Question

A mass \(m\) on a spring with constant \(k\) satisfies the differential equation (see Section 3.8 ) \(m u^{\prime \prime}+k u=0\) where \(u(t)\) is the displacement at time \(t\) of the mass from its equilibrium position. (a) Let \(x_{1}=u\) and \(x_{2}=u^{\prime}\); show that the resulting system is \(\mathbf{x}^{\prime}=\left(\begin{array}{rr}{0} & {1} \\ {-k / m} & {0}\end{array}\right) \mathbf{x}\) (b) Find the eigenvalues of the matrix for the system in part (a). (c) Sketch several trajectories of the system. Choose one of your trajectories and sketch the corresponding graphs of \(x_{1}\) versus \(t\) and of \(x_{2}\) versus \(t\), Sketch both graphs on one set of axes. (d) What is the relation between the eigenvalues of the coefficient matrix and the natural frequency of the spring-mass system?

Short answer

Question: Rewrite the second-order ordinary differential equation governing the motion of a mass-spring system as a system of two first-order ODEs, find the eigenvalues of the system matrix, and determine the relationship between the eigenvalues and the natural frequency of the system.

Step-by-step solution

Define the variables

Define \(x_1 = u\) and \(x_2 = u'\). The goal is to rewrite the given second-order differential equation in terms of these new variables.

Calculate derivatives

Compute the derivatives of \(x_1\) and \(x_2\) with respect to time: $$ x_1' = u' = x_2 $$ $$ x_2' = u'' = -\frac{k}{m}u = -\frac{k}{m}x_1 $$

Write the system

Combine the expressions for \(x_1'\) and \(x_2'\) into a matrix equation: $$ \begin{pmatrix} x_1' \\ x_2' \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ -\frac{k}{m} & 0 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \end{pmatrix} $$ So, we have \(\mathbf{x}' = \begin{pmatrix} 0 & 1 \\ -\frac{k}{m} & 0 \end{pmatrix} \mathbf{x}\). #b. Eigenvalues of the matrix#

Determine the characteristic equation

Determine the characteristic equation for the given matrix \(\begin{pmatrix} 0 & 1 \\ -\frac{k}{m} & 0 \end{pmatrix}\): $$ \det\left(\begin{pmatrix} 0-\lambda & 1 \\ -\frac{k}{m} & 0-\lambda \end{pmatrix}\right) = \lambda^2 + \frac{k}{m} = 0 $$

Solve for the eigenvalues

Solve the characteristic equation to find the eigenvalues: $$ \lambda^2 = -\frac{k}{m} \Rightarrow \lambda = \pm \sqrt{-\frac{k}{m}} i $$ #c. Trajectories and graphs of solutions# This part of the problem requires graphical analysis, which cannot be accurately shown here in text form. For reference, the following should be noticed for the solutions of the system: 1. The trajectories of the solutions will be ellipses in the \(x_1 - x_2\) plane, representing simple harmonic motion. 2. Both \(x_1(t)\) and \(x_2(t)\) will be sinusoidal functions, with \(x_1(t)\) representing the displacement of the mass and \(x_2(t)\) representing the velocity of the mass. #d. Eigenvalues and natural frequency#

Identify the natural frequency

The natural frequency \(\omega_n\) of the spring-mass system is given by: $$ \omega_n = \sqrt{\frac{k}{m}} $$

Compare with the eigenvalues

The eigenvalues are \(\pm \sqrt{-\frac{k}{m}}i\). Comparing the two expressions, we have: $$ \pm i \omega_n = \pm \sqrt{-\frac{k}{m}} i $$

Determine the relationship

The real components of the eigenvalues are zero, and the imaginary components are directly related to the natural frequency of the system: $$ \operatorname{Im}(\lambda) = \pm \omega_n $$

