Question

Consider an underdamped oscillator (such as a mass on the end of a spring) that is released from rest at position \(x_{\mathrm{o}}\) at time \(t=0 .\) (a) Find the position \(x(t)\) at later times in the form $$x(t)=e^{-\beta t}\left[B_{1} \cos \left(\omega_{1} t\right)+B_{2} \sin \left(\omega_{1} t\right)\right]$$ That is, find \(B_{1}\) and \(B_{2}\) in terms of \(x_{\mathrm{o}}\). (b) Now show that if you let \(\beta\) approach the critical value \(\omega_{\mathrm{o}}\) your solution automatically yields the critical solution. (c) Using appropriate graphing software, plot the solution for \(0 \leq t \leq 20,\) with \(x_{\mathrm{o}}=1, \omega_{\mathrm{o}}=1,\) and \(\beta=0,0.02,0.1,0.3,\) and 1

Short answer

(a) \( B_1 = x_0, B_2 = \frac{\beta x_0}{\omega_1} \). (b) Critical solution shows as \( \beta \to \omega_o \). (c) Plot shows damping effects.

Step-by-step solution

Understanding the Differential Equation

The system can be described by the second-order differential equation \( \frac{d^2 x}{dt^2} + 2\beta \frac{dx}{dt} + \omega_o^2 x = 0 \). For an underdamped oscillator, \( \beta < \omega_o \), and the solution takes the form \( x(t) = e^{-\beta t} (B_1 \cos(\omega_1 t) + B_2 \sin(\omega_1 t)) \), where \( \omega_1 = \sqrt{\omega_o^2 - \beta^2} \).

Apply Initial Conditions

Given that the system is released from rest at position \( x_0 \), this implies \( x(0) = x_0 \) and \( \frac{dx}{dt}(0) = 0 \). First, substitute \( t = 0 \) into the position equation to find \( B_1 \): \( x(0) = e^{0} (B_1 \cdot 1 + B_2 \cdot 0) = B_1 = x_0 \).

Differentiate and Apply Rest Condition

Differentiate \( x(t) \) with respect to \( t \) to find the velocity: \( \frac{dx}{dt} = -\beta e^{-\beta t}(B_1 \cos(\omega_1 t) + B_2 \sin(\omega_1 t)) + e^{-\beta t}(-B_1 \omega_1 \sin(\omega_1 t) + B_2 \omega_1 \cos(\omega_1 t)) \). At \( t = 0 \), this simplifies to \( -\beta x_0 + B_2 \omega_1 = 0 \). Solve for \( B_2 \): \( B_2 = \frac{\beta x_0}{\omega_1} \).

Confirm the Critical Solution

As \( \beta \to \omega_o \), the system approaches critical damping. Here, \( \omega_1 \to 0 \), and the behavior changes to a critically damped oscillator. In the critically damped case, the solution is often \( x(t) = (A + Bt)e^{-\omega_o t} \), which can be related by limits to the form \( e^{-\beta t}(B_1 \cos(\omega_1 t) + B_2 \sin(\omega_1 t)) \) as \( \omega_1 \to 0 \).

Graphing the Solution

Plot \( x(t) = e^{-\beta t}(B_1 \cos(\omega_1 t) + B_2 \sin(\omega_1 t)) \) for different values of \( \beta \) using software for the given conditions: \( x_0 = 1 \), \( \omega_o = 1 \), and the specified \( \beta \) values. Each plot shows how the damping affects the amplitude and oscillatory behavior over the range \( 0 \leq t \leq 20 \).

Key concepts

Differential Equation

A differential equation is a mathematical expression that relates a function to its derivatives. In the context of an underdamped oscillator, the system is governed by a second-order differential equation:

• \( \frac{d^2 x}{dt^2} + 2\beta \frac{dx}{dt} + \omega_o^2 x = 0 \)

This equation describes how the position \( x(t) \) of the oscillator evolves over time under the influence of damping, represented by \( \beta \), and the natural frequency, \( \omega_o \). The equation is called 'second-order' because it involves second derivatives.
The solution to this differential equation helps predict the behavior of the oscillator, specifying how it moves over time. It's crucial for modeling systems where damping causes the system to lose energy, leading to a decrease in amplitude over time.
Understanding differential equations allows us to make predictions about physical systems, offering insights into phenomena like oscillation and damping.

Initial Conditions

Initial conditions in differential equations provide the specific values needed to find a particular solution from a family of solutions. For the underdamped oscillator, the initial conditions are:

• \( x(0) = x_0 \), the position of the oscillator at time \( t = 0 \).
• \( \frac{dx}{dt}(0) = 0 \), indicating the oscillator is released from rest.

These conditions allow us to determine the constants \( B_1 \) and \( B_2 \) in the solution. By inserting these initial conditions into our position and velocity equations, we can solve for these constants:
\( B_1 = x_0 \)
\( B_2 = \frac{\beta x_0}{\omega_1} \)
The initial conditions ensure that our mathematical model accurately reflects the physical scenario we're analyzing. They are pivotal for tailoring the general solution of the differential equation into one that fits the specific situation of the oscillator at the start of its motion.

Critical Damping

Critical damping refers to the specific condition where the system returns to equilibrium without oscillating and as quickly as possible. It occurs when the damping coefficient \( \beta \) reaches the critical value \( \omega_o \), the natural frequency:

• When \( \beta = \omega_o \), we achieve critical damping.
• The system does not oscillate but returns to its equilibrium position smoothly.

In a mathematical sense, as \( \beta \rightarrow \omega_o \), \( \omega_1 \rightarrow 0 \) which alters the form of the solution. The underdamped solution converges to a critically damped form:
\( x(t) = (A + Bt)e^{-\omega_o t} \)
This transformation highlights how the damping impacts the system’s behavior, shifting from oscillatory to a smooth return to equilibrium as damping increases. Critical damping is crucial in systems where a rapid return to stability is needed, such as in automotive suspensions or building structures to prevent excessive movement or damage.

Second-order Differential Equation

Second-order differential equations are fundamental in modeling physical systems involving acceleration, such as oscillators. They involve the second derivative of a function, indicating how the rate of change of the rate of change (acceleration) is described in terms of other variables:

• Our equation \( \frac{d^2 x}{dt^2} + 2\beta \frac{dx}{dt} + \omega_o^2 x = 0 \) is second-order because it includes \( \frac{d^2 x}{dt^2} \).

In the context of the underdamped oscillator, this equation captures:

• The natural motion described by \( \omega_o^2 x \).
• The damping effect through \( 2\beta \frac{dx}{dt} \).

Second-order differential equations are widespread in engineering and physics for modeling all kinds of dynamical systems—from simple harmonic oscillators to complex mechanical systems. They allow for the precise description and prediction of system behaviors, making them invaluable in design, analysis, and control processes. Understanding these equations is key to unlocking how various systems react under different forces and constraints.