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Machine Design

Introduction

Machine design is the engineering process of turning a required function into a safe, reliable, manufacturable, maintainable, and economically reasonable mechanical system. In this aiMOOC, you study the design of common machine elements and, more importantly, the reasoning that connects requirements, loads, materials, geometry, failure modes, manufacturing, and verification.

Machine design is iterative. You rarely calculate a component once and finish. You define requirements, propose a concept, model loads, size parts, check failure and deformation, select standard components, consider manufacturing and assembly, evaluate risk, and revise the design until the evidence supports the requirements. Good design therefore combines Engineering mechanics, Mechanics of materials, Materials science, Tribology, Manufacturing engineering, Computer-aided design, and engineering judgment.

The gear train above is a compact example of a system-level design problem: tooth geometry, shaft strength, bearing reactions, lubrication, alignment, housing stiffness, tolerances, manufacturability, noise, cost, and service life all interact.

Learning Goals

After completing this aiMOOC, you should be able to explain and apply a systematic machine-design process; convert functional requirements into quantitative design criteria; construct useful free-body diagrams; calculate basic axial, bending, torsional, and combined stresses; distinguish yielding, fracture, fatigue, wear, buckling, and excessive deformation; select materials and standard machine elements; analyze shafts, bearings, gears, belts, springs, fasteners, and power screws; use safety factors and reliability concepts appropriately; and justify design decisions with calculations, models, tests, and documented assumptions.

You should also be able to recognize that a component can be strong enough yet still be a poor design because it is too flexible, difficult to manufacture, difficult to assemble, unsafe to maintain, sensitive to contamination, noisy, inefficient, or vulnerable to a different failure mode.

The Machine-Design Process

Requirements, Functions, and Constraints

A design begins with a need, not with a dimension. Translate the need into measurable requirements: transmitted power, speed, torque, load spectrum, motion, life, stiffness, accuracy, temperature range, environment, mass, envelope, noise, cost, maintenance interval, allowable leakage, and safety constraints. Separate requirements from preferences so that later trade-offs are transparent.

A useful functional statement describes what the system must accomplish without prematurely choosing a solution. For example, “transmit 5 kW between parallel shafts at a 3:1 speed reduction” is more useful at the beginning than “use a particular gear pair.” The first statement leaves room for comparing gears, belts, chains, and other concepts.

Constraints can come from physics, interfaces, standards, manufacturing capability, available materials, regulation, sustainability targets, or existing equipment. In university design work, document every important assumption. An unsupported assumption can dominate the result even when the algebra is flawless.

Concept Generation and Iteration

Concept generation asks which physical principles and machine elements can satisfy the required functions. You compare alternatives using criteria such as efficiency, stiffness, size, weight, controllability, serviceability, reliability, cost, and risk. The best concept is not necessarily the one with the fewest parts; it is the one that best satisfies the complete set of requirements.

Iteration is expected because changes propagate. Increasing a shaft diameter may reduce stress and deflection but require larger bearings, a larger gear bore, more mass, and a different housing. Increasing bolt preload may improve joint separation resistance but can increase bolt stress and assembly-control requirements. A good design workflow therefore keeps interfaces and assumptions visible.

Design and Manufacturing Must Be Considered Together

Geometry that is theoretically strong may be expensive or impossible to produce with the intended process. Tolerances influence machining time, inspection effort, interchangeability, and assembly yield. Surface finish can affect sealing, friction, contact fatigue, and fatigue strength. Material selection must consider available product forms, heat treatment, joining, and dimensional stability.

Loads, Stress, and Deformation

Load Paths and Free-Body Diagrams

A load path describes how forces and moments travel from where they enter a machine to the supports and ultimately to the environment. Before calculating stress, identify the load path. Draw a free-body diagram for each relevant component or assembly and include external forces, moments, support reactions, contact forces, and important geometric distances.

Common load types include axial force, transverse force, bending moment, torque, pressure, contact force, impact, and thermal load. Real machines often combine several of these. Loads may also vary with time, which can make fatigue more important than static strength.

For idealized members, useful nominal relations include axial normal stress $σ = F / A$ , bending stress $σ = M c / I$ , and torsional shear stress $τ = T r / J$ . These equations are powerful only when their assumptions fit the geometry and load state.

