ISO Standards Intermediate Kinematics & Machine Dynamics

Planar 4-Bar Grashof Kinematics & Coupler Curves

Synthesize crank-rocker vs double-crank inversions and trace continuous coupler point trajectories.

Calculates closed-loop loop equations using Freudenstein coordinates. Verifies the Grashof condition (s + l ≤ p + q), reveals transmission angles, and plots the kinematic path of an attached coupler point.

Governing Physical Law & Equations

s + l \le p + q, \quad K_1 \cos\theta_4 - K_2 \cos\theta_2 + K_3 = \cos(\theta_2 - \theta_4)

Grashof mobility criterion and Freudenstein analytical loop-closure equation.

Law: Grashof Mechanism Inversion & Loop-Closure Constraint | Standard Reference: ISO 13348 / ASME B105 / Freudenstein Analytical Kinematics

Adjustable System Parameters

Parameter	Nominal Value	Dynamic Range	Physical Role
Crank Length r2 (r_2)	40 mm	20 to 80 mm	Input rotating crank link length
Coupler Length r3 (r_3)	120 mm	60 to 180 mm	Floating coupler connector link
Rocker Length r4 (r_4)	90 mm	40 to 140 mm	Output oscillating rocker link
Ground Frame r1 (r_1)	130 mm	80 to 200 mm	Fixed base distance between frame pivots
Input Crank Speed (ω)	20 RPM	5 to 60 RPM	Crank rotational angular velocity

Analytical Proof & Derivation

Grashof’s Theorem establishes that for a planar four-bar link assembly with link lengths s, l, p, q, at least one link can make a complete 360° revolution if and only if s + l ≤ p + q. Vector loop closure r₂ + r₃ - r₄ - r₁ = 0 leads to Freudenstein’s equation: K₁ cos(θ₄) - K₂ cos(θ₂) + K₃ = cos(θ₂ - θ₄), where K₁ = r₁/r₂, K₂ = r₁/r₄, and K₃ = (r₁² + r₂² - r₃² + r₄²) / (2 r₂ r₄).

Verification Benchmark

Kinematic benchmark: r₁ = 130 mm, r₂ = 40 mm, r₃ = 120 mm, r₄ = 90 mm. s + l = 40 + 130 = 170 mm ≤ p + q = 90 + 120 = 210 mm (Grashof Class I crank-rocker verified). Transmission angle μ remains within safe operating limits 40° ≤ μ ≤ 140° throughout full cycle.

Field Engineering Insights

Transmission angle μ between coupler and output rocker dictates mechanical force transmission efficiency. If μ drops below 35° in industrial pick-and-place linkages, the mechanism can toggle or bind due to joint pin friction.