Sources and References

• Norton, R. L. Design of Machinery: An Introduction to the Synthesis and Analysis of Mechanisms and Machines. McGraw-Hill.
• Uicker, J. J., Pennock, G. R., and Shigley, J. E. Theory of Machines and Mechanisms. Oxford University Press.
• Waldron, K. J., and Kinzel, G. L. Kinematics, Dynamics, and Design of Machinery. Wiley.
• Budynas, R. G., and Nisbett, J. K. Shigley’s Mechanical Engineering Design. McGraw-Hill.
• Meriam, J. L., and Kraige, L. G. Engineering Mechanics: Dynamics. Wiley.
• Hibbeler, R. C. Engineering Mechanics: Dynamics. Pearson.
• Myszka, D. H. Machines and Mechanisms: Applied Kinematic Analysis. Pearson.
• MIT OpenCourseWare, 2.72 Elements of Mechanical Design.
• Stanford ME 203 Design and Manufacturing.
• Cambridge University Engineering Tripos, Part IB Mechanisms and Machine Elements.

Chapter 1: Introduction to Mechanisms

A machine is a collection of resistant bodies arranged so that forces from available energy sources can be transformed, transmitted, or modified to perform useful work. A mechanism, by contrast, is the kinematic skeleton of a machine: the geometric arrangement of links and joints that dictates how motion is routed, independent of the forces involved. The study of mechanisms therefore begins with pure geometry and proceeds only later to forces, torques, and inertial effects.

A link is treated as a rigid body, and a joint (or kinematic pair) is a connection between two links that constrains their relative motion. The two most common lower pairs in planar machinery are the revolute (pin) joint, which permits one relative rotation, and the prismatic (slider) joint, which permits one relative translation. Higher pairs such as the cam–follower contact permit a mixture of rolling and sliding.

The mobility, or degree of freedom (DOF), of a planar mechanism is counted using the Kutzbach–Gruebler criterion:

\[ F = 3(n-1) - 2 j_1 - j_2, \]

where \( n \) is the number of links including the ground, \( j_1 \) is the number of lower (one-DOF) pairs, and \( j_2 \) the number of higher (two-DOF) pairs. For a classic four-bar linkage with four links and four pin joints one obtains \( F = 3(4-1) - 2(4) = 1 \), confirming the well-known result that a four-bar has a single independent input angle. When the count gives a value lower than expected, one searches for redundant constraints, parallel links, or special geometries that grant additional freedom.

Any closed kinematic chain can be reframed by grounding a different link; this operation is called inversion. The four inversions of the slider-crank chain yield, respectively, the ordinary engine mechanism, the Whitworth quick-return used on shapers, the oscillating-cylinder engine, and a hand pump. Inversion preserves relative motion between links but dramatically changes absolute motion and therefore the engineering function.

A four-bar linkage is said to obey the Grashof condition when

\[ s + \ell \le p + q, \]

with \( s \) the shortest link, \( \ell \) the longest, and \( p, q \) the remaining two. Under strict inequality at least one link can rotate continuously, giving crank-rocker, double-crank, or double-rocker motion depending on which link is grounded. When equality holds, the mechanism is at a change-point and can momentarily lose constraint.

The Whitworth and offset slider-crank quick-return mechanisms exploit these geometric facts to make the forward stroke slower than the return. The time ratio

\[ Q = \frac{\alpha}{360^\circ - \alpha} \]

is set by the angle \( \alpha \) subtended at the driving crank as the output moves through its working stroke, giving the designer a direct geometric handle on shaper, slotter, and pump cycles.

Chapter 2: Kinematic Analysis of Mechanisms

Kinematic analysis answers three questions for every link: where is it, how fast is it moving, and how fast is that velocity changing, given the input. The modern analytical approach closes the mechanism into one or more vector loops and differentiates the closure equations.

For a single-loop four-bar with link vectors \( \vec{r}_2, \vec{r}_3, \vec{r}_4, \vec{r}_1 \) arranged head-to-tail, the loop closure equation is

\[ \vec{r}_2 + \vec{r}_3 - \vec{r}_4 - \vec{r}_1 = \vec{0}. \]

Projecting onto the \( x \) and \( y \) axes yields two scalar equations in the two unknown angles \( \theta_3, \theta_4 \) once the input \( \theta_2 \) is prescribed. These are transcendental in the unknowns, and Freudenstein’s equation provides a compact closed form; for arbitrary mechanisms a numerical Newton–Raphson iteration is more robust and generalises directly to multi-loop systems.

