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Kinematic Inversion
Every mechanism has moving members which move relative to each other about the joints connecting them. These relative motions result in the trajectories of the points on members of the mechanism. In any mechanism one link or member is fixed and acts as the frame. The trajectories and motion characteristics of mechanism depend on the choice of the reference frame link.
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Inversions of a mechanism are the different configurations of the mechanism with change of the fixed reference link called frame. For different inversions of a mechanism although the motion characteristics are entirely different but the relative angular displacements of the members remain unchanged irrespective of the link chosen as frame.
The information obtained from one inversion of the linkage can be used to study other inversions of that linkage. Inversion technique is used extensively for analysis and synthesis of mechanisms.
Determining the Inversions of a Mechanism
Before going into details of obtaining inversions of a