Linkage and Cam Kinematics: Method

Linkage and Cam Kinematics: Method and Sources

The tool works four planar mechanisms as geometry and calculus: the four-bar linkage, the slider-crank, the plate cam with a translating follower, and the external Geneva wheel. Nothing in it is an empirical rating; every result follows from the dimensions entered, and each formula below was checked against an independent numerical solution (finite differences, root finding, quadrature) before it was used. Lengths may be entered in any unit of the form; angles are in degrees, counter-clockwise positive; angular speeds are entered in rev/min and printed in rad/s; angular accelerations are in rad/s². Linear velocities and accelerations are printed in the length unit of the results (inches or millimetres) per second.

Four-bar linkage

Notation. The fixed pivots are O₂ (origin) and O₄ at the distance d along the x axis. The crank of length a runs from O₂ to pin A at the angle θ₂, the coupler of length b from A to pin B at θ₃, and the output link of length c from O₄ to B at θ₄. The vector loop is a e^iθ2 + b e^iθ3 = d + c e^iθ4.

Grashof's condition. With s and l the shortest and longest links and p, q the other two, s + l < p + q allows the shortest link to turn fully relative to its neighbours: a crank-rocker when the shortest link is the crank, a double-crank (drag link) when it is the ground, a double-rocker with a fully turning coupler when it is the coupler, and a rocker-crank when it is the output link. s + l = p + q is the change-point case (the linkage can fold flat), and s + l > p + q a triple-rocker. A linkage whose longest link is not shorter than the other three together cannot be assembled.

Position. Pin A is known from θ₂. The diagonal from A to O₄ has the length e, e² = a² + d² − 2ad cos θ₂, and closes the triangle A B O₄, so the angle at A between the diagonal and the coupler is ψ = acos[(b² + e² − c²)/(2be)]. The coupler direction is the direction of the diagonal plus ψ or minus ψ: the two assemblies. The tool calls "open" the one with B to the left of the directed line from A to O₄, which is the uncrossed quadrilateral while the crank is above the ground line and stays the same branch through the whole rotation of a Grashof linkage. θ₄ then follows from the coordinates of B. This is the solution of Freudenstein's equation in closed form; the test compares it with the half-angle form of that equation on 2,126 positions of 20 random linkages (largest difference 3 × 10⁻¹⁴ rad, loop closure 4 × 10⁻¹⁶ of the perimeter). A position exists only while |b − c| ≤ e ≤ b + c; the ends of that range are the toggle positions of the crank, where the coupler and the output link are in line.

Velocity and acceleration. The time derivative of the loop equation is linear in ω₃ and ω₄: ω₃ = a ω₂ sin(θ₄ − θ₂) / [b sin(θ₃ − θ₄)] and ω₄ = a ω₂ sin(θ₂ − θ₃) / [c sin(θ₄ − θ₃)]. The second derivative gives two linear equations of the same matrix for α₃ and α₄, with the centripetal terms a ω₂², b ω₃², c ω₄² and the crank term a α₂ on the right. A coupler point P at the distance p from A and the angle δ from the coupler line has r_P = a e^iθ2 + p e^{i(θ3 + δ)}; its velocity and acceleration are the derivatives of that expression.

Transmission angle. The angle γ between coupler and output link at B satisfies cos γ = (b² + c² − e²)/(2bc); the transmission angle μ is γ or 180° − γ, whichever is acute. Because γ depends on the diagonal alone, its extremes fall at θ₂ = 0 (e = |d − a|) and θ₂ = 180° (e = d + a), or at the toggle positions when the crank cannot reach those angles.

Limit positions and time ratio. The output link stops and reverses where crank and coupler are in line, extended (O₂B = a + b) or overlapped (O₂B = |b − a|); each is the triangle with the sides O₂B, c and d, solved by the law of cosines. For a crank-rocker the crank turns through unequal angles between the two limits, and at constant crank speed the ratio of the larger to the smaller is the time ratio of the slow and the quick stroke.

