Theory of Machines: Kinematic Joints & Degrees of Freedom

Examine how revolute joints constrain motion between rigid bodies and derive the DOF as three times the number of bodies minus two times the number of revolute joints.

understand how the prismatic joint provides a one dof kinematic constraint, illustrated with crank-slider and slider-on-bar mechanisms, including ground sliders, and how dof is computed as 3n minus 2j1.

Examine the rigid joint and its effect on degrees of freedom by analyzing a system with rigid, revolute, and prismatic joints, including welded connections and degrees of freedom calculations.

Determine the degrees of freedom for six mechanical systems using the DOF formula 3n - 2j1 - 3j0 or by assigning three DOF per body and reducing via joints.

Apply the caption's degrees-of-freedom formula to determine the degrees of freedom in mechanisms, distinguishing revolute and prismatic joints from rigid joints, with multiple example systems.

Explore revolute joints between convergent links and learn to compute the dof using the (n-1) rule for multiple pinned bodies, with practical examples.

Practice analyzing revolute joints between convergent links to determine degrees of freedom across configurations from ground to multi-link assemblies; download the exercise and view the next lecture for the solution.

Apply the kinematic counting rule to 13 rigid bodies with convergent links, tally 19 revolute joints, and conclude the mechanism's DOF is 1.

simplify a triangle of three links pinned two by two to a single rigid body, then compute dof using 3 times the number of bodies minus 2 times revolute joints.

Identify over-constrained cases by recognizing when simple formulations miscount degrees of freedom for rigid-body systems, and use reconstruction to correctly determine the DOF of pinned frames.

Apply the DOF formula to detect over-constrained cases, using simplifications (double pins, convergent links, triangles) to show the system becomes a single rigid body with 3 DOF.

Learn how the pin on slot joint constrains motion to a single DOF, combining revolute and prismatic behavior, and replace such joints with an equivalent pin on slot formulation.

Analyze four pin on slot joint setups to determine their degrees of freedom, including straight and curved slot paths, ground involvement, and moving versus stationary pins.

Compute the degrees of freedom for pin on slot joint configurations with revolute and prismatic joints. Show that original and simplified models—ground-connected and slider-on-slot reductions—yield 1 dof.

Explore sliding and rolling contact between rigid bodies, defining normal and tangential constraints, and determine DOF with j1 and j2 constraints while identifying and resolving over-constrained cases.

Determine the degrees of freedom for four systems featuring sliding and rolling contacts, including a pin-on-slot joint, a cam-follower rolling contact, and a ground-connected gray link with three revolute joints.

Compute degrees of freedom for multiple mechanisms using sliding and rolling contacts, revolute and prismatic joints, and subsystem decomposition to determine final dof values.

Review the course concepts by analyzing the dof of rigid bodies in 2d and 3d, and explore how revolute, prismatic, rigid, and pin on slot joints influence dof.

Count the bodies and joints to determine the degrees of freedom for four systems, apply the standard formulation, and note rolling and sliding contacts, pins, sliders, cams, and revolute joints.

The lecture reviews course review solutions, tallying bodies, revolute and prismatic joints, and rolling or sliding contacts to compute DOF for four systems, yielding 4 DOF.