Kinematics Studio: linkage mechanism calculator

The page opens with a running crank-rocker; “New” clears the canvas and the library loads ready-made linkages. In Quick build each click places a joint and connects it to the previous one, and a click on the first point closes the loop; in Single part you place links, joints, sliders, cams, gears, racks, Geneva drives, cylinders, springs and loads one at a time, and for a slider-crank you click the free end of the coupler first with “Slider” and then the crank pivot, and the guide is as long as the slider can travel: its own drive’s stroke if it has one, otherwise up to the dead centres. Dragging a joint changes the dimensions, which then appear in the properties panel; “Assemble the other branch” switches to the second assembly configuration. “Choose image” puts a sketch or scan under the drawing, and the file never leaves your computer. Any planar chain is computed as soon as it is constrained: select the link that stands still and choose “Ground (G)”, then the driven link with “Drive (A)”.

Kinematics Studio runs free of charge in the browser, with no installation and no sign-up: build linkages, run them as an animation, inspect forces and key figures and share them by link. Only the outputs are paid, that is the PDF report, DXF, value table, cam contour and video.

Yes. The linkage runs on the drawing area, optionally with coupler curve, traces and instantaneous centre; the space bar starts and stops it. You save one cycle as a WebM video, evenly timed and cut so that it loops seamlessly. The video is one of the paid outputs, like the report and the DXF.

A linkage is constrained when its degree of freedom according to Grübler equals the number of drives; the drives then fix the motion completely. If it is larger, a constraint or a drive is missing and the linkage is open; while building, that is normal. If it is smaller, one constraint or drive too many is in the model and it is overconstrained. The chip at the top right shows the state, a click on it names the remedy, and the characteristics card names loose or redundant parts. If links hang loose, it also gives the remedy, tried out on the model: on which links another drive helps and which joints, made rigid, join two links into one.

“from φ” and “to φ” are absolute angles of the driven link against the horizontal; the arc at the pivot shows the range. For continuous running the linear law fits, for an indexing cycle with a stop the 3-4-5 polynomial, the inclined sine for jerk-sensitive and the modified sine for torque-limited drives (VDI 2143). Motion duration or speed sets the time axis, from which velocities, accelerations and torques follow. A cylinder or a driven sliding joint travels a distance instead of an angle; its support points can be dragged in the “Motion function” sheet.

The transmission angle μ measures how favourably the coupler acts on the output link: at 90° the whole force goes into motion, at 0° the linkage locks in a dead centre. The smallest value over the cycle is rated against the empirical value of 40° for slow-running linkages (Kerle). At a cam joint the same limit applies as the pressure angle α = 90° − μ. For a four-bar the Grashof condition, s + l < p + q, says whether a crank rotates fully and which class the linkage belongs to.

Yes, as soon as there is a load in the model: mass and inertia of a link, gravity, a spring or an external load. From that follow drive torque, drive power and the forces in all joints over the cycle. The motion stays that of the drive; springs and loads change the forces, not the kinematics. Through a profile a load acts in only part of the cycle, such as a pressing force. Elasticity and play are not modelled.

With exactly one rotary drive and a load in the model, the card gives the RMS torque M_rms over the motion time, the reduced inertia J_red at the drive and the shaking force on the frame. Under “Motor and drive train” at the drive you enter gear ratio i, efficiency η, motor inertia J_M, rated torque and peak torque; from these follow motor torque, RMS motor torque, motor speed and the inertia ratio J_red,max/(i²·J_M). If the motor sits on a link other than the one whose motion is prescribed, choose it under “Motor sits on”; the master/slave table for the controller is then available as CSV, per degree of the uniformly running master axis. With a coefficient of fluctuation δ the flywheel follows VDI 2149 part 1 from ΔW_max/(δ·ω²), with the torsional stiffness c the natural frequency f_e = √(c/J_red)/2π, which should be at least ten times the cycle frequency.

Select the link point that carries the product. Under “Product and belt”, a_perm limits the acceleration of the product, for instance 9.81 m/s² for an open can; the card gives the largest acceleration at the point and the permissible cycle rate n_perm = n·√(a_perm/a_max), because on the same path every acceleration grows with the square of the cycle rate. With belt speed, direction and tolerance the tool finds the longest range in which the point's speed along the belt stays within the tolerance (synchronism according to VDI 2740 part 3), and gives its share of the cycle, the travel within it and the largest transverse speed. The diagrams plot the speed along and across the belt, and dashed in the colour of the point the canvas draws its path relative to the belt, as the travelling product sees it.

Friction is computed as soon as a joint carries a coefficient of friction µ; none does by default. At a revolute joint the friction circle r_R = r·µ/√(1+µ²) with the journal radius r applies (Kerle, eq. 5.6 and 5.7); at a sliding joint µ and the guide length count. If a joint also carries a coefficient of static friction µ₀, the calculator verifies self-locking according to VDI 2158: the linkage is self-locking when the locking factor r exceeds 1 in every position.

Draw the curve a point is to follow in Target path mode. The largest deviation of the path of a link point or joint is measured, either as a pure shape comparison (nearest point) or time-correct (equal arc length). Enter an admissible deviation under “Limits”; there is no default, because it depends on the task alone. Path search looks for a four-bar or a slider-crank that follows a target path, or improves the drawn linkage, whatever its type. If it stays far off at corners, it offers point guidance with two cams, which passes exactly through the corners.

