MECH 335
Theory of Mechanisms
Spring 2010
Instructor: Dr. Ron Podhorodeski
Introduction to Mechanisms
• What is a mechanism?
– A set of rigid bodies, connected so as to move
with definite relative motion
• Mechanism sub-types:
– Planar: All bodies move in parallel planes (and all
forces act in parallel planes)
– Spherical: All bodies move on a sphere, and rotate
about normals to the sphere
– Spatial: Up to 3 translational and 3 rotational
Degrees Of Freedom (DOF) are possible
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Introduction to Mechanisms
• Knowledge of the kinematic & dynamic
performance of mechanisms is critical to the
design of mechanical components
• Tasks of mechanisms can be classified in three
groups:
– Function generation
– Path generation
– Motion generation
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Introduction to Mechanisms
• Example: 4-bar mechanism
Path Generation:
The point P follows a
defined path (red
curve)
P
Motion Generation:
The motion of link 3
relative to the input link
(link 2) may be of
interest (e.g. toggle
clamp)
3
2
Function Generation:
θ4 = f(θ2)
4
θ2
1
θ4
1
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Introduction to Mechanisms
• Mechanisms are depicted schematically using
kinematically equivalent diagrams
– Also called skeleton sketches
– Used to simplify visualization and analysis
MECH 335 Lecture Notes
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Introduction to Mechanisms
• Binary link: A link with two connection points
MECH 335 Lecture Notes
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Introduction to Mechanisms
• Ternary link: link with three connection points,
fixed relative to each other
MECH 335 Lecture Notes
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Introduction to Mechanisms
• In-line ternary link: need to distinguish from
two connected, aligned binary links
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Introduction to Mechanisms
• Base revolute joint: revolute joint on the base
(fixed) link
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Introduction to Mechanisms
• Prismatic Slider: link slides tangent to a
reference path (ground, in this case)
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Introduction to Mechanisms
• Mechanisms are made up of Kinematic chains:
– More than one link, with links connected by
pairing elements (joints)
– Chains can be open or closed
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Introduction to Mechanisms
• Examples of kinematic chains
Open chain:
Planar manipulator
MECH 335 Lecture Notes
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Closed chain:
4-Bar mechanism
Mobility Analysis
• Degrees of Freedom (DOF):
– Number of coordinate values required to
completely describe the position of all links in a
mechanism
• Total DOF ≡ Mobility:
– Number of inputs required to determine the
position of all links of a mechanism
– Pairing elements (e.g. joints) in a chain remove
DOF (i.e. reduce mobility) by constraining the
position of two or more links at once
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Mobility Analysis
• EXAMPLE: Consider a single link in the plane:
θ1
x1
y
3 DOF
y1
x
MECH 335 Lecture Notes
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Mobility Analysis
• Adding another free link adds another 3 DOF:
θ2
x2
θ1
x1
y2
3+3=6 DOF
y
y1
x
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Mobility Analysis
• But joining the two links at a revolute joint
reduces the total DOF by 2:
θ2
θ1
x1
y
3+1=4 DOF
y1
x
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Mobility Analysis
• Planar pairing elements (cf. Text – Table 1.2)
Pairing element
Schematic View
Pin (revolute) joint
DOF
1
Prismatic Slider
1
Rolling Contact (no slip)
1
Rolling Contact (with slip)
2
Gear Teeth – 2 (roll+slip)
Pitch Circles – 1 (roll)
Gear Contact
Spring
3
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Mobility Analysis
• Consider DOF contributions in a planar chain
of n links:
– DOF of free links  3n
– Fixed base link  -3 (base link’s DOF are removed)
– Each 1 DOF joint  -2 (cf. revolute joint example)
– Each 2 DOF joint  -1
• Let the number of 1 DOF joints = f1
• Let the number of 2 DOF joints = f2
MECH 335 Lecture Notes
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Mobility Analysis
• Summing contributions gives:
F
3(n 1) 2 f1
f2
Gruebler’s Equation
• F > 0 : F is the number of required inputs
• F = 0: Statically determinate structure
• F < 0: Statically indeterminate structure
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
Mobility Analysis: Examples
• 4-Bar Mechanism:
n= 4
f1 = 4
f2 = 0
F= 1
F
3(n 1) 2 f1
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
f2
Mobility Analysis: Examples
• 5-Bar Mechanism:
n= 5
f1 = 5
f2 = 0
F= 2
F
3(n 1) 2 f1
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
f2
Mobility Analysis: Examples
• 3-Bar Structure:
n= 3
f1 = 3
f2 = 0
F= 0
F
3(n 1) 2 f1
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
f2
Mobility Analysis: Examples
• 4-Bar Structure:
n= 4
f1 = 6
f2 = 0
F = -3
F
3(n 1) 2 f1
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
f2
Mobility Analysis: Examples
• Cam-Follower System:
n= 3
f1 = 2
f2 = 1
F= 1
F
3(n 1) 2 f1
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
f2
Mobility Analysis: Examples
• Planar Positioning Stage:
n= 8
f1 = 9
f2 = 0
F= 3
F
3(n 1) 2 f1
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
f2
Mobility Analysis: Examples
• Planar Manipulator:
n= 5
f1 = 4
f2 = 0
F= 2
F
3(n 1) 2 f1
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
f2
Mobility Analysis: Examples
• Interesting Case: Mechanism with F = 1
n= 5
f1 = 6
f2 = 0
F = 0 (!)
F
3(n 1) 2 f1
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
f2
Mobility Analysis: Examples
• Gruebler’s eqn may fail for special geometries!
n= 5
f1 = 6
f2 = 0
F = 0 (!)
F
3(n 1) 2 f1
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
f2
Mobility Analysis
• Gruebler’s Eqn can be extended to spatial
mechanisms (up to 6 DOF / link)
• Compare to planar version
MECH 335 Lecture Notes
© R.Podhorodeski, 2009
End of Lecture Pack 1
MECH 335 Lecture Notes
© R.Podhorodeski, 2009