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Flywheel & Inertia Calculator — Required Inertia, Limit 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, peak centrifugal stress and the speed margin to the limit speed. The governing stress is the envelope σ = ρ(ω·R_outer)², evaluated at the outer fibre — for a rim that is the mean radius plus half the derived rim thickness. This envelope is the thin-rim limit of Timoshenko's rotating-annulus solution and provably bounds the exact peak for every bore fraction and every Poisson's ratio, so no geometry the calculator can describe is ever under-predicted.

Beyond the core flywheel design tab, the calculator includes a moment-of-inertia library for seven standard shapes (solid cylinder, hollow cylinder, thin ring, rectangular bar, thin rod, cone and sphere) with the parallel-axis theorem, 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, a rotating-disc stress explorer that draws the exact radial and hoop stress field for a declared bore, 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
  • Limit-speed analysis on a conservative peak-stress envelope, with the governing speed margin and the total margin on yield
  • Moment of inertia for 7 standard shapes with the parallel-axis theorem
  • 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 explorer: the exact radial, hoop and von Mises field across the radius for a bore you declare

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

Peak centrifugal stress and the limit speed

A rotating rim or disc develops centrifugal stress that limits its maximum speed. For a rotating annulus or bored disc with free surfaces the exact peak sits at the bore and is σ_θ(a) = (3+ν)/4·ρω²[b² + (1−ν)/(3+ν)a²]. Setting that equal to ρω²b² gives a/b = 1 identically — the (1−ν) cancels for every Poisson's ratio — and the peak rises monotonically with bore, so ρω²·R_outer² bounds the exact peak for every bore fraction and every material. MechanixCalc uses that envelope as the governing stress. It is also a bound on von Mises, because the radial stress is zero at the bore.

This matters because a flywheel is bored: a shaft passes through it, and even a vanishingly small hole doubles the peak stress compared with a hypothetical boreless disc. Sizing against the boreless centre value would understate the real peak by a factor of two. The limit speed N_limit is found by setting the envelope equal to the material's allowable stress, and the governing result is the speed margin N_limit / N_operating.

Two honest caveats. First, this is a limit speed on an allowable-stress basis, not a fracture burst speed — a ductile disc bursts by plastic collapse at ultimate strength, which is higher, and the calculator does not model ultimate strength. Second, the envelope covers rotational load only: it excludes interference-fit hub pressure, keyseats, bolt holes, spoke-root restraint and thermal gradients, any of which can raise the bore stress well above it. Use the rotating-disc stress explorer to see the exact field for a declared bore.

Peak stress envelope (governing)
σ_max = ρ · (ω · R_outer)²

where σ_max = peak stress envelope (Pa); ρ = material density (kg/m³); ω = angular velocity (rad/s); R_outer = outer radius (m) — the disc radius, or the mean rim radius plus half the derived rim thickness.

Exact rotating-annulus peak (Timoshenko & Goodier §32)
σ_θ(a) = (3 + ν) / 4 · ρ · ω² · [ b² + (1 − ν)/(3 + ν) · a² ]

where σ_θ(a) = peak tangential stress at the bore (Pa); a = bore radius, b = outer radius (m); ν = Poisson's ratio. Equals the envelope only in the thin-rim limit a → b, and is below it for every smaller bore.

Limit speed and governing margin
ω_limit = √(σ_allowable / ρ) / R_outer ; margin = ω_limit / ω

where ω_limit = angular velocity at which the envelope reaches the material allowable (rad/s). The stress ratio σ_allow/σ_max is exactly this margin squared — it is an identity, not a second safety factor.

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 governing stress is the peak-stress envelope σ = ρ(ω·R_outer)², which is the thin-rim limit of Timoshenko's rotating-annulus solution and bounds the exact peak for every bore fraction and every Poisson's ratio. The rotating-disc explorer draws that exact solution for a bore you declare. 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, 0.003–0.01 for generators and alternators, 0.02–0.05 for pumps and compressors, and 0.10–0.20 for punching or shearing machines. A smaller Cs requires a heavier flywheel but gives smoother speed regulation. Note that on a rim design a very small Cs drives the derived rim thickness towards the axis; the calculator rejects a rim thicker than its own radius rather than sizing an impossible part.

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. Both are governed by the same stress envelope σ = ρ(ω·R_outer)², but R_outer differs: for a solid disc it is simply the radius you enter, while for a rim it is the mean rim radius plus half the rim thickness the calculator derives from your inertia requirement. A rim therefore carries higher stress than a disc of the same nominal radius, and the calculator accounts for that automatically.

What is the limit speed and how large should the margin be?

The limit speed N_limit is the rotational speed at which the peak centrifugal stress reaches the material's allowable stress, and the governing result is the speed margin N_limit / N_operating. Because stress goes as the square of speed, a speed margin of 3 is a stress margin of 9 on the allowable — and since the allowable already embeds a design margin on yield, the total margin on yield is larger still, which the calculator prints explicitly. The minimum margin is set at 3.0 deliberately: a bursting flywheel is a fragmentation hazard, and the model excludes hub interference pressure, keyseats and bolt holes, each of which raises the real bore stress. Note this is a limit speed on an allowable-stress basis, not a fracture burst speed — the calculator does not model ultimate strength and does not quote a burst speed.

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 included in the free 14-day trial and in every paid plan.

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