Yes. The “Design” sheet computes in closed form: slider-crank to VDI 2130, crank-rocker for equal stroke times, four-bar for a transmission ratio, function synthesis after Freudenstein, relative position synthesis to VDI 2121 for two or three position assignments with crank or slider at input and output, approximate straight-line linkage, dwell mechanism and point guidance with two cams to VDI 2741: for the support points of a path and their cam angles it designs a five-bar with two roller levers and computes both cam profiles backwards; pivot spacing, pivot depth, lever and coupler can be given, and left empty the design searches for them. For two to five prescribed positions of a plane, position synthesis after Burmester supplies the joints, and for prescribed path points, path synthesis does; both start in the same sheet at the tiles “Linkage from positions” and “Linkage from path points”, drawn even without a linkage (a position with two clicks for place and direction, a path point with one) or set with a link, for example a triangle whose copy you drag into every further position, and “Compute linkage” in the panel on the right opens the synthesis with the best proposals; under “Choose by hand” the joints can also be clicked on the link, as in the three-position construction, and after building, a click on a position shows how exactly the linkage reaches it and goes to it. Path search, by contrast, tries: it finds a good four-bar for a drawn target path, not provably the best, and its run is repeatable through the random seed. Every result then stands on the canvas as an ordinary linkage, with its derivation in the report.

For every four-bar with a point on its coupler there are two further four-bars that trace the same coupler curve, with other link lengths and a third frame point C₀. They help when your own linkage does not fit the installation space or hits a blocked area. In the “Design” sheet the tile “Roberts cognates” computes them from the linkage on the drawing area or from entered dimensions; if the coupler carries a link point, the evaluation and the properties panel lead there too. The preview shows all three linkages in their own colour on the same curve, highlights the selected one and runs them together. The selected one is adopted; once you have viewed them, they also appear in the report.

In Blocked area mode you draw a region no link may pass through. Every moving link is checked over the whole cycle; the first contact and the deepest penetration are reported. A link has no width here, its centre line is checked: widen the area by half the link width. Links are not checked against each other.

The instantaneous centre is the point about which a link is turning at this moment; when the link translates it lies at infinity and is drawn as a direction stroke. Fixed and moving centrode are its trace in the frame and in the link, one rolling on the other. A trace draws the path of a joint, the evolute the centres of curvature of a coupler curve; its smallest radius of curvature limits, for example, a roller that is to follow the path. The position series overlays several positions of the cycle.

At a joint or link point they show velocity v, acceleration a and joint force F in the position shown. Their length follows from the scale below the drawing, such as “m_v = 12.5 (mm/s) per mm”; in round brackets it is derived from the cycle, without brackets you set it yourself. The acceleration arrow has a double shaft, and the force arrow has its tip on the point, because a force acts on it.

In Measure mode two clicks on joints or link points give a distance, three an angle; the second click is the vertex. Angles are measured counter-clockwise, and the other click order gives the angle on the other side. The value follows the position, and the report adds the smallest and the largest over the cycle. The lock next to a length or angle in the properties panel keeps that dimension fixed while dragging.

With the part “Cam”: a click on empty space creates disc, roller lever and drive, and the “Motion plan” sheet opens. The segments must cover 0° to 360° without gaps, and rises and returns must balance. As in the cam calculator, pressure angle, curvature against the roller, cutter radius and roller lift-off are checked. Under “Form closure” it becomes a groove cam or gets a conjugate cam with a second roller, and with “Disc follows the path” its profile is recomputed from the target path with timing after every change. The contour goes to CAD as a SolidWorks curve, Excel or CSV; with two cams, “Profile of” chooses which, and with form closure “Profile” chooses the outer flank or the conjugate cam.

Each has its own part in Single part mode, and a click on empty space creates a running mechanism. Two clicks on existing rotation axes set a gear pair, a click on the pinion axis and one on the carrier link a rack; radii and type then appear in the properties panel. The cylinder takes two clicks, base and head, and drives the linkage like a hydraulic cylinder. The tooth force enters the bearing forces with its tangential and radial parts; the pressure angle, preset to 20°, is set in the properties panel.

The PDF report contains the complete model, the system sketch of the position shown, characteristics, ratings and the derivation including model limitations. Ratings are only made against limits you enter; the smallest transmission angle is checked against 40° by default. In addition there is the drawing as DXF, to scale in millimetres with separate layers, the value table as Excel and CSV, the cam contour and an animation as WebM. Strength verifications are not included.

The browser saves the linkage continuously without sending it to us, and offers it for continuing the next time you open the page. “Copy link” puts the whole linkage into an address you can pass on; “Save as project” stores it in your account. A background image does not travel along, only its placement; the recipient chooses the same file and gets it in the same place.

Yes. Tab reaches the tools, the tree and the handles on the drawing area; Enter on a handle selects it, the arrow keys move it, with Alt in tenth steps. In Quick build and Single part the arrow keys move a keyboard pointer, and Enter places at its position like a click. Delete removes the selection, Ctrl+Z and Ctrl+Y undo and redo, the space bar starts and stops the run.