.

Resonance

Resonance occurs when a system is driven at a frequency that matches its natural frequency \( \omega_{\mathrm{o}} \), causing it to oscillate with larger amplitude. In the context of the damped oscillator problem, resonance is observed when the driving frequency \( \omega \) equals the natural frequency \( \omega_{\mathrm{o}} \). This results in a maximum average power transfer, maximizing the amplitude of the oscillation.
Resonance can lead to significant consequences, such as:

• Mechanical resonance in bridges or buildings, potentially leading to structural failure if not adequately handled.
• Electrical resonance in circuits, which can increase voltages and current, sometimes causing failures or inefficiencies.
• Acoustic resonance in musical instruments, affecting tone and volume.

Understanding resonance is critical in designing systems to either avoid or harness it safely. For systems where resonance is desired, like in tuning radios or musical instruments, precise control and calculations are necessary to maintain efficient and safe operation.