Degrees-of-Freedom and Mobility

This topic reviews:

Degrees-of-Freedom on Parts and Joints
Degrees-of-Freedom of a Kinematic-Chain and the Gruebler Equation
Mobility of kinematic-chains
Degrees-of-Freedom of Open and Closed kinematic-chains

tog_minus        Degrees-of-Freedom and Parts

Image from SOLIDWORKS

Image from SOLIDWORKS

DOF of a Part [General]

Parts free to move in space have six[6] degrees-of-freedom [D.O.F.].

Why six? Take the image to the left, you can see that a Part, in general 'Global Coordinates:

can 'translate' [move linearly] along the three axis directions and coordinates: X, Y, Z
can 'rotate' about the three linear axes, say with rotation coordinates: α, β, γ

 

KinematcsBasics-FreePart

DOF of a Free-Part on a Mechanism-Plane

In MechDesigner, Parts lie on a Mechanism-Plane.

A Free-Part has three[3] D.O.F.

The Part can translate along the: x and y directions.

The Part can rotate about axis that is normal to the Mechanism-Plane: θ

tog_minus        Degrees-of-Freedom removed by a single Joint

KinematcsBasics-PinJoint

KinematcsBasics-SlideJoint

Pin-Joint:

Requires a Line in the moving Part to remain co-linear with a Line in the Mechanism-Plane, and a plane perpendicular to this Line in the moving Part maintain contact with a similar perpendicular plane in the Mechanism-Plane.

A Pin-Joint removes two degrees-of-freedom - the Part has one degree-of-freedom remaining - it can rotate.

Slide-Joint

Requires that a Line in the moving Part remains co-linear with a line in the Mechanism-Plane, and a plane parallel to this Line in the moving Part maintain contact with a similar parallel plane in the Mechanism-Plane.

A Slide-Joint removes two degrees-of-freedom - the Part has one degree-of-freedom remaining - it can slide.

tog_minus        DOF and Gruebler Equation

DOF of a Kinematic-Chain?

Kinematic-chains may have many Parts and Joints. Fortunately, we can use the Gruebler Equation to calculate the number of degrees-of-freedom of most kinematic-chains.

Gruebler Equation

For a Planar kinematic-chain, the Gruebler Equation is:

F = 3*(N-1) – 2*J – H

MechDesigner automatically calculates the number of Degrees-of-Freedom in a Kinematic-Chain.

Note: if the Degrees-of-Freedom = Zero, then the kinematic-chain is a structure.

where:

F = degrees-of-freedom for the kinematic-chain

N = number of parts - include the Base-Part

J = number of one degree-of-freedom joints

H = number of 'higher-pair' joints - see Joints

It is important to notice that the Gruebler Equation does not need to know the lengths or shape of parts or the type of joint.

tog_minus        Mobility of a Kinematic-Chain

DOF-MD1

MOBILITY = Number of Motion DimensionsNumber of Degrees-of-Freedom

When the Mobility = 0  the Kinematic-Chain is a kinematically-defined chain.

The Target:

Number of Motion-Dimension FBs = Number of D.O.F. in the Kinematic-Chain.

The Kutzbach Criterion states that for a kinematic-chain to be useful, the number of 'drives' (motion-dimension) must equal the number of degrees-of-freedom.

tog_minus        Degrees-of-Freedom and Mobility of 'Open' and 'Closed' Kinematic-Chains

Degrees-of-Freedom and Mobility: without Motion Dimensions

minus        Open Kinematic Chain

KinChain-2a

The image shows an 'open' kinematic-chain.

There are four Parts (the Base-Part + 3 Added-Parts), with three Joints.

Gruebler Equation: F = 3* (4-1) – 2*3 – 0 = 3 . Three degrees-of-freedom.

Mobility =  Number of Motion Dimensions - Number of Degrees-of-Freedom

Mobility = 0 – 3 = -3

minus        Closed Kinematic Chain

KinChain-3a

The image shows a 'closed' kinematic-chain.

There are still four Parts, but now with four Joints.

Gruebler Equation: F = 3* (4-1) – 2*4 – 0 = 1. One Degree-of-Freedom

Mobility =  Number of Motion Dimensions - Number of Degrees-of-Freedom

Mobility = 0 – 1 = -1

Degrees-of-Freedom and Mobility: with Motion Dimensions

minus        Open Kinematic Chain

KinChain-finger2

Add three Motion-Dimension FBs. It had three Degrees-of-freedom.

Mobility =  Number of Motion Dimensions - Number of Degrees-of-Freedom

Mobility = 3 – 3 = 0

minus        Closed Kinematic Chain

KinChain-4bar1

Add on Motion Dimension to the kinematic-chain

It had one Degree-of-freedom, .

Mobility = Number of Motion Dimensions - Number of Degrees-of-Freedom

Mobility = 1 – 1 = 0

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