Natural Frequency Calculator

Natural frequency is the rate at which a system tends to oscillate after it is displaced and released without a continuing external force. Every elastic mechanical system has one or more natural frequencies determined by its mass distribution, stiffness, geometry, and boundary conditions. This calculator covers two foundational models: an ideal spring carrying a concentrated mass and a simple pendulum making small swings. These models are useful starting points for physics lessons, vibration analysis, machine design, and preliminary engineering estimates. For a spring-mass system, angular natural frequency equals the square root of spring stiffness divided by mass. Frequency in hertz is angular frequency divided by two pi, while the period is the reciprocal of frequency. Increasing stiffness raises the frequency because the restoring force becomes stronger. Increasing mass lowers the frequency because greater inertia resists changes in motion. The ideal equation assumes a linear spring, a rigid supporting structure, negligible damping, and a mass that can be represented by one concentrated value. For a simple pendulum, angular frequency equals the square root of gravitational acceleration divided by pendulum length. A longer pendulum therefore swings more slowly, while stronger gravity produces faster oscillation. Mass does not appear because gravitational force and inertia scale together and cancel. The familiar pendulum equation is a small-angle approximation; it is highly accurate for modest amplitudes but increasingly underestimates the period for large swings. Natural frequency should not be confused with forcing frequency. When a periodic input approaches a lightly damped system's natural frequency, resonance can amplify motion substantially. Real systems include damping, distributed mass, flexible supports, nonlinear materials, and multiple vibration modes, so measured values may differ from an ideal calculation. Use the result to understand trends, check hand calculations, or establish an initial design target. Safety-critical structures and rotating machinery require a detailed modal analysis and experimental validation.