A gyrating lever transmission connects two shafts on the same axis. A lever passes through a spherical support at its midpoint, and its ends connect to off-centre journals on the two discs. As the input turns, the lever sweeps a cone about the fixed centre and turns the other disc. Hiscox illustrates this arrangement in figure 138f.
Gyrating Lever Transmission Interactive Calculator
Set the full end-to-end lever length and the ratio of crank radius to that length. The symmetric model calculates its pin orbit and cone angle, then places the two disc planes at the required separation.
Equation Used
L is the full distance between end-joint centres. The central support bisects the lever and both pin orbits have radius r. Disc-plane separation is D = sqrt(L² − 4r²). The ideal construction requires 0 < k < 0.5. End joints permit angular articulation. Clearances, friction, torque capacity and bearing life are not calculated.
A lever between two rotating discs
The two shaft centres and the central spherical support lie on one straight axis. Each disc carries an off-centre joint. Those two joints connect to opposite ends of a rigid lever, whose midpoint stays in the fixed spherical support.
The drawing uses equal lever halves and equal crank radii. The end joints remain diametrically opposed when projected onto a plane perpendicular to the shaft axis. Their radial directions differ by 180 degrees, while their angular speeds are equal. Viewed along the same axis, the shafts rotate in the same sense.
The lever changes its direction in space. Its ends therefore need articulated connections to the discs; plain rigid corner connections would bind. The animation represents those connections by spherical joints and shows the fixed central socket separately from the moving lever.
Length, radius and shaft separation
Let the central support be O=(0,0,0). Put the left disc plane at x=−D/2 and the right at x=D/2. For input angle θ, the lever ends are A=(−D/2,r cos θ,r sin θ) and B=(D/2,−r cos θ,−r sin θ).
Their midpoint is O for every angle. The lever length obeys L²=D²+4r², so D=√(L²−4r²). The lever makes a constant half-angle β with the shaft axis, where sin β=2r/L. During a revolution that tilted lever direction travels around the axis: this is the gyrating motion.
There is one output revolution per input revolution in this symmetric geometry. The quantity L/r describes dimensions, not a speed reduction. Increasing k increases the cone angle and reduces disc separation for the same lever length.
Example: a 300 mm lever
For L=300 mm and k=0.25, each end journal orbits at r=75 mm. Its orbit diameter is 150 mm and L/r=4. The cone half-angle is arcsin(0.5)=30 degrees.
The disc-plane separation is √(300²−4×75²)=259.81 mm. Both discs complete a revolution in the same time. These values describe an ideal layout; they do not specify its operating speed or transmitted load.
Studying a coaxial spatial linkage
This arrangement demonstrates how a nutating lever can transfer rotary motion between separated coaxial shafts. It is useful for comparing spatial joints with planar crank-and-slider or four-bar linkages. The discs rotate continuously; neither lever end is constrained to a straight-line guide.
A practical assembly needs a rigid support, suitable angular travel at every joint, lubrication and adequate separation between the lever, discs and frame. Selecting those details requires the actual load and duty cycle.
Geometric reach and physical design
The relation D=√(L²−4r²) requires r<L/2 for two distinct disc planes. At r=L/2 their separation would be zero, so that limiting geometry does not describe the separated-shaft arrangement. The calculator uses k from 0.10 to 0.35.
The drawing fits the selected geometry to the screen and runs at a fixed explanatory speed. It retains lever length, disc radii and the central joint throughout each cycle. Display proportions outside these defined dimensions are illustrative.
The cone angle alone does not establish acceptable joint articulation, stress, efficiency, wear or vibration. Those depend on the selected bearings, lever section, housing and operating conditions.
Frequently asked questions
Does this produce a straight output stroke?
No. Each end joint follows a circle on its disc. The lever gyrates around its fixed midpoint while the output shaft rotates.
Does a 4:1 lever-to-crank ratio reduce speed fourfold?
No. It compares lengths. Equal end radii and equal lever halves give a one-to-one angular-speed relationship in this model.
Why are the two disc pins on opposite sides?
The rigid lever passes through its midpoint support. With equal halves, the endpoint vectors from that midpoint are equal and opposite.
What happens when I increase the crank factor?
The end-orbit radius and cone angle increase. For a fixed full lever length, the calculated separation between the two disc planes decreases.
Are the end joints ordinary rigid attachments?
No. They must permit angular articulation as the lever direction changes. The animation represents this freedom with spherical joints.
Does the calculator choose a safe load or speed?
No. It calculates geometry only. Loads, bearing ratings, fatigue, lubrication and dynamic behaviour require a defined physical design.
Mechanism references
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
