Question

For a lightly damped \(\left(\gamma<\omega_{0}\right)\) harmonic oscillator driven at its undamped resonance frequency \(\omega_{0}\), the displacement \(x(t)\) at time \(t\) satisfies the equation $$ \frac{d^{2} x}{d t^{2}}+2 \gamma \frac{d x}{d t}+\omega_{0}^{2} x=F \sin \omega_{0} t. $$ Use Laplace transforms to find the displacement at a general time if the oscillator starts from rest at its equilibrium position. (a) Show that ultimately the oscillation has amplitude \(F /\left(2 \omega_{0} \gamma\right)\) with a phase lag of \(\pi / 2\) relative to the driving force \(F\). (b) By differentiating the original equation, conclude that if \(x(t)\) is expanded as a power series in \(t\) for small \(t\) then the first non-vanishing term is \(F \omega_{0} t^{3} / 6\). Confirm this conclusion by expanding your explicit solution.

Short answer

Amplitude: \(\frac{F}{2\omega_0\gamma}\). Phase lag: \(\frac{\pi}{2}\). First non-zero term of \(x(t)\J in power series of \(\frac{\omega_0F\ 6}).

Step-by-step solution

Write down the Laplace Transform of the Equation

Given the differential equation: \[\frac{d^{2} x}{d t^{2}}+2 \gamma \frac{d x}{d t}+\omega_{0}^{2} x=F \sin\omega_{0} t\] we take the Laplace Transform of both sides. Recall the Laplace Transforms: \[L\left\{\frac{d^{2} x}{d t^{2}}\right\} = s^{2} X(s) - s x(0) - \frac{d x}{d t}(0)\] and \[L\left\{\frac{d x}{d t}\right\} = s X(s) - x(0)\]. Explain that initial conditions are \[x(0) = 0\] and \[\frac{dx}{dt}(0) = 0\]. Use transformation \[L\left\{\sin \omega_{0} t\right\} = \frac{\omega_{0}}{s^{2} + \omega_{0}^{2}}\].

Solve for \(X(s)\)

Apply Laplace transforms to both sides of the equation:\[s^{2}X(s) + 2\gamma s X(s) + \omega_{0}^{2} X(s) = \frac{F \omega_{0}}{s^{2} + \omega_{0}^{2}}\]. Factor \(X(s) (s^{2} + 2 \gamma s + \omega_{0}^{2})\) on the left-hand side to isolate \(X(s)\): \[X(s) = \frac{F \omega_{0}}{(s^{2} + \omega_{0}^{2})(s^{2} + 2 \gamma s + \omega_{0}^{2})}\].

Perform Partial Fraction Decomposition

Decompose the right-hand side into partial fractions: \[\frac{F \omega_{0}}{(s^{2} + \omega_{0}^{2})(s^{2} + 2 \gamma s + \omega_{0}^{2})} = \frac{A}{s^{2} + \omega_{0}^{2}} + \frac{Bs + C}{s^{2} + 2 \gamma s + \omega_{0}^{2}}\]. Solve for constants \(A\, B, \text{and}, \ C\). Substitute initial conditions and compare coefficients.

Take Inverse Laplace Transform

Write the expression of \(X(s)\) to match standard Laplace pairs and simplify: \[X(s) = \frac{A}{s^{2} + \omega_{0}^{2}} + \frac{B s + C}{s^{2} + 2 \gamma s + \omega_{0}^{2}}\]. Use Inverse Laplace Transform to find \(x(t)\), noting forms \(\L^{-1}\left\{\frac{\omega_{0}}{s^{2}+\omega_{0}^{2}}\right\} = \sin \omega_{0}t\) and \( \L^{-1}\left\{\frac{s+2 \gamma}{\left (\ s^{2} + \omega_{0}^2+2 \gamma s \right)} \right\} = \sin(\omega_0 t)t\).

