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Flywheel & Inertia Calculator — Required Inertia, Burst Speed & MOI (Classical Mechanics)

The MechanixCalc flywheel and inertia calculator covers every stage of flywheel design from first principles: enter the energy fluctuation, coefficient of fluctuation and operating speed and the tool instantly returns the required moment of inertia, flywheel mass, rim or disc geometry, hoop stress and burst safety factor. The burst speed uses the Rankine thin-ring model for rim-type flywheels and the Timoshenko equal-biaxial disc formula for solid-disc cross-sections, so the governing stress is always the correct one for the chosen geometry.

Beyond the core flywheel design tab, the calculator includes a moment-of-inertia library for more than ten standard shapes (solid cylinder, hollow cylinder, thin ring, rectangular bar, cone, sphere and more) with the parallel-axis theorem for compound assemblies, a multi-body rotational system acceleration solver, a motor run-up time analyser with a speed–time chart, an energy storage and material comparison panel, and a spoked-flywheel stress module. All results feed a single-click branded PDF engineering report.

What this calculator does

  • Flywheel sizing: required inertia, disc mass and rim/disc geometry from energy fluctuation, Cs and speed
  • Burst speed analysis using the Rankine thin-ring model (rim) and Timoshenko rotating-disc formula (solid disc), with safety factor
  • Moment of inertia for 10+ standard shapes with the parallel-axis theorem for compound assemblies
  • Multi-body rotational system dynamics: total reflected inertia, net torque and angular acceleration
  • Motor run-up time with 50 %/90 %/100 % speed milestones and a speed–time chart
  • Energy storage analysis with cross-material specific-energy comparison chart
  • Rotating disc stress distribution (radial, hoop, von Mises) across the radius with a burst-speed N_burst output

Method & formulas

Required inertia and flywheel sizing

For a machine with a cyclic load — a punch press, engine or compressor — the flywheel must absorb and release the energy fluctuation ΔE between peak and mean torque. The required moment of inertia follows directly from ΔE, the coefficient of speed fluctuation Cs (the fractional speed variation the application tolerates) and the mean angular velocity ω. Once I_req is known the tool back-calculates disc thickness or rim cross-section for the chosen geometry and material.

Required moment of inertia
I_req = ΔE / (Cs · ω²)

where I_req = required moment of inertia (kg·m²); ΔE = energy fluctuation per cycle (J); Cs = coefficient of speed fluctuation (dimensionless, e.g. 0.02 for engines, 0.1–0.2 for punching machines); ω = mean angular velocity (rad/s) = 2π·N/60

Hoop stress and burst safety factor

A rotating rim or disc develops centrifugal stress that limits its maximum speed. For a thin rim the hoop stress is the classic rotating-ring formula; for a solid disc the peak stress is the Timoshenko equal-biaxial value at the centre, which is lower than the thin-ring value by the factor 8/(3+ν). MechanixCalc applies the correct formula for the selected cross-section so a solid-disc flywheel is never penalised with the (overly conservative) rim formula. The burst speed N_burst is then found by setting the peak stress equal to the material's allowable stress, and the safety factor SF = N_burst / N_op.

Rim hoop stress (Rankine thin-ring)
σ_hoop = ρ · (ω · R)²

where σ_hoop = hoop stress (Pa); ρ = material density (kg/m³); ω = angular velocity (rad/s); R = mean rim radius (m). Burst occurs when σ_hoop = σ_allowable.

Solid-disc peak stress (Timoshenko, at centre)
σ_max = (3 + ν) / 8 · ρ · ω² · R²

where σ_max = peak (equal-biaxial) stress at disc centre (Pa); ν = Poisson's ratio; ρ = density (kg/m³); ω = angular velocity (rad/s); R = outer radius (m).

Moment of inertia and multi-body system dynamics

The MOI library computes the second moment of mass for each standard shape about its own centroidal axis, then adds any offset mass via the parallel-axis theorem (I = I_cm + m·d²). For a drivetrain with several rotating bodies at different gear ratios the system inertia is the sum of each body's inertia reflected to the reference shaft (I_ref = Σ Iᵢ · nᵢ²), which together with the net torque gives the angular acceleration and run-up time for the system.

