Historical Context & Motivation
The study of oscillatory motion is one of the oldest branches of mechanics, stretching back to antiquity when scholars first observed the rhythmic swing of pendulums and the plucking of stringed instruments. The concept of undamped free vibration — a system oscillating at its natural frequency without external forcing or energy dissipation — represents the simplest and most fundamental model in vibration theory. Understanding this idealized case is essential because it establishes the mathematical backbone upon which all more complex analyses, including damped, forced, and nonlinear vibrations, are built. In engineering practice, every structure from a skyscraper to a micro-electromechanical sensor possesses natural frequencies, and predicting them begins with this very model.
The central question this lesson addresses is deceptively simple: if a spring-mass system is displaced from equilibrium and released with no friction or external driving force, what is the subsequent motion? Answering this question precisely — determining the natural frequency, period, and displacement as a function of time — provides the essential toolkit for every subsequent topic in vibration analysis.
Core Principles & Definitions
Before diving into the mathematics, it is important to establish the physical assumptions and terminology that define undamped free vibration. The system we consider consists of a rigid body (or point mass) connected to a linearly elastic restoring element (a spring) with no energy-dissipating mechanism (no dashpot, no air resistance). Once displaced from its static equilibrium position and released, the system oscillates indefinitely with constant amplitude. While no real system is truly undamped, this idealization is remarkably useful: many lightly damped engineering systems behave nearly identically to the undamped model over short time horizons, and the natural frequency predicted by the undamped model closely approximates the damped natural frequency when damping is small.
Free Vibration
Undamped
Natural Frequency (ωₙ)
Simple Harmonic Motion (SHM)
Degrees of Freedom (DOF)
Visual Explanation — The Spring-Mass System
The diagram above illustrates the fundamental setup for every undamped free vibration problem. A mass m is attached to a spring of stiffness k whose other end is fixed to a rigid wall. When the mass is displaced from its static equilibrium position by a distance x and released, the spring exerts a restoring force equal to −kx. The negative sign is critical: it indicates that the force always acts to return the mass toward equilibrium, opposing the direction of displacement. With no damping and no external force, Newton's second law ΣF = ma yields the second-order ordinary differential equation mẍ + kx = 0, which is the governing equation of undamped free vibration. The solution to this equation is a sinusoidal function of time, confirming that the motion is simple harmonic.
Mathematical Framework
We now develop the complete mathematical solution to the undamped free vibration problem. Starting from the equation of motion derived via Newton's second law, we obtain the general solution, and then apply initial conditions to determine the specific response. Two equivalent forms of the solution are presented, each useful in different contexts.
This is a linear, second-order, homogeneous ODE with constant coefficients. Assuming a trial solution x(t) = Cest, substitution yields the characteristic equation s² + ωₙ² = 0, giving purely imaginary roots s = ±iωₙ. This confirms oscillatory (non-decaying) motion. Using Euler's formula, the general solution can be written in two standard forms.
Detailed Breakdown — Displacement, Velocity & Acceleration
A thorough understanding of undamped free vibration requires examining how displacement, velocity, and acceleration relate to one another over time. Because the solution is sinusoidal, these three quantities are phase-shifted versions of the same oscillation. The displacement x(t) peaks when velocity is zero (the mass reverses direction), and the velocity peaks when the mass passes through equilibrium (where the spring force and hence acceleration are zero). Acceleration is exactly 180° out of phase with displacement, always acting to pull the mass back toward equilibrium.
| Quantity | Expression (for x₀ > 0, v₀ = 0) | Max Value | Phase Relative to x(t) |
|---|---|---|---|
| Displacement x(t) | x₀ cos(ωₙt) | x₀ | 0° (reference) |
| Velocity ẋ(t) | −x₀ωₙ sin(ωₙt) | x₀ωₙ | Leads by 90° (−sin) |
| Acceleration ẍ(t) | −x₀ωₙ² cos(ωₙt) | x₀ωₙ² | 180° out of phase |
An important energy perspective reinforces these observations. The total mechanical energy E = ½kx² + ½mẋ² remains constant throughout the motion. At the extremes of displacement where x = ±x₀ and ẋ = 0, all energy is stored as elastic potential energy in the spring. At the equilibrium crossing where x = 0 and ẋ is maximum, all energy has converted to kinetic energy. The continuous exchange between potential and kinetic energy is the engine that sustains the oscillation indefinitely in the absence of damping.
