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.
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Velocidade crítica de eixo
Um eixo rotativo que carrega um rotor (disco, engrenagem, impulsor) não é infinitamente rígido — ele flexiona. E como todo sistema elástico com massa, tem uma frequência natural de flexão. A velocidade crítica é a rotação em que a frequência de giro coincide com essa frequência natural: ω_c = √(k ÷ m), onde k é a rigidez do eixo à flexão e m a massa do rotor. O perigo é a ressonância: nessa velocidade, o pequeno desbalanceamento inevitável do rotor (que gira na mesma frequência do eixo) excita exatamente a frequência natural, e a deflexão do eixo dispara — o rotor 'incha' radialmente, as forças nos mancais explodem, e o equipamento pode ser destruído em segundos. Por isso o projeto de máquinas rotativas gira (literalmente) em torno das velocidades críticas. Há duas estratégias. Rotores rígidos operam com folga abaixo da primeira velocidade crítica (rotação de trabalho < ~0,7·ω_c) — solução comum em máquinas mais lentas. Rotores flexíveis (turbinas a vapor e a gás, compressores de alta rotação) operam acima da primeira crítica, o que exige atravessar a ressonância rapidamente na partida e na parada, contando com amortecimento para limitar o pico durante a passagem — e às vezes lidando com uma segunda e terceira velocidades críticas. Curiosamente, acima da crítica o rotor tende a se 'auto-centrar' (o centro de massa se aproxima do eixo de rotação), tornando a operação mais suave. Calcular ω_c, junto com o mapa de Campbell que mostra como as críticas variam com a velocidade, é etapa obrigatória no projeto de qualquer máquina rotativa rápida. Informe a rigidez do eixo e a massa do rotor.
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).
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.
Vibration Transmissibility
Calculate the transmissibility of an undamped vibration isolator, TR = 1 ÷ |r² − 1|, from the frequency ratio r = f ÷ f_n (excitation frequency over natural frequency). The dimensionless result is the fraction of force (or motion) transmitted through the isolator: TR < 1 means isolation (the transmitted vibration is less than the applied one), which only occurs for r > √2. Near r = 1 (resonance), TR spikes; the higher r, the lower the transmissibility and the better the isolation. It is the key criterion in designing antivibration mounts. Enter the frequency ratio.
Belt Span Natural Frequency
Calculate the natural vibration frequency of a belt's free span, f_n = (1 ÷ (2·L))·√(T/m), from the free span length L (m, the distance between pulleys), the belt tension T (N) and the mass per unit length m (kg/m). A belt's free span, between two pulleys, behaves like a stretched STRING (like a guitar string): when disturbed, it vibrates at a natural frequency depending on its tension and mass. The HIGHER the tension, the HIGHER the frequency (tighter string, higher pitch); the higher the mass per metre, the lower the frequency. This relation is the basis of a clever, widely used method to MEASURE belt tension in the field: the SONIC (or frequency) tension meter — the technician 'plucks' the belt to make it vibrate, and a sensor (or phone app) measures the sound frequency; knowing the span length and belt mass, the tension is computed back (inverting the formula). It is far more practical and accurate than the old methods of measuring deflection under a force. Keeping the correct tension is essential: a slack belt slips (loses power, heats, wears) and an over-tight belt overloads the bearings and shortens belt life. Enter the span length, the tension and the mass per unit length.
Half-Wave Dipole Length
Calculate the physical length of a half-wave dipole, L = (150 ÷ f)·VF, from the frequency f (MHz) and the velocity factor VF (typically ~0.95 for wires, correcting the end effect). The result, in metres, is the total length of the dipole antenna resonant at the desired frequency — each arm is half this value. The half-wave dipole is the most used reference antenna, with 2.15 dBi gain. The velocity factor makes the antenna slightly shorter than a half wavelength in vacuum. Enter the frequency and the velocity factor.
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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.