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These notes from ME 3610 provide an overview of kinematics, its relationship with mechanics, and the study of motion in mechanical systems. Topics include the definition of kinematics, the history of its relationship with mechanics, key definitions, motion of rigid bodies, kinematic pairs, and joint classification. The notes also discuss the importance of kinematics and its future applications.
Typology: Lecture notes
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Part II: Kinematics, the fundamentals
This section will provide an overview of the fundamental concepts in kinematics. This will include the following topics:
1: What is Kinematics?
“Kinematics is the study of _________________________________________________
Kinetics is the study of _________ on systems in motion:
Underlying Assumptions in Kinematics:
Rigid bodies
Ignore forces (applied, friction, etc)
Bodies are connected by Joints – Kinematic pairs
The nature of connection b/n kinematic pairs is maintained
This is kinematics,
and this is kinematics,
This is NOT kinematics,
2. Kinematics within Mechanics Kinematics and the theory behind machines have a long history. Kinematics has evolved to become a unique component within Mechanics as demonstrated in this figure 3. A few key definitions:
Kinematics
Dynamics
Kinematic Chain
Mechanism
Machine
Crank
Rocker
Coupler
Degrees of freedom
Constraint
Mobility
Mechanics
Statics Dynamics
Kinematics Kinetics
Mechanics of materials
4. Motion (of a rigid body): displacement of a rigid body w.r.t. a fixed frame or reference frame (for dynamics, needs to be an IRF).
Translation:
Rotation:
Planar:
Spatial:
Kinematic Pairs: Two members (links) are jointed through a connection (joint) that defines the relative motion b/n the two.
Links Joints
More about joint Classically classified into a couple classes: Higher pair (point contact) and Lower pair (line contact).
Various types of joints:
Revolute:
Prismatic (slider)
Cam or gear
Rolling contact
Spring
Others?
5. Transmission of Motion: The motion of a mechanism is defined by its constraints (kinematic). The following example shows one of the most general cases of motion between two bodies and demonstrates some key elements in understanding the behavior of motion. Consider two general kinematic bodies (rigid bodies, known geometric properties) in contact at point P. Each body rotates about a fixed point, O2 and O3.
Notes:
𝜔 3 𝜔 2 =^
𝑂 2 𝐾 𝑂 3 𝐾
Requirement for constant velocity:
Requirement for no sliding:
t
6. Mobility Analysis: Mobility is defined as the number of dof. Mobility is calculated as the total number of possible degrees of freedom, minus the number of constraints. The following diagrams will demonstrate the process:
Item Diagram DOF One body
Two Bodies
Two bodies connected by a revolute
Ground (it is a body)
Writing these rules as equations yields:
Which is known as Grubler’s or the Kutzbach equation.
Note: when M = 0 Structure, statically determinant M < 0 Indeterminant structure M > 0 Mechanism with M dof
Number Synthesis: This is the determination of the number and the order of links and joints necessary to produce motion of a particular degree of freedom
Example of a 1 DOF planar mechanism with Revolute Joints and up to 8 links
Total Links Binary Ternary Quaternary Pentagonal Hexagonal 4 4 0 0 0 0 6 4 2 0 0 0 6 5 0 1 0 0 8 7 0 0 0 1 8 4 4 0 0 0 8 5 2 1 0 0 8 6 0 2 0 0 8 6 1 0 1 0
Statically indeterminate DOF = -
Statically determinate DOF = 0
Mechanism DOF = 1
Linkage Transformation This technique gives the designer a toolkit to basic linkages of a particular DOF. In this technique, the designer is not constrained to using only full, or rotating joints. So, the designer can transform basic linages to a wider variety of mechanisms with greater usefulness.
Several techniques that can be applied are:
Examples
In this example, a fourbar crank-rocker linkage had been transformed into a fourbar slider-crank by the application of rule #1. It is still a fourbar linkage. Link 4 has become a sliding block. The Gruebler’s equation is unchanged at one degree of freedom because the slider block provides a full joint against link1 (ground), as did the pin joint it replaces. Note that this transformation from a rocking output link to a slider output link is equivalent to increasing the length (radius) of rocker link 4 until its arc motion at the joint between links 3 and 4 becomes a straight line. Thus the slider block is equivalent to an infinitely long rocker link 4, which is pivoted at infinity along a line perpendicular to the slider axis as shown in the figure above.
Grashof crank-rocker
Slider block
Effective Link 4
Effective rocker pivot is at infinity
Grashof slider-crank
Transforming a fourbar crank- rocker to a slider- crank
Intermittent Motion This is a sequence of motions of dwells. A dwell is a period in which the output link remains stationary while the input link continues to move. There are many applications in machinery which require intermittent motion. Some examples of mechanism that exhibits such motions are:
Inversion This is created by grounding a different link in the kinematic chain. There are many inversions of a given linkage as has links and the motions resulting from each inversion can be quite different. However, some inversions of a linkage may yield motions similar to other inversions of the same linkages. In these cases, only some of the inversions may have distinctly different motions. Those inversion which have distinctly different motions are denoted as Distinct Inversions.
A) Inversion # Slider block translates
B) Inversion #2 Slider block has complex motion
C) Inversion #3 Slider block rotates
D) Inversion #4 Slider block is stationary