Stiffness Matters as Much as Strength

Strength asks whether a component will yield, fracture, buckle, wear out, or fail by another mechanism. Stiffness asks how much it deforms under load. Excessive shaft deflection can misalign gears and bearings even when shaft stress is below yield. Excessive bolt-joint deformation can allow separation or leakage. Excessive spring deflection can cause coil contact or loss of control.

For linearly elastic behavior, stress is linked to strain through elastic constants such as Young's modulus and shear modulus. In design, stiffness constraints often determine geometry before strength constraints do.

The stress-strain comparison above emphasizes that material response is not described by one “strength” number. Elastic modulus, yield behavior, ultimate strength, ductility, and fracture behavior support different design decisions.

Materials and Failure Modes

Material Selection

Material selection should connect properties to failure modes and manufacturing. Important properties can include elastic modulus, yield strength, ultimate tensile strength, fatigue strength, fracture toughness, hardness, density, thermal expansion, thermal conductivity, corrosion resistance, wear resistance, and temperature capability.

A high-strength material is not automatically the best choice. A lower-strength material may be preferable if it is easier to machine, more corrosion resistant, less notch sensitive, cheaper, weldable, or sufficiently stiff at lower cost. Likewise, changing material may not improve a stiffness-limited design because stiffness depends strongly on geometry and elastic modulus.

Static Failure

For ductile metals under multiaxial loading, yielding is commonly evaluated with criteria such as the Von Mises yield criterion or the maximum-shear-stress criterion. Brittle materials require different attention because fracture can occur with little plastic deformation and because tensile and compressive strengths may differ substantially.

A design factor compares a limiting condition with the calculated or expected condition. It is not a substitute for understanding uncertainty. Its selection should reflect variability in loads, material data, analysis accuracy, environment, consequences of failure, quality control, and the reliability target.

Stress Concentration

Discontinuities such as holes, grooves, threads, shoulders, keyways, and sharp fillets disturb the nominal stress field. A theoretical stress-concentration factor $K_{t}$ relates a local elastic peak stress to a nominal stress for a specified geometry and loading. In fatigue, notch sensitivity matters, so the effective fatigue concentration may differ from the purely elastic value.

Reducing a stress concentration can involve a larger fillet radius, smoother geometry transition, relief feature, improved surface condition, or a different load path. A local finite-element model can help, but mesh refinement and boundary conditions must be checked before trusting a peak value.

Fatigue and Variable Loading

A component can fail after many cycles at stresses below its monotonic yield strength. Fatigue is therefore central to rotating shafts, gears, springs, bearings, fasteners, and many structures subject to repeated load.

For a cyclic normal stress with maximum $σ_{m a x}$ and minimum $σ_{m i n}$ , the mean stress is $σ_{m} = (σ_{m a x} + σ_{m i n}) / 2$ and the alternating stress is $σ_{a} = (σ_{m a x} - σ_{m i n}) / 2$ . Similar definitions apply to shear stress.

An S-N approach relates stress amplitude to cycles to failure for a specified material condition and test basis. Real components require corrections or design allowances for effects such as surface condition, size, temperature, reliability, notches, residual stress, corrosion, and load spectrum. For finite-life design, the expected number and magnitude of cycles must be considered explicitly.

A Goodman-type diagram combines alternating and mean stress. A common linear design relation for tensile mean stress is written conceptually as $σ_{a} / S_{e} + σ_{m} / S_{u t} \leq 1 / n$ , where $S_{e}$ represents an endurance-strength quantity, $S_{u t}$ the ultimate tensile strength, and $n$ a chosen design factor. The exact data and correction procedure must match the material, geometry, loading, and design standard you use.

Shafts, Keys, and Couplings

Shafts transmit torque and often also carry gears, pulleys, sprockets, or rotors. They can experience combined bending and torsion, axial load, stress concentrations, fatigue, and deflection. A shaft design therefore requires more than a torsional-strength calculation.

For a solid circular shaft, nominal torsional shear stress at the surface is $τ = 16 T / (π d^{3})$ . Nominal bending stress at the surface is $σ = 32 M / (π d^{3})$ . When bending and torsion act together, an equivalent-stress criterion can be used for static yielding, while fatigue analysis should distinguish mean and alternating components.

