Historical Context & Motivation
The study of vibrating systems is one of the oldest branches of mechanics, yet it remains central to modern engineering design. From the pendulums of early clockmakers to the structural resonance that collapsed the Tacoma Narrows Bridge in 1940, understanding natural frequency has been essential for ensuring that machines, structures, and instruments operate safely and predictably. The single-degree-of-freedom (SDOF) mass–spring system is the simplest idealization that captures this vibratory behavior, and virtually every vibration analysis course begins with it because it introduces the fundamental concepts—restoring force, inertia, and oscillation—without the algebraic complexity of multi-degree-of-freedom models.
The fundamental question that the SDOF mass–spring model addresses is deceptively simple: if a mass is displaced from equilibrium and released, at what frequency will it oscillate? Answering this question rigorously requires combining Hooke's linear restoring force with Newton's second law, yielding a second-order ordinary differential equation whose solution reveals the system's natural frequency—the intrinsic rate at which the system prefers to vibrate in the absence of external forcing or damping.
Core Principles & Definitions
Before diving into the mathematics, it is critical to internalize the physical ingredients that give rise to oscillatory motion. Every SDOF mass–spring system consists of exactly two energy-storage elements—a mass that stores kinetic energy and a spring that stores potential (elastic) energy. The interplay between these two forms of energy is what sustains the oscillation: as the mass moves away from equilibrium, the spring absorbs kinetic energy and converts it to potential energy; as the spring pulls the mass back, potential energy converts to kinetic energy. This continuous exchange repeats at a fixed rate dictated entirely by the ratio of stiffness to mass.
Degree of Freedom
Hooke's Law (Linear Spring)
Inertia & Newton's Second Law
Natural Frequency (ω_n)
Free Vibration
Visual Explanation — The SDOF Mass–Spring Model
The diagram above captures the essential features of the SDOF model. On the left, the mass slides on a frictionless surface (we neglect damping for now) and is constrained to move only along the horizontal axis—hence one degree of freedom. The spring, depicted by the zigzag element, exerts a force that is always directed toward the equilibrium position; this restoring force is the mechanism that drives the mass back whenever it is displaced. Observe from the free-body diagram that the only horizontal force acting on the mass is −kx, which, when substituted into Newton's second law, immediately yields the governing equation of motion: mẍ + kx = 0. The lower-right panel shows the solution—a pure sinusoid—confirming that undamped free vibration is simple harmonic motion at the natural frequency ωn.
Mathematical Framework
We now derive the equation of motion from first principles and extract the natural frequency. Consider a mass m attached to a linear spring of stiffness k, constrained to translate along a single axis. Define x(t) as the displacement measured from the static equilibrium position so that x = 0 corresponds to the natural length configuration (gravity effects, if present, are already accounted for in defining the equilibrium). Applying Newton's second law in the x-direction gives the governing ordinary differential equation.
Dividing through by m and defining the quantity ωn² = k/m puts the equation into standard form. This normalization is not merely cosmetic—it reveals that the parameter ωn completely characterizes the free response.
To solve, assume x(t) = Aest. Substitution yields the characteristic equation s² + ωn² = 0, giving purely imaginary roots s = ±iωn. By Euler's formula the general solution is a superposition of sine and cosine at the natural frequency.
Energy Methods & Equivalent Spring Constants
In many practical situations the effective stiffness of a system is not immediately obvious because springs may be arranged in series or parallel configurations. Identifying the equivalent spring constant keq is essential before computing the natural frequency. Additionally, an energy-based approach offers a powerful alternative to Newton's-law derivation and provides a consistency check on equations of motion.
An alternative and often more elegant route to the equation of motion is the energy method. For conservative systems, the total mechanical energy E = T + U (kinetic plus potential) is constant. Differentiating E with respect to time and setting dE/dt = 0 yields d/dt[ ½mẋ² + ½kx² ] = mẋẍ + kxẋ = ẋ(mẍ + kx) = 0. Since ẋ is not identically zero, we recover mẍ + kx = 0. This technique—sometimes called the Rayleigh energy method—scales naturally to more complex systems. For instance, Rayleigh's quotient ωn² = keq/meq generalizes the SDOF result by interpreting 'stiffness' and 'mass' as effective quantities derived from energy expressions, even for distributed-parameter systems.
| Configuration | Equivalent Stiffness k_eq | Natural Frequency ωₙ |
|---|---|---|
| Single spring | k | √(k/m) |
| n springs in parallel | k₁ + k₂ + … + kₙ | √(Σkᵢ / m) |
| n springs in series | 1/(1/k₁ + 1/k₂ + … + 1/kₙ) | √(k_eq / m) |
| Two identical springs in parallel | 2k | √(2k/m) |
| Two identical springs in series | k/2 | √(k / 2m) |
Worked Example — Engine Mount Vibration
An engine of mass 150 kg is supported on four identical rubber mounts, each with stiffness 60 kN/m, arranged in parallel. The engine is displaced 5 mm downward from equilibrium and released from rest. Determine (a) the equivalent spring constant, (b) the natural frequency in Hz, (c) the period, and (d) the displacement at t = 0.05 s.