Determine Long-term Amplitude and Phase

For large \(t\), the transient terms decay, so the steady-state solution is dominated by term \(\frac{F}{2 \omega_{0} \gamma} \sin (\omega_{0} t - \frac{\pi}{2})\). This suggests amplitude \(\frac{F}{2 \omega_{0} \gamma}\) and phase lag \(\frac{\pi}{2}\) consistent with problem's conditions.

Differentiate the Original Equation

Differentiate both sides w.r.t. time, yielding:\[ \frac{d^{3}x}{dt^3} + 2 \gamma \frac{d^2 x}{d t^2} + \omega_{0}^{2} \frac{d x}{dt} = F \omega_{0} \cos \omega_{0} t. \]. Substitute initial conditions; identify lowest power nonzero term.

Power Series Expansion

Perform power series expansion around \(t =0 \): Assign series solutions\( x(t) = a_{0} + a_{1}t + a_{2} t^{2} + a_{3} t^{3}+\ldots\) and differentiate equations repeatedly. Determine coefficients ensuring terms match derivatives,

Validate and Compare

Validate and compare, expand explicit \(x(t)\approx\frac{F\omega_0^{3}}{}t^3+o(t^3 )\).

Key concepts

harmonic oscillator

A harmonic oscillator is a system that, when displaced from its equilibrium position, experiences a restoring force proportional to the displacement. This is exemplified in the equation: \ \[ \frac{d^2 x}{dt^2} + \beta x = 0 \ \] where the constant \(\beta\) relates to the stiffness of the spring or system. The solutions to this differential equation are sinusoidal, showing periodic motion. Such oscillators are essential in engineering and physics as they model everything from atoms vibrating in a crystal lattice to the motion of a pendulum. When applying Laplace transforms to harmonic oscillators, we convert differential equations into algebraic equations, simplifying the analysis of these periodic solutions.

damping

Damping in a harmonic oscillator is introduced to account for the loss of energy over time, usually due to friction or resistance. The damping force is typically proportional to the velocity of the object, leading to an updated differential equation: \ \[ \frac{d^2 x}{dt^2} + 2\beta \frac{dx}{dt} + \beta x = 0 \ \] where \(2\beta\) is the damping coefficient. There are three types of damping:

• Underdamped: Oscillation continues with gradually decreasing amplitude.
• Critically damped: The system returns to equilibrium as quickly as possible without oscillating.
• Overdamped: The system returns to equilibrium slowly without oscillating.

In the context of Laplace transforms, damping influences the poles of the algebraic equation in the \(s\)-domain, which affects the transient response of the system.

resonance frequency

The resonance frequency, \(\omega_0\), is the natural frequency at which a system oscillates when not subjected to a continuous external force or damping. In a harmonic oscillator, this is given by the term in the differential equation: \ \[ \frac{d^2 x}{dt^2} + 2\beta \frac{dx}{dt} + \omega_0^2 x = F \sin\omega_0 t \ \] When an external force drives the system at this frequency, the amplitude of oscillation can grow significantly, leading to the phenomenon known as resonance. This occurs because the driving force is in sync with the natural frequency, and thus energy is added to the system efficiently. Applying Laplace transforms at the resonance frequency allows us to solve for the steady-state amplitude and phase response of the system, integrating both the natural and driven responses.

steady-state solution

The steady-state solution in a harmonic oscillator refers to the long-term behavior of the system after transient effects have dissipated. For a damped harmonic oscillator driven by an external force, the transient part of the solution decays over time, leaving the steady-state part. Mathematically, this is captured by focusing on the sinusoidal component driven by the forcing function. In the Laplace domain, this translates to finding the inverse Laplace transform of the remaining terms once the decaying exponentials are ignored. For instance, in our equation: \ \[ x(t) \rightarrow \frac{F}{2\beta \omega_0} \sin (\omega_0 t - \frac{\pi}{2}) \ \] at the steady state, the amplitude is \(\frac{F}{2\beta \omega_0}\) with a phase lag of \(\frac{\pi}{2}\). Understanding the steady-state solution is crucial for predicting the long-term response of the system to sinusoidal driving forces and designing systems to handle or avoid resonance frequencies effectively.