🚀 Natural Frequency Calculator & Simulator
- Define System Components: Input the mass (m) of the suspended vibrating weight, the spring stiffness (k), and the damping coefficient (c) of the hydraulic damper.
- Analyze Free Vibration: Pull down and release the weight with your mouse, or observe the amplitude decay trajectory of the transient damping curves (underdamped, critically damped, overdamped) in the 'Free Oscillation' state.
- Input External Force Frequency (f_e): Apply a harmonic external force to observe the continuous oscillation amplitude of the forced vibration based on the frequency ratio (r).
- Induce Resonance & Observe Amplification: Match the external force frequency with the system's natural frequency (fn) (r = 1.0) to monitor the resonance phenomenon, where the amplitude catastrophically surges by tens of times or more, and track changes in the Dynamic Magnification Factor (DMF).
📚 Detailed Mechanical Vibration: Mathematical Equations of Motion & Resonance Formulas Explained ▼
1. Second-Order Linear Ordinary Differential Equation of a Spring-Mass-Damper System
The dynamic equilibrium state of a single-degree-of-freedom (1-DOF) spring-mass-damper system along its degree of freedom is described by Newton's second law as the following second-order linear ordinary differential equation (2nd order ODE):
m·x''(t) + c·x'(t) + k·x(t) = F(t)
Where m is the mass (kg), c is the damping coefficient (N·s/m), k is the spring constant (N/m), and F(t) is the applied harmonic external force.
2. Determination of Transient Damping State Based on Natural Frequency (f_n) and Damping Ratio (ζ)
The formulas for the undamped natural angular frequency (ω_n) and natural frequency (f_n, Hz), at which the system naturally tends to oscillate in the absence of damping, are as follows:
ω_n = √(k / m) [rad/s], f_n = ω_n / (2π) [Hz]
The critical damping coefficient C_c, which defines the maximum damping limit of the system, and the actual damping ratio (Damping Ratio, ζ) are expressed as follows:
C_c = 2·√(m·k) [N·s/m], ζ = c / C_c
Depending on the magnitude of the calculated damping ratio ζ, the physical vibration state strictly transitions into one of four regimes:
- ζ = 0: Undamped Oscillation - Perpetually continues to swing
- 0 < ζ < 1: Underdamped - Decaying spiral-type oscillation
- ζ = 1: Critically Damped - Returns to equilibrium the fastest without oscillating
- ζ > 1: Overdamped - Converges extremely slowly due to excessive damping resistance
3. Frequency Ratio (r), Dynamic Magnification Factor (DMF), and Catastrophic Resonance
When an external force acts periodically as F(t) = F_0 · sin(ω_e · t), the Dynamic Magnification Factor (DMF), which represents the ratio of the maximum dynamic response to the static displacement, is defined by the following formula:
DMF = 1 / √[ (1 - r²)² + (2·ζ·r)² ]
Here, r = f_e / f_n is the frequency ratio. When the external force frequency matches the system's natural frequency at the resonance point (r ≈ 1.0), the denominator approaches zero as the damping ratio ζ decreases, causing the DMF to spike vertically by tens of times or even infinitely. All catastrophic structural vibrations, such as the Tacoma Narrows Bridge collapse and the seismic fatigue failure of buildings, stem from this destructive resonant action.