7.1.1 Degrees of freedom in the plane

A free rigid link in the plane has three freedoms, two translations and one rotation. A mechanism with n links includes one frame that is fixed, so n minus 1 links are movable. Without joints the movable system has 3 times n minus 1 freedoms. Each joint removes freedoms by imposing constraints.

7.1.2 Full joints and half joints

A full joint, denoted j1, is a revolute pin or a prismatic slider. It leaves one relative freedom, rotation for a pin and sliding for a slider, and therefore removes two freedoms. A half joint, denoted j2, is a rolling or sliding contact such as a cam and roller contact or a gear contact. It leaves two relative freedoms, rolling plus sliding or separation constrained only in the normal direction, and therefore removes one freedom.

7.1.3 Kutzbach relation derived

Subtracting the constraints from the free count gives the mobility M, the number of independent inputs needed to fix all positions:

M = 3 times (n minus 1) minus 2 times j1 minus j2.

The derivation is a count, not a dynamic law. Start from 3 times (n minus 1) free motions. Each j1 joint deletes two motions. Each j2 joint deletes one motion. The remainder is the number of actuators required. M equal to 1 means one input drives the whole chain. M equal to 2 means two inputs are needed, as in a differential. M less than or equal to 0 means a locked structure, not a mechanism.

7.1.4 Joint counting discipline in prose

The frame counts as a link. A pin joining k links counts as k minus 1 revolute joints, because the first link provides the reference and each added link adds one relative rotation. A slider in a slot counts as one j1. A cam and roller contact counts as one j2, not as a pin, because rolling preserves an extra freedom.

In prose, a four-bar chain has four links including the frame and four pins, so n is 4, j1 is 4, j2 is 0, and M is 3 times 3 minus 8, which is 1. One crank input fixes the motion, as expected for a crank rocker drive. A slider crank engine chain likewise has four links and four full joints, three pins plus one slider, so M is again 1. A cam with a translating roller follower has three links, frame plus cam plus follower, with one cam pin and one follower slider giving j1 equal to 2, plus one cam roller contact giving j2 equal to 1, so M is 6 minus 4 minus 1, which is 1. Cam rotation alone fixes follower position. Counting the roller contact as a pin would wrongly give M equal to 0, and that miscount is the standard error to avoid.

Quick illustration — DOF count for a four-bar

Given: $n = 4$, $j_1 = 4$, $j_2 = 0$.

Step 1: free motions $3(n-1) = 3 \times 3 = 9$.

Step 2: constraints $2j_1 + j_2 = 2 \times 4 + 0 = 8$.

Step 3: mobility $M = 9 - 8 = 1$.

Result: $M = 1$, so one crank input fixes all positions.

Trap: omitting the frame from $n$ or counting a rolling contact as $j_1$ corrupts the count, so apply pin rule $k-1$ and keep $j_2$ separate.

7.2.1 Grashof inequality stated and explained

Let a four-bar have lengths s for shortest, l for longest, and p and q for the intermediate links. If s plus l is less than or equal to p plus q, the chain is Grashof and at least one link can make a full rotation relative to the others. If s plus l is greater than p plus q, the chain is non-Grashof and no link can fully rotate. The logic is geometric continuity. The shortest link can only swing fully around if the sum of the shortest and longest can be bridged by the other two links through the folded and extended positions. The boundary case s plus l equal to p plus q is a folding or change-point chain in which all links line up at one instant, and practical drives avoid dwelling at that singular lineup.

7.2.2 Inversion principle

Inversion means fixing a different link of the same chain as frame. Relative motions are unchanged, but absolute motions change because a new link is held still. Therefore one chain generates a family of mechanisms, one per fixed link.

7.2.3 Four-bar inversions

For a Grashof chain, fixing the shortest link gives a double crank or drag crank, in which both side links rotate fully, suited to coupling two rotating shafts. Fixing a link adjacent to the shortest gives a crank rocker, in which the crank rotates and the opposite link swings, suited to converting rotation into swing for drives and pumps. Fixing the link opposite the shortest gives a double rocker with both side links swinging and the coupler capable of full rotation in the change-point sense, suited to swing to swing transfer with a floating coupler. For a non-Grashof chain, fixing any link gives a double rocker with no full rotation, suited to limited swing linkages such as steering approximations.

