Reading the dimensions
Measure crank, coupler and rocker lengths between pivot centers. The ground length is the spacing between the two fixed pivots. A tracer point also needs a defined location on the coupler; it is not determined by the four link lengths alone.
Hoecken Linkage Interactive Calculator
Multiply crank length by the entered coupler, rocker and ground ratios. Velocity variation is not calculated. The animation follows fixed link lengths and flags crank angles where the two connecting links cannot meet.
Equation Used
The three length outputs are products of crank length and entered ratios. The default proportions are an illustrative starting point. No working interval, trajectory-error tolerance or velocity analysis is supplied.
- Default crank:coupler:rocker:ground proportions are 1:2.5:2.5:2; changing ratios changes the geometry.
- The schematic places a tracer beyond the coupler-rocker pivot by 2.5 crank-length units; this fixed choice is not an optimized tracer location.
- Velocity variation and straightness error are not calculated. Geometric closure is checked for ideal pivots; physical clearances and loads are not.
A coupler curve from four links
Two fixed pivots support the crank and rocker. The coupler joins their moving pivots, and a tracer attached to the coupler follows a curve as the linkage moves. Appropriate geometry can make part of that curve approximately straight.
Each pose must satisfy both connecting-link lengths. Some entered proportions cannot close at every crank angle. At an impossible pose, the animation shows the crank and fixed pivots, hides the unconnected links, and displays a no-assembly message. Full-turn closure means the ideal connecting-link circles can meet at every crank angle; singular positions and collisions still need separate assessment.
The schematic tracer extends beyond the coupler-rocker pivot by a fixed 2.5 times the crank length. The green curve follows this same tracer through the available crank positions. It is not an exact straight line. Playback has a fixed display speed, and the view scales to fit. Crossings in the front view represent bars in different planes.
Studying rotary-to-path motion
Approximate straight-line linkages are useful teaching examples for coupler curves and all-rotary motion constraints. A practical application requires its own path tolerance, working interval, load and speed analysis.
This calculator supports only dimensional scaling. It does not establish a walking speed, print-head performance, payload, motor requirement or a substitute for a linear guide.
Link-length scaling
Let a be the crank length and rc, rr and rg the entered ratios. Then coupler b = arc, rocker c = arr, and ground d = arg. All four dimensions use the same chosen length unit.
These products do not determine a usable stroke or velocity variation. A path assessment requires a consistent assembly branch, tracer location and working interval. Velocity additionally requires the input motion and differentiation of the tracer trajectory.
Worked dimensional example
For a crank length of 1 unit and ratios 2.5, 2.5 and 2, the outputs are 2.5 units coupler, 2.5 units rocker and 2 units ground. Doubling the crank to 2 units doubles those lengths to 5, 5 and 4 units.
Uniform scaling preserves an ideal geometric curve’s shape when the tracer location is scaled too. It does not preserve stresses, deflection, inertia or capacity. No stroke or velocity tolerance follows from these length outputs alone.
Approximate paths and guided motion
An approximate straight-line linkage constrains motion with revolute joints and accepts some path curvature over a selected interval. A guided slider imposes a straight path through a prismatic constraint. Accuracy, friction, loads and service life depend on the actual design.
Compare candidate mechanisms using their complete trajectories, assembly limits, transmission angles and loads. This calculator assigns no universal accuracy, speed, cost or maintenance advantage to one linkage.
Frequently asked questions
Why is velocity variation not calculated?
The calculator only scales link lengths. It has no specified working interval or verified velocity analysis from which to calculate a meaningful percentage.
Does the default geometry give an exact straight line?
No. A Hoecken linkage is an approximate straight-line mechanism. The relevant error depends on tracer location and the portion of the curve selected.
Can every slider combination make a complete revolution?
No. The drawing flags angles where the connecting links cannot meet. Even when ideal full-turn closure is possible, singularities, link collisions and bearing clearances need separate checks.
Where is the tracer in this drawing?
It is placed along the coupler line beyond the coupler-rocker pivot by 2.5 crank-length units. That choice is fixed in the illustration and is not optimized by the calculator.
Does scaling the lengths change velocity?
A complete answer needs the input angular speed and trajectory. At a fixed angular motion, uniformly scaled ideal coordinates and their derivatives scale together; no absolute speed is computed here.
Can this size a motor, bearing or link thickness?
No. Loads, material properties, dynamic forces and allowable deflection are absent.
Linkage analysis references
MechRef: Planar mechanisms explains planar linkage constraints and kinematics. MechRef: Graphical linkage synthesis describes constructing motion from geometric requirements.
The calculator’s numerical outputs are the three length-scale products. A straightness or velocity claim requires a separate trajectory analysis with a declared tracer and interval.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
