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The equation of motion of a damped single-degree-of-freedom system under a force \( p(t) \) is

  • A.\( m\ddot{u} + c\dot{u} + ku = p(t) \)
  • B.\( m\ddot{u} + ku = c\dot{u} \) with the forcing omitted entirely
  • C.\( c\dot{u} + ku = m\ddot{u} \) with the signs reversed throughout
  • D.\( m\dot{u} + c\ddot{u} + ku = p(t) \) with the two derivatives interchanged

Answer: A

Correct answer: \( m\ddot{u} + c\dot{u} + ku = p(t) \)

The equation states dynamic equilibrium among four forces: the inertia force \( m\ddot{u} \), the damping force \( c\dot{u} \), the restoring force \( ku \), and the applied load.

Each term pairs a property with the derivative it acts on - mass with acceleration, damping with velocity, stiffness with displacement - and that pairing is what the distractors disturb. Interchanging the derivatives, or moving a term to the other side, gives an equation with no physical meaning.

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