Grashof’s condition compares the lengths of a planar four-bar chain to determine whether full relative rotation is possible. Enter four pivot-to-pivot lengths here to check assembly, compare the shortest-plus-longest sum with the other two, and view one chosen arrangement. The result describes ideal geometry; it does not establish motor torque, clearances or durability.
Grashof Linkage Interactive Calculator
Enter four planar link lengths; the calculator sorts them, checks quadrilateral assembly and evaluates Grashof's inequality before solving exact four-bar closure.
Equation Used
- Four positive rigid planar links with ideal revolute joints.
- Lengths are sorted internally before applying Grashof's condition.
- Strict quadrilateral assembly requires the longest link to be shorter than the sum of the other three.
- No interference, branch-change, clearance, force, speed, acceleration, stress or tolerance model.
Ideal planar rigid-link geometry only. Links are sorted internally, so the result remains correct regardless of input order. The visualization fixes the longest link and places the shortest adjacent link as input; other inversions have different classifications.
Lengths first, then the fixed link
Sort the lengths as s ≤ p ≤ q ≤ l. First check l < s + p + q: without this strict inequality, the four links cannot make a nondegenerate quadrilateral. For an assemblable chain, a positive value of p + q − s − l gives the strict Grashof case; a negative value gives the non-Grashof case. Zero is the change-point boundary.
The fixed link determines the inversion. In the strict Grashof case, fixing the shortest link gives a double-crank; fixing an adjacent link gives a crank-rocker; fixing the opposite link gives a double-rocker. The shortest link can rotate relative to a neighbor even when the chosen fixed-link arrangement has no ground-connected crank.
This drawing fixes the longest link and places the shortest link next to it. The two remaining links close through a circle intersection. In a non-Grashof case, the input rocks within the feasible angle interval.
Reading the drawing
The ground joins the two fixed pivots. The input joins one fixed pivot to the coupler, which connects onward to the output and its fixed pivot. Measurements are between pivot centers. Joint clearance and link thickness are not included.
What the test helps you decide
A crank-rocker can turn continuous input rotation into a swinging output. A double-crank gives two revolving ground-connected links, while a double-rocker produces bounded motion at both. These distinctions are useful when choosing a four-bar arrangement for a feeder, an oscillating arm or a motion demonstration.
The inequality alone does not specify output swing, timing or coupler path. Once a length set and fixed link are chosen, examine the full motion, transmission angles and clearances under the intended load.
Sorted Grashof and assembly checks
Sort the four lengths as s≤p≤q≤l. A nondegenerate quadrilateral requires l<s+p+q. For a valid four-bar, Grashof's condition is s+l≤p+q. Equality is the change-point case.
The condition means at least one link can rotate relative to another; which link is fixed determines whether an inversion is crank-rocker, double-crank or double-rocker.
Example: a positive and negative margin
Enter 40, 140, 120 and 100 mm. The sorted values are s = 40, p = 100, q = 120 and l = 140 mm.
- Assembly check: 140 < 40 + 100 + 120 = 260 mm.
- Shortest plus longest: 40 + 140 = 180 mm.
- Other two: 100 + 120 = 220 mm.
- Grashof margin: 220 − 180 = +40 mm.
- Ratio: 220 / 180 = 1.222.
With the longest link fixed and shortest adjacent, this is a crank-rocker. The test does not establish a particular output angle or load capacity.
If the longest link increases to 200 mm, assembly remains possible, but the margin becomes 220 − 240 = −20 mm. The illustrated input must now rock rather than rotate fully.
For a separate tolerance check on the first set, let s and l each increase by 0.3 mm while p and q each decrease by 0.3 mm. The margin becomes 219.4 − 180.6 = +38.8 mm. This preserves the strict inequality in that scenario; it does not verify joint fits, interference or strength.
Scope and limitations
This is ideal planar kinematics. A buildable mechanism also requires a selected inversion and assembly branch, joint and link clearances, collision checks, transmission-angle and singularity analysis, forces, bearing loads, strength, fatigue and manufacturing tolerances.
Frequently asked questions
Must I enter the shortest link first?
No. All four values are sorted before the test. Changing their input order does not choose a different physical arrangement in this drawing.
Why can a positive ratio appear with an invalid assembly?
The ratio is only a comparison of two sums. Always check assembly status first; the longest link must be shorter than the other three combined.
Does passing Grashof mean every moving link rotates?
No. The fixed link and link order determine the inversion. This view fixes the longest link beside the shortest.
What happens at zero margin?
The linkage is at the change-point boundary, where collinear configurations can make the continuation of motion ambiguous. The displayed branch does not establish a reliable physical passage through that position.
Is Grashof Margin a manufacturing tolerance?
No. It is the difference between the two length sums. You can recalculate with tolerance extremes, but it is not a clearance, wear or reliability allowance.
Does the ratio predict mechanical advantage?
No. It compares sorted length sums. Mechanical advantage depends on the instantaneous geometry and applied loads.
Why does the mechanism disappear for some inputs?
Those lengths fail the strict quadrilateral assembly check. Separate bars are displayed instead of a closed mechanism.
References & Further Reading
- Wikipedia contributors. Four-bar linkage. Wikipedia
Building or designing a mechanism like this?
Explore the precision-engineered motion control hardware used by mechanical engineers, makers, and product designers.
