Short Answer:

A single-degree-of-freedom (SDOF) system is modeled by a mass, spring and (optionally) a damper. Its equation of motion comes from Newton’s second law: sum of forces = mass × acceleration. Taking upward/right as positive, the inertial force  plus damping force  plus restoring spring force  balance any external force .

So the standard time-domain equation of motion is

which describes undamped, damped and forced cases depending on  and . This single second-order ODE completely defines the SDOF dynamics.

Detailed Explanation :

Derivation of Equation of Motion

Consider the common SDOF mechanical model: a lumped mass  attached to a linear spring of stiffness  and a viscous damper of coefficient . The mass can move only along one coordinate  measured from equilibrium. An external force  may act on the mass. We want a simple, clear derivation of the governing differential equation.

1. Choose sign convention and free-body diagram
   • Let  be positive to the right (or upwards).
   • Draw the mass and show three forces acting on it: spring force , damper force , and external force .
   • The spring force opposes displacement: .
   • The viscous damping force opposes velocity: .
   • The inertial force (from Newton) is  and acts in the direction of acceleration.
2. Apply Newton’s second law (translational form)
   Sum of forces acting on the mass  acceleration:

Substitute the expressions for spring and damper forces:

Rearranging to collect terms on the left gives the standard form:

1. Interpretation of terms
   •  is the inertial term (resists acceleration).
   •  is the damping term (energy dissipation).
   •  is the restoring (elastic) term.
   •  is the applied excitation (can be zero for free vibration).

Special Cases and Important Parameters

• Undamped free vibration ():

Solution is sinusoidal:  with natural angular frequency

• Damped free vibration ():

Define the critical damping  and damping ratio . The response depends on :

• • Underdamped (): oscillatory with exponentially decaying amplitude; damped frequency .
  • Critically damped (): fastest return to equilibrium without oscillation.
  • Overdamped (): slow, non-oscillatory return.
• Harmonically forced steady-state (): steady-state amplitude and phase are

and the phase lag  satisfies

These formulas show resonance behavior when forcing frequency  approaches , moderated by damping.

Energy and Physical Meaning

• The terms in the equation reflect energy storage and dissipation:
  •  stores potential (elastic) energy .
  •  relates to kinetic energy .
  •  dissipates energy at a rate .
• For  and , total mechanical energy is conserved; with  energy decays.

Modeling Notes and Assumptions

• Linear model assumes Hooke’s law for the spring and viscous linear damping. Good for small displacements and many engineering problems.
• Real systems may require nonlinear springs/dampers or multiple degrees of freedom; SDOF is the first and most useful approximation.
• The sign convention must be consistent when writing the equation; reversing signs changes the algebra but not physics.

Conclusion

Deriving the equation of motion for an SDOF system is straightforward: draw the free-body diagram, write expressions for spring and damping forces, and apply Newton’s second law. The standard second-order ODE

captures inertia, damping, stiffness and external forcing. From this single equation we obtain natural frequency, damping ratio, transient and steady-state responses — fundamental quantities used to design, analyze and control vibrating mechanical systems.