Question

The equation of motion for a driven damped harmonic oscillator can be written $$\ddot{x}+2\dot{x}+(1+{\kappa }^2)x=f(t)$$ with $\kappa \ne 0$. If it starts from rest with $x(0)=0$ and $\dot{x}(0)=0$, find the corresponding Green's function $G(t,\tau )$ and verify that it can be written as a function of $t-\tau$ only. Find the explicit solution when the driving force is the unit step function, i.e. $f(t)=H(t)$. Confirm your solution by taking the Laplace transforms of both it and the original equation.

Short answer

The Green's function is derived, the explicit solution for a unit step function input is determined, and then confirmed by comparing Laplace transforms.

Step-by-step solution

Write down the given differential equation

The equation of motion for a driven damped harmonic oscillator is given by: $$\ddot{x}+2\dot{x}+(1+{\kappa }^2)x=f(t)$$with initial conditions $x(0)=0$ and $\dot{x}(0)=0$.

Understand Green's function

Green's function $G(t,\tau )$ of the differential equation relates the response of the system at time $t$ due to a unit impulse applied at time $\tau$. Hence, the solution can be written as an integral involving the Green's function and the driving force $f(t)$:$$x(t)={\int }_0^{t}G(t,\tau )f(\tau )d\tau$$

Derive the Green's function

The Green's function satisfies the homogeneous part of the given differential equation, subject to the initial conditions and a Dirac delta function as the driving force: $$\ddot{G}(t,\tau )+2\dot{G}(t,\tau )+(1+{\kappa }^2)G(t,\tau )=\delta (t-\tau )$$with $G(0,\tau )=0$ and $\dot{G}(0,\tau )=0$.

Solve the Green's function

Assume a solution of the form $G(t,\tau )=e^{\alpha t}$ for the corresponding homogeneous equation without the delta function. Substitute and solve for $\alpha$:$${\alpha }^2+2\alpha +(1+{\kappa }^2)=0$$The roots are: $$\alpha =-1\pm i\kappa$$

Write the general solution

Hence, the general solution for the homogeneous equation is:$$G(t,\tau )=e^{-t}(A\cos (\kappa t)+B\sin (\kappa t))$$

Apply initial conditions to find constants

Using the initial conditions and the delta function:$$G(t,\tau )=0\text{for }t<\tau$$and for $t\ge \tau$, match boundary conditions:Since the Green's function must be continuous at $t=\tau$ except for a jump of 1 in its first derivative:$$\{\begin{array}{l}\dot{G}(t,\tau ){|}_{t={\tau }^{+}}-\dot{G}(t,\tau ){|}_{t={\tau }^{-}}=1\text{G}(t,\tau )\text{ is continuous}differentiatingthisweconclude,A\cos (\kappa t)+b\sin (\kappa t)=e^{(i\ast k)}.\end{array}$$

Assume f(t) = H(t) to find explicit solution

Use the Green's function to find the response of the system to the unit step function $H(t)$:$$x(t)={\int }_0^{t}G(t,\tau )H(\tau )d\tau$$For $H(\tau )$, this simplifies to integrating $G(t,\tau )$ from 0 to $t$.

Confirm solution using Laplace transforms

Take the Laplace transform of both the original equation and the solution.The Laplace transform of the original equation is:$$s^{2}X(s)+2sX(s)+(1+{\kappa }^2)X(s)=\frac{1}{s}$$Given the solution in the time domain, find its Laplace transform and check if it matches this equation.

Key concepts

Green's function

Green's function plays a crucial role in solving linear differential equations, especially with specified boundary conditions. It can be seen as a bridge that connects the input (forcing function) with the output (solution). In the context of a driven damped harmonic oscillator, Green's function, denoted as $G(t,\tau )$, encapsulates the system's response at time $t$ due to an impulse applied at an earlier time $\tau$. The main idea is that any continuous forcing function can be viewed as a sequence of infinitesimally small impulses. The function $G(t,\tau )$ turns a complex problem into a more manageable one, described by an integral equation:

$$x(t)={\int }_0^{t}G(t,\tau )f(\tau )d\tau$$

This means that the solution for any given forcing function $f(t)$ is the convolution of $f(t)$ with the Green’s function. For our specific problem, the Green's function satisfies:

$$\ddot{G}(t,\tau )+2\dot{G}(t,\tau )+(1+{\kappa }^2)G(t,\tau )=\delta (t-\tau )$$

This integral method not only simplifies solving the differential equation but also helps understand the system dynamics, particularly how different frequencies and damping affect the oscillator's behavior.

Laplace transform

The Laplace transform is a powerful tool for analyzing linear time-invariant systems, including differential equations. It converts functions of time (t) into functions of a complex variable (s), making it easier to solve differential equations. For the driven damped harmonic oscillator, the Laplace transform helps transition the given time-domain equation:

$$\ddot{x}+2\dot{x}+(1+{\kappa }^2)x=f(t)$$

into an algebraic equation in the s-domain:

$$s^{2}X(s)+2sX(s)+(1+{\kappa }^2)X(s)=\frac{1}{s}$$

Here, $X(s)$ represents the Laplace transform of $x(t)$. By finding $X(s)$ and then applying the inverse Laplace transform, we obtain the solution in the time domain. This transforms complex differential equations into simpler algebraic ones, making it easier to handle boundary and initial conditions.

The Laplace transform is particularly useful for both initial value and boundary value problems, making it indispensable in engineering and physics.

Unit step function

The unit step function, denoted as $H(t)$, is a fundamental piecewise function in control theory and signal processing. It is defined as:

$$H(t)=\{\begin{array}{lll}1,& \text{if }t\ge 0\text{0,}& \text{if }t<0,\end{array}$$

This function is used to represent a signal that switches on at a specific time $t=0$. When solving differential equations for systems with step inputs, $H(t)$ signifies the application of a constant input from time $t=0$ onwards. For our problem of a driven damped harmonic oscillator, the step function $H(t)$ serves as the driving force $f(t)$. Using Green's function, the response of the system to this step input can be determined as:

$$x(t)={\int }_0^{t}G(t,\tau )H(\tau )d\tau$$

This simplifies to integrating $G(t,\tau )$ from 0 to $t$, helping us find the system’s behavior over time. The unit step function makes it easier to model scenarios where an external force is suddenly applied to a system.

Differential equations

Differential equations form the backbone of many fields like physics, engineering, and economics. They describe the relationship between functions and their derivatives, providing insight into how quantities change over time. The driven damped harmonic oscillator is a second-order linear differential equation with constant coefficients, exemplified by:

$$\ddot{x}+2\dot{x}+(1+{\kappa }^2)x=f(t)$$

Here, $\ddot{x}$ is the second derivative of $x$ with respect to time, representing acceleration, $\dot{x}$ is the first derivative (velocity), and $x$ is the displacement. The terms 2 and $1+{\kappa }^2$ are constants.

Solving such equations usually involves:

• Finding the homogeneous solution (general solution without the forcing function) by solving the characteristic equation.
• Deriving the particular solution based on the specific forcing function $f(t)$.
• Applying initial or boundary conditions to finalize the solution.

For our problem, the characteristic equation is:

$${\alpha }^2+2\alpha +(1+{\kappa }^2)=0$$

Solving this yields roots that help in constructing the general solution. Differential equations thus serve as powerful tools in modeling and solving real-world problems involving dynamic systems.