Mechanical advantage and static torque. By the principle of virtual work the torques and forces in equilibrium do no net work in a small displacement: T₂ ω₂ + T_L ω₄ + F · v_P = 0, so the crank torque that holds an output torque T_L and a force F at the coupler point is T₂ = −(T_L ω₄ + F · v_P)/ω₂, and the torque ratio of the bare linkage is ω₂/ω₄. The test compares this with the free-body equations of the three moving links. Inertia, friction and joint forces are not computed.

Slider-crank

The crank of radius r turns about O; the rod of length l joins the crank pin A to the wrist pin B, which slides on a line at the distance e from O. With θ measured from the line of stroke, x = r cos θ + √(l² − (r sin θ − e)²). Velocity and acceleration are the exact derivatives, v = (dx/dθ) ω and a = (d²x/dθ²) ω² + (dx/dθ) α. Expanding the square root to first order, √(l² − u²) ≈ l − u²/(2l), gives x ≈ l + r cos θ − (r sin θ − e)²/(2l) and, at constant speed, a ≈ −r ω² [cos θ + (r/l) cos 2θ + (e/l) sin θ]; with no offset this is the two-term form of engine practice, a ≈ −r ω² (cos θ + (r/l) cos 2θ), whose error is of the order (r/l)³. The dead centres are the positions with crank and rod in line, x = √((l ± r)² − e²); with an offset the stroke exceeds 2r and the two strokes take unequal crank angles (quick return). The inertia force of the reciprocating mass is −m a; a force F on the slider directed toward the crank produces the crank torque −F dx/dθ.

Cam and follower

Motion laws. A segment moves the follower by h over the cam angle β with s = h y(u), u = θ/β. The laws offered and their peak factors C_v = max |y′|, C_a = max |y″|, C_j = max |y‴|, each obtained by calculus:

• constant velocity, y = u: C_v = 1; the velocity steps at both ends unless the neighbours match it;
• parabolic (constant acceleration and deceleration), y = 2u² then 1 − 2(1 − u)²: C_v = 2, C_a = 4, the lowest acceleration of any law, but it reverses in a step at mid-stroke;
• simple harmonic, y = (1 − cos πu)/2: C_v = π/2 = 1.5708, C_a = π²/2 = 4.9348, C_j = π³/2 = 15.503; the acceleration is not zero at the ends;
• cycloidal, y = u − sin(2πu)/(2π): C_v = 2, C_a = 2π = 6.2832, C_j = 4π² = 39.478; zero acceleration at both ends;
• 3-4-5 polynomial, y = 10u³ − 15u⁴ + 6u⁵: C_v = 15/8 = 1.875, C_a = 10/√3 = 5.7735 at u = (3 − √3)/6, C_j = 60 at the ends;
• 4-5-6-7 polynomial, y = 35u⁴ − 84u⁵ + 70u⁶ − 20u⁷: C_v = 35/16 = 2.1875, C_a = 7.5132, C_j = 52.5 at mid-stroke; zero jerk at both ends;
• modified trapezoid: the acceleration rises as a quarter sine over the first eighth, is constant to three eighths, reverses through a half cosine to five eighths, and mirrors; integrating twice and requiring y(1) = 1 gives the amplitude C_a = 4.8881, with C_v = 2 and C_j = 4π C_a = 61.426;
• modified sine: a quarter sine over the first eighth, a half cosine of three times the period to seven eighths, and a quarter sine to the end; the same integration gives C_a = 5.5280, C_v = 1.7596, C_j = 69.466.

The last two are defined here by the shape of their acceleration, and their amplitudes are computed, not quoted; the values agree with the figures printed in cam design texts. The peak values of a segment are C_v h ω/β, C_a h ω²/β² and C_j h ω³/β³ with β in radians. The tool compares velocity and acceleration on the two sides of every segment boundary and reports a step in either.

