Torsional Oscillator Calculator
Calculate natural frequency, damping ratio, and response of torsional oscillators. Analyze shaft torsional vibrations for engineering applications.
Torsional Oscillator Calculator
Calculator
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Formula: omega_n = sqrt(kappa / I), zeta = c / (2 sqrt(kappa I))
Worked example — Natural: 15.81 rad/s (2.517 Hz) | Damping ratio: 0.079 | Damped: 15.76 rad/s
Formula
omega_n = sqrt(kappa / I), zeta = c / (2 sqrt(kappa I))
Where omega_n is the natural frequency in rad/s, kappa is the torsional stiffness (Nm/rad), I is the mass moment of inertia (kg m^2), zeta is the damping ratio, and c is the damping coefficient (Nm s/rad). The damped frequency is omega_d = omega_n x sqrt(1 - zeta^2) for underdamped systems.
Worked Examples
Example 1: Engine Crankshaft Torsional Vibration
Problem:A crankshaft has torsional stiffness of 200 Nm/rad, moment of inertia 0.8 kg m^2, and damping coefficient 2.0 Nm s/rad. Find the natural frequency, damping ratio, and damped frequency.
Solution:Natural frequency: omega_n = sqrt(200/0.8) = sqrt(250) = 15.81 rad/s = 2.517 Hz Period = 1/2.517 = 0.3973 s Critical damping = 2 x sqrt(200 x 0.8) = 2 x 12.649 = 25.298 Nm s/rad Damping ratio = 2.0 / 25.298 = 0.0791 (underdamped) Damped frequency = 15.81 x sqrt(1 - 0.0791^2) = 15.81 x 0.9969 = 15.76 rad/s = 2.508 Hz
Result:Natural: 15.81 rad/s (2.517 Hz) | Damping ratio: 0.079 | Damped: 15.76 rad/s
Example 2: Torsion Pendulum for Physics Lab
Problem:A torsion pendulum has a wire with stiffness 0.5 Nm/rad and a disk with I = 0.01 kg m^2. Damping is negligible (c = 0.001). Initial displacement is 30 degrees.
Solution:Natural frequency: omega_n = sqrt(0.5/0.01) = sqrt(50) = 7.071 rad/s = 1.125 Hz Period = 0.889 s Damping ratio = 0.001 / (2 x sqrt(0.5 x 0.01)) = 0.001 / 0.1414 = 0.00707 Max energy = 0.5 x 0.5 x (30 x pi/180)^2 = 0.5 x 0.5 x 0.2741 = 0.0685 J Max angular velocity = 0.5236 x 7.071 = 3.703 rad/s
Result:Period: 0.889 s | Damping ratio: 0.007 (underdamped) | Max energy: 0.069 J
Frequently Asked Questions
What is a torsional oscillator and where is it used?
A torsional oscillator is a mechanical system that undergoes rotational vibrations about its axis. It consists of a rotating mass (disk, flywheel, or rotor) connected to a restoring element such as a shaft or torsion spring that resists angular displacement. When the mass is displaced from its equilibrium angle and released, it oscillates back and forth around the rest position. Torsional oscillators appear in many engineering contexts including crankshafts in internal combustion engines, drive shafts in vehicles, turbine rotors, quartz crystal oscillators in watches and electronics, and torsion pendulums used in precision measurements. Understanding torsional oscillations is critical for preventing resonance failures in rotating machinery.
How is the natural frequency of a torsional oscillator calculated?
The natural frequency of a torsional oscillator is determined by the ratio of torsional stiffness to the moment of inertia. The formula is omega_n equals the square root of kappa divided by I, where kappa is the torsional stiffness in Newton-meters per radian and I is the mass moment of inertia in kilogram-meters squared. The result is in radians per second. To convert to Hertz (cycles per second), divide by 2 times pi. For example, a system with kappa = 50 Nm/rad and I = 0.5 kg m squared has omega_n = sqrt(50/0.5) = 10 rad/s, which equals 1.592 Hz with a period of 0.628 seconds. Higher stiffness increases frequency while greater inertia decreases it.
What is the damping ratio and what do different values mean?