Once positions are known, velocity analysis follows by differentiating the loop equation with respect to time. Because each link length is constant, only the angles vary, and the velocity loop becomes linear in the unknown angular velocities:

\[ \left[ J(\boldsymbol{\theta}) \right] \dot{\boldsymbol{\theta}} = \vec{b}(\dot{\theta}_2). \]

The matrix \( J \) is the Jacobian of the loop-closure equations; when it is singular the mechanism is at a dead-centre or toggle configuration, where finite input velocity produces zero output or, conversely, where infinitesimal input force produces unbounded output force. A second differentiation gives an equally linear system for angular accelerations, with a right-hand side that now contains centripetal terms of the form \( \dot{\theta}_i^2 \).

Multi-loop mechanisms such as the Stephenson and Watt six-bars are treated by writing one independent loop equation per independent loop, as identified by the cyclomatic number \( L = j_1 - n + 1 \). The resulting block of equations can be assembled and solved in MATLAB with minimal changes from the single-loop case, making computer-aided analysis the default workflow for any mechanism beyond a basic four-bar.

Chapter 3: Synthesis of Mechanisms

Analysis treats the mechanism as given; synthesis treats the task as given and asks for a mechanism that performs it. Three classical subproblems are recognised. Function generation requires the output angle to be a prescribed function of the input angle at a finite number of positions. Path generation requires a point on a coupler to pass through a sequence of prescribed points in space. Motion generation, or rigid-body guidance, requires the coupler itself to adopt a sequence of prescribed positions and orientations.

Three-precision-point synthesis is the workhorse technique for MTE 321. For function generation of a four-bar, Freudenstein’s equation

\[ K_1 \cos\theta_4 - K_2 \cos\theta_2 + K_3 = \cos(\theta_2 - \theta_4) \]

is written at each of the three pairs \( (\theta_2^{(i)}, \theta_4^{(i)}) \) to give three linear equations in the three constants \( K_1, K_2, K_3 \). Inverting these yields the link-length ratios directly. For motion generation one uses the pole-based or dyadic approach: given three positions of the coupler, the centre-point and circle-point curves degenerate to unique points, so the fixed and moving pivots are obtained in closed form as the centres of circles through three prescribed image points.

The designer then checks whether the synthesised linkage is Grashof-compliant, whether the transmission angle (the angle between coupler and output at their common joint) stays within the conventional \( 40^\circ \) to \( 140^\circ \) band, and whether the mechanism avoids branch and circuit defects that would keep the coupler on the wrong assembly mode. The crank-slider can be synthesised by the same method with \( \theta_4 \) replaced by a slider displacement. This machinery underlies the physical prototypes built in the first course project.

Chapter 4: Kinetic Analysis of Mechanisms

Kinetic analysis relates the motion already computed to the forces and torques that cause it. Two complementary formulations are taught. Newton–Euler analysis isolates each moving link as a free body, writes

\[ \sum \vec{F} = m\, \vec{a}_G, \qquad \sum M_G = I_G\, \boldsymbol{\alpha}, \]

about its own centre of mass, and imposes equal-and-opposite pin-joint reactions between adjacent links. For a planar \( n \)-link mechanism this gives \( 3(n-1) \) scalar equations in the same number of unknown reaction components plus the required input torque. The governing equations are assembled into a single linear system of the form \( [A]\, \vec{x} = \vec{b} \) and solved one input configuration at a time as the mechanism sweeps through a cycle.

An alternative is the principle of virtual work or, equivalently, the power-balance method: at any instant the net power delivered by external forces and torques, including inertial \( -m\vec{a}_G \) and \( -I_G\boldsymbol{\alpha} \) terms introduced per d’Alembert, must vanish. This avoids computing internal joint reactions when only the input torque is desired and is particularly economical when the mechanism has few external loads.

A central output of kinetic analysis is the set of shaking forces and shaking moments, defined as the resultant force and moment that the moving links transmit to the ground frame at a given instant. Because these excite the foundation, balancing them is essential in high-speed machinery. For a rotating single mass, static balance eliminates the shaking force; for multi-plane systems dynamic balance is additionally required so that the principal inertia axis coincides with the spin axis.