Critical details include shoulders, fillets, retaining-ring grooves, keyways, spline roots, press fits, and threaded regions. Bearing spans and gear locations affect both bending moment and deflection. The final diameter may be controlled by stiffness, fatigue, bearing-seat requirements, manufacturability, or standard sizes rather than static yield.

Keys and splines transmit torque through bearing and shear stresses but also reduce shaft cross-section and create stress concentration. Couplings connect shafts and may accommodate some combination of axial, angular, or parallel misalignment. Their torque capacity, stiffness, balance, allowable misalignment, and failure behavior must match the system.

Bearings and Lubrication

Bearings support and locate moving members while controlling friction and motion. Rolling-element bearings are selected by bearing type, load direction, equivalent dynamic load, required life, speed, lubrication, contamination, mounting, stiffness, noise, temperature, and available space.

The exploded bearing view helps you identify the rings, rolling elements, races, and cage. These features must work together with shaft and housing fits, internal clearance, lubricant, and sealing.

For rolling bearings, basic rating life is often expressed in millions of revolutions by $L_{10} = (C / P)^{p}$ , where $C$ is a basic dynamic load rating, $P$ an equivalent dynamic bearing load, and $p$ an exponent determined by bearing type. This is a standardized statistical rating concept, not a guarantee that every bearing will last exactly that long. Real selection also checks static capacity, reliability requirements, lubrication life, contamination, alignment, temperature, and installation.

Lubrication separates surfaces, reduces friction and wear, removes heat, and protects against corrosion. Too little lubricant can cause damage; too much grease can increase churning and temperature. Seal selection is part of bearing-system design because contamination can dominate bearing life.

Gears and Gear Trains

Gears transmit rotary motion and torque with a defined velocity ratio. Spur, helical, bevel, worm, and planetary arrangements serve different shaft layouts and performance needs. Important geometric parameters include tooth count, module or diametral pitch, pressure angle, face width, pitch diameter, backlash, and center distance.

Involute teeth are widely used because properly meshing involute profiles maintain a constant angular velocity ratio despite small center-distance changes. The line of action and pressure angle determine the direction of tooth force.

For a simple external gear pair, the magnitude of the speed ratio is set by tooth counts. If gear 1 drives gear 2, $ω_{1} / ω_{2} = z_{2} / z_{1}$ . The tangential tooth force at pitch radius $r$ follows $F_{t} = T / r$ . Gear design must then check tooth-root bending, contact stress, surface durability, scuffing risk, lubrication, alignment, manufacturing quality, and dynamic effects.

The reducer animation shows a fundamental machine-design trade: speed is reduced while torque is increased approximately in inverse proportion to speed, apart from losses. The surrounding shafts, bearings, keys, housing, and lubricant must all be sized for the resulting loads.

Belts, Pulleys, and Flexible Drives

Belt drives can transmit power over larger center distances, isolate some vibration, and tolerate modest misalignment. Their behavior depends on belt type, pulley diameter, wrap angle, tension, speed, installation, and environment. V-belts use wedge action in pulley grooves, while synchronous belts use meshing teeth to minimize slip.

A belt drive must be checked for transmitted power, belt speed, tension, pulley size, shaft load, belt life, and guarding. Small pulley diameters increase belt bending and can shorten life. Excessive pretension raises bearing loads, while insufficient tension can promote slip or tooth jump depending on belt type.

Fasteners, Bolted Joints, and Power Screws

A bolted joint is a coupled elastic system. Tightening creates preload in the bolt and compression in the clamped members. Under a separating external load, only part of that load is added to the bolt because the bolt and joint have finite stiffness. Maintaining clamping force can prevent separation, leakage, fretting, and loss of alignment.

Bolt design can involve tensile stress, thread stress concentration, fatigue, proof strength, tightening method, friction uncertainty, embedment, relaxation, vibration, temperature, and joint stiffness. Torque-controlled tightening is convenient but uncertain because much of the applied torque is consumed by friction. Critical joints may require a more controlled preload method.

Power screws convert rotary motion into linear motion or force. Their design can be controlled by thread stresses, efficiency, self-locking behavior, wear, column buckling, bearing pressure, and the torque required to raise or lower a load.

Springs and Energy Storage

Mechanical springs store elastic energy, provide restoring force, maintain contact, absorb shock, and control motion. A helical compression spring is characterized by wire diameter, mean coil diameter, active coil count, free length, end condition, material, and allowable stress.