7.2.4 Slider-crank inversions and workshop applications

The slider crank is a four-bar with one infinitely long link replaced by a slider. Fixing the frame link gives the reciprocating engine and compressor, with crank rotation driving piston sliding. Fixing the crank gives the crank and slotted lever quick-return shaper drive, with the fixed crank centre offset from the slotted lever pivot so forward and return strokes subtend different crank angles. Fixing the connecting rod equivalent gives the Whitworth quick-return form with the slider block driving, also with unequal stroke timing. Fixing the slider block gives the hand pump or oscillating cylinder inversion with the cylinder rocking about a trunnion. The quick-return character in shaping applications comes directly from choosing the inversion with crank centre offset, so cutting time exceeds return time by the crank-angle ratio.

7.3.1 Instant-centre theorem

At any instant, the velocity field of a body in plane motion is a pure rotation about a point called the instant centre, denoted IC. Velocity magnitude at any point equals angular velocity times distance from the IC, directed perpendicular to the line from the IC. A pin joint is the IC of the two joined bodies. A rolling-without-slipping contact has its IC at the contact point. A slider has its IC at infinity in the direction normal to sliding.

7.3.2 Kennedy rule and location procedure

Kennedy rule states that for any three bodies in relative plane motion, their three pairwise ICs lie on one straight line. The practical procedure is fixed. First mark the obvious ICs at pins and rolling contacts. Then draw Kennedy lines through known ICs to intersect at the unknown IC. Then read unknown velocities from v equal to omega times distance from the relevant IC. The number of ICs for n bodies is n times (n minus 1) divided by 2, so a four-bar with four bodies has six ICs, of which four are pins and two are found by Kennedy lines.

7.3.3 Coriolis acceleration derived in words

When a slider moves along a link that is itself rotating, as in the crank and slotted lever or Whitworth drive, the acceleration of the coincident point contains an extra term beyond the sliding acceleration and the link acceleration. Let omega be the link angular velocity and v the sliding velocity along the slot. In a short time the sliding mass moves outward while the link turns, so the transverse velocity omega times radius changes both because radius grows and because direction turns. The combined rate gives twice omega times v, directed perpendicular to the sliding direction, turned 90 degrees ahead of the sliding velocity in the sense of rotation. Magnitude is 2 times omega times v. Any acceleration analysis of a slider on a rotating link must add this Coriolis term, or the transverse balance is incomplete and the computed pin forces are wrong.

7.4.1 Module as meshing condition

Module m is pitch diameter D divided by tooth count T. Module fixes tooth size, including addendum, dedendum, and circular pitch pi times m. Two gears mesh only when module and pressure angle match, because only equal tooth size allows successive teeth to enter and leave contact without wedging or backlash drift.

7.4.2 Velocity ratio and centre distance

For external mesh, speed ratio is inverse to tooth ratio, N2 over N1 equals T1 over T2, with directions reversed. Internal mesh keeps the same sense of rotation. Centre distance for external spur mesh is m times (T1 plus T2) divided by 2. Power crosses the mesh through tangential tooth load Wt equal to 2 times torque divided by pitch diameter.

7.4.3 Contact ratio meaning

Contact ratio is arc of contact divided by circular pitch, equivalently the average number of tooth pairs in contact. It must exceed 1 so that at least one pair is always engaged. Typical spur values lie near 1.4 to 1.7. A value below 1 means gaps between engagements and therefore impact loading. Higher contact ratio gives smoother transfer and lower load per tooth, at the cost of more sliding.

7.4.4 Interference and minimum teeth reasoning

Interference is tip digging into the mating flank below the active involute, caused when the addendum circle extends beyond the interference point on the line of action. It is avoided by limiting addendum, increasing centre distance, or keeping enough teeth so the involute starts low enough on the flank. Undercut at the root is the manufacturing counterpart that weakens the tooth. For a 20-degree full-depth involute system the minimum pinion teeth against a rack is 17, with more teeth needed against a small mating pinion. Fewer teeth mean a smaller base circle and a more curved, weaker flank, so the interference limit is simultaneously a strength limit.

7.4.5 Gear characters in prose

Spur gears connect parallel shafts with straight teeth, simple to make but noisy at speed because engagement starts abruptly along the full face. Helical gears use inclined teeth that engage gradually, so they run quieter and carry more load for the same size, but the helix pushes an axial thrust that bearings must carry, with herringbone pairs canceling that thrust. Bevel gears connect intersecting shafts and turn the corner of the drive, with straight bevels for low speed and spiral bevels for smooth high-speed transfer. Worm drives pair a screw-like worm with a wheel for a huge single-stage reduction through sliding contact, compact and capable of self-locking, but with sliding friction and therefore lower efficiency that demands attention to heat and lubrication.

7.5.1 Simple trains

A simple train has one gear per shaft. Overall ratio is product of driven teeth divided by product of driving teeth, with sign set by the number of external meshes. Idlers between input and output bridge distance and flip direction but cancel out of the magnitude, because each idler appears once as driven and once as driver.