Key concepts

Spring-Mass System

A spring-mass system is an essential physics model used to describe oscillatory motion. The system involves a mass attached to a spring, where the mass is free to move back and forth due to the elastic properties of the spring. Such a system is governed by Hooke's law, where the force exerted by the spring is proportional to the displacement from equilibrium: \[ F = -ku \] Here, \(k\) denotes the spring constant, a measure of the spring's stiffness, and \(u\) represents the displacement of the mass from its equilibrium position.
In the exercise, the equation given, \(m u'' + ku = 0\), depicts the motion of the mass. The equation is a second-order differential equation, which is a typical way to describe systems involving forces and motion. Here:

• \(m\) is the mass of the object
• \(u''\) is the acceleration of the mass (second derivative of displacement with respect to time)
• \(u\) is the displacement

The goal is to analyze this system using the concept of state variables \(x_1\) and \(x_2\), leading to insights into the system's dynamic nature.

Eigenvalues

Eigenvalues are a fundamental part of understanding systems of differential equations, like the spring-mass system in this exercise. Simply put, they help determine the nature of the system's solutions, especially their stability and oscillatory behavior. The equation transforms into a matrix form: \[ \begin{pmatrix} x_1' \ x_2' \end{pmatrix} = \begin{pmatrix} 0 & 1 \ -\frac{k}{m} & 0 \end{pmatrix} \begin{pmatrix} x_1 \ x_2 \end{pmatrix} \] Here, \(x_1 = u\) and \(x_2 = u'\). To find the eigenvalues of the matrix \( \begin{pmatrix} 0 & 1 \ -\frac{k}{m} & 0 \end{pmatrix} \), we solve the characteristic equation: \[ \lambda^2 + \frac{k}{m} = 0 \] Which gives us: \[ \lambda = \pm \sqrt{-\frac{k}{m}} i \] Eigenvalues tell us that the motion is well-behaved and periodic. The imaginary components suggest oscillations, typical of a stable spring-mass system. An important take-away: for real oscillating systems, eigenvalues indicating sinusoidal motion are a clear sign that the system is neither growing nor decaying exponentially.

Natural Frequency

The natural frequency, denoted as \( \omega_n \), represents the frequency at which a system tends to oscillate in the absence of any driving or damping force. For the spring-mass system, the natural frequency is calculated using the formula: \[ \omega_n = \sqrt{\frac{k}{m}} \] This frequency is significant because it characterizes the inherent oscillation speed of the system. The greater the stiffness of the spring (larger \(k\)), or the lighter the mass (smaller \(m\)), the higher the natural frequency becomes.
It's crucial in systems design – in engineering, for example, when designing buildings or vehicles, avoiding resonant frequencies can prevent structural failures. In the context of the eigenvalues, which we found to be \( \pm i \omega_n \), this shows their relation to the natural frequency. The imaginary part of the eigenvalues is precisely this \( \omega_n \), tying back to the oscillatory nature and stability of the motion described by the spring-mass system: \[ \operatorname{Im}(\lambda) = \pm \omega_n \]

Phase Plane Analysis

Phase plane analysis is a visual approach to understanding the behaviors of dynamic systems like the spring-mass system. By plotting the trajectory of the system in a coordinate plane with axes \(x_1\) (displacement) and \(x_2\) (velocity), we can visualize how the system evolves over time. In this analysis:
- Each point on the phase plane represents a specific state of the system. - Trajectories (or paths) depict how the state changes over time.
In this exercise, the trajectories formed are ellipses, indicating periodic, bounded movements consistent with harmonic oscillators. Because the system is not subject to external forces or damping, these closed loops signify continuous, repetitive motion without loss of energy.
This graphical representation helps:

• Identify stable and unstable behaviors instantly.
• Understand the relationship between displacement and velocity intuitively.
• Predict long-term behavior without solving the differential equations repeatedly.

Phase plane analysis often accompanies mechanical systems analysis, making it a powerful tool to study spring-mass oscillators and many other dynamics-based models.