For an idealized round-wire helical compression spring, a common stiffness relation is $k = G d^{4} / (8 D^{3} N_{a})$ , where $G$ is shear modulus, $d$ wire diameter, $D$ mean coil diameter, and $N_{a}$ the number of active coils. Real spring design also considers curvature and direct-shear effects, fatigue, set, solid height, buckling, surge, end conditions, tolerances, surface condition, and corrosion.

A force-deflection requirement should be paired with packaging and life requirements. A spring that gives the correct rate but reaches solid height in service is not acceptable.

Fits, Tolerances, Surface Finish, and Assembly

Nominal dimensions alone do not define a manufacturable machine. Tolerances specify permissible variation, and fits describe the resulting relationship between mating features. A clearance fit supports free relative motion or easy assembly, a transition fit may provide accurate location with limited assembly force, and an interference fit can transmit load or locate parts through contact pressure.

Geometric tolerancing controls form, orientation, and location in ways that size tolerances alone cannot. Surface texture matters because real surfaces contact at asperities. Finish can influence friction, wear, sealing, fatigue, lubrication retention, and appearance.

Tolerance choices should follow function. Tightening every tolerance increases cost without necessarily improving performance. A tolerance stack-up should show whether worst-case or statistical variation can violate assembly, alignment, preload, clearance, or motion requirements.

Design for Manufacturing and Verification

Manufacturing processes impose characteristic limits and opportunities. Machining favors accessible features and realistic tool radii. Casting requires consideration of draft, section thickness, shrinkage, and defect control. Welding introduces residual stress, distortion, heat-affected material, and inspection needs. Additive processes can create complex geometry but may introduce anisotropy, surface roughness, support requirements, and post-processing needs.

A production drawing or model-based definition should communicate geometry, material, heat treatment, surface requirements, tolerances, and inspection-critical features clearly enough that another person can manufacture and verify the part.

CAD helps define geometry and interfaces. Finite-element analysis can estimate stress, deformation, temperature, vibration, or contact behavior, but it does not automatically validate a design. You must examine mesh sensitivity, constraints, load introduction, contact assumptions, material models, and whether the model represents the real failure mode.

Verification asks whether the design output satisfies the specified requirements. Methods include hand calculations, simulation, dimensional inspection, material certification, prototype testing, fatigue testing, vibration testing, thermal testing, leak testing, and field data. Validation asks whether the resulting machine satisfies the intended user need in its real context.

Reliability, Safety, and Sustainability

Reliability is the probability that a system performs its required function for a specified period under stated conditions. A machine may contain many individually reliable parts yet have poor system reliability if there are many series-critical failure points, weak interfaces, or unaddressed common-cause failures.

Risk analysis should identify hazards and failure consequences early. A failure mode and effects analysis can help teams ask how each function might fail, what causes the failure, what the effects are, how the failure might be detected, and what design action can reduce risk. Engineering safety should prioritize eliminating or reducing hazards through design before relying only on warnings or procedures.

Sustainable machine design considers material and energy use, efficiency, durability, repairability, remanufacture, recyclability, and the environmental effects of manufacturing and operation. Longer life is useful only when it is achieved without creating disproportionate impacts elsewhere in the system.

Formula Toolbox

The following relations are starting points, not automatic design rules. Always check assumptions, units, geometry, material behavior, stress concentrations, load variation, and the relevant design standard.

Design quantity	Useful relation	Typical interpretation
Axial normal stress	$σ = F / A$	Average normal stress in a prismatic member
Bending stress	$σ = M c / I$	Elastic flexural stress for beam-like behavior
Torsional shear stress	$τ = T r / J$	Elastic torsional stress in a circular shaft
Mechanical power	$P = T ω$	Power transferred by torque at angular speed
Gear speed ratio	$ω_{1} / ω_{2} = z_{2} / z_{1}$	Magnitude of ratio for a simple external pair
Bearing basic rating life	$L_{10} = (C / P)^{p}$	Statistical rolling-bearing life relation
Helical spring rate	$k = G d^{4} / (8 D^{3} N_{a})$	Idealized compression-spring stiffness
Goodman-type fatigue check	$σ_{a} / S_{e} + σ_{m} / S_{u t} \leq 1 / n$	One common linear mean-stress design relation

Integrated Design Example: A Gearbox Output Shaft

Consider an output shaft carrying a gear between two rolling bearings. The shaft transmits torque while the gear mesh produces tangential and radial forces. A defensible design sequence starts by defining torque, speed, duty cycle, life, space, material, environment, and allowable deflection. You then determine bearing reactions from free-body diagrams and construct bending-moment diagrams in the relevant planes.