7.5.2 Compound trains

A compound train has at least one shaft carrying two gears fixed together. The overall ratio is the product of the stage ratios, each stage computed as driven over driver. Compounding achieves large reductions in short centre distance because each stage multiplies.

7.5.3 Epicyclic trains and the tabular method explained

An epicyclic train has gears orbiting on a moving arm, typically a sun gear, one or more planets on the arm, and a ring gear. Fixed-axis formulas fail directly because planet centres move. The tabular method restores a fixed axis in three declarative steps. First, give the whole train a rotation y with the arm, recording arm y, sun y, planet y, ring y. Second, lock the arm and apply the fixed-axis ratio to the gears, for example with arm fixed at 0, sun plus 1 gives ring minus TS over TR for external sun to internal ring mesh, with planet value carried as needed. Third, add a uniform rotation x to every entry to represent arm motion, so final speeds are arm x plus y equivalents. Two known speeds fix x and y, and all remaining speeds follow. In prose, a sun with 20 teeth and a fixed ring with 80 internal teeth gives, with arm fixed, a sun to ring ratio of minus 80 over 20, and adding back the arm motion yields an arm speed that is a small forward fraction of the sun speed, which is the familiar planetary reduction.

7.6.1 Energy fluctuation derived

A flywheel stores kinetic energy E equal to one half times mass moment of inertia I times angular velocity omega squared. Over one cycle the load torque rises above and falls below the mean torque. The largest energy the wheel must absorb and later release is the fluctuation Delta E, the area between the turning-moment curve and the mean-torque line. With mean speed omega and extreme speeds omega-max and omega-min, the energy change between extremes is one half times I times (omega-max squared minus omega-min squared), which factors as I times omega squared times Cs, where Cs is defined below. Hence Delta E equals I times omega squared times Cs. Turning-moment area gives Delta E, the driven machine specifies Cs, and then I follows, with rim sizing from I near m times R squared for a rim-dominated wheel.

7.6.2 Coefficient of fluctuation Cs

Cs equals (omega-max minus omega-min) divided by mean omega. It is a tightness specification, not a material property. Small Cs means tight speed regulation and a larger wheel for the same Delta E. Large Cs means loose regulation and a smaller wheel.

7.6.3 Application table by order of magnitude

Punching and shearing presses tolerate large swing, Cs near 0.1 to 0.2, because the stroke is intermittent and violent. Single-cylinder engines use Cs near 0.03 to 0.06. Multi-cylinder automobile engines use Cs near 0.01 to 0.02 because overlapping power strokes smooth the torque. Generator and electric drives need Cs near 0.005 to 0.01 for frequency and process stability. The trend is monotonic, steadier load or tighter electrical tolerance calls for smaller Cs and therefore larger I at given speed and Delta E.

7.7.1 Function contrasted with flywheel

A flywheel smooths torque by storing energy, with no action on fuel supply. A governor regulates mean speed by moving a sleeve that acts on fuel supply. One handles cyclic swing within a cycle, the other handles load change across cycles.

7.7.2 Constructions and speed ranges

A Watt governor is a simple pendulum with two balls on arms pivoted to a spindle, sleeve motion taken from the ball height. It works at low speed only, because gravity alone provides the restoring effect and the required height grows rapidly with speed. A Porter governor adds a central dead weight on the sleeve to the Watt geometry, raising the speed for the same ball radius into the medium range and increasing response for the same size. A Hartnell governor replaces gravity and dead weight with a spring acting on a bell-crank lever carrying the balls, with sleeve motion compressing the spring. It works at high speed in compact form, and spring stiffness plus preload set the speed range.

7.7.3 Sensitiveness compared

Sensitiveness is speed change per sleeve travel, commonly expressed as (N2 minus N1) over mean N across the sleeve stroke. It measures eagerness to respond. Watt is least sensitive. Porter is more sensitive than Watt because the central load steepens the speed to radius relation. Hartnell is most sensitive and, uniquely, adjustable through spring stiffness and bell-crank ratio. High sensitiveness gives fast fuel correction but spends stability margin.

7.7.4 Stability, hunting, and isochronism explained

Stability means one sleeve position per speed, so the governor settles after a disturbance. Hunting means overcorrection and sustained oscillation around the set speed, caused by excessive sensitiveness or friction with lag. Isochronism is the limit of infinite sensitiveness, the same equilibrium speed at every ball radius, achievable at one Hartnell spring setting only. True isochronism is inherently unstable in practice because any displacement then lacks a restoring speed difference, so practical governors are set slightly stable rather than exactly isochronous.

