Hart's W-frame is the crossed-link form of Hart's straight-line inversor. Two equal long bars and two equal short bars form a crossed cell. A fifth moving link constrains an input point to a circle through a fixed inversion centre, and a corresponding point traces a straight line. The animation shows the midpoint construction with both grounded pivots visible.
Hart's W-frame Interactive Calculator
Choose the long-bar length, long-to-short ratio and requested input half-swing. The motion stays inside the crossed cell’s geometric reach. The bar-match percentage is a separate dimensional allowance, not a prediction of straightness or pivot wear.
Equation Used
L and l are full bar lengths. The crank radius is L/4. The actual half-swing θ is the smaller of the requested half-swing, 65°, and 80% of the circle-closure angular limit. The reduced swing avoids a singular display pose; it is not a hardware safety factor. Ideal rigid links and zero-clearance joints are assumed.
A crossed cell with a grounded midpoint
Two long bars CD and EF have equal length L; short bars CE and DF both have length l. The long bars cross without a pivot at their crossing. A is the midpoint of CE and is fixed to the base. G is the midpoint of EF, and P is the midpoint of CD.
The red crank BG connects G to a second fixed pivot B. Its radius equals AB, so the circle traced by G passes through A. The complete straight-line mechanism has five moving links and the fixed frame, with seven revolute joints. P is a marked tracing point on CD, not a separate sliding joint.
Kempe’s 1877 lecture explains Hart’s crossed cell using corresponding division points on its bars. The midpoint version shown here is one particular choice of those points. The A-frame is another Hart straight-line arrangement; its connectivity differs from this crossed cell.
Define the lengths before calculating the constant
For the midpoint construction and full bar lengths used here, K=(L²−l²)/4. The points A, G and P are collinear and satisfy |AG|×|AP|=K on this branch. If half-bar lengths were used instead, the same constant would be written as their squared difference. The length definition matters.
Set A=(0,0), B=(R,0) and G=(R+R cos θ,R sin θ), with R=L/4. Circle closure determines E at distances l/2 from A and L/2 from G. Then C=−E and F=2G−E. D is the crossed-branch intersection of a circle of radius L about C and one of radius l about F. P=(C+D)/2.
Inversion gives P=K G/|G|². Substituting the input circle gives x_P=K/(2R) and y_P=[K/(2R)]tan(θ/2). Thus the path is a vertical line offset by d=K/(2R) from A. Its stroke between symmetric input limits ±θ_max is 2d tan(θ_max/2).
Example: a 200 mm long bar and 4:1 ratio
With L=200 mm and long-to-short ratio r=4, each short bar is 50 mm. K=(200²−50²)/4=9,375 mm². The input crank radius is 50 mm and the straight trace is d=93.75 mm from the fixed inversion centre.
A requested half-swing of 45° gives a total input excursion of 90°. The ideal tracing stroke is 2×93.75×tan(22.5°)=77.67 mm. This describes the motion between the two endpoints, not the distance from the inversion centre to either endpoint.
A separate comparison allowance of 0.05% of the long bar gives 200×0.05/100=0.100 mm. This does not establish that the trace is accurate to 0.100 mm or to any other value.
Limit swing without changing the linkage lengths
Circle closure requires |AG|≥(L−l)/2. With the chosen crank radius, the corresponding angular boundary is 2 arccos(1−l/L). The actual half-swing is limited to 80% of that boundary and at most 65°. The readout shows the actual value used by the animation.
For example, increasing the long-to-short ratio reduces the short-bar length. It can also reduce the allowed input swing. When the requested angle exceeds the displayed limit, the animation uses that limit and the stroke calculation follows it.
A physical build also needs joint clearances, adequate stiffness and axial spacing between crossing links. Predicting straightness requires a defined model of dimensional errors, joint clearances, loading and the complete linkage geometry.
Explore geometry before making a mechanism
The W-frame demonstrates exact straight-line generation from revolute joints. The green trace is the computed locus of a marked point, not a rail. Change the size ratio and swing to see how the same crossed-cell principle gives different strokes.
The matched pairs are essential to the ideal construction, but the machining accuracy needed for a particular application depends on the complete design and load. This tool supplies geometric dimensions and a percentage-to-length comparison, not a specified fit, operating speed, lifetime or metrology accuracy.
Frequently asked questions
How many links and joints are shown?
Five links move relative to one fixed frame. The complete mechanism has six links including ground and seven revolute joints. The long-bar crossing has no pivot.
What does the long-to-short ratio control?
Each short bar is l=L/r. The equal long bars remain L. This changes the inversion constant, trace offset and available input swing.
Why can actual swing be smaller than requested swing?
The chosen crossed-cell dimensions have a finite reach. The display stays below its geometric limiting angle and reports the actual half-swing used.
Is half-swing the total crank movement?
No. The crank moves from minus that angle to plus that angle about its central position. Total angular excursion is twice the half-swing.
Does the bar allowance predict trace wander?
No. It converts a chosen percentage of L into millimetres. Manufacturing and clearance effects require a separate geometric and mechanical error analysis.
Does an exact path mean no friction or wear?
No. Exactness describes the ideal geometry. A real pin joint can have friction, clearance and wear even though no sliding guide is used.
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