Question

\(m x^{\prime \prime}+\frac{2 k \delta}{\pi} \tan \left(\frac{\pi x}{2 \delta}\right)=F(t), \quad x(0)=0, \quad x^{\prime}(0)=0 ; \quad 0 \leq t \leq 15\) This problem was used to model a nonlinear spring-mass system (see Exercise 18 in Section 6.1). The motion is assumed to occur on a frictionless horizontal surface. In this equation, \(m\) is the mass of the object attached to the spring, \(x(t)\) is the horizontal displacement of the mass from the unstretched equilibrium position, and \(\delta\) is the length that the spring can contract or elongate. The spring restoring force has vertical asymptotes at \(x=\pm \delta\). Time \(t\) is in seconds. Let \(m=100 \mathrm{~kg}, \delta=0.15 \mathrm{~m}\), and \(k=100 \mathrm{~N} / \mathrm{m}\). Assume that the spring-mass system is initially at rest with the spring at its unstretched length. At time \(t=0\), a force of large amplitude but short duration is applied: $$ F(t)=\left\\{\begin{array}{ll} F_{0} \sin \pi t, & 0 \leq t \leq 1 \\ 0, & 1

Short answer

Question: Solve the following second-order nonlinear differential equation with initial conditions and plot the displacement \(x(t)\) for two values of the force amplitude \(F_0\): 4 N and 40 N. \(m\frac{d^2x}{dt^2}=-2k\delta\tan\left(\frac{\pi x}{2\delta}\right) + F(t)\) with \(m=100\,\text{kg}\), \(\delta=0.15\,\text{m}\), \(k=100\,\text{N/m}\), and initial conditions \(x(0)=0\) and \(x'(0)=0\). The external force \(F(t)\) is defined as a piecewise function: \(F(t) = \begin{cases}F_{0} \sin \pi t, & 0 \leq t \leq 1\\ 0, & 1<t<15\end{cases}\)

Step-by-step solution

Define the parameters

First, define the given parameters: \(m=100\,\text{kg},\ \delta=0.15\,\text{m},\ k=100\,\text{N/m}\).

Rewrite the second-order differential equation as a system of first-order equations

Let \(x_1(t)=x(t)\) and \(x_2(t)=x'(t)\). Then we get the following system of first-order differential equations: $$ \begin{aligned} \frac{dx_1}{dt} &= x_2\\ \frac{dx_2}{dt} &= -\frac{2k\delta}{m\pi} \tan\left(\frac{\pi x_1}{2\delta}\right) + \frac{F(t)}{m} \end{aligned} $$

Define the external force \(F(t)\) as a piecewise function

Based on the given definition of \(F(t)\), we have: $$ F(t) = \begin{cases} F_{0} \sin \pi t, & 0 \leq t \leq 1\\ 0, & 1<t<15 \end{cases} $$ For the two cases \(F_0=4\,\text{N}\) and \(F_0=40\,\text{N}\), we will have two different piecewise functions.

Solve the system numerically for both cases, \(F_0=4\,\text{N}\) and \(F_0=40\,\text{N}\)

Use a numerical method, such as Python's scipy.solve_ivp or MATLAB's ode45, to solve the first-order system. Provide the initial conditions \(x_1(0)=x(0)=0\) and \(x_2(0)=x'(0)=0\) and solve the system for both given values of \(F_0\).

Extract the displacement \(x(t)\) and plot it

Extract the solution for \(x(t)\) from the numerical solution of the first-order system obtained in the previous step - this corresponds to \(x_1(t)\). Plot the displacement versus time for both cases on the same graph. Observe how they differ based on the applied force amplitude \(F_0\).

Key concepts

Differential Equations

Differential equations form the backbone of many engineering and physics problems, including the analysis of spring-mass systems. In the study of a nonlinear spring-mass system, a differential equation is a mathematical expression that relates the function of one or more variables, its derivatives, and the time. The given exercise presents a second-order nonlinear differential equation, which is characterized by the presence of the term \(\tan\left(\frac{\pi x}{2\delta}\right)\) and describes the motion of a mass attached to a nonlinear spring.

In simpler terms, a differential equation like the one given in the exercise means that the acceleration of the mass (represented by the second derivative of the displacement, \(x''\)) depends on the position of the mass in a nonlinear way. The nonlinearity comes from the tangential function, which makes the spring force infinite at certain points (the asymptotes at \(x=\pm\delta\)). This nonlinearity introduces complexity into solving the equation, often requiring approximation methods when an analytical solution is not feasible.

Moreover, the initial conditions \(x(0)=0\) and \(x'(0)=0\) indicate the starting point of the system, which is crucial for determining the unique solution to the differential equation. These initial conditions represent a system that begins at rest, with the mass at the equilibrium position. Understanding how to work with differential equations and how to identify and apply initial conditions is essential for students to solve dynamic system problems effectively.

Numerical Methods

When tackling complex nonlinear differential equations, numerical methods come into play. These methods provide a way to approximate the solutions of equations that cannot be solved analytically. Numerical methods include a variety of algorithms that can iteratively reach closer to the solution.

In the context of the nonlinear spring-mass system, the step-by-step solution involves the use of numerical integration to solve the system of first-order differential equations derived from the original second-order equation. This transformation is necessary because most numerical solvers are designed to handle first-order equations. For instance, methods like Euler's method, the Runge-Kutta method, and others are well-established numerical techniques that compute the values of the unknown function at discrete points.

Choosing the Right Numerical Method

The selection of the numerical method depends on the required precision and computational resources available. Some methods are faster but less accurate, while others offer a higher precision at the cost of more computing power. For educational purposes, software packages such as MATLAB or Python's SciPy library provide built-in functions like \(\text{ode45}\) or \(\text{solve\_ivp}\), which are sophisticated enough to handle a wide range of problems with user-friendly commands.

Boundary Value Problems

A boundary value problem (BVP) is a type of differential equation coupled with a set of boundary conditions. Unlike initial value problems, which specify conditions at one point (typically the start of a time domain), boundary value problems require conditions at two or more points. In the exercise provided, although the problem presented is initially described as governed by initial conditions, the nature of the spring system with limits on the elongation invokes the concept of boundary constraints, such as the impossibility of the spring to be stretched or compressed beyond certain physical limits (as indicated by the asymptotes at \(x=\pm\delta\)).

BVPs are common in the study of steady-state situations, where conditions at the spatial endpoints must be satisfied. They are also crucial in areas like quantum mechanics, heat conduction, and in the study of steady fluid flows. In the more theoretical sense, the asymptotic behavior of the spring force, becoming infinite as the displacement approaches \(\pm\delta\), can be thought of metaphorically as a boundary in the physical model though the actual numeric solution does not usually directly incorporate this concept since it focuses on the dynamic behavior well within these boundaries.

Understanding boundary value problems is crucial for students as it provides the context for realistic constraints in physical systems, emphasizing that while numerical methods can find solutions, these are bound by the physical realities dictated by the system's properties.