A first diameter estimate can come from combined bending and torsion. The geometry is then refined around bearing seats, shoulders, seals, retaining features, and the gear connection. Stress concentrations are added at shoulders and keyways. If loading is repeated, mean and alternating stresses are evaluated using an appropriate fatigue criterion. Deflection and slope at the gear and bearings are checked because alignment may control performance.

Next, bearings are selected for radial and axial loads, target life, speed, lubrication, and mounting constraints. Fits and tolerances are assigned to control bearing creep, assembly, and alignment. The housing stiffness, seal arrangement, lubricant path, and service access are then considered. Finally, calculations and simulations are verified against test or inspection evidence. If any requirement fails, the design is revised. This integrated example shows why machine design is a system of interacting decisions rather than a collection of isolated formulas.

Interactive Tasks

Quiz: Test Your Knowledge

What is the best first step when a new machine-design problem is assigned? (Translate the need into measurable requirements) (!Choose the strongest available material) (!Create a detailed finite element mesh) (!Select the smallest standard bearing)

Which criterion is commonly used to assess yielding of ductile metals under multiaxial stress? (Von Mises criterion) (!Archimedes principle) (!Snell law) (!Fourier law)

What is the main effect of a geometric stress concentration? (It raises local stress above nominal stress) (!It guarantees plastic collapse) (!It eliminates fatigue damage) (!It makes material density increase)

Which pair of quantities is especially useful for describing a fluctuating normal stress? (Mean stress and alternating stress) (!Density and thermal conductivity) (!Pressure angle and module) (!Pitch and lead)

What does L10 represent in basic rolling-bearing life calculations? (A statistical basic rating life) (!The exact life of every bearing) (!The lubricant viscosity grade) (!The bearing outside diameter)

Why are involute gear teeth widely used? (They maintain a constant velocity ratio in proper mesh) (!They remove all contact forces) (!They require no lubrication) (!They make every gear ratio equal to one)

What is preload in a bolted joint? (The initial tensile force created in the bolt by tightening) (!A crack formed before service) (!The free clearance under a bolt head) (!The weight of the joint before assembly)

Why must shaft deflection be checked even when stress is acceptable? (Deflection can misalign gears and bearings) (!Deflection always increases material strength) (!Deflection removes stress concentrations) (!Deflection makes fatigue impossible)

Which change strongly increases the stiffness of an idealized helical compression spring? (Increasing wire diameter) (!Increasing active coil count) (!Increasing mean coil diameter) (!Removing all material strength)

What is a central principle of design for manufacturing? (Geometry and tolerances should suit the intended process) (!Every tolerance should be as tight as possible) (!Manufacturing should be considered only after testing) (!Standard components should always be avoided)

Memory Game

Yield strength	Stress level associated with the onset of permanent deformation in a material
Endurance strength	Fatigue resistance used for high-cycle design under specified conditions
Stress concentration	Local amplification of stress near a geometric discontinuity
Preload	Initial clamping-related tensile force created in a fastener during tightening
Backlash	Relative clearance between mating gear teeth along the direction of motion
Bearing life	Statistical measure of rolling-bearing durability under stated loading
Module	Metric gear-tooth size parameter equal to pitch diameter divided by tooth count
Spring rate	Change in spring force per unit deflection

Drag and Drop

Match the correct terms.	Topic
Increase fillet radius	Reduce a shoulder stress concentration
Add joint preload	Improve resistance to separation in a bolted joint
Increase shaft diameter	Reduce nominal bending stress and deflection
Improve sealing	Reduce contamination entering a bearing system
Increase pulley diameter	Reduce belt bending severity

...

Crossword Puzzle

Fatigue	What failure process can occur under repeated loading below monotonic yield strength?
Bearing	Which machine element supports relative motion and carries load between moving parts?
Involute	Which common gear-tooth profile maintains a constant velocity ratio in proper mesh?
Preload	What initial fastener force is created by tightening a bolted joint?
Torsion	What loading mode twists a shaft about its longitudinal axis?
Clearance	What fit condition provides intentional space between mating parts?

