Dyads [Assur Groups]

Dyads always have:

Two Parts
Three Joints

There is no limit to the number of Dyads in a single kinematic-chain.

The joints specify the relative motion of the Parts in the Dyad.

Dyads are also known as Assur Groups

A Dyad, or Assur group, is a kinematic chain with zero[0] degree of mobility, which added or subtracted from a kinematic-chain do not alter its original number of degrees-of-freedom.

They were first described by the Russian engineer Leonid Assur [es] (1878–1920) in 1914.

tog_minus        Dyads: Degrees-of-Freedom

The Gruebler Equation shows that a Dyad has Zero[0] Degrees-of-Mobility.

Gruebler Equation:

F = 3*(N-1) – 2*J1 – J2

F = 3 * (3-1) – 2*3 –0

F = 6 – 6

F= 0

N = 3

There are two Parts in a Dyad. You must join these two other Parts. However, when they are already kinematically-defined, you, and the Gruebler equation, take them to be one Part. Hence N= 2+1 =3

J1 = 3

There are three joints

J2 = 0

There are not any higher pairs [e.g' Cams and Follower Joint which has two degrees of freedom]

Therefore, a Dyad does not change the degrees-of-freedom of a kinematic-chain.

It also means you can add as many Dyads as you want to a kinematic-chain. The number of Degrees-of-Freedom does not change.

tog_minus        Dyads: The Basic Construction.

Dyads are Joints

We do not identify a Dyad by its number of Parts. This is because ALL dyads have two Parts.

We identify a Dyad with three letters, because ALL dyads have three Joints.

Each letter represent a Joint in the Dyad.
The letters are R, P and S.
Why R, P and S?
oR is for Revolute Joint. This is the same as a Pin-Joint.
oP is for Prismatic Joint. This is the same as a Slide-Joint.
oS is for Spherical Joint. This is the same as a Ball-Joint

Dyads and Parts

We can re-write the dyad with dashes[—] between each letter so that the dash[—] represents a Part.

For example, re-write the RRR Dyad as the R—R—R Dyad.

Then, it is easier to see that the middle letter represents the joint with which the two Parts are joined together.

The other two letters represent the joints with which the two Parts are joined to two other Parts in the kinematic-chain.

Dyads and Kinematically Defined Chains.

If the kinematic-chain is a kinematically-defined chain BEFORE you add a new dyad, it will also be kinematically-defined chain AFTER you add the dyad.

KDP1 > {<JOINT1> PART1 <JOINT2> PART2 <JOINT3>} < KDP2

            {------------------ DYAD---------------------------}

KDP1, KDP2 : Kinematically Defined Parts

PART1, PART2 : Parts in the Dyad

JOINT, JOINT, JOINT3 : Joints in the Dyad

AN EXAMPLE: RRR Dyad

RRRrocker1-1

Preparation: Two different Kinematically Defined Parts that already exist in the model

A Rocker1s-red (KDP1) joined to the Base-Part2s (KDP2).

RRR Dyad two Parts

Add Two Parts for the Dyad:

Part13s and Part24s

RRR Dyad - two joints-one more Pin-Joint needed

Add Three Joints:

Joint15s, Joint27s, Joint36s


Joint15s & Joint36s are the 'first and last' Joints.

These join the two Parts in the Dyad to the Kinematically Defined Parts (KDP1 & KDP2) that already exist in the model.

Simple RRR Dayd

Joint27s is the 'middle' Joint.

It joins the two Parts in the Dyad together.

tog_minus        Dyads: Suggested Steps to Add a new Dyad

RRRrocker1-1RRRrocker2-1RRRrocker3-1RRRrocker4-1

STEP 1: Start with a Kinematic-Chain with a minimum of two Kinematically Defined Parts: here a Rocker1s-red and Base-Part2s
STEP 2: Add a Two new Parts - Part13s and Part24s
STEP 3: Add 'Outside' Joint15s between Part14s and Kinematically Defined Part11s-red
STEP 4: Add 'Outside' Joint36s between Part23s and Kinematically Defined Part22s
STEP 5: Add 'Inside' Joint27s between Part13s and Part24s

tog_minus        Dyads: in the Kinematic Tree

A Dyad is identified with a three letter acronym. Each letter is a Joint.  The letters are R, P and S.

Why R, P and S?

The letter R is a Revolute Joint. It is exactly the same as a Pin-Joint.
The letter P is a Prismatic Joint. It is exactly the same as a Slide-Joint.
The letter S is a Spherical Joint. It is exactly the same as a Ball-Joint.

There are five possible Dyads in Planar Kinematics:

Dyad Acronym

Dyad Icon

Kinematic Name

MechDesigner Name

R R R

DYAD-RRR

Revolute – Revolute – Revolute

Pin – Pin – Pin

R R P

DYAD-RRP

Revolute – Revolute – Prismatic

Pin – Pin – Slide

R P R

DYAD-RPR

Revolute – Prismatic – Revolute

Pin – Slide – Pin

P R P

Dyad-PRP

Prismatic – Revolute – Prismatic

Slide – Pin – Slide

R P P

Dyad-RPP

Revolute – Prismatic – Prismatic

Pin – Slide – Slide

We have two 'Ram' type Dyads - useful for Air Cylinders

Ram - R

DYAD-RRR

Revolute – Revolute – Revolute

Pin +Motion-Point - Pin - Pin

Ram - P

DYAD-RRP

Revolute – Revolute – Prismatic

Pin +Motion-Point - Pin - Slide

We have two Dyads for Spacial Kinematics. More are possible but not frequently found in Packaging Machines.

Dyad Acronym

Dyad Icon

Kinematic Name

MechDesigner Name

S S R

DYAD-SSR

Spherical – Spherical – Revolute

Ball – Ball – Pin

S S P

DYAD-SSP

Spherical – Spherical – Prismatic

Ball – Ball – Pin

Tutorial and Reference Help Files for MechDesigner and MotionDesigner 13.2 + © Machine, Mechanism, Motion and Cam Design Software by PSMotion Ltd