Chapter 5: Mechanics of Deformable Solids Review

The design half of the course opens by re-establishing the stress analysis foundation. Internal force resultants are found from shear-and-moment diagrams drawn using the load-intensity relations

\[ \frac{dV}{dx} = -w(x), \qquad \frac{dM}{dx} = V(x), \]

where \( w \) is the distributed load. Once \( V \) and \( M \) are known, axial stress from bending is \( \sigma = My/I \) and transverse shear from the resultant is \( \tau = VQ/(Ib) \). Torsion of a circular shaft adds a shear stress \( \tau = Tr/J \), and axial loads contribute \( \sigma = P/A \). For combined loading the three-dimensional stress state at the critical point is assembled and then reduced to principal stresses using Mohr’s circle, which offers both a computational shortcut and the geometric intuition that maximum shear and maximum normal stresses occur at orthogonal orientations on the circle. The critical element on a typical rotating shaft is the one experiencing the largest bending stress and the largest torsional shear simultaneously, and that element is what all subsequent failure criteria act on.

Chapter 6: Static Yield Failure of Ductile and Brittle Materials

Once principal stresses are known the question becomes whether the material will yield or fracture. For ductile metals two theories dominate industrial practice. The maximum-shear-stress (Tresca) criterion predicts yield when

\[ \sigma_1 - \sigma_3 \ge S_y, \]

with \( \sigma_1 \ge \sigma_2 \ge \sigma_3 \) and \( S_y \) the uniaxial yield strength. The distortion-energy (von Mises) criterion, derived from the energy stored in shape-change rather than volume-change, predicts yield when

\[ \sigma' = \sqrt{\tfrac{1}{2}\left[(\sigma_1-\sigma_2)^2 + (\sigma_2-\sigma_3)^2 + (\sigma_3-\sigma_1)^2\right]} \ge S_y. \]

Von Mises is slightly less conservative than Tresca, matches experiment more closely, and is the default choice in MTE 321’s design project for ductile shafts.

Brittle materials such as grey cast iron show very different tensile and compressive strengths and fail by fracture without appreciable plastic flow. The maximum-normal-stress theory is the simplest model; more accurate for cast irons is the Coulomb–Mohr theory, which in its modified form gives

\[ \frac{\sigma_A}{S_{ut}} - \frac{\sigma_B}{S_{uc}} \ge 1 \]

when \( \sigma_A > 0 > \sigma_B \), with separate tensile and compressive ultimate strengths. The design factor \( n = S_y/\sigma' \) or its brittle analogue is chosen to reflect uncertainty in loads, material variability, and consequences of failure.

Geometric discontinuities such as fillets, holes, and grooves amplify local stress far above nominal values. This amplification is captured by a static stress-concentration factor \( K_t \), obtained from charts (Peterson) or finite-element analysis, and applied as \( \sigma_{\max} = K_t\, \sigma_{\text{nom}} \). Ductile materials under static load can generally redistribute stress by local yielding so the full \( K_t \) is not used; brittle materials cannot, and \( K_t \) must be applied directly.

Chapter 7: Stress-Life Fatigue and the Endurance Limit

Most machine-element failures in service occur not from a single static overload but from the accumulation of damage under fluctuating loads. The stress-life method, taught in this course for its design transparency, characterises the material by an S–N diagram plotting fatigue strength against the logarithm of cycles to failure. For wrought steels one observes a distinct plateau below which failure does not occur within any practical life, called the endurance limit \( S_e' \); for non-ferrous alloys no such plateau exists and a fatigue strength at a specified life (typically \( 5\times 10^8 \) cycles) is used instead.

The uncorrected endurance limit from a rotating-beam specimen is idealised and must be modified for realistic components. The Marin equation collects all the correction factors into a single product:

\[ S_e = k_a\, k_b\, k_c\, k_d\, k_e\, k_f\, S_e'. \]

Each factor has physical meaning. The surface factor \( k_a \) penalises rougher finishes; the size factor \( k_b \) accounts for the greater number of defects sampled by larger cross-sections; the load factor \( k_c \) distinguishes bending, axial, and torsional loading; the temperature factor \( k_d \) degrades the endurance limit at elevated temperatures; the reliability factor \( k_e \) shifts the allowable stress to the desired survival percentile; and the miscellaneous factor \( k_f \) covers effects such as residual stresses, corrosion, case hardening, and frettage.