Motor run-up time follows from Newton's second law for rotation: α = T_net / I_total, t_runup = Δω / α. The run-up panel also back-calculates the motor torque required to reach a user-specified target time, which is useful when specifying the drive motor for a flywheel-assisted machine.

Solid cylinder MOI
I = ½ · m · R²

where I = moment of inertia about the spin axis (kg·m²); m = mass (kg); R = radius (m).

Motor run-up time
t_runup = (ω_final − ω_initial) / α where α = T_net / I_total

where t_runup = time to reach final speed (s); T_net = motor torque minus load torque (N·m); I_total = total system inertia reflected to the motor shaft (kg·m²); α = angular acceleration (rad/s²).

Worked example

Size a flywheel for a small punch press. The press has an energy fluctuation per stroke of ΔE = 1000 J, the allowable coefficient of speed fluctuation is Cs = 0.05, and the operating speed is N = 1000 rpm. Find the required moment of inertia.

Given

  • Energy fluctuation ΔE1000 J
  • Coefficient of fluctuation Cs0.05
  • Operating speed N1000 rpm

Result

  • Required moment of inertia I_req≈ 1.82 kg·m²
  1. Convert the operating speed to angular velocity: ω = 2π·N/60 = 2π × 1000/60 ≈ 104.72 rad/s.
  2. Square the angular velocity: ω² ≈ 104.72² = 10966 rad²/s².
  3. Apply the flywheel inertia formula: I_req = ΔE / (Cs · ω²) = 1000 / (0.05 × 10966) = 1000 / 548.3.
  4. I_req ≈ 1.82 kg·m². A solid steel disc of radius R = 0.25 m with I = ½mR² would need m = 2 × 1.82 / 0.25² = 58.2 kg.

This is an illustrative example — verify ΔE from a torque–angle diagram for your specific machine. The calculator also computes the disc mass, thickness, hoop stress and burst safety factor from your actual geometry and material selection.

Frequently asked questions

Which standard does this flywheel calculator use?

The core design tabs use classical mechanics: the Rankine thin-ring model for rim hoop stress and burst speed, and the Timoshenko rotating-disc formula for peak stress in a solid disc. The moment-of-inertia library uses standard textbook second-moment-of-mass derivations with the parallel-axis theorem. The spoked-flywheel panel uses the Bhandari design method and is labelled an engineering estimate. There is no single ISO or AGMA standard that governs the full scope of flywheel design; the governing method is shown in the generated PDF report.

What is the coefficient of speed fluctuation (Cs) and how do I choose it?

Cs is the fractional variation in speed over one cycle: Cs = (ω_max − ω_min) / ω_mean. Typical values are 0.002–0.003 for precision machine tools, 0.01–0.02 for generators, 0.03–0.05 for pumps and compressors, and 0.1–0.3 for punching or shearing machines. A smaller Cs requires a heavier flywheel but gives smoother speed regulation.

What is the difference between the rim and solid-disc cross-sections?

A rim flywheel concentrates mass at the outer radius (highest peripheral speed) for maximum energy storage per kilogram, while a solid disc stores energy more uniformly across the radius. The burst-stress formula differs between the two: a thin rim uses the rotating-ring hoop stress σ = ρ(ωR)², while a solid disc uses the Timoshenko centre stress σ = (3+ν)/8·ρω²R², which is roughly 2.4× lower for steel. The calculator applies the correct formula automatically for the selected cross-section.

What is the burst speed and how large should the safety factor be?

The burst speed N_burst is the rotational speed at which the centrifugal stress in the flywheel equals the material's allowable stress. The safety factor SF = N_burst / N_operating. A minimum SF of 2–3 is typical for industrial flywheels; energy-storage applications, automotive cranks and high-speed spindles usually require SF ≥ 3. The calculator flags SF < 3 with a warning.

Is the flywheel calculator free?

You can run calculations during a free 30-minute preview with no sign-up required, and a free 14-day account trial (no credit card) unlocks every calculator on the platform. The branded PDF engineering report with full method shown and the ability to save and reload calculations are part of a paid plan.

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