Machine Design formula solvers

Bearing Basic Rating Life (L₁₀)

L_{10} = \left( \frac{C}{P} \right)^{p}

Machine DesignHow long a rolling bearing lasts before one in ten of a large group of identical bearings has failed by fatigue, in millions of revolutions. The load ratio raised to a power — cubed for ball bearings, 10/3 for roller bearings — which is why halving the load multiplies life eightfold.

Belt Length — Open Drive

L = 2C + \frac{\pi}{2}(D + d) + \frac{(D - d)^2}{4C}

Machine DesignThe length of a belt running open — both pulleys turning the same way — from the two pulley diameters and the centre distance. Two straight spans, two arcs, and a small correction for the fact that the arcs are not half-circles when the pulleys differ.

Belt Power from Tight and Slack Tensions

P = (F_1 - F_2) \, v

Machine DesignThe power a belt actually delivers: the difference between the tight-side and slack-side tensions — the net pull — multiplied by the belt speed. The tensions themselves do no work; only their difference does.

Belt Speed

v = \pi \, d \, n

Machine DesignHow fast the belt itself is travelling: the pulley's circumference times its rotational speed. The figure that decides how much power a belt of a given section can carry, and the one printed in feet per minute on every North American drive chart.

Brake Torque from Friction

T = \mu F r n

Machine DesignThe torque a friction brake develops: the coefficient of friction times the actuating force times the effective radius, once for each rubbing surface. The starting point for every brake and holding-clamp calculation.

Capstan Equation (Belt Tension Ratio)

\frac{F_1}{F_2} = e^{\mu \theta}

Machine DesignThe limit on how hard a belt or rope can pull before it slips: the tension ratio a wrapped flexible member can hold grows exponentially with the coefficient of friction and the wrap angle. Why three turns of line round a bollard hold a ship, and why the wrap angle must be in radians.

Chain Length in Pitches

L = \frac{2C}{p} + \frac{N_1 + N_2}{2} + \left( \frac{N_2 - N_1}{2\pi} \right)^{2} \frac{p}{C}

Machine DesignHow many pitches of roller chain a drive needs, from the two sprocket tooth counts, the chain pitch and the centre distance. The answer must be rounded up to a whole number — and preferably an even one, so the chain closes without an offset link.

Compound Gear Train Value

e = \frac{N_2 \, N_4}{N_1 \, N_3}

Machine DesignThe overall ratio of a two-stage compound gear train: the product of the driven tooth counts over the product of the driver tooth counts. The rule that lets a 40:1 reduction be built from two ordinary meshes instead of one enormous wheel.

Critical Speed of a Shaft

N_c = \frac{1}{2\pi} \sqrt{\frac{48 E I}{m L^{3}}}

Machine DesignThe running speed at which a shaft's own bending resonance is reached and it begins to whirl: the first critical of a simply supported shaft carrying a central rotor. A speed to pass through quickly, or to stay well away from.

Damped Natural Frequency

f_d = f_n \sqrt{1 - \zeta^{2}}

Machine DesignThe frequency a damped system actually rings at, which is always a little below the undamped one. For the damping levels found in real structures the difference is negligible — and knowing exactly how negligible is the useful part.

Damping Ratio from the Damping Coefficient

\zeta = \frac{c}{2\sqrt{k m}}

Machine DesignThe dashpot's rating measured against the one value that would stop the oscillation dead: ζ = c/c_c, where c_c = 2√(km). The number that decides whether a system rings, creeps back, or does something in between.

Disc Clutch Torque (Uniform Wear)

T = N \pi \mu p_{a} r_{i} \left( r_{o}^{2} - r_{i}^{2} \right)

Machine DesignThe torque an annular disc clutch transmits on the uniform-wear assumption, where the pressure is highest at the INNER radius and the product p·r is constant. The conservative and standard basis for clutch and disc-brake design.

External Load That Separates a Preloaded Joint

P_{0} = \frac{F_{i}}{1 - C}

Machine DesignThe external tensile load at which the clamped members lose all their compression and the joint opens. Past this point the bolt carries the whole load on its own, and every advantage of preload disappears at once.

Fillet Weld Size for a Load per Unit Length

w = \frac{f}{0.707 \, \tau_{a}}

Machine DesignThe leg size an equal-leg fillet weld needs to carry a stated load per unit length at a stated allowable stress. The form weld design is actually done in, because a continuous weld's length is set by the joint, not by the designer.

Flywheel Energy Fluctuation

E = I \omega^{2} C_{s}

Machine DesignThe energy a flywheel exchanges as its speed swings across one cycle, set by its inertia, its mean speed and the coefficient of speed fluctuation allowed. The relation that sizes every flywheel on a press, engine or compressor.

Force Transmissibility

TR = \sqrt{\frac{1 + (2\zeta r)^{2}}{(1 - r^{2})^{2} + (2\zeta r)^{2}}}

Machine DesignThe fraction of a machine's shaking force that reaches the floor through its mounts, as a function of the frequency ratio and the damping. Below a ratio of √2 it is greater than 1 — the isolator amplifies.

