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A single-degree-of-freedom vibrating system consists of a mass, a restoring element (spring), and optionally a damping element (dashpot). For undamped free vibration, Newton's second law yields the equation of motion m*x'' + k*x = 0, with the natural frequency omega_n = sqrt(k/m) and period tau = 2*pi/omega_n. The general solution x(t) = A*sin(omega_n*t + phi) describes simple harmonic motion. When viscous damping is added, the equation becomes m*x'' + c*x' + k*x = 0, characterized by the damping ratio zeta = c/(2*m*omega_n). If zeta < 1 (underdamped), the system oscillates with exponentially decaying amplitude at the damped frequency omega_d = omega_n*sqrt(1 - zeta2). If zeta = 1 (critically damped) or zeta > 1 (overdamped), the system returns to equilibrium without oscillation. The logarithmic decrement delta = ln(x_n/x_{n+1}) = 2*pi*zeta/sqrt(1 - zeta2) provides a practical way to measure damping from experimental decay data.
Derive the equation of motion from Newton's second law for a spring-mass-dashpot system, identify the natural frequency and damping ratio from the coefficients, and then write the solution form based on the damping regime. Work problems that ask for period, frequency, maximum displacement, and the number of cycles for amplitude to decay by a given factor. For rotational systems, draw the analogy: I*theta'' + c_t*theta' + k_t*theta = 0, with omega_n = sqrt(k_t/I).
From your study of simple harmonic motion, you know that a restoring force proportional to displacement produces sinusoidal oscillation. Single-degree-of-freedom vibration analysis is what happens when you apply Newton's second law to a spring-mass system systematically, extract the governing parameters, and then generalize to damped systems. The architecture of the whole subject flows from a single equation of motion.
For an undamped spring-mass system, Newton's second law gives m*x'' = −k*x, which rearranges to m*x'' + k*x = 0. Every coefficient carries physical meaning: m is inertia (resisting acceleration) and k is stiffness (providing the restoring force). Dividing through by m gives x'' + (k/m)*x = 0, and the quantity ω_n = √(k/m) is the natural frequency — the rate at which the system oscillates if displaced and released. You should read ω_n physically: a stiffer spring (larger k) or lighter mass (smaller m) gives higher natural frequency. The solution x(t) = A sin(ω_n t + φ) is exactly the simple harmonic motion you already know, with amplitude A and phase φ set by initial conditions.
Adding a dashpot (viscous damper with constant c) introduces a velocity-proportional force: the equation becomes m*x'' + c*x' + k*x = 0. Two new parameters emerge. The critical damping coefficient c_c = 2m*ω_n is the value at which the system returns to equilibrium in minimum time without oscillating. The damping ratio ζ = c / c_c compares the actual damping to critical. The three regimes flow directly from ζ: if ζ < 1 (underdamped), the system oscillates at the reduced damped natural frequency ω_d = ω_n√(1−ζ²) with amplitude that decays as e−ζω_n t; if ζ = 1 (critically damped), the system returns to equilibrium as fast as possible with no overshoot; if ζ > 1 (overdamped), return is even slower and also non-oscillatory. Automobile suspensions are tuned to ζ ≈ 0.3–0.7 to balance ride comfort against unwanted oscillation.
The logarithmic decrement δ = ln(x_n / x_{n+1}) bridges theory and experiment. Because successive peaks of an underdamped oscillation decay by the factor e−ζω_n T_d per cycle, the log of consecutive peak amplitudes directly yields ζ. This makes it practical to measure damping from recorded vibration data without ever knowing m, k, or c individually — a tool used routinely in structural health monitoring, modal testing, and machine diagnostics. The entire single-DOF framework — equation of motion, natural frequency, damping ratio, response regimes — repeats structurally for every vibrating system from a guitar string to a skyscraper; only m, k, and c take different physical forms.