Between the low-cycle region near \( 10^3 \) cycles and the endurance limit at \( 10^6 \) cycles the S–N curve is modelled as a straight line on log-log axes of the form \( S_f = a\, N^b \), with \( a \) and \( b \) fit from the corrected endurance limit and a fraction of ultimate strength at \( 10^3 \) cycles. This gives a design formula for finite-life problems.

Chapter 8: Fatigue Failure Criteria Under Fluctuating Loads

Real components rarely see fully reversed sinusoidal loading. A general fluctuating stress is described by its midrange and alternating components,

\[ \sigma_m = \frac{\sigma_{\max}+\sigma_{\min}}{2}, \qquad \sigma_a = \frac{\sigma_{\max}-\sigma_{\min}}{2}. \]

On a Haigh diagram of \( \sigma_a \) against \( \sigma_m \), experimental failure points for non-zero mean stress fall below the zero-mean endurance limit. Empirical envelopes capture this. The Goodman line, joining \( S_e \) on the alternating axis to \( S_{ut} \) on the mean axis, is linear and gives

\[ \frac{\sigma_a}{S_e} + \frac{\sigma_m}{S_{ut}} = \frac{1}{n}. \]

Gerber’s parabolic envelope fits data better for ductile steels, the ASME-elliptic criterion balances fatigue and yielding, and the Soderberg line substitutes \( S_y \) for \( S_{ut} \) and so also prevents yield on the first cycle. Which criterion is used depends on conservatism required and on whether yielding must be separately checked via the Langer line \( \sigma_a + \sigma_m = S_y \).

Stress concentration in fatigue is handled through the fatigue stress-concentration factor \( K_f \), which is less than the static \( K_t \) because small plastic zones and material sensitivity blunt the effect. Neuber’s rule gives

\[ K_f = 1 + q(K_t - 1), \]

with a notch sensitivity \( q \) between zero and one read from charts as a function of notch radius and material ultimate strength. The fatigue factor is applied to the alternating component, while the mean component may use either \( K_f \) or \( K_{fm} \) depending on whether local yielding is expected.

Under combined bending, torsion, and axial loading the alternating and mean components are each reduced to an equivalent von Mises stress by combining the concentration-corrected normal and shear components. These equivalent alternating and mean stresses are then fed into whichever failure criterion governs. Cumulative damage from variable-amplitude histories is estimated using the Palmgren–Miner linear damage rule,

\[ D = \sum_i \frac{n_i}{N_i}, \]

failure predicted when \( D \) reaches unity. Although simplistic, the rule underlies most practical fatigue life calculations in shaft and structural design.

Chapter 9: Shaft Design

Shaft design is the capstone topic because it combines every preceding tool. A rotating shaft carrying gears or pulleys experiences bending moments that reverse each revolution, giving fully alternating normal stresses at the critical fibre; a steady torque produces mean shear stress. Axial components from helical gears contribute a mean normal stress. The designer must simultaneously avoid yielding on the first cycle, avoid fatigue failure over the design life, keep deflections and slopes within bearing and gear-mesh tolerances, and keep critical whirling speeds well away from operating speeds.

The DE-Goodman diameter equation, derived by substituting the equivalent alternating and mean stresses for a rotating round shaft into the Goodman criterion, gives a closed-form expression for the required diameter as a function of bending moment, torque, stress concentration factors, endurance limit, ultimate strength, and factor of safety. Analogous DE-Gerber, DE-Elliptic, and DE-Soderberg forms are tabulated in Shigley; the choice matches the fatigue criterion being used and is applied at every critical section along the shaft.

Shoulders are needed to locate bearings and gears axially; keys and keyways in standard dimensions transmit torque between the shaft and the hub. Each introduces its own fatigue stress-concentration factor that dominates the critical-section check at that location. Retaining rings, tapered collets, splines, press fits, and setscrews are additional component choices whose selection depends on load level and required repeatability.

Deflection analysis is performed using singularity functions or the elementary beam equation

\[ EI\, \frac{d^2 y}{dx^2} = M(x), \]

integrated twice with boundary conditions at the bearings, to ensure that slopes at bearings remain below vendor limits (typically a fraction of a degree for ball bearings) and that deflections at gear meshes remain below about one-thousandth of the tooth module. Because deflection is proportional to the fourth power of diameter, modest increases in shaft size produce large improvements in stiffness, and iteration between strength and stiffness criteria quickly converges on a workable design. The full design problem solved in the course project draws these ideas together by taking a loaded three-dimensional shaft and sizing every section to survive both static and fatigue loading.