Feature	Watt	Porter	Hartnell
Construction	Pendulum balls, gravity restored	Watt geometry plus central dead weight	Spring-loaded balls on bell-crank lever
Speed range	Low speed only	Medium speed	High speed
Sensitiveness	Least	Higher than Watt	Highest and adjustable
Size for same duty	Bulky at speed	Bulkier due to dead weight	Compact
Preferred use	Laboratory models and low-speed engines	Medium-speed engines needing cheap sensitivity gain	High-speed automotive and small engines with space limits

7.8.1 Static balance versus dynamic balance

Static balance means the mass centre lies on the shaft axis, so there is no net centrifugal force at speed. It is corrected in one plane by adding or removing mass opposite the heavy spot. Dynamic balance additionally kills the centrifugal couple caused by heavy spots in different planes along the shaft. A rotor can be statically balanced yet dynamically unbalanced, rocking end to end at speed. Full correction needs two planes, with masses sized and phased to cancel both net force and net moment.

7.8.2 Reciprocating unbalance harmonics

Reciprocating masses can never be fully balanced by rotation alone. With crank radius denoted r, connecting-rod ratio n equal to l over r, reciprocating mass m, crank speed omega, and crank angle theta from inner dead centre, the unbalanced force along the stroke is approximately m times r times omega squared times (cos theta plus cos 2 theta divided by n). The first term is the primary harmonic at crank speed. The second term is the secondary harmonic at twice crank speed arising from rod obliquity. Counterweights opposite the crank partly balance the primary at the cost of a perpendicular shake of the same order, while the secondary is normally accepted as-is in single-cylinder practice or canceled by layout in multi-cylinder and opposed designs.

7.8.3 Whirling and critical speed preview

A shaft with static deflection delta under its disc weight whirls with growing lateral amplitude when running speed approaches omega-c equal to sqrt(g over delta). The form matches transverse natural frequency because whirling is transverse resonance seen from a rotating frame. Operation must stay well below or well above omega-c, with rapid transit and added damping if the crossing cannot be avoided.

7.9.1 Equation of motion derived for translation

Consider mass m on a massless spring of stiffness k with displacement x from static equilibrium and no damping or forcing. Newton law gives m times x-double-dot plus k times x equal to 0, because the spring force opposes displacement while inertia opposes acceleration. Assume harmonic motion x equal to A times sin(omega-n times t). Substitution gives minus m times omega-n squared plus k times A equal to 0, so omega-n equals sqrt(k over m). Motion is an endless trade between kinetic energy of the mass and strain energy of the spring at that circular frequency. Cyclic frequency fn is omega-n over 2 pi, and period is 2 pi over omega-n.

7.9.2 Torsional form and beam transverse form

For inertia I on a shaft of torsional stiffness Kt with twist theta, the same derivation gives I times theta-double-dot plus Kt times theta equal to 0 and omega-n equal to sqrt(Kt over I). For a mass on a beam, replace k with the beam transverse stiffness at the mass location, and the same square-root form holds. Stiffness up or mass down raises frequency, and the scaling is square root, so quadrupling stiffness doubles frequency.

7.9.3 Static-deflection shortcut proved

Static sag delta under weight W equals W over k, which is m times g over k. Rearranged, k over m equals g over delta. Substitution into omega-n equals sqrt(k over m) gives omega-n equals sqrt(g over delta). This is exact for a massless spring model and highly useful because sag is often known from statics. With delta in millimetres and g equal to 9810 millimetres per second squared, omega-n is 99 over sqrt(delta in mm) radians per second, and fn is 15.76 over sqrt(delta in mm) hertz. The constants 99 and 15.76 are simply sqrt(g) with unit conversion and division by 2 pi.

7.9.4 Spring and shaft combinations

Springs in parallel share the same stretch, so forces add and k equals k1 plus k2. Springs in series carry the same force, so stretches add and 1 over k equals 1 over k1 plus 1 over k2. Shaft segments in series and parallel follow the same rules with Kt in place of k. A soft element dominates a series chain, while a stiff element dominates a parallel set.

7.10.1 Critical damping and zeta

Add viscous force c times velocity to the free model, giving m times x-double-dot plus c times x-dot plus k times x equal to 0. Divide by m and define omega-n as sqrt(k over m). The characteristic roots are governed by c relative to 2 times m times omega-n. Critical damping cc is 2 times m times omega-n, equivalently 2 times sqrt(k times m). Damping ratio zeta is c over cc. Zeta less than 1 gives underdamped oscillation with decay. Zeta equal to 1 gives critically damped return without overshoot in minimum time. Zeta greater than 1 gives overdamped creep without oscillation.

