mech 335 theory of mechanisms - ariel to mechanisms •what is a mechanism? –a set of rigid...

MECH 335Theory 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


• 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



• Example: 4-bar mechanism


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)


• Mechanisms are depicted schematically using kinematically equivalent diagrams

– Also called skeleton sketches

– Used to simplify visualization and analysis



• Binary link: A link with two connection points



• Ternary link: link with three connection points, fixed relative to each other



• In-line ternary link: need to distinguish from two connected, aligned binary links



• Base revolute joint: revolute joint on the base (fixed) link



• Prismatic Slider: link slides tangent to a reference path (ground, in this case)



• Mechanisms are made up of Kinematic chains:

– More than one link, with links connected by pairing elements (joints)

– Chains can be open or closed



• Examples of kinematic chains


Open chain:Planar manipulator

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


Mobility Analysis

• EXAMPLE: Consider a single link in the plane:


y

x

θ1

x1

y1

3 DOF

Mobility Analysis

• Adding another free link adds another 3 DOF:


y

x

θ1

x1

y1

θ2

x2

y2

3+3=6 DOF

Mobility Analysis

• But joining the two links at a revolute joint reduces the total DOF by 2:


y

x

θ1

x1

y1

θ2

3+1=4 DOF

Mobility Analysis

• Planar pairing elements (cf. Text – Table 1.2)


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

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


Mobility Analysis

• Summing contributions gives:

• F > 0 : F is the number of required inputs

• F = 0: Statically determinate structure

• F < 0: Statically indeterminate structure


212)1(3 ffnF

Gruebler’s Equation

Mobility Analysis: Examples

• 4-Bar Mechanism:


n =

f1 =

f2 =

F =

212)1(3 ffnF

4

4

0

1


• 5-Bar Mechanism:


n =

f1 =

f2 =

F =

212)1(3 ffnF

5

5

0

2


• 3-Bar Structure:


n =

f1 =

f2 =

F =

212)1(3 ffnF

3

3

0

0


• 4-Bar Structure:


n =

f1 =

f2 =

F =

212)1(3 ffnF

4

6

0

-3


• Cam-Follower System:


n =

f1 =

f2 =

F =

212)1(3 ffnF

3

2

1

1


• Planar Positioning Stage:


n =

f1 =

f2 =

F =

212)1(3 ffnF

8

9

0

3


• Planar Manipulator:


n =

f1 =

f2 =

F =

212)1(3 ffnF

5

4

0

2


• Interesting Case: Mechanism with F = 1


n =

f1 =

f2 =

F =

212)1(3 ffnF

5

6

0

0 (!)


• Gruebler’s eqn may fail for special geometries!


n = 5

f1 = 6

f2 = 0

F = 0 (!)

212)1(3 ffnF

Mobility Analysis

• Gruebler’s Eqn can be extended to spatial mechanisms (up to 6 DOF / link)

• Compare to planar version


End of Lecture Pack 1


mech 335 theory of mechanisms - ariel to mechanisms •what is a mechanism? –a set of rigid...

Documents