Question

Systems that can be modelled as damped harmonic oscillators are widespread; pendulum clocks, car shock absorbers, tuning circuits in television sets and radios, and collective electron motions in plasmas and metals are just a few examples. In all these cases, one or more variables describing the system obey(s) an equation of the form $$ \ddot{x}+2 \gamma \dot{x}+\omega_{0}^{2} x=P \cos \omega t $$ where \(x=d x / d t\), etc. and the inclusion of the factor 2 is conventional. In the steady state (i.e. after the effects of any initial displacement or velocity have been damped out) the solution of the equation takes the form $$ x(t)=A \cos (\omega t+\phi) $$ By expressing each term in the form \(B \cos (\omega t+\epsilon)\) and representing it by a vector of magnitude \(B\) making an angle \(\epsilon\) with the \(x\)-axis, draw a closed vector diagram, at \(t=0\), say, that is equivalent to the equation. (a) Convince yourself that whatever the value of \(\omega(>0) \phi\) must be negative \((-\pi<\phi \leq 0)\) and that $$ \phi=\tan ^{-1}\left(\frac{-2 \gamma \omega}{\omega_{0}^{2}-\omega^{2}}\right) $$ (b) Obtain an expression for \(A\) in terms of \(P, \omega_{0}\) and \(\omega\).

Short answer

\(\phi = \tan^{-1}\left(\frac{-2 \gamma \omega}{\omega_{0}^{2} - \omega^2}\right)\) and \(A = \frac{P}{\sqrt{(\omega_{0}^2 - \omega^2)^2 + (2 \gamma \omega)^2}}\).

Step-by-step solution

- Steady State Solution

Consider the given differential equation in the form \(\ddot{x} + 2 \gamma \dot{x} + \omega_{0}^{2} x = P \cos(\omega t)\). In the steady state, the solution is given by \(x(t) = A \cos(\omega t + \phi)\).

- Express Each Term as a Cosine Function

Rewrite each term in the differential equation in the form \(B \cos(\omega t + \epsilon)\). The terms are \(\ddot{x} = -A \omega^{2} \cos(\omega t + \phi)\), \(2 \gamma \dot{x} = -2 \gamma A \omega \sin(\omega t + \phi)\), and \(\omega_0^2 x = A \omega_0^2 \cos(\omega t + \phi)\).

- Represent Terms as Vectors

Represent these cosine functions as vectors at \(t = 0\). The term \(\ddot{x} = -A \omega^2\) is a vector in the negative x-direction, \(2 \gamma \dot{x}\) is a vector at an angle \(\frac{\pi}{2}\) minus \(\phi\), and \(\omega_0^2 x\) is a vector at an angle \(\phi\). Draw these vectors to form a closed vector diagram.

- Convince Yourself the Phase Must Be Negative

Observe that in the steady state, the equation must hold for all times \(t\). At \(t = 0\), \(x(0) = A \cos(\phi)\), leading to a closed loop if and only if \(\phi\) is between \(-\pi\) and 0.

- Prove the Expression for \(\phi\)

Using trigonometric identities, solve for \(\phi\). Take the imaginary (sine) and real (cosine) parts separately to form two equations, then solve \(\phi = \tan^{-1}\left(\frac{-2 \gamma \omega}{\omega_{0}^{2} - \omega^2}\right).\).

- Obtain Expression for A

Using the known form of the solution,\(x(t) = A \cos(\omega t + \phi)\), substitute back into the original differential equation and equate terms to solve for \(A\). The expression becomes \(A = \frac{P}{\sqrt{(\omega_{0}^2 - \omega^2)^2 + (2 \gamma \omega)^2}}\).

Key concepts

differential equations

A differential equation is a mathematical equation involving derivatives of an unknown function. In the context of damped harmonic oscillators, the differential equation helps describe the system's behavior over time. For example, let's consider the differential equation:

\[ \frac{d^2x}{dt^2} + 2\beta \frac{dx}{dt} + \theta_{0}^{2}x = P \text{cos}(\theta t) \]

Here, the terms include second and first derivatives with respect to time, along with the unknown function \( x \). This equation models the motion of oscillating systems like pendulum clocks or car shocks. The parameters \( 2\beta \) and \( \theta_{0}^{2} \) represent the damping coefficient and the natural frequency squared, respectively, while \( P \text{cos}(\theta t) \) represents an external driving force. To solve this equation, we need to find a function \( x(t) \) that satisfies it for all times.

One typical approach is to consider solutions of the form \( x(t) = A \text{cos}(\theta t + \text{{\phi}}) \) for the steady state, meaning the behavior after the initial transients have died out. This helps simplify the analysis and understand the system's long-term behavior.

steady state solution

The steady state solution of a differential equation refers to the condition where the transients have decayed, and the system’s response is purely driven by external forces. For our damped harmonic oscillator, we assume the solution in the form:

\[ x(t) = A \text{cos}(\theta t + \text{\phi}) \]

This represents an oscillation with amplitude \( A \) and phase angle \( \text{\phi} \). Inserting this form back into the differential equation helps isolate and understand the system’s consistent behavior over time. In the steady state, the solution captures the sustained oscillations without considering initial conditions. Importantly, it allows us to analyze the phase relationships and amplitudes directly influenced by the external driving force. Instead of the full general solution that might include decaying exponential terms from initial conditions, the steady state focuses on the long-term periodic solution.

vector representation

Vector representation provides a graphical way to visualize complex oscillatory behavior. When modeling damped harmonic oscillators, each term in the differential equation can be interpreted as a vector. For instance:

- The acceleration term \( \frac{d^2x}{dt^2} \) becomes a vector pointing along the negative x-axis.
- The velocity term \( 2\beta \frac{dx}{dt} \) represents a vector rotated by \( \frac{\text{\pi}}{2} \) minus the phase angle \( \text{\phi} \).
- The position term \( \theta_{0}^{2}x \) aligns with the phase angle \( \text{\phi} \).

By drawing these vectors at a specific time, say \( t = 0 \), we can form a closed vector diagram. This approach simplifies understanding how different forces and components interact in the oscillatory system. The vector lengths correspond to magnitudes like amplitude and driving force, providing an intuitive visualization of the relationships.

phase angle calculation

The phase angle, denoted as \( \text{\phi} \), describes the shift between the driving force and the system's response. Its calculation helps understand the time lag introduced by damping and the system's natural resistance. The phase angle for our system is given by:

\[ \text{\phi} = \text{tan}^{-1} \bigg(\frac{-2\beta \theta}{\theta_{0}^{2} - \theta^2}\bigg) \]

This formula reveals the relationship between damping (\( 2\beta \)), the driving frequency (\( \theta \)), and the natural frequency (\( \theta_{0} \)). By ensuring \( -\text{\pi} < \text{\phi} \text{\leq 0} \), we understand the phase angle must be negative, indicating the system lags behind the driving force. Calculating \( \text{\phi} \) helps predict the timing of the maximum responses relative to the applied forces. It ultimately aids in fine-tuning the oscillator’s design, ensuring optimal performance in practical applications like tuning circuits or shock absorbers.