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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

Expert verified
(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

01

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 \( \omega_1 = \sqrt{\omega_o^2 - \beta^2} \).
02

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 \( B_1 \): \( x(0) = e^{0} (B_1 \cdot 1 + B_2 \cdot 0) = B_1 = x_0 \).
03

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 \( B_2 \): \( B_2 = \frac{\beta x_0}{\omega_1} \).
04

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 \( \omega_1 \to 0 \).
05

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 \( 0 \leq t \leq 20 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

Suppose that you have found a particular solution \(x_{\mathrm{p}}(t)\) of the inhomogeneous equation (5.48) for a driven damped oscillator, so that \(D x_{\mathrm{p}}=f\) in the operator notation of \((5.49) .\) Suppose also that \(x(t)\) is any other solution, so that \(D x=f .\) Prove that the difference \(x-x_{\mathrm{p}}\) must satisfy the corresponding homogeneous equation, \(D\left(x-x_{\mathrm{p}}\right)=0 .\) This is an alternative proof that any solution \(x\) of the inhomogeneous equation can be written as the sum of your particular solution plus a homogeneous solution; that is, \(x=x_{\mathrm{p}}+x_{\mathrm{h}}\).

(a) If a mass \(m=0.2 \mathrm{kg}\) is tied to one end of a spring whose force constant \(k=80 \mathrm{N} / \mathrm{m}\) and whose other end is held fixed, what are the angular frequency \(\omega\), the frequency \(f\), and the period \(\tau\) of its oscillations? (b) If the initial position and velocity are \(x_{\mathrm{o}}=0\) and \(v_{\mathrm{o}}=40 \mathrm{m} / \mathrm{s},\) what are the constants \(A\) and \(\delta\) in the expression \(x(t)=A \cos (\omega t-\delta) ?\)

An undamped oscillator has period \(\tau_{\mathrm{o}}=1.000 \mathrm{s},\) but \(\mathrm{I}\) now add a little damping so that its period changes to \(\tau_{1}=1.001\) s. What is the damping factor \(\beta\) ? By what factor will the amplitude of oscillation decrease after 10 cycles? Which effect of damping would be more noticeable, the change of period or the decrease of the amplitude?

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).

Let \(f(t)\) be a periodic function with period \(\tau\). Explain clearly why the average of \(f\) over one period is not necessarily the same as the average over some other time interval. Explain why, on the other hand, the average over a long time \(T\) approaches the average over one period, as \(T \rightarrow \infty\)

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