7.10.2 Logarithmic decrement derived

For zeta less than 1, successive peak amplitudes decay geometrically because each cycle dissipates a fixed fraction of energy. Let x0 and xn be peaks n cycles apart. The log decrement delta is defined as (1 over n) times ln(x0 over xn). Analysis of the damped sine gives delta equal to 2 pi times zeta divided by sqrt(1 minus zeta squared). Inversion gives zeta equal to delta divided by sqrt(4 pi squared plus delta squared). The damped frequency is omega-d equal to omega-n times sqrt(1 minus zeta squared), barely below omega-n at the light damping levels common in machines.

7.10.3 Small-angle approximation justified

When decay per cycle is modest, zeta is small, zeta squared is negligible beside 1, and the denominator sqrt(1 minus zeta squared) is near unity. Then delta is near 2 pi times zeta, so zeta is near delta over 2 pi. In prose, a trace falling from 12.0 mm to near 4.95 mm over two cycles gives delta near 0.44 and zeta near 0.07 by both exact and approximate forms, differing only in the third decimal. The approximation is therefore sound for light damping and saves algebra, while the exact form remains available for heavy decay.

Quick illustration — log decrement from two peaks

Given: $x_0 = 12.0$ mm, $x_2 = 4.95$ mm, $n = 2$ cycles.

Step 1: decrement $\delta = (1/2)\ln(x_0/x_2) = (1/2)\ln(12.0/4.95) = 0.443$.

Step 2: exact ratio $\zeta = \delta/\sqrt{4\pi^2 + \delta^2} = 0.0703$.

Step 3: light-damping check $\delta/(2\pi) = 0.0705$, matching to the third decimal.

Result: $\delta = 0.443$, $\zeta = 0.070$, lightly damped.

Trap: forgetting division by $n$ doubles the decrement here, so always normalize by the counted cycles.

7.11.1 Magnification curve explained

With harmonic excitation F0 times sin(omega times t), the steady equation is m times x-double-dot plus c times x-dot plus k times x equal to F0 times sin(omega times t). Define frequency ratio r equal to omega over omega-n and static deflection X-static equal to F0 over k. Steady amplitude X equals X-static times MF, where magnification MF equals 1 divided by sqrt((1 minus r squared) squared plus (2 times zeta times r) squared). At r near 0 the response is quasi-static near unity. At r near 1 the denominator collapses to 2 times zeta, so the peak is limited only by damping and is violent when zeta is small. This is resonance. At large r the response falls as 1 over r squared, with mass inertia dominating stiffness.

7.11.2 Transmissibility derived

Force transmitted to the foundation is the vector sum of spring force k times X and damper force c times omega times X, in quadrature. Dividing transmitted magnitude by exciting magnitude F0 gives transmissibility TR equal to sqrt(1 plus (2 times zeta times r) squared) divided by sqrt((1 minus r squared) squared plus (2 times zeta times r) squared). The numerator captures the damper path that grows with frequency, while the denominator is the same magnification denominator. Damping lowers the resonance peak but raises high-frequency transmission, which is the central trade in mount selection.

7.11.3 Isolation threshold r greater than sqrt(2) proved from the curve

Isolation means TR less than 1. For light damping the numerator is near 1, so TR less than 1 requires (1 minus r squared) squared greater than 1, which simplifies to r squared greater than 2, hence r greater than sqrt(2). Below sqrt(2) the mount amplifies, mildly at low r and violently near r equal to 1. Above sqrt(2) the mount attenuates with increasing r, more strongly for lower zeta. Design reading is therefore a threshold, not a trend. A soft mount with low omega-n makes r large at operating speed and isolates well, but the machine must pass through resonance during start-up and shutdown, so enough damping is added to survive the crossing even though it slightly degrades high-frequency isolation.

7.12.1 Nodes in two-rotor shafts

In torsional vibration the node is the cross-section that stays still while inertias on either side oscillate in opposition. For two rotors Ia and Ib joined by shaft segments, torsional moments must balance at the node, giving Ia times la equal to Ib times lb when stiffness is uniform, where la and lb are distances from each rotor to the node. The node therefore divides the length in inverse ratio of the inertias, lying closer to the larger inertia. Natural frequency follows from either side as omega-n equal to sqrt(Kt-a over Ia), where Kt-a is the torsional stiffness from rotor A to the node. Both sides give the same value when the node is correctly placed.

