A ten-bar linkage is a planar kinematic chain containing ten links, including its frame. This calculator evaluates the nominal Kutzbach mobility count from entered link and revolute-joint counts; it does not define geometry or prove assembly, motion or usefulness.
Ten-bar Linkage Interactive Calculator
Enter total links including one ground link and the number of ideal full revolute joints. Results are a topological count under independent-constraint assumptions.
Equation Used
- All entered links are rigid planar bodies and one link is ground.
- Every entered joint is an ideal full revolute lower pair joining the intended connected chain.
- Counted constraints are independent and there are no redundant passive freedoms.
- Link lengths, topology, assembly branch, interference, singularities, forces, tolerances, wear and actuation are omitted.
Ten-link Planar Chain and Mobility Count
A linkage is a collection of rigid links connected by joints. For an ideal planar body, three coordinates describe position and orientation. Fixing one link as ground leaves 3(n−1) free-link coordinates before joint constraints are counted.
The illustration is a conceptual multi-link network, not a solved ten-bar geometry. Changing counts does not synthesize links, establish connectivity or prove that a physical assembly can traverse a desired path.
What This Explanatory Calculator Does
MIT OpenCourseWare presents the planar Kutzbach criterion M = 3(n−1)−2j for bodies connected by full pinned joints. This tool exposes the two terms and their difference.
Use the number only as an initial topology check under the stated assumptions. Geometry and a full kinematic analysis are still required.
Planar Kutzbach Count
n is total links including one ground link, and j is the number of ideal full revolute joints. The calculator also reports max(0,M) as the positive part of the index; it is not an actuator prescription.
Worked Count
For n = 10 links and j = 13 ideal revolute joints, 3(n−1) = 27 and 2j = 26, so the nominal mobility index is M = 1 and its positive part is 1.
That arithmetic alone does not prove that any particular ten-bar geometry assembles, avoids singularities, follows a chosen branch or produces a desired path.
Model Boundary
- Calculated: a count of planar free-link coordinates, nominal revolute-joint constraints, their difference and its positive part.
- Required assumptions: one connected ideal planar chain, one ground link and independent constraints.
- Not calculated: graph topology, link lengths, pivot coordinates, loop closure, assembly branch, range of motion, singularities, interference, coupler path, velocity, acceleration, force, stress, actuator sizing, tolerance, clearance, wear, reliability or suitability for any medical or safety-critical use.
Frequently Asked Questions
Does M = 1 prove a mechanism will move?
No. It is a nominal count subject to topology, independence and geometry assumptions.
Does a negative index mean the assembly is impossible?
No. Redundant or dependent constraints and special geometry can make the simple count misleading.
Does the formula detect singular positions?
No. It contains no dimensions or configuration variables.
Is max(0,M) the exact number of motors?
No. It is only the positive part of the nominal index; actuation depends on the real mechanism and control objective.
Does entering ten links create a ten-bar design?
No. The calculator does not create topology or dimensions.
Can this validate a prosthetic or production machine?
No. Those require complete kinematic, structural, control, safety and regulatory analysis.
References
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