The damping ratio (zeta) is a dimensionless measure that describes how oscillations decay over time. It equals the actual damping coefficient divided by the critical damping coefficient (2 times the square root of kappa times I). When zeta is less than 1, the system is underdamped and oscillates with gradually decreasing amplitude. When zeta equals exactly 1, the system is critically damped and returns to equilibrium as quickly as possible without oscillating. When zeta exceeds 1, the system is overdamped and slowly creeps back to equilibrium without oscillation. Most engineering structures have damping ratios between 0.01 and 0.10. Automobile suspension systems are typically designed near critical damping around 0.7 to 1.0.
What is the logarithmic decrement and how is it measured?
The logarithmic decrement is the natural logarithm of the ratio of two successive peak amplitudes in a free vibration response. It provides a practical way to measure the damping ratio from experimental data. The formula relates the log decrement delta to the damping ratio: delta = 2 pi zeta divided by the square root of (1 minus zeta squared). To measure it experimentally, record the oscillation, identify two consecutive peaks of the same sign, measure their amplitudes A1 and A2, and compute delta = ln(A1/A2). The damping ratio can then be back-calculated as zeta = delta divided by the square root of (4 pi squared plus delta squared). This method is widely used in structural health monitoring.
How does shaft geometry affect torsional stiffness?
Torsional stiffness of a solid circular shaft depends on its material shear modulus, diameter, and length according to the formula kappa = G times J divided by L, where G is the shear modulus (79.3 GPa for steel), J is the polar moment of inertia (pi times d to the fourth power divided by 32), and L is the shaft length. Stiffness increases with the fourth power of diameter, meaning a small increase in diameter dramatically raises stiffness. Doubling the shaft diameter increases stiffness by a factor of 16. Conversely, doubling the length halves the stiffness. For hollow shafts, the polar moment of inertia is pi times (outer diameter to the fourth minus inner diameter to the fourth) divided by 32. This relationship makes shaft sizing a critical engineering design decision.
What happens when an excitation frequency coincides with the torsional natural frequency?
At resonance the forcing torque feeds energy in step with the motion and the amplitude is limited only by damping. The steady-state amplification factor is approximately 1 / (2 zeta), so a system with zeta = 0.05 magnifies the static twist about ten times, while zeta = 0.01 magnifies it fifty times. In reciprocating engines the firing pulses arrive at fixed multiples of shaft speed, called orders, so a critical speed exists wherever an order crosses omega_n. Rubber or viscous torsional dampers are fitted to raise zeta and detune those crossings.
How much does damping actually shift the oscillation frequency?
The damped frequency is omega_d = omega_n sqrt(1 - zeta^2), and because zeta appears squared the shift is tiny for lightly damped machinery. At zeta = 0.05 the factor is 0.99875, so the frequency drops by about one eighth of one percent, far too little to measure on a typical rig. Damping's real effect is on amplitude, not timing: the envelope decays as exp(-zeta omega_n t). Only as zeta approaches 1 does omega_d collapse toward zero and the motion stop oscillating altogether.
How do you handle a shaft with a rotor at each end?
A two-rotor system vibrates about a stationary node somewhere along the shaft, with the two discs always twisting in opposite directions. Its frequency is omega_n = sqrt(kappa (1/I1 + 1/I2)), which is the same as using an equivalent inertia I_eq = I1 I2 / (I1 + I2) in the single-disc formula. The node sits closer to the larger inertia, and with a light uniform shaft the torque is carried equally by every section between the rotors, so a step, keyway, or fillet decides where fatigue cracking starts. A generator set with a heavy alternator and a lighter engine flywheel is the classic case.
How can a torsion pendulum be used to measure a material's shear modulus?
Hang a disc of known moment of inertia from a wire of the material, time many oscillations to get an accurate period T, then invert the period formula to get the stiffness: kappa = 4 pi^2 I / T^2. Because kappa = G J / L for a circular wire, the shear modulus follows as G = kappa L / J, with J = pi d^4 / 32. The fourth-power dependence on diameter means the wire diameter dominates the uncertainty, so it should be measured with a micrometer at several points and averaged.
How does a torsional oscillator map onto an ordinary spring-mass system?
Every term has a rotational counterpart: torsional stiffness kappa replaces the spring constant k, moment of inertia I replaces mass m, angle theta replaces displacement x, and torque replaces force. The governing equation I theta-double-dot + c theta-dot + kappa theta = 0 has exactly the same form as m x-double-dot + c x-dot + k x = 0, so every result for damping ratio, logarithmic decrement, and resonance carries straight over. The practical difference is detectability, since a twisting shaft produces little audible noise or casing movement yet can still fail in fatigue.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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