Damped Natural Frequency
Calculate the damped natural frequency of a vibrating system, ω_d = ω_n·√(1 − ζ²), from the undamped natural frequency ω_n and the damping ratio ζ. The result, in the unit of ω_n (rad/s or Hz), is the actual frequency at which an underdamped system oscillates freely after a disturbance — always lower than the undamped natural frequency, since damping slows the oscillation. For small ζ (lightly damped systems), ω_d ≈ ω_n; as ζ → 1 (critical damping), ω_d → 0 and the system stops oscillating. Enter the natural frequency and the damping ratio.
Resultado
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Frequência natural amortecida
Um sistema massa-mola-amortecedor, ao ser perturbado e solto, oscila livremente — mas o amortecimento não só faz a amplitude decair como também desacelera a própria oscilação. A frequência real dessa oscilação amortecida é ω_d = ω_n·√(1 − ζ²), onde ω_n é a frequência natural não amortecida (a que o sistema teria sem atrito) e ζ é a razão de amortecimento (a fração do amortecimento crítico). O fator √(1 − ζ²) é sempre ≤ 1, então a oscilação amortecida é sempre mais lenta que a não amortecida. Para sistemas pouco amortecidos (ζ < 0,1, comum em estruturas metálicas), a diferença é desprezível e ω_d ≈ ω_n. Mas conforme ζ aumenta, ω_d cai: em ζ = 0,6, a frequência amortecida é 80% da natural. No limite do amortecimento crítico (ζ = 1), ω_d chega a zero — o sistema não oscila mais, apenas retorna lentamente ao equilíbrio sem ultrapassá-lo. Acima disso (ζ > 1, superamortecido), o movimento é puramente exponencial. A frequência amortecida é o que se mede experimentalmente no decaimento livre de uma estrutura, e dela, junto com o decremento logarítmico, extrai-se o amortecimento real do sistema. Informe a frequência natural e a razão de amortecimento.
Related Tools
Natural Frequency from Static Deflection
Calculate a system's natural frequency from its static deflection, f_n = (1 ÷ 2π)·√(g ÷ δ), where δ is the static deflection caused by self-weight and g the gravitational acceleration (9.81 m/s²). The result, in Hz, is a practical and elegant way to estimate the natural frequency without separately knowing mass and stiffness — you just measure how much the system sags under its own weight. Larger deflections (more flexible systems) give lower natural frequencies, desirable in vibration isolators. It is widely used in spring and mount design. Enter the static deflection (in metres).
Shaft Critical Speed
Calculate the critical speed of a rotating shaft, ω_c = √(k ÷ m), from the shaft stiffness k and the rotor mass m. The result, in rad/s, is the rotational speed that coincides with the shaft's bending natural frequency — at it, any small unbalance causes large-amplitude resonant vibration that can damage the equipment. Shafts should run with a safe margin below the first critical speed (rigid rotors) or pass through it quickly to a range above (flexible rotors). It is an essential calculation in designing high-speed turbines, pumps and motors. Enter the stiffness and the mass.
Amplification Factor (Q)
Calculate the resonance amplification factor (quality factor Q), Q = 1 ÷ (2·ζ), from the damping ratio ζ. The dimensionless result shows how many times the vibration amplitude at resonance exceeds the equivalent static deflection: lightly damped systems (small ζ) have high Q and sharp, dangerous resonance peaks; well-damped systems have low Q and smooth response. It is central to designing structures, machines and instruments to avoid destructive amplification and to characterizing the selectivity of filters and resonators. Enter the damping ratio.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.