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Incoming links: Middle-mass-independent mode of a fixed three-mass spring chain
Past exam of the mathematical methods II course of the University of Oxford 2019 11 c Solution 2026-10-25
To specify “initial motion” unambiguously, choose each mode to pass through equilibrium at with proportional to its listed mode shape. The arrows in the figure then describe relative initial velocities:
- In mode 1, all three masses move in the same direction; the central mass has times either outer velocity.
- In mode 2, the two outer masses move with equal and opposite velocities, while the central mass remains stationary.
- In mode 3, the outer masses move together while the central mass moves oppositely with times their speed.
Reversing every arrow changes only the common phase. A quarter-cycle later these same ratios describe the extremal displacements, with zero instantaneous velocity.
Relative initial velocities of the three fixed-end normal modes, starting as the masses pass through equilibrium.
Arrow lengths follow the mode shapes. The antisymmetric mode has equal and opposite outer velocities and zero central velocity; its angular frequency is unchanged by any positive change in central mass. Details.
Now replace the central mass by , leaving the outer masses equal to . The antisymmetric mode shape gives zero net force on the central mass, since . Each outer mass still obeys . ConsequentlyThis middle-mass-independent mode of a fixed three-mass spring chain is exact, not merely a large- approximation. The other two modes move the central mass and generally change. For example, their symmetric-sector frequencies satisfywhich explicitly depends on .