Gear Centre Distance

C = \frac{d_1 + d_2}{2}

Machine DesignHow far apart two meshing gear shafts sit: the mean of the two pitch diameters, which is the sum of the two pitch radii. The number that gets drilled into a housing and cannot be changed afterwards.

Gear Pitch Diameter

d = m \, N

Machine DesignThe pitch diameter of a spur gear from its module and tooth count — the diameter of the imaginary rolling cylinder the teeth are cut around. The metric module form, d = m N, with the North American diametral pitch convention P = N/d explained and bridged rather than silently substituted.

Gear Tooth Tangential Force

W_t = \frac{2T}{d}

Machine DesignThe tangential load a gear tooth carries: torque divided by the pitch radius, which is twice the torque over the pitch diameter. The starting number for every bending and contact stress calculation, and for sizing the shaft and bearings behind the gear.

Helical Compression Spring Rate

k = \frac{G d^{4}}{8 D^{3} n}

Machine DesignThe stiffness of a round-wire helical spring from its wire diameter, mean coil diameter, active coil count and the shear modulus of the wire. The fourth power on the wire is the whole design lever.

Helical Spring Shear Stress (with the Wahl Factor)

\tau = K_{W} \frac{8 F D}{\pi d^{3}}

Machine DesignShear stress in the wire of a helical spring under load, corrected by the Wahl factor for coil curvature and direct shear. The stress peaks on the INSIDE of the coil, which is where springs actually break.

Joint Stiffness Ratio of a Bolted Joint

C = \frac{k_{b}}{k_{b} + k_{m}}

Machine DesignThe share of an external load that lands on the bolt rather than on the clamped members, set by the two stiffnesses in parallel. Usually between 0.1 and 0.3, and the reason a properly preloaded bolt barely notices the load it is holding.

Logarithmic Decrement

\delta = \frac{1}{n} \ln\!\left(\frac{x_1}{x_2}\right)

Machine DesignDamping measured the way it is actually measured on site: hit the machine, record the decay, and take the natural log of how much the amplitude shrank over n cycles. No dashpot rating required.

Magnification Factor of a Forced Vibration

M = \frac{1}{\sqrt{(1 - r^{2})^{2} + (2\zeta r)^{2}}}

Machine DesignHow much bigger the steady-state motion is than the deflection the same force would cause if it were applied slowly. At resonance it collapses to 1/(2ζ), which is why lightly damped structures amplify so violently.

Natural Frequency from Static Deflection

f_n = \frac{1}{2\pi} \sqrt{\frac{g}{\delta_{st}}}

Machine DesignThe natural frequency read off the amount the mount sinks under the weight it carries — no spring rate needed, because the weight already measured it. The fastest sanity check in isolation work.

Rotating Unbalance Force

F = m_u \, e \, \omega^{2}

Machine DesignThe rotating force a small unbalanced mass throws off, growing with the SQUARE of speed. The reason a rotor that runs smoothly at half speed can tear its mounts apart at full speed.

Shaft Diameter from Allowable Torsional Shear

d = \sqrt[3]{\frac{16 T}{\pi \tau}}

Machine DesignThe diameter a solid round shaft needs so that the torsional shear stress at its surface stays within an allowable value. The torsion formula rearranged, and the reason a shaft carrying eight times the torque only has to be twice as thick.

Shaft Torque from Power and Angular Speed

T = \frac{P}{\omega}

Machine DesignThe torque in a rotating shaft: transmitted power divided by angular speed, with ω in radians per second. The relation that decides how thick every shaft in a machine has to be, because a shaft feels the torque and never the power.

Shear Stress in a Parallel Key

\tau = \frac{2T}{d \, w \, L}

Machine DesignThe average shear stress across a parallel key in a shaft keyway: the tangential force at the shaft surface spread over the key's width times its length. The check that decides whether the key or the shaft gives way first — and it is meant to be the key.

Shear Stress on a Fillet Weld Throat

\tau = \frac{F}{0.707 \, w L}

Machine DesignShear stress on the effective throat of an equal-leg fillet weld: the load divided by 0.707 times the leg size times the weld length. A fillet weld is always checked in shear on the throat, whichever way the load pulls.

Thread Tensile Stress Area

A_{t} = \frac{\pi}{4} \left( d - k_{t} p \right)^{2}

Machine DesignThe effective cross-section a threaded fastener pulls on: a circle of diameter partway between the pitch and minor diameters, not the shank and not the root. The area every bolt strength figure is quoted against.

Undamped Natural Frequency

f_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}}

Machine DesignThe frequency a mass on a spring returns to when nothing is driving it: stiffness over mass, square-rooted, divided by 2π. Every other number in vibration work is measured against this one.

Vibration Isolation Efficiency

I = 1 - TR

Machine DesignTransmissibility restated the way a specification writes it: the percentage of the shaking force the mounts keep off the floor. A negative answer is the honest report that the mounts are making it worse.