Question

A spring-mass system with a hardening spring (Problem 32 of Section 3.8 ) is acted on by a periodic external force. In the absence of damping, suppose that the displacement of the mass satisfies the initial value problem $$ u^{\prime \prime}+u+\frac{1}{5} u^{3}=\cos \omega t, \quad u(0)=0, \quad u^{\prime}(0)=0 $$ (a) Let \(\omega=1\) and plot a computer-generated solution of the given problem. Does the system exhibit a beat? (b) Plot the solution for several values of \(\omega\) between \(1 / 2\) and \(2 .\) Describe how the solution changes as \(\omega\) increases.

Short answer

Question: Based on the given initial value problem for a spring-mass system with a hardening spring and an external periodic force, describe the general behavior of the system as the driving frequency ω increases from 1/2 to 2.

Step-by-step solution

Define the initial value problem (IVP)

The given initial value problem is: $$ u^{\prime \prime} + u + \frac{1}{5} u^3 = \cos \omega t, \quad u(0) = 0, \quad u^{\prime}(0) = 0 $$

Convert the second-order ODE into a system of first-order ODEs

Let \(v = u'\), then: $$ u' = v $$ and $$ v' = u'' = -u - \frac{1}{5}u^3 + \cos \omega t $$ Now, we have a system of first-order ODEs: $$ \begin{cases} u' = v \\ v' = -u - \frac{1}{5}u^3 + \cos \omega t \end{cases} $$ with initial conditions \(u(0) = 0\) and \(v(0) = 0\).

Solve the system of first-order ODEs numerically

To solve the system of first-order ODEs numerically, we can use the Runge-Kutta method. We can implement this method using Python programming and the scipy.integrate library (odeint function).

Plot the computer-generated solutions

After obtaining the numerical solution, we plot the displacement u(t) against time t. For part (a), set ω = 1, and analyze the plot to see if the system exhibits a beat. For part (b), plot the solutions for different values of ω in the interval [1/2, 2] and describe how the solution changes as ω increases.

Analyze the plots

Based on the graphs, we can make the following observations: (a) With ω = 1, the system may or may not exhibit a beat depending on the chosen numerical method's accuracy and the system conditions. (b) As ω increases, analyze the plots and describe the changes in the solution's behavior. Generally, the amplitude and frequency of oscillation will be affected by the changes in ω values. This concludes the step-by-step solution for the spring-mass system with a hardening spring and external periodic force.

Key concepts

Initial Value Problem

When we talk about an initial value problem (IVP), we're addressing a scenario where we have a differential equation along with the values of the solution at a starting point, usually time zero. In the case of our spring-mass system, the differential equation given is:
\[ u^{{\prime \prime}} + u + \frac{1}{5} u^{3} = \cos \omega t \]
with the initial conditions being \( u(0) = 0 \) and \( u^{\text{prime}}(0) = 0 \). This IVP defines the motion of the mass attached to a nonlinear, hardening spring under the influence of an external periodic force. Solving an IVP helps us predict the system's future behavior based on initial conditions and the underlying physics, encapsulated by the differential equation.

Second-Order Differential Equations

Second-order differential equations are prevalent in physics and engineering, often emerging in systems with acceleration, like our spring-mass system. Such equations, like the one we have here, relate an unknown function with its derivatives:
\[ u^{{\prime \prime}}(t) = -u(t) - \frac{1}{5}u(t)^{3} + \cos \omega t \]
In this equation, \( u^{{\prime \prime}} \) is the second derivative of the displacement \( u \) with respect to time, symbolizing the mass's acceleration. Solving these can be quite complex, particularly with nonlinear terms like \( \frac{1}{5}u^{3} \), requiring advanced analytical skills or numerical methods for a solution.

Numerical Methods for ODEs

Numerical methods for Ordinary Differential Equations (ODEs) enable us to approximate solutions when they're impossible to obtain analytically. They work by converting continuous problems into discrete ones, which we can tackle with computational power. Implementing these methods is often done using programming languages like Python or MATLAB, and they allow us to explore how the system behaves under varying conditions without an exact formula.

Runge-Kutta Method

The Runge-Kutta method, particularly the fourth-order version often referred to simply as 'RK4', is one of the most commonly used techniques for solving ODEs numerically. It's prized for its balance between computational efficiency and accuracy. The method works by estimating the slope at several points within each step and then taking a weighted average of these slopes to determine the next value of the variable. This process is iteratively repeated to build a solution across the desired range.

System Behavior with Varying Omega

In our spring-mass system, omega (\( \omega \)) represents the angular frequency of the periodic driving force. Studying the system behavior with varying \( \omega \) values is particularly interesting because it relates to how the mass resonates. When \( \omega \) aligns closely with the system's natural frequency, we can expect significant oscillations due to resonance. As we vary \( \omega \), the character of the motion changes, possibly leading to diverse phenomena like beats or chaotic motion. Plotting this relationship allows us to visually capture these changes and understand the physical implications more concretely.