For planar rigid links connected only by independent one-degree-of-freedom lower pairs, the Grübler count subtracts two scalar constraints per joint from three coordinates per moving link.
Twelve-bar Linkage Interactive Calculator
Enter total links including ground and total one-degree-of-freedom revolute or prismatic joints. Results are a count criterion—not proof of actual assembly mobility or a twelve-bar motion solution.
Equation Used
- n includes one fixed ground link.
- Every other link is a planar rigid body with three unconstrained coordinates.
- Every entered joint is an ideal revolute or prismatic one-DOF lower pair that removes two independent coordinates.
- All constraints are independent and the assembly is connected.
- No higher pairs, multiple-joint equivalence, special geometry, redundant constraints, singularity, link dimensions, position, velocity, force or stress are modeled.
Coordinates Minus Assumed Constraints
MIT mechanism notes give the planar Kutzbach criterion F=3(n-1)-2j. Each moving rigid link contributes two translations and one rotation; each ideal revolute or prismatic lower pair leaves one relative motion and therefore contributes two scalar constraints.
What This Calculator Represents
The tool counts coordinates and constraints for entered planar links and one-DOF lower-pair joints. It does not know which links connect, whether a pin joins more than two links, or whether the constraints are independent.
The default is twelve links including ground. To determine actual motion, the connection layout, joint types, dimensions and constraint independence are also needed.
Planar Lower-Pair Mobility Count
assumed joint constraints = 2j
Grübler count M = 3(n-1)-2j
1-DOF joint count = [3(n-1)-1]/2
signed joint difference = entered j - 1-DOF joint count
Worked Count Example
For n=12 total links including ground and j=16 ideal one-DOF lower-pair joints, the free-body count is 33 and the assumed constraint count is 32, so M=1. This arithmetic does not prove that an unspecified assembly moves with one degree of freedom.
Model Boundary
- Calculated: a planar lower-pair mobility count from totals.
- Not calculated: connection graph, multiple-joint counting, constraint rank, redundant constraints, special geometry, singularity, assembly feasibility, positions, coupler path, velocity, acceleration, force, torque or stress.
Research on mechanism structure distinguishes assemblies satisfying the count from non-isomorphic mechanisms and emphasizes further structural analysis. NIST work likewise uses constraint Jacobian rank and invertibility when evaluating actual mechanism stability.
Frequently Asked Questions
Does M=1 prove the linkage has one physical degree of freedom?
No; it is a criterion under stated independence and topology assumptions.
Does a negative count prove an assembly cannot move?
No; constraints may be redundant or special geometry may change rank.
Is ground included in n?
Yes.
Can one pin joining three links be entered as one joint?
Not generally; multiple-joint equivalence must be counted correctly before using this tool.
Is a twelve-bar coupler path calculated?
No.
Are higher pairs included?
No.
References
- MIT 2.72, 2D Mechanisms: Mobility, Kinematic Analysis and Synthesis: F=3(n-1)-2j and the one-DOF criterion.
- Rothenhofer et al., Precision Engineering: planar link coordinates, revolute/prismatic joints and Kutzbach-Grübler equation.
- NIST, Analysis and Design of Parallel Mechanisms With Flexure Joints: constraint Jacobian rank, overconstraint and underconstraint distinctions.
- University of Maryland, Structural Synthesis and Analysis of Planar and Spatial Mechanisms Satisfying Grübler's Degrees of Freedom Equation: structural analysis beyond equation satisfaction.
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