Strengths & Limitations of the SDOF Undamped Model
The SDOF undamped mass–spring model is one of the most widely used idealizations in engineering, but like every model it has a domain of validity. Understanding what it captures well and where it breaks down is critical for making sound engineering judgments about when to use it and when to reach for a more sophisticated tool.
| Strengths | Limitations |
|---|---|
| Closed-form analytical solution—no numerical methods required | Neglects damping; real oscillations always decay due to friction, material hysteresis, or air resistance |
| Accurately predicts natural frequency ωₙ, which is the most important design parameter in vibration isolation | Assumes linear spring behavior; real springs exhibit nonlinearity at large displacements |
| Provides physical intuition via the energy exchange between KE and PE | Single DOF cannot capture mode shapes of distributed or multi-body systems |
| Serves as the building block for MDOF analysis (modal superposition) | No external forcing term; cannot model forced vibration or resonance directly |
| Easily extended: add a dashpot (damping) or forcing function to handle real-world conditions | Massless spring assumption may fail when the spring's mass is comparable to the attached mass |
Connection to Damped & Forced Vibration
The SDOF undamped model is the foundation upon which the entire superstructure of vibration theory is built. The most immediate extension is the addition of viscous damping, represented by a dashpot with damping coefficient c. This changes the equation of motion to mẍ + cẋ + kx = 0 and introduces the damping ratio ζ = c / (2√(km)) as a second system parameter. When the system is also subjected to a harmonic external force F₀ sin(ωt), the equation becomes mẍ + cẋ + kx = F₀ sin(ωt), and the phenomenon of resonance—where the driving frequency ω approaches ωn—emerges. In all these extensions, ωn = √(k/m) remains the reference frequency against which all dynamic behavior is measured.
| Feature | Undamped Free (This Lesson) | Damped Free | Damped Forced |
|---|---|---|---|
| Equation of Motion | mẍ + kx = 0 | mẍ + cẋ + kx = 0 | mẍ + cẋ + kx = F₀ sin(ωt) |
| Key Parameters | ωₙ = √(k/m) | ωₙ, ζ = c/(2√km) | ωₙ, ζ, ω/ωₙ (frequency ratio) |
| Response Character | Perpetual sinusoid | Decaying sinusoid (ζ < 1) | Transient + steady-state sinusoid |
| Amplitude Behavior | Constant forever | Exponential decay e^(−ζωₙt) | Peaks near ω = ωₙ (resonance) |
| Design Application | Natural frequency identification | Decay rate, logarithmic decrement | Vibration isolation, frequency response |
Beyond single-degree-of-freedom systems, the SDOF concepts generalize elegantly through modal analysis. A multi-degree-of-freedom (MDOF) system with N masses and springs can be decomposed into N independent SDOF oscillators, each vibrating at its own natural frequency (eigenvalue). The associated mode shapes (eigenvectors) describe the spatial pattern of each mode. Continuous structures—beams, plates, shells—possess infinitely many modes, but the lowest few natural frequencies typically dominate the response. In every case, the ability to solve the basic SDOF problem is the prerequisite to understanding the more complex analysis.
Practice Problems
Lesson Summary
The single-degree-of-freedom mass–spring system is the foundational model in vibration analysis. By combining Hooke's law (F = −kx) with Newton's second law (F = mẍ), we derive the governing equation mẍ + kx = 0, whose solution is a pure sinusoid at the natural frequency ωn = √(k/m). This intrinsic frequency depends only on the stiffness and mass, not on initial conditions, and it can be expressed in hertz as fn = ωn/(2π) or as a period T = 2π/ωn.
The energy method (equating kinetic and potential energy exchanges) provides an alternative derivation and extends naturally to equivalent spring systems—springs in parallel (keq = Σkᵢ) and in series (1/keq = Σ1/kᵢ). While the undamped model predicts perpetual oscillation, it accurately captures ωn, which remains the key reference parameter when damping and forced excitation are later introduced. Mastering this model prepares you for modal analysis of multi-DOF systems, vibration isolation design, and resonance avoidance in real-world structures.