LearningApps

Cloze Text

Machine design begins by translating needs into measurable

. A free-body diagram helps you identify external forces and

. Local geometric discontinuities can create a stress

. Repeated loading makes

an important failure mode. A rotating shaft may be governed by strength, fatigue, or excessive

. Rolling-element bearing selection uses load, speed, lubrication, and required

. In a bolted joint, tightening creates fastener

. Involute gear teeth are used to maintain a constant velocity

. A helical compression spring stores elastic

. Final verification must show that the design satisfies its stated

.

Open-Ended Tasks

Easy

Design brief: Choose a familiar mechanical device, write five measurable requirements for one subsystem, and explain why each requirement matters.
Free-body diagram: Select one loaded component in a bicycle, vise, hand tool, or similar mechanism and create a labeled free-body diagram with a short explanation of the load path.
Machine element: Photograph or sketch at least six different machine elements in one real product and explain the function, likely load, and likely failure mode of each.
Material selection: Compare two candidate materials for a simple shaft or bracket using at least five relevant properties and justify your preferred material.

Standard

Shaft design: Design a stepped shaft for a specified torque and transverse load, check combined stress and deflection, and produce a dimensioned concept drawing with assumptions.
Bolted joint: Develop a bolted-joint concept for a removable cover, estimate an appropriate preload strategy, and explain how you would reduce the risks of separation, loosening, or leakage.
Bearing selection: Select a rolling-element bearing for a rotating shaft using a manufacturer catalog, calculate a basic rating-life estimate, and discuss speed, lubrication, fit, and contamination.
Spring design: Design a helical compression spring for a specified force-deflection requirement, check solid height and stress, and create a simple prototype or digital model.

Advanced

Fatigue analysis: Build a fatigue assessment for a notched rotating shaft under variable bending and steady torque, state all correction factors or assumptions, and compare at least two mean-stress criteria.
Gearbox design: Create a compact single-stage reduction gearbox concept that includes gears, shafts, bearings, housing, lubrication, and assembly strategy, then justify the main dimensions and interfaces.
Design for manufacturing: Redesign a machined component to reduce manufacturing effort without losing function, document the original and revised process plans, and estimate the effect on cost, tolerance risk, and material waste.
Failure analysis: Interview a technician, engineer, or laboratory supervisor about a real mechanical failure or inspect an approved failed component, then reconstruct the likely failure chain and propose design changes supported by evidence.

Learning Assessment

Requirements traceability: Given a mechanical design brief, create a traceability matrix linking each requirement to at least one analysis, inspection, or test that could verify compliance.
Combined loading: Analyze a shaft section under bending and torsion, choose an appropriate static-failure criterion, and explain whether stress or deflection is likely to control the design.
Fatigue design: Evaluate how a shoulder fillet, surface finish, mean stress, and desired reliability change the fatigue assessment of a rotating component.
Machine element selection: Compare a gear drive and a belt drive for the same power-transmission task and defend your choice using speed ratio, efficiency, maintenance, alignment, noise, cost, and life.
Joint design: Explain how bolt stiffness and clamped-member stiffness influence the change in bolt load when an external separating force is applied, then propose a verification test.
Design review: Critique a complete mechanical assembly for strength, stiffness, manufacturability, assembly, lubrication, safety, maintenance, and sustainability, and prioritize the three most important revisions.

Evidence of Learning

Knowledge: You can explain load paths, nominal and local stress, stiffness, yielding, fatigue, bearing life, gear action, fastener preload, spring behavior, fits, tolerances, lubrication, reliability, and design-for-manufacturing principles.

Skills: You can translate requirements into engineering checks, create free-body diagrams, perform order-of-magnitude estimates and detailed calculations, select standard components, build and critique CAD models, interpret simulation results, evaluate uncertainty, and communicate assumptions.

Products: Strong evidence includes a requirements specification, calculation notebook, annotated free-body diagrams, material-selection rationale, CAD assembly, manufacturing drawing, component-selection record, risk analysis, test plan, and design-review presentation.

Transfer: You can apply the same reasoning to unfamiliar machines by identifying functions, interfaces, load paths, dominant failure modes, uncertainty, and verification evidence rather than relying only on memorized formulas.