Pressure angle, roller follower. The cam turns counter-clockwise and the follower translates along a line at the offset ε from the cam centre. Cam and follower have their instant centre on the line through the cam centre perpendicular to the follower motion, at the distance s′ = ds/dθ from the centre (the cam point there moves with the follower speed ω s′). The common normal at the contact passes through that point and through the roller centre, which is at the height R = √(R_p² − ε²) + s, where R_p = R_b + R_r is the prime circle radius. Hence tan φ = (s′ − ε)/R. An offset toward the side where the cam surface moves in the direction of the rise lowers the pressure angle of the rise and raises that of the return.

Profile and curvature. Holding the cam still and turning the follower back by θ gives the pitch curve Z = (ε + iR) e^−iθ in axes fixed to the cam. Differentiating twice, its radius of curvature is ρ_p = [R² + (s′ − ε)²]^3/2 / [R² + (s′ − ε)(2s′ − ε) − R s″], positive where the curve is convex. The cam surface is the curve one roller radius inside the pitch curve along its normal, with the radius of curvature ρ_p − R_r; where the convex pitch curve is sharper than the roller the surface is undercut and the follower cannot follow the program. The test differentiates the pitch curve numerically and compares the pressure angle, the curvature and the surface points.

Flat-faced follower. A face perpendicular to the follower motion touches the cam at the distance s′ from the cam centre line, so the surface point is (s′, R_b + s) turned back by θ, the pressure angle is zero, and the radius of curvature of the cam is ρ = R_b + s + s″. The smallest base circle is the one that keeps ρ positive (or above a required value); the face must span the largest and the most negative s′.

Base circle sizing. The smallest prime circle that keeps the pressure angle within a limit is found by bisection on the scanned maximum. The classical estimate, printed beside it, puts the pitch circle through the point of steepest slope: r = C_v h/(β tan φ_max), which is Furman's formula r = 57.3 h f/b (b in degrees) with his cam factor f = C_v cot φ_max. Furman's table of cam factors for the straight line, the crank curve (simple harmonic) and the parabola at 20° to 60° is reproduced by this expression (one cell, the parabola at 20°, is printed as 5.50 where 2 cot 20° = 5.495), and his Problem 3 (3 units in 90° by the parabola, 30°: chart length 41.52, pitch circle radius 6.61) is the default case of this analysis (with the pitch circle rounded up to 6.62, since 6.61 gives 30.02°). The estimate is exact for the parabola, whose steepest slope is at mid-lift, and slightly low for the other laws.

Geneva wheel

For the pin to enter the slot without shock the crank must be perpendicular to the slot at entry, which fixes the crank radius r = C sin(180°/N) and the wheel radius R = C cos(180°/N) for N slots and the centre distance C. With m = r/C and the crank angle α from the line of centres, the wheel angle is β = atan[m sin α/(1 − m cos α)]; its derivatives are dβ/dα = m(cos α − m)/(1 − 2m cos α + m²) and d²β/dα² = −m(1 − m²) sin α/(1 − 2m cos α + m²)². The speed ratio peaks at m/(1 − m) on the line of centres, the acceleration where cos α = [−(1 + m²) + √((1 + m²)² + 32m²)]/(4m), and the acceleration at entry and exit is tan(180°/N) ω², not zero. The wheel moves for the fraction (N − 2)/(2N) of the crank revolution.

Sources and what was checked against them

No government handbook with worked numerical examples of these mechanisms was located in the public record searched (the NASA technical reports server holds applications of linkages and a 1974 patent for a Geneva mechanism, but no kinematics reference with worked numbers), so the acceptance values are closed forms and independent numerical solutions, plus one public-domain text: Furman's Cams, Elementary and Advanced (1921), whose formula for the pitch circle, table of cam factors (p. 18) and Problem 3 (p. 29) are reproduced. The two-term slider-crank acceleration is printed in the public-domain Kinematics and Kinetics of Machinery of Dent and Harper (1921). The modern texts in the reference list are cited for the names of the motion laws and for general reading; nothing was copied from them.

Limits: rigid links, ideal joints, planar motion; no inertia of the links in the four-bar torque; translating followers only; one driving pin on the Geneva wheel.