Dynamic Simulation:
Degrees of Freedom and Joints
Objective
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The objective of this module is to introduce the concepts associated
with degrees of freedom, joints, and kinematic constraints used in
multi-body dynamic simulation.
The material is presented for planar (2D) mechanisms typically
studied in an undergraduate engineering curriculum.
The concepts are universal and can be extended to three-dimensional
(3D) mechanisms.
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Section 4 – Dynamic Simulation
Degrees of Freedom (DOF)
Module 2 – DOF & Joints
Page 2
• The independent parameters
used to uniquely define the
position and orientation of a
part in space are called the
degrees-of-freedom (DOF).
• In three-dimensional space six
DOF are required; three
coordinates to define a location
and three orientation angles.
• Each DOF can have motion
associated with it.
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y
x
z
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Section 4 – Dynamic Simulation
Planar Systems
Module 2 – DOF & Joints
Page 3
A planar mechanism is one that
has all parts constrained to move
in a plane.
 Constraining a part to a plane
removes three degrees of
freedom.
 The part in the figure is part of a
planar mechanism. It can have no
motion in the global Z direction
and no rotations about the local x
& y axes.
 A part constrained to move in a
plane has three degrees-offreedom.

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Y
y
x
θ
X
Z
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Section 4 – Dynamic Simulation
Body Fixed Coordinate System
Module 2 – DOF & Joints
Page 4
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A body fixed coordinate system is
used to define the DOF’s of a part.
A body fixed coordinate system is
rigidly fixed to the body.
In a rigid body no point in the body
moves relative to the body fixed
coordinate system.
The origin of the coordinate
system is located at the center-ofgravity.
The axes are oriented along the
principle axes of inertia.
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Principal Axes of Inertia
x
y
θ
C.G.
X
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Section 4 – Dynamic Simulation
Position Constraints
Module 2 – DOF & Joints
Page 5
Position constraints impose
conditions on the location and
orientation of the part.
Y
 In the figure, a constraint is
given to each DOF.
 The constraints can be written
Ycg=15
in equation form as:
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y
x
θ  0.5 rad
X cg  20  0
Ycg  15  0
X
  0 .5  0

In this example, the constraint
equations are not a function of
time and the part is fixed in
space (grounded).
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Xcg=20
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Section 4 – Dynamic Simulation
Kinematic (Joint) Constraints
Module 2 – DOF & Joints
Page 6
Kinematic constraints impose conditions on
the relative motion between a pair of bodies.
Kinematic constraints
remove DOF’s from the
assembly.
Unconstrained
Link with 3 DOF
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Constrained Link
with 1 DOF
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In this example, the addition
of constraints at the end of
the link changes it from an
object that is free to move in
two directions and rotate to
an object that can only
rotate.
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Section 4 – Dynamic Simulation
Mobility
Module 2 – DOF & Joints
Page 7
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Mobility is the number of
unconstrained DOF’s in a mechanism.
Each planar body starts out with
three DOF’s.
Constraints eliminate some of these
DOF’s until only a small subset are
left.
The number of DOF’s in the subset is
the mobility of the mechanism.
The number of actuators required to
control the mechanism is equal to the
mobility.
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coupler
follower
crank
This four-bar mechanism has a
mobility of one. The position of
the coupler and follower can be
computed if the angular position
of the crank is specified.
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Section 4 – Dynamic Simulation
Common Joints
Module 2 – DOF & Joints
Page 8
A revolute and prismatic joint are commonly used joints in planar
mechanisms.
A revolute and
prismatic joint
remove two DOF
from a pair of parts in
a planar mechanism.
Revolute Joint
Allows two parts to
rotate relative to each
other about a shared
axis.
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Prismatic Joint
Allows two parts to
translate relative to
each other along a
shared axis.
Revolute and
prismatic joints are
also common in 3D
mechanisms.
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Section 4 – Dynamic Simulation
Gruebler’s Equation
Module 2 – DOF & Joints
Page 9
Gruebler’s equation can be used
to determine the mobility of planar
mechanisms.
