Torsional Oscillator Calculator
Our rotational & periodic motion calculator computes torsional oscillator accurately. Enter measurements for results with formulas and error analysis.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
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.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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