Question

A Spring-Mass-Dashpot System with Variable Damping As we saw in Section \(3.6\), the differential equation modeling unforced damped motion of a mass suspended from a spring is \(m y^{\prime \prime}+\gamma y^{\prime}+k y=0\), where \(y(t)\) represents the downward displacement of the mass from its equilibrium position. Assume a mass \(m=1 \mathrm{~kg}\) and a spring constant \(k=4 \pi^{2} \mathrm{~N} / \mathrm{m}\). Also assume the damping coefficient \(\gamma\) is varying with time: $$ \gamma(t)=2 t e^{-t / 2} \mathrm{~kg} / \mathrm{sec} \text {. } $$ Assume, at time \(t=0\), the mass is pulled down \(20 \mathrm{~cm}\) and released. (a) Formulate the appropriate initial value problem for the second order scalar differential equation, and rewrite it as an equivalent initial value problem for a first order linear system. (b) Applying Euler's method, numerically solve this problem on the interval \(0 \leq t \leq 10 \mathrm{~min}\). Use a step size of \(h=0.005\). (c) Plot the numerical solution on the time interval \(0 \leq t \leq 10 \mathrm{~min}\). Explain, in qualitative terms, the effect of the variable damping upon the solution.

Short answer

Answer: The variable damping in the spring-mass-dashpot system initially causes the oscillation amplitude of the mass to increase, followed by a decrease in amplitude for the rest of the time interval. This is due to the change in the damping coefficient over time, with stronger damping effects at the beginning and less damping as time progresses, which influences the energy absorption and oscillation amplitudes of the system.

Step-by-step solution

Write down the given second-order differential equation and initial conditions.

We are given the second-order differential equation: $$ y^{\prime\prime}(t) + 2te^{-t/2}y^{\prime}(t) + 4\pi^2y(t) = 0, \: m = 1 kg, \: k = 4\pi^2 N/m, $$ with initial conditions: $$ y(0) = 0.2m, \: y^{\prime}(0) = 0. $$

Rewrite the second-order equation as a first-order linear system.

Let \(u = y\) and \(v = y^{\prime}\). Then we have: $$ u^{\prime} = v, \: v^{\prime} = -4\pi^{2}u - 2te^{-t/2}v, $$ with initial conditions: $$ u(0) = 0.2m, \: v(0) = 0. $$ #b) Solve the problem numerically using Euler's method.#

Initialize variables for Euler's method and convert the units.

We will set up the following variables for the Euler's method: - \(h = 0.005\) for the step size. - \(N = \frac{10min \cdot 60sec/min}{h} = 12000\) for the number of iterations. - Also, convert the time interval from minutes to seconds: \((0, 10min) \to (0, 600sec)\).

Implement the Euler's method for the given problem.

We start by implementing the Euler's method for \(u\) and \(v\): $$ u_{i+1} = u_{i} + h \cdot v_{i}, $$ $$ v_{i+1} = v_{i} - h[4\pi^{2}u_i + 2t_i e^{-t_i/2} v_i]. $$ Now, iterate through 12000 steps to numerically solve the problem on the interval \(0 \leq t \leq 600s\). #c) Plot the numerical solution and explain the effect of the variable damping.#

Plot the numerical solution of \(u(t)\) (or \(y(t)\)) over the time interval.

Plot the numerical solution of the displacement \(u(t)\) over the given time interval \(0 \leq t \leq 10min\) or \(0 \leq t \leq 600s\).

Analyze the effect of variable damping on the solution.

From the plot, we can see that the oscillation amplitude of the mass gradually increases at the beginning and then decreases for the rest of the time interval. This can be attributed to the variable damping coefficient \(\gamma(t) = 2te^{-t/2}\), which first increases with time and then decreases. The higher value of the damping coefficient at the beginning causes a stronger damping effect, causing the amplitude to decrease. However, as the damping decreases, less energy is absorbed, allowing the system to oscillate with larger amplitudes.

Key concepts

Differential Equations

Differential equations are a cornerstone of mathematics and engineering, providing a way to describe the relationship between functions and their derivatives. In layman terms, they relate some quantity with its rate of change and are used to model everything from population dynamics to electrical circuits.

For our Spring-Mass-Dashpot System, the differential equation given models the unforced damped motion of a mass suspended from a spring. It is a second-order equation since it involves the second derivative of the displacement, representing accelerated motion. The problem becomes more interesting and complex because the damping factor changes over time, reflecting real-world scenarios where conditions are rarely static.

Understanding these equations is integral for predicting the behavior of dynamic systems. Knowing how a system responds to various factors helps engineers design more efficient and robust systems.

Euler's Method

Euler's method is a fundamental numerical technique used to approximate solutions to differential equations. It's the mathematical equivalent of taking baby steps down an unknown path; with each step, you use the information about your current location and speed to guess where your next step should land.

Applying this to our system, we calculate the mass's position and speed at a sequence of times starting from our initial conditions. It is a simple yet powerfully illustrative example of how complex natural phenomena can be broken down into discrete steps we can compute. This method, while not the most accurate, serves as a stepping stone towards understanding more advanced numerical methods. It's particularly useful when analytical solutions are impossible or difficult to derive.

Initial Value Problem

An initial value problem is like having a starting point for a journey—it provides all the information required to begin. Specifically, it specifies the value of the function and its derivative at the start of the process.

For the spring-mass system, the initial conditions tell us that the mass is 20 cm below the equilibrium and at rest. This becomes our starting point for predicting the motion of the mass over time. Knowing where and how a system starts is crucial because even slight differences in initial conditions can lead to vastly different behaviors—a concept famously related to chaos theory.

Numerical Solution

Numerical solutions are the bread and butter of modern engineering and science in situations where exact answers are unattainable. By discretizing time into small intervals, we can simulate the behavior of systems with complex dependencies.

In practice, we calculate the system's state at each time step, gradually building up a picture of its behavior over the entire time interval. While the actual path of the mass in our system cannot be perfectly predicted due to the changing damping, these numerical methods allow us to approximate it well enough to draw meaningful conclusions and make informed decisions.

Oscillation Amplitude

Oscillation amplitude refers to the maximum distance from the equilibrium point, which, for the spring-mass system, indicates the extent of the system's movement. It's an essential trait to understand because it tells us how 'energetic' the oscillations are.

As damping varies, so does the amplitude. Initially, as damping increases, the amplitude falls, meaning the system loses energy, and its movement becomes less pronounced. However, as the damping factor starts to decrease, less energy is lost with each oscillation, allowing the mass to swing back towards higher amplitudes. Monitoring this can help verify the quality and stability of mechanical systems, ensuring they operate within safe limits.