L=2
J=1
G=1
DOF = 1
Link 1
3 DOF
Gruebler’s Equation
DOF
L
J
G
= mobility
= number of links
= number of revolute joints or
prismatic joints
= number of grounded links
1 DOF
Link 2
3 DOF
DOF = 3*L – 2* J – 3 *G
= 3 (L-1) – 2 * J
The revolute joint removes 2 DOF and
the grounded link removes 3 DOF.
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Section 4 – Dynamic Simulation
Mobility of Vise Grip Pliers
Module 2 – DOF & Joints
Page 10
This example applies Gruebler’s equation to
the determine the mobility of a vise grip plier.
1
4
5
1
3
4
3
2
2
Each revolute joint
removes two DOF.
The screw joint
removes two DOF.
L=5
J = 4 (revolute)
J = 1 (screw)
G = 1 (your hand)
DOF = 3*5 - 2*5 - 1*3 = 2
The mobility of the plier is two. Link 3 can be moved relative link1
when you squeeze your hand and the jaw opening is controlled by
rotating link 5.
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Section 4 – Dynamic Simulation
Welded Joints
Module 2 – DOF & Joints
Page 11
It is common for parts in a
mechanism to be designed to
Upper Retainer
move together.
 In the figure, the bottom retainer,
valve guide, and valve guide seal
do not move relative to each
other.
 Similarly, the valve stem cup,
valve cap, top retainer, keepers,
and valve do not move relative to
each other.
 A joint that allows no relative
motion is called a welded joint.
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Valve Stem Cup
Valve Cap
Keepers
Valve
Valve Spring
Valve Guide Seal
Valve Guide
Bottom Retainer
Engine Block
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Section 4 – Dynamic Simulation
Ground Joints
Module 2 – DOF & Joints
Page 12
One or more parts in a mechanism
must be anchored to something
rigid that holds it in place.
Upper Retainer
 The anchored parts have all
degrees of freedom removed.
 A joint that removes all DOF’s by
setting the coordinates and
orientation angles equal to
constant values are called Ground
Joints.
 In the figure, all moveable parts
will move relative to the engine
block that is fixed by a Ground
Joint.
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Valve Stem Cup
Valve Cap
Keeper x 2
Valve
Valve Spring
Valve Guide Seal
Valve Guide
Bottom Retainer
Engine Block
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Section 4 – Dynamic Simulation
Joint Types in Dynamic Simulation
Module 2 – DOF & Joints
Page 13
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The Dynamic Simulation browser
contains information about the
types of joints used in a
simulation.
Grounded parts are fixed in space
and cannot move.
Mobile groups are parts or groups
of parts that can move relative to
each other
Welded groups contain a list of
parts that are joined together.
Standard joints include common
joints such as revolute, prismatic,
etc.
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Section 4 – Dynamic Simulation
Automatic Creation of Joints
Module 2 – DOF & Joints
Page 14
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Dynamic Simulation operates on
assemblies created in the Assembly
environment.
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The parts in an assembly have
constraints that keeps them in the
correct position relative to each
other.
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The assembly constraints are
interpreted and converted to
kinematic constraints in the Dynamic
Simulation environment.
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The Insert constraint would be
used in the Assembly
environment to create this joint.
It would automatically be
converted to a revolute joint in
the Dynamic Simulation
environment.
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Section 4 – Dynamic Simulation
Automatic Conversion Control
Module 2 – DOF & Joints
Page 15
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The Dynamic Simulation
environment allows joints
automatically created from the
assembly constraints to be turned
off and on.
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When the automatic conversion is
turned off, all standard joints are
deleted (including any that were
manually created).
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Section 4 – Dynamic Simulation
Module Summary
Module 2 – DOF & Joints
Page 16
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This module presented the concepts associated with degrees-offreedom, mobility of a mechanism, and joints used in multi-body
dynamics.
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It showed examples of how the concepts are incorporated into
Dynamic Simulation.
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These concepts will be used in subsequent modules that deal with
multi-body dynamics theory and its practical implementation in
Dynamic Simulation.
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