Worked Example — Horizontal Spring-Mass System
A 4 kg block rests on a frictionless horizontal surface and is attached to a spring of stiffness k = 900 N/m. The block is pulled 0.05 m to the right of its equilibrium position and released from rest. Determine the natural frequency, the period, the displacement as a function of time, and the velocity and acceleration when x = 0.03 m.
Assumptions, Strengths & Limitations
Every mathematical model is only as good as its assumptions. The undamped free vibration model is the simplest vibration model available, and its power lies in providing closed-form analytical solutions that offer direct physical insight. However, practicing engineers must understand when the model is applicable and when its idealizations break down.
| Aspect | Strengths | Limitations |
|---|---|---|
| Damping | Predicts natural frequency ωₙ = √(k/m) accurately even for lightly damped systems (ζ < 0.1), since ωd = ωₙ√(1 − ζ²) ≈ ωₙ. | Cannot predict amplitude decay, settling time, or energy loss — all real systems dissipate energy. |
| Linearity | Linear ODE yields superposition: solutions for different initial conditions can be added. Analysis stays tractable. | Assumes spring force is exactly proportional to displacement. Large deflections, nonlinear springs, and geometric nonlinearity are not captured. |
| External forcing | Provides the homogeneous solution needed to solve forced vibration problems (complementary function). | Cannot model resonance or steady-state response to external loads on its own. |
| Degrees of freedom | SDOF model is the building block for multi-DOF modal analysis. | Real structures have infinitely many DOFs; SDOF captures only the dominant mode. |
| Practical use | Quick hand calculations, preliminary design, natural frequency estimation, modal testing interpretation. | High-fidelity predictions require damped, forced, and potentially nonlinear models. |
Connection to Damped & Forced Vibration Theory
The undamped free vibration model is the departure point for a hierarchy of increasingly realistic models. Adding a viscous damping term cẋ (proportional to velocity) yields the damped free vibration model mẍ + cẋ + kx = 0, whose solution decays exponentially at a rate governed by the damping ratio ζ = c/(2mωₙ). Adding an external time-varying force F(t) leads to forced vibration, where resonance phenomena become critical. Multi-degree-of-freedom systems are decomposed into sets of SDOF equations using modal analysis, each governed by its own undamped natural frequency. The table below summarizes how each extension builds on the undamped free vibration foundation.
| Feature | Undamped Free (This Lesson) | Damped Free | Forced (Undamped or Damped) |
|---|---|---|---|
| Governing Equation | mẍ + kx = 0 | mẍ + cẋ + kx = 0 | mẍ + cẋ + kx = F(t) |
| Amplitude over time | Constant | Decays exponentially | Steady-state + transient |
| Oscillation frequency | ωₙ = √(k/m) | ωd = ωₙ√(1−ζ²) | ωₙ (transient) + ω (forcing) |
| Energy | Conserved | Dissipated by damper | Input by external force |
| Key new concept | Natural frequency ωₙ | Damping ratio ζ | Resonance, magnification factor |
As you advance in dynamics and structural analysis, you will find that the natural frequency ωₙ computed from the undamped model appears in virtually every formula: the damped frequency ωd, the frequency response function, the dynamic magnification factor, and modal participation factors all reference ωₙ. Mastering the undamped case therefore gives you a transferable tool that you will use throughout your engineering career, from earthquake engineering to aerospace flutter analysis.
Practice Problems
Lesson Summary
This lesson introduced undamped free vibration — the oscillatory motion of a single-degree-of-freedom system subject to no damping and no external forcing. Beginning with the physical setup of a spring-mass system, we derived the equation of motion mẍ + kx = 0 using Newton's second law, identified its solution as simple harmonic motion, and expressed the general response in both the sine-cosine form x(t) = A cos(ωₙt) + B sin(ωₙt) and the amplitude-phase form x(t) = X sin(ωₙt + φ). The natural frequency ωₙ = √(k/m) depends solely on the stiffness and mass of the system, not on initial conditions, and governs the period τₙ = 2π/ωₙ and cyclic frequency fₙ = ωₙ/(2π).
We examined the phase relationships among displacement, velocity, and acceleration and connected them to the conservation of energy between kinetic and elastic potential forms. Through a worked example, we demonstrated the complete solution process from identifying parameters to computing the response at a specific displacement. While the undamped model is an idealization, it accurately predicts natural frequencies for lightly damped systems and serves as the foundation upon which damped, forced, and multi-DOF vibration analyses are constructed.