7.12.2 Multi-rotor modes

Three rotors give two natural modes. The first mode has one node with all outer motions phased to balance about it. The second mode has two nodes with the middle rotor opposing the outer pair. Each higher mode adds one node and one frequency. Exact placement uses the Holzer or transfer-matrix balance in full analysis, but the node concept remains the same, zero-motion sections separating opposing swings.

7.12.3 Critical speed as transverse resonance

Critical speed of a shaft carrying discs is the running speed matching a transverse natural frequency. Value follows omega-n equal to sqrt(g over delta) computed with the disc weight and the shaft transverse stiffness at the disc. A shaft with distributed mass has multiple criticals matching its beam modes. Operating speed must avoid dwelling at any critical, with rapid transit and damping control through the first critical if the machine runs supercritical.

7.13.1 Three failure paths per pitch

Per pitch length p with rivet diameter d and plate thickness t, tearing of the plate between rivets gives strength (p minus d) times t times allowable tensile stress. Shearing of the rivets gives number of shear planes times (pi over 4) times d squared times allowable shear stress, with one plane for lap and single-strap butt and two planes for double-strap butt. Crushing of plate against rivet gives d times t times allowable crushing stress. Each path is a different physical limit, so all three must be evaluated.

7.13.2 Efficiency as weakest over solid

Joint efficiency eta is weakest of the three strengths divided by solid plate strength p times t times allowable tensile stress. The minimum governs because the chain breaks at its weakest link, and averaging would overstate capacity. Design then selects pitch, diameter, and strap sizing so the three values are balanced with the tearing value slightly controlling, avoiding wasteful excess in any one mode.

7.14.1 Throat 0.707 explained

A fillet weld of leg size s has a triangular cross section with the throat as the shortest shear plane at 45 degrees. Throat thickness is s times sin 45 degrees, which is 0.707 times s. Strength lives in that throat area, so shear stress is load divided by 0.707 times s times weld length. The factor 0.707 is geometry, not an empirical knockdown.

7.14.2 Concentric shear

Under concentric load the stress is uniform over the total throat area. Capacity is allowable shear times total throat area. Length is added in parallel, so longer or multiple welds share load proportionally.

7.14.3 Eccentric load vector method

An eccentric load P with eccentricity e imposes direct shear plus torsion on the weld group. Direct shear stress is P over total throat area, uniform in the load direction. Torsional shear arises from torque T equal to P times e about the weld-group centroid, with stress at radius vector rr equal to T times rr divided by polar moment J of the throat group about its centroid. At the farthest point the two vectors add as vectors, accounting for angle between uniform direct shear and perpendicular torsional shear. The resultant is checked against the allowable. The procedure is fixed in order, find centroid, compute J, evaluate both stresses at the extreme fibre, take the vector sum.

7.15.1 Preload and load sharing

A preloaded bolted joint clamps members with an initial tension that keeps the joint closed under external load. External tensile load is shared between bolt and clamped members by their stiffnesses, with the bolt taking only the fraction set by bolt stiffness over total stiffness. Preload therefore moves most of the alternating component into the members and leaves the bolt with a high mean load plus a small alternating increment, which greatly improves fatigue life and prevents separation and fretting.

7.15.2 Mean and alternating stresses

Under fluctuating load the bolt stress cycles between max and min. Mean stress sigma-m is the average. Alternating stress sigma-a is half the range. Endurance limit Se applies to the alternating part with stress-concentration, size, surface, and reliability corrections, while yield Sy and ultimate Su bound the mean part.

7.15.3 Goodman, Soderberg, and Gerber compared

Soderberg line is sigma-m over Sy plus sigma-a over Se equal to 1 over n. It uses yield on both axes and is most conservative, suited to demanding safety or uncertain overload. Goodman line is sigma-m over Su plus sigma-a over Se equal to 1 over n. It is the standard for ductile design practice and the default choice for bolt fatigue. Gerber parabola is (sigma-m times n over Su) squared plus sigma-a times n over Se equal to 1. It fits ductile test data best and is least conservative. Ordering by conservatism is Soderberg then Goodman then Gerber. Selection is therefore a margin decision, Soderberg where conservatism is mandated, Goodman as the routine ductile rule, Gerber where best-fit data correlation is wanted with full awareness of lower margin.

7.16.1 Combined bending and torsion

A shaft carrying bending moment M and torque T together sees normal stress from bending plus shear stress from torsion at the outer fibre. Maximum-shear-stress theory combines them into an equivalent torque Te equal to sqrt((kb times M) squared plus (kt times T) squared), where kb and kt are shock and fatigue factors for bending and torsion that inflate the moments for sudden application and keyway effects.

