Past exam of the mathematical methods II course of the University of Oxford / 2019 / 11

A normal mode is a motion of a linear oscillatory system in which every coordinate has the same angular frequency and a fixed relative amplitude and phase. For a conservative small-displacement system it has the formwhere $q$ is the mode shape, $K$ the stiffness matrix, and $M$ the mass matrix. Superposing normal modes gives the general small-oscillation motion when their eigenvectors form a complete basis.

Two common examples are standing transverse oscillations of a string fixed at both ends, and oscillations of coupled masses and springs. The string's mode shapes are sinusoidal along its length; the coupled system's mode shapes are fixed patterns of particle displacements. The following parts determine those patterns for the three-mass system in the source figure.

Let $x_1,x_2,x_3$ be longitudinal displacements from equilibrium, with positive direction to the right. The fixed walls have zero displacement. The first mass feels forces $-k{x}_1$ and $k(x_2-x_1)$; the central mass feels $k(x_1-x_2)$ and $k(x_3-x_2)$; the last mass feels $k(x_2-x_3)$ and $-k{x}_3$. Hooke's law and Newton's second law therefore giveEquivalently,All four springs contribute: the wall springs supply one of the two diagonal terms in the first and third rows. This is a fixed-end chain, not a freely ended chain with a zero-frequency translation mode.

Put $\mathbf{x}=q{e}^{i\omega t}$ in solution. The normal mode equation is $Aq=\lambda q$, with $\lambda =m{\omega }^2/k$. Its characteristic polynomial isThe three eigenvalues are $2-\sqrt{2}$, $2$, and $2+\sqrt{2}$. Solving the corresponding linear equations yieldsEach mode shape may be multiplied by any nonzero scalar. Their squared Euclidean norms are $4,2,4$, respectively, so unit-normalized eigenvectors are $q_1/2,q_2/\sqrt{2},q_3/2$. They are mutually orthogonal, as expected for the real symmetric stiffness matrix with equal masses.

The entries in the table are angular frequencies; the corresponding ordinary frequencies in cycles per unit time are $f_j={\omega }_j/(2\pi )$. All three are positive because the fixed walls remove uniform translation as a free motion.

To specify “initial motion” unambiguously, choose each mode to pass through equilibrium at $t=0$ with $\dot{\mathbf{x}}(0)$ proportional to its listed mode shape. The arrows in the figure then describe relative initial velocities:

Reversing every arrow changes only the common phase. A quarter-cycle later these same ratios describe the extremal displacements, with zero instantaneous velocity.

Now replace the central mass by $M$, leaving the outer masses equal to $m$. The antisymmetric mode shape $q=(1,0,-1{)}^T$ gives zero net force on the central mass, since $k{x}_1+k{x}_3=0$. Each outer mass still obeys $m\ddot{x}=-2kx$. ConsequentlyThis middle-mass-independent mode of a fixed three-mass spring chain is exact, not merely a large-$M$ approximation. The other two modes move the central mass and generally change. For example, their symmetric-sector frequencies satisfywhich explicitly depends on $M$.