Question

Consider a linear array of \(n\) masses, each equal to \(m,\) connected by \(n+1\) springs, all massless and having spring constant \(k\), with the outer ends of the first and last springs fixed. The masses can move without friction in the linear dimension of the array. a) Write the equations of motion for the masses. b) Configurations of motion for which all parts of a system oscillate with the same angular frequency are called normal modes of the system; the corresponding angular frequencies are the system's normal-mode angular frequencies. Find the normal-mode angular frequencies of this array.

Short answer

a) The equations of motion for the linear array of masses can be written as: For mass 1: $$m\ddot{x}_1 = k(x_1-x_2) - kx_1$$ For mass 2: $$m\ddot{x}_2 = k(x_2 - x_1) - k(x_2-x_3)$$ For mass i: $$m\ddot{x}_i = k(x_i - x_{i-1}) - k(x_i - x_{i+1})$$ For mass n: $$m\ddot{x}_n = k(x_n-x_{n-1})-kx_n$$ b) The normal-mode angular frequencies for the linear array of masses connected by springs are given by: $$\omega = \sqrt{\frac{k}{m}} \sqrt{1-\cos\frac{(2j-1)\pi}{2(n+1)}}$$ Where \(j = 1, 2, \dots, n\).

Step-by-step solution

Derive the equations of motion for the masses

Let us consider the displacement of each mass along the linear dimension as \(x_1, x_2, \dots, x_n\). Now, we can find the net force acting on each mass using Hooke's law and Newton's second law: For mass \(1\): $$F_1 = k(x_1-x_2) - kx_1 = m\ddot{x}_1$$ For mass \(2\): $$F_2 = k(x_2 - x_1) - k(x_2-x_3) = m\ddot{x}_2$$ Similarly, for mass \(i\): $$F_i = k(x_i - x_{i-1}) - k(x_i - x_{i+1}) = m\ddot{x}_i$$ For mass \(n\): $$F_n = k(x_n-x_{n-1})-kx_n = m\ddot{x}_n$$

Write the equations of motion in matrix form

We can represent the above equations of motion in the matrix form: $$ \begin{bmatrix} k & -k & 0 & \dots & 0 \\ -k & 2k & -k & \dots & 0 \\ 0 & -k & 2k & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & k \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \\ \end{bmatrix} = m \begin{bmatrix} \ddot{x}_1 \\ \ddot{x}_2 \\ \ddot{x}_3 \\ \vdots \\ \ddot{x}_n \\ \end{bmatrix} $$

Find the normal modes

In order to find the normal modes of oscillation, we assume a trial solution of the following form: $$x_i(t) = A_i \cos(\omega t + \phi_i)$$ Substitute the trial solution into the matrix equation derived above. This will give us: $$ \begin{bmatrix} k & -k & 0 & \dots & 0 \\ -k & 2k & -k & \dots & 0 \\ 0 & -k & 2k & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & k \end{bmatrix} \begin{bmatrix} A_1 \\ A_2 \\ A_3 \\ \vdots \\ A_n \end{bmatrix} = m\omega^2 \begin{bmatrix} A_1 \\ A_2 \\ A_3 \\ \vdots \\ A_n \end{bmatrix} $$

Find normal-mode angular frequencies

To find the normal-mode angular frequencies, we need to find the eigenvalues of the mass-spring matrix: $$ \begin{bmatrix} (k-m\omega^2) & -k & 0 & \dots & 0 \\ -k & (2k-m\omega^2) & -k & \dots & 0 \\ 0 & -k & (2k-m\omega^2) & \dots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \dots & (k-m\omega^2) \end{bmatrix} \begin{bmatrix} A_1 \\ A_2 \\ A_3 \\ \vdots \\ A_n \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \\ \vdots \\ 0 \\ \end{bmatrix} $$ We now have to solve this eigenvalue problem. The eigenvalues will give us the normal-mode angular frequencies: $$\omega = \sqrt{\frac{k}{m}} \sqrt{1-\cos\frac{(2j-1)\pi}{2(n+1)}}$$ Where \(j = 1, 2, \dots, n\). These are the normal-mode angular frequencies for the linear array of masses connected by springs.

Key concepts

Equations of Motion

Understanding the equations of motion is crucial for analyzing systems in classical mechanics. These equations describe how the position of an object changes in time due to the forces acting upon it. In our exercise, we consider a linear array of masses connected by springs, obeying Newton's second law: the acceleration of an object is proportional to the net force acting on it and inversely proportional to its mass.

To derive the equations of motion for each mass, we look at the forces applied by the neighboring masses through the connecting springs. Each spring force is determined by Hooke's law, where the force is proportional to the displacement from the equilibrium position with a proportionality constant known as the spring constant. By writing out these forces for each mass and assuming that the ends of the array are fixed, we end up with a set of second-order differential equations. These equations can be conveniently expressed in matrix form, facilitating the use of linear algebra to solve for the system's behavior.

Hooke's Law

At the heart of our problem lies Hooke's law, a principle stating that the force exerted by a spring is directly proportional to the amount it is stretched or compressed from its relaxed position. Mathematically, Hooke's law is expressed as \( F = -kx \), where \( F \) is the force exerted by the spring, \( k \) is the spring constant, and \( x \) is the displacement. This linear relationship is fundamental to understanding oscillatory motion, as it provides a predictive basis for the force present at any given displacement within the elastic limit of the spring.

In the given exercise, Hooke's law is used to calculate the forces exerted on each mass by the springs. This allows us to find the net force acting on each mass, which in turn is used to establish the equations of motion for the masses.

Eigenvalue Problem

An eigenvalue problem is a type of mathematical problem that arises in many fields of science and engineering. In the context of our exercise, it is related to determining the normal-mode angular frequencies of an oscillatory system. When we represent the system of masses and springs in matrix form and substitute a trial solution for the motion of the masses, we transform our problem into seeking values of angular frequency \( \omega \) that satisfy the characteristic equation.

This leads us to a classic eigenvalue problem, where we are looking for eigenvalues of the matrix that relates to the spring constants and mass. Solving this problem yields values for \( \omega \), which correspond to the natural frequencies at which the system tends to oscillate without external forces. These are the normal-mode angular frequencies.

Oscillatory Motion

Oscillatory motion is a type of repetitive movement back and forth about an equilibrium position, the central point of the motion. This is exhibited by systems like our mass-spring array when they're displaced from equilibrium. An important feature of such systems is that they can experience normal-mode oscillations, where all parts oscillate with the same frequency and in a fixed pattern. These modes are particularly simple forms of motion that the system can undergo.

To find the normal-mode angular frequencies for our array of masses, we seek solutions to the equations of motion where all masses move in sync. We achieve this by predicting a form for the motion, substituting it into our matrix equations, and then solving the resulting eigenvalue problem. The resulting frequencies from this calculation denote the specific frequencies at which our system prefers to vibrate and are a key concept in understanding the behavior of such oscillatory systems.