mech 335 theory of mechanisms - ariel to mechanisms •what is a mechanism? –a set of rigid...
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MECH 335Theory of Mechanisms
Spring 2010
Instructor: Dr. Ron Podhorodeski
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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
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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
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Introduction to Mechanisms
• Example: 4-bar mechanism
MECH 335 Lecture Notes © R.Podhorodeski, 2009
1 1
2 4
3
P
θ4θ2
Path Generation:The point P follows a defined path (red curve)
Function Generation:θ4 = f(θ2)
Motion Generation:The motion of link 3 relative to the input link (link 2) may be of interest (e.g. toggle clamp)
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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 © R.Podhorodeski, 2009
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Introduction to Mechanisms
• Binary link: A link with two connection points
MECH 335 Lecture Notes © R.Podhorodeski, 2009
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Introduction to Mechanisms
• Ternary link: link with three connection points, fixed relative to each other
MECH 335 Lecture Notes © R.Podhorodeski, 2009
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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
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Introduction to Mechanisms
• Base revolute joint: revolute joint on the base (fixed) link
MECH 335 Lecture Notes © R.Podhorodeski, 2009
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Introduction to Mechanisms
• Prismatic Slider: link slides tangent to a reference path (ground, in this case)
MECH 335 Lecture Notes © R.Podhorodeski, 2009
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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
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Introduction to Mechanisms
• Examples of kinematic chains
MECH 335 Lecture Notes © R.Podhorodeski, 2009
Open chain:Planar manipulator
Closed chain:4-Bar mechanism
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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
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Mobility Analysis
• EXAMPLE: Consider a single link in the plane:
MECH 335 Lecture Notes © R.Podhorodeski, 2009
y
x
θ1
x1
y1
3 DOF
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Mobility Analysis
• Adding another free link adds another 3 DOF:
MECH 335 Lecture Notes © R.Podhorodeski, 2009
y
x
θ1
x1
y1
θ2
x2
y2
3+3=6 DOF
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Mobility Analysis
• But joining the two links at a revolute joint reduces the total DOF by 2:
MECH 335 Lecture Notes © R.Podhorodeski, 2009
y
x
θ1
x1
y1
θ2
3+1=4 DOF
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Mobility Analysis
• Planar pairing elements (cf. Text – Table 1.2)
MECH 335 Lecture Notes © R.Podhorodeski, 2009
Pairing element Schematic View DOF
Pin (revolute) joint
Prismatic Slider
Rolling Contact (no slip)
Rolling Contact (with slip)
Gear Contact
Spring
1
1
1
2
Gear Teeth – 2 (roll+slip)
Pitch Circles – 1 (roll)
3
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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 © R.Podhorodeski, 2009
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Mobility Analysis
• Summing contributions gives:
• 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
212)1(3 ffnF
Gruebler’s Equation
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Mobility Analysis: Examples
• 4-Bar Mechanism:
MECH 335 Lecture Notes © R.Podhorodeski, 2009
n =
f1 =
f2 =
F =
212)1(3 ffnF
4
4
0
1
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Mobility Analysis: Examples
• 5-Bar Mechanism:
MECH 335 Lecture Notes © R.Podhorodeski, 2009
n =
f1 =
f2 =
F =
212)1(3 ffnF
5
5
0
2
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Mobility Analysis: Examples
• 3-Bar Structure:
MECH 335 Lecture Notes © R.Podhorodeski, 2009
n =
f1 =
f2 =
F =
212)1(3 ffnF
3
3
0
0
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Mobility Analysis: Examples
• 4-Bar Structure:
MECH 335 Lecture Notes © R.Podhorodeski, 2009
n =
f1 =
f2 =
F =
212)1(3 ffnF
4
6
0
-3
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Mobility Analysis: Examples
• Cam-Follower System:
MECH 335 Lecture Notes © R.Podhorodeski, 2009
n =
f1 =
f2 =
F =
212)1(3 ffnF
3
2
1
1
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Mobility Analysis: Examples
• Planar Positioning Stage:
MECH 335 Lecture Notes © R.Podhorodeski, 2009
n =
f1 =
f2 =
F =
212)1(3 ffnF
8
9
0
3
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Mobility Analysis: Examples
• Planar Manipulator:
MECH 335 Lecture Notes © R.Podhorodeski, 2009
n =
f1 =
f2 =
F =
212)1(3 ffnF
5
4
0
2
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Mobility Analysis: Examples
• Interesting Case: Mechanism with F = 1
MECH 335 Lecture Notes © R.Podhorodeski, 2009
n =
f1 =
f2 =
F =
212)1(3 ffnF
5
6
0
0 (!)
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Mobility Analysis: Examples
• Gruebler’s eqn may fail for special geometries!
MECH 335 Lecture Notes © R.Podhorodeski, 2009
n = 5
f1 = 6
f2 = 0
F = 0 (!)
212)1(3 ffnF
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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
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End of Lecture Pack 1
MECH 335 Lecture Notes © R.Podhorodeski, 2009