7.16.2 Diameter sizing

For a solid shaft of diameter d with allowable shear tau-allow, Te equals (pi over 16) times d cubed times tau-allow. Diameter follows directly by inversion. Bending alone and torsion alone are special cases of the same root with the other moment set to zero. Hollow shafts replace the solid section modulus with the hollow value, and deflection and slope checks follow separately for stiffness-critical drives.

7.17.1 L10 idea and load-life exponents

Rolling-bearing life is statistical. L10 is the life in million revolutions that 90 percent of a bearing population survives under given load. With dynamic capacity C and equivalent load P, L10 equals (C over P) raised to p million revolutions. Exponent p is 3 for ball bearings with point contact and 10 over 3 for roller bearings with line contact. The cubic form reflects Hertzian stress to life scaling integrated over the contact, with rollers showing a stronger load dependence due to line-contact geometry. Doubling load therefore cuts ball-bearing life by a factor of 8, and a roller bearing outlives an identical-geometry ball bearing by (C over P) raised to one third whenever C over P exceeds 1.

7.17.2 Life-hours conversion

Catalog Mrev values must be converted to operating hours at speed N in rpm before judging adequacy, via Lh equal to L10 times 10 raised to 6 divided by (60 times N). A large Mrev number at turbine speed can still mean short calendar life. In prose, C equal to 30 kN with P equal to 5 kN gives C over P equal to 6 and L10 equal to 216 million revolutions for a ball bearing, which at 1500 rpm is 2400 hours, adequate for intermittent duty but short of continuous-duty targets near 8000 hours, calling for higher capacity or a roller type with recomputation.

Quick illustration — L10 life in hours

Given: $C = 30$ kN, $P = 5$ kN, $N = 1500$ rpm, ball exponent $p = 3$.

Step 1: load ratio $C/P = 30/5 = 6$.

Step 2: rating life $L_{10} = 6^3 = 216$ million revolutions.

Step 3: calendar life $L_h = 216 \times 10^6 / (60 \times 1500) = 2400$ h.

Result: $L_h = 2400$ h, short of continuous duty near $8000$ h, so raise $C$ or switch to roller type.

Trap: reading $216$ as hours without dividing by speed overstates life by orders of magnitude, so always convert Mrev at speed.

7.17.3 Sommerfeld meaning for sliding bearings

A hydrodynamic journal bearing rides on its own oil wedge. Sommerfeld number S equals (mu times Ns over P) times (R over c) squared, where mu is viscosity, Ns is speed in revolutions per second, P is unit pressure load over projected area, R is journal radius, and c is radial clearance. Same S means same operating point and same eccentricity ratio, so S is the scaling law that carries a tested bearing to new size, speed, oil, or clearance. Low S runs eccentric and stiff with thin-film risk. High S runs centred but hot and leaky with power loss. Design sits in the stable middle of the Stribeck curve with margin against seizure at start-up and against whirl at speed.

7.18.1 Lewis bending reasoning

Bending breaks teeth at the root. Lewis models the tooth as a cantilever with load Wt at the tip, giving bending stress sigma-b equal to Wt divided by (b times m times Y), where b is face width, m is module, and Y is the Lewis form factor. Y grows with tooth count because larger tooth counts give fuller, less curved flanks. Fewer teeth therefore mean weaker teeth for the same module, and below the interference limit the undercut root is weaker still. The 17-tooth minimum is thus both a geometry rule and a strength rule.

7.18.2 Buckingham dynamic increment

At speed, manufacturing error hammers the mesh with an extra dynamic increment on top of Wt. Buckingham load adds velocity, error, and compliance effects to the static tangential load, so high-speed gears are checked against dynamic plus wear loads rather than static Wt alone. Precision manufacture, profile correction, and balanced hardness reduce the increment.

7.18.3 Wear and surface durability

Contact stress and sliding cause pitting and scoring. Wear load capacity scales with face width, ratio factor, and surface endurance, with harder, well-lubricated pairs carrying more load. Bending and wear checks are both required, with bending sizing the module and wear sizing hardness and face proportions.

7.19.1 Uniform-pressure derivation for new linings

A new friction lining beds evenly, so pressure p is uniform over the annulus from inner radius small r to outer radius capital R. Axial force W is pressure times area, pi times p times (capital R squared minus small r squared). Torque per surface integrates mu times p times radius over area, giving T equal to (2 over 3) times mu times W times (capital R cubed minus small r cubed) divided by (capital R squared minus small r squared). This is the uniform-pressure torque used to design a new clutch.

7.19.2 Uniform-wear derivation for worn linings

A worn-in lining wears proportionally to pressure times sliding distance, and sliding distance grows with radius, so even wear requires p times radius constant, denoted capital C. Axial force W is 2 pi times capital C times (capital R minus small r). Torque per surface is mu times W times mean radius Rm equal to (capital R plus small r) divided by 2. For the same spring force this torque is always lower than the pressure-theory value, by a few percent for typical radius ratios. This is the uniform-wear torque used to rate an old clutch.

Quick illustration — uniform-wear clutch torque

Given: $\mu = 0.30$, $W = 2000$ N, $R = 0.15$ m, $r = 0.10$ m, one surface.

Step 1: mean radius $R_m = (R + r)/2 = (0.15 + 0.10)/2 = 0.125$ m.

Step 2: torque $T = \mu W R_m = 0.30 \times 2000 \times 0.125 = 75$ N m.

Step 3: two-plate pack $T_{total} = 2 \times 75 = 150$ N m.

Result: $T = 75$ N m per surface, $150$ N m for two surfaces.

Trap: applying uniform-pressure torque to a worn lining overstates capacity, so use mean-radius form for run-in service.

7.19.3 New-versus-old rule explained

Design a new clutch with uniform pressure because the unworn surface starts flat. Rate a worn clutch with uniform wear because running has already tapered the contact toward constant p times radius. Using pressure theory on a worn clutch overstates capacity and risks slip. Multiply single-surface torque by the number of active surfaces for multi-plate packs.

7.19.4 Band brake and capstan analogy

A flexible band wrapped over a drum with contact angle theta in radians and friction mu multiplies tension exponentially, T1 over T2 equal to e raised to (mu times theta), with T1 the tight side. Braking torque is (T1 minus T2) times drum radius. The analogy is the capstan rope, where a small hold force controls a large load through wrap friction. Differential pivots can assist or oppose the application force depending on moment direction, with assisting geometry reducing effort and over-assisting geometry risking self-energizing grab, so moment sense must be checked before signing the effort.

7.19.5 Power-screw self-lock and thread comparison

A power screw with lead l equal to pitch times starts on mean diameter dm has helix angle alpha with tan alpha equal to l divided by (pi times dm), against friction angle phi equal to arctan mu. Raising torque overcomes both load and friction, lowering torque is assisted by load. Self-lock holds when tan alpha is less than or equal to mu, equivalently alpha less than or equal to phi, with efficiency below 50 percent. Above that threshold the screw overhauls and needs a brake to hold load. Self-lock is wanted in a jack and avoided in a feed drive expected to back-drive. Thread comparison is declarative, square thread has highest efficiency and is suited to power transmission with a split nut for wear take-up, Acme thread is easier to cut and allows a split nut for engagement as in lead screws, Buttress thread carries heavy one-direction thrust as in presses, with sliding friction and collar friction added in full torque accounting.

Motion is governed by counting and limits. Kutzbach count sets inputs, Grashof sets rotatability, inversion selects the mechanism, Kennedy plus Coriolis complete velocity, module plus contact plus tooth count govern gears, tabular addition governs planetaries, Delta E with Cs sizes flywheels, spring versus weight construction selects governors with sensitiveness traded against stability, two-plane correction plus harmonic acceptance handle balance, and sqrt(g over delta) anticipates whirling.

Shaking is governed by the square-root frequency and the ratio r. Omega-n equals sqrt(k over m) in translation and sqrt(Kt over I) in torsion, with sqrt(g over delta) as the static-sag shortcut. Zeta sets decay through the log decrement near delta over 2 pi for light damping. Magnification peaks at r equal to 1 limited only by damping. Transmissibility falls below 1 only for r greater than sqrt(2), which is the isolation threshold carried forward to mount selection.

Life is governed by weakest-link and statistical laws. Rivet efficiency takes the minimum of tear, shear, and crush over solid strength. Weld strength lives in 0.707 times leg throat with eccentric load as vector sum of direct and torsional shear. Preloaded bolts share load by stiffness and check mean plus alternating stress against Soderberg, Goodman, or Gerber in decreasing conservatism. Shafts combine M and T into Te with shock factors under maximum shear. Bearings convert C over P to L10 with exponent 3 for balls and 10 over 3 for rollers, then to hours at speed, while Sommerfeld S scales sliding bearings at constant eccentricity. Gears check Lewis bending with form factor Y plus Buckingham dynamic and wear loads. Clutches use uniform pressure when new and uniform wear with mean radius when worn. Band brakes multiply tension by the capstan exponential. Power screws self-lock when helix angle does not exceed friction angle.