Machine Design and Power Transmission Systems Formula Sheet

This formula sheet collects first-pass relationships used in machine design and power transmission. Use it to size and review torque paths, shaft sections, gear or belt stages, keys, bearings, reflected inertia, tolerance stacks, vibration separation, service factors, and validation evidence.

These formulas do not replace detailed machinery standards, supplier ratings, finite element analysis, gear design codes, bearing catalog rules, shaft fatigue design, lubrication analysis, or machine-safety requirements. They are most useful when the goal is to build an auditable calculation trail before detailed component selection.

Unit Conventions and Notation

Use consistent SI units unless a formula states a practical unit form.

Symbol	Meaning	Typical unit
P	power	\text{W} or \text{kW}
T	torque	\text{N m} or \text{N mm}
n	rotational speed	\text{rpm}
\omega	angular speed	\text{rad/s}
i	speed reduction ratio	dimensionless
\eta	efficiency	dimensionless
K_s	service factor	dimensionless
d	shaft or pitch diameter	\text{mm} or \text{m}
M	bending moment	\text{N mm}
F_t	tangential force	\text{N}
C	bearing dynamic capacity	\text{N}
P_e	equivalent bearing load	\text{N}
J	mass moment of inertia	\text{kg m}^2

State whether a torque is nominal, design, peak, braking, stall, or proof-test torque. Many machine design errors come from mixing these cases.

Operating Profile and Service Factor

Design torque from nominal torque:

If several factors are applied separately:

where K_o may represent overload, K_s service severity, and K_m mounting or alignment condition. Do not multiply factors unless their meanings are independent and documented.

Utilization:

Margin:

Factor against a limit:

Engineering Comment

A service factor is not a substitute for an operating profile. Record starts per hour, jam events, reversing duty, braking events, shock loads, temperature, contamination, inspection interval, and consequence of failure.

Power, Torque, and Speed

Angular speed:

Power relation:

Practical torque relation:

Design torque:

For a reduction stage:

Output speed:

Output torque:

Power loss:

Validity

These relations assume steady rotating power transmission. Transients, torsional vibration, acceleration torque, clutch slip, brake torque, and variable-speed drive limits require separate checks.

Gear, Belt, and Chain Forces

Tangential force at pitch diameter:

For a spur gear with pressure angle \phi:

Pitch-line velocity:

Effective belt or chain tension difference:

Speed ratio for a belt or chain drive, neglecting slip:

Engineering Comment

These equations give load path forces, not full gear or belt ratings. Tooth bending, contact stress, wear, lubrication, pitch accuracy, misalignment, shock, start-stop duty, and supplier derating must be checked before release.

Shaft Screening

For a solid circular shaft under torsion:

Required diameter from allowable shear stress:

Elastic bending stress:

Combined von Mises screening for bending plus torsion:

Static yield utilization:

Engineering Comment

This is a screening model. A real shaft review also checks shoulders, grooves, keyways, splines, press fits, surface finish, residual stress, heat treatment, corrosion, bearing seats, deflection, critical speed, fatigue, and inspectability.

Keyways, Pins, and Hub Interfaces

Approximate key shear stress:

Approximate key bearing pressure:

where d is shaft diameter, b is key width, h is key height, and L is engaged length.

Hub interface torque from tangential force:

Torque margin for an interface:

Engineering Comment

Key formulas are approximate because real load sharing depends on fit, hub stiffness, key clearance, material strength, stress concentration, reversals, fretting, and assembly quality. A keyway also weakens the shaft, so the shaft and key must be checked together.

Bearings

Equivalent dynamic bearing load:

Basic rating life in millions of revolutions:

where:

and:

Life in hours:

Static safety factor:

Validity

Catalog bearing life depends on bearing type, lubrication, contamination, temperature, mounting, misalignment, internal clearance, reliability adjustment, load spectrum, and minimum load. The formula is a starting point for selection, not a release certificate.

Reflected Inertia and Acceleration Torque

For a reduction ratio:

Load inertia reflected to the motor:

Total inertia at the motor:

Acceleration torque at the motor:

Load torque reflected to the motor:

Total required motor torque during acceleration:

Engineering Comment

Reflected inertia is sensitive to the definition of gear ratio. Always state whether i is motor speed divided by load speed or the inverse. Servo drives, hoists, indexing mechanisms, and robots can be torque-limited by acceleration even when steady power is acceptable.

Tolerance Stack-Up and Thermal Growth

Worst-case linear stack:

Root-sum-square stack for independent centered tolerances:

Thermal growth:

Runout utilization:

Assembly clearance:

Validity

RSS assumptions require independent, stable, centered contributors. Use worst-case logic for safety-critical interference, minimum clearance, sealing, bearing preload, and conditions where manufacturing distributions are unknown.

Vibration and Rotating Machinery Checks

Single-degree natural frequency:

Frequency ratio:

Separation margin from a critical frequency:

Unbalance force:

where m_ee is the unbalance mass times eccentricity.

Engineering Comment

Frequency separation rules are design-basis dependent. For many machines, avoid continuous operation close to resonance and validate with run-up, coast-down, vibration spectra, phase, bearing temperature, and inspection of looseness or rub marks.

Heat, Efficiency, and Lubrication Screening

Power rejected as heat:

Temperature rise from thermal resistance:

Lubricant viscosity margin:

Thermal utilization:

Engineering Comment

Gearboxes, bearings, clutches, brakes, and seals can fail thermally even when static strength is adequate. Validate heat rejection with a thermal run, lubricant condition, bearing temperature, housing temperature, and duty-cycle evidence.

Reliability and Maintainability Quantities

Series reliability:

Steady-state availability:

Risk priority number:

where S is severity, O is occurrence, and D is detection rating.

Engineering Comment

Reliability formulas must be tied to the operating environment and maintenance policy. A bearing life calculation, inspection interval, lubrication task, spare part strategy, and failure-mode review should be consistent with the same duty assumptions.

Worked Example 1: Gearbox Output Torque and Shaft Screening

A gearbox output shaft transmits:

at:

The service factor is:

Use allowable torsional shear stress:

Nominal torque:

Design torque:

Convert to \text{N mm}:

Required solid-shaft diameter:

A first-pass selection is:

Engineering Comment

The 40 mm shaft is only a torsion screen. The next checks are keyway weakening, bending from gear forces, bearing reactions, shoulder stress concentration, fatigue, deflection, runout, heat treatment, and inspection access.

Worked Example 2: Key Shear and Bearing Pressure

Use the design torque from the previous example:

Select a key on a 40 mm shaft with:

Key shear stress:

Key bearing pressure:

Engineering Comment

The shear number alone is not enough. Hub bearing pressure, shaft stress concentration, fretting, reverse torque, fit quality, and material hardness can control the design. A longer hub or a different shaft-hub connection may be better than simply using a stronger key material.

Worked Example 3: Bearing Rating Life

A ball bearing has:

The equivalent dynamic load is:

The shaft speed is:

For a ball bearing:

Rating life:

Life in hours:

Engineering Comment

3650 h may be acceptable for a replaceable auxiliary machine and unacceptable for a continuous-duty production gearbox. The release decision also needs lubrication, contamination, temperature, mounting, alignment, load spectrum, and maintenance evidence.

Worked Example 4: Reflected Inertia and Motor Torque

A motor drives a load through a reduction ratio:

Load inertia:

Motor inertia:

Gear efficiency:

The load torque is:

The motor accelerates from rest to:

in:

Reflected load inertia:

Total motor-side inertia:

Motor angular speed:

Acceleration:

Acceleration torque:

Load torque reflected to motor:

Required motor torque before extra losses:

Engineering Comment

The motor is not selected from steady load torque alone. Acceleration torque is almost 44 percent of the required value in this move. The drive current limit, thermal limit, brake release timing, and gearbox peak torque rating must all be checked.

Worked Example 5: Tolerance Stack and Thermal Clearance

A bearing arrangement needs at least:

of axial clearance after assembly and warm operation. Nominal cold clearance is:

Three independent spacer tolerances are:

Worst-case tolerance stack:

Thermal growth difference closes the clearance by:

Minimum clearance:

The requirement is not met:

Engineering Comment

The design is short by 0.01 mm before considering measurement uncertainty, preload error, housing distortion, or bearing internal clearance. The review should change the tolerance allocation, nominal clearance, thermal design, or bearing arrangement before release.

Common Mistakes

1. Using nominal torque instead of design, braking, jam, or stall torque.
2. Checking shaft torsion while ignoring bending, keyway weakening, fatigue, and deflection.
3. Treating catalog bearing life as guaranteed field life.
4. Applying a gear or belt ratio without tracking efficiency and heat loss.
5. Reflecting inertia through the wrong ratio convention.
6. Using RSS tolerance stacks where worst-case clearance is required.
7. Separating vibration checks from stiffness, bearing support, and operating speed range.
8. Releasing a calculation without test evidence, inspection limits, or maintenance assumptions.

Validation Evidence

A machine design calculation package should be tied to evidence such as:

• torque path and load-case definition;
• shaft, key, bearing, gear, belt, and coupling calculations;
• supplier ratings and derating assumptions;
• tolerance stack and runout measurements;
• alignment and bearing temperature records;
• vibration spectra during run-up and operating speed;
• thermal run or lubricant condition evidence;
• overload, brake, or proof-test records when relevant;
• inspection plan for wear, cracks, looseness, leakage, and corrosion;
• controlled release limits and requalification triggers.

The formulas are useful only when they support an engineering decision. A machine is released when the torque path, life assumptions, thermal behavior, alignment, vibration, inspection evidence, and maintenance plan are all consistent with the required duty.