Combination crank-motion curves, as illustrated in Hiscox figure 1127, are drawn by a pencil on the extended connecting arm of a crank-rocker linkage. Crank AD revolves about A, link BE rocks about B, and their connecting arm DE carries the pencil F. Changing the link proportions or the position of F changes the closed curve.
Combination Crank-motion Curves Interactive Calculator
Adjust the crank, rocker, connecting arm and pen position to draw a four-bar coupler curve. The crank turns about A, link BE rocks about B, and pencil F stays fixed to the rigid connecting arm. Frame spacing is 100 mm. Start angle changes the starting point; cycle time changes speed.
Equation Used
- Fixed frame spacing AB = 100 mm. All adjustable lengths are illustrative.
- One rigid coupler carries both E and the extended pen F. Offset sign selects the side of DE.
- Perfect planar pins and rigid links; no force, clearance or collision prediction.
- Uniform crank speed and one crank revolution per entered cycle time.
Single crank-rocker reconstruction of Hiscox figure 1127, replacing the unrelated two-speed rotary model. Ideal rigid geometry; illustrative dimensions and fixed 100 mm frame.
One crank, one rocker and a rigid pen arm
Frame pivots A and B stay fixed. The driven crank AD turns through a full revolution. Point E is constrained by both the fixed length DE and the rocker length BE, so it follows the intersection of two circles. The model consistently chooses one assembly branch.
The pen is carried on a bent extension of the connecting arm. Its along-arm distance is measured from E in the D-to-E direction; its signed perpendicular offset is measured normal to that direction. These two dimensions remain fixed while the arm moves, so the pen trace is a coupler curve rather than an independently imposed path.
The pale line shows the complete curve. Dark ink grows from the selected initial crank angle. The example uses adjustable illustrative proportions with a fixed 100 mm frame spacing. Hiscox’s engraving supplies the mechanism arrangement, not the numerical dimensions used here.
What this curve generator demonstrates
A point on a moving rigid link can follow a complex path even though the input is uniform rotation. This makes coupler curves useful for studying path generation and linkage synthesis. This demonstration is not a claim that any named packaging or textile machine uses the selected proportions.
Four-bar closure and pen position
Set A = (0,0) and B = (100,0), in millimeters. For crank length a and angle θ, D = a(cos θ, sin θ). The position E must satisfy |E − D| = b and |E − B| = c, where b is the coupler length and c is the rocker length. Their circle intersection supplies the exact linkage pose.
Let u = (E − D)/b and n = (−uy, ux). The pen is F = E + e u + h n, where e is the along-arm extension and h the signed perpendicular offset. All three links and both pen-arm dimensions remain constant throughout a revolution.
Crank speed is 60/T rpm for cycle time T seconds. Pen-curve width and height and rocker angular sweep are sampled every 0.5° of crank rotation. These are approximate geometric envelopes, not force, speed-limit or dwell guarantees.
Explore a closed coupler curve
Start with a 30 mm crank, 105 mm coupler, 75 mm rocker and 100 mm frame. Set the pen extension to 55 mm and its perpendicular offset to −60 mm. One four-second revolution corresponds to 15 rpm.
Move the perpendicular offset toward zero and the trace approaches the connecting-arm line. Set both pen dimensions to zero and F coincides with E, tracing the rocker endpoint’s arc. Change the initial angle from 90° to 0° to start elsewhere on the same full curve. Doubling the cycle time halves the crank rpm while leaving that curve unchanged.
Geometry and model limits
The available link ranges maintain a connected crank-rocker assembly through a full turn: the input crank is shorter than the other moving links and the two closure circles continue to intersect. A change in link proportions changes the path; a change in speed does not change the ideal path.
The drawing places links in separate ideal planes at crossings. It does not check physical clearance, link thickness, pencil loading, bearing play or dynamic deflection. A desired production path requires additional synthesis and mechanical design.
Crank-curve questions
Why is there no second crank-speed ratio?
The referenced arrangement has a rotating crank and a rocking link. The rocker’s angle is set by linkage closure; it is not an independently geared rotating shaft.
Does the initial angle change the full curve?
No. It changes the pen’s starting point and the current-cycle ink sequence. The complete curve is fixed by the link lengths and pen-arm dimensions.
What does a negative pen offset mean?
It places the pen on the negative-normal side of the directed D-to-E coupler. Positive offsets put it on the opposite side.
What happens when both pen dimensions are zero?
The pen coincides with E, so it follows the rocker endpoint’s circular arc instead of an extended coupler curve.
Reference and scope
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances, printed page 276, figure 1127 shows crank AD, rocking link BE and a pencil F on an extended connecting arm. The original text explicitly varies proportions to produce different curves. The numerical example here is an ideal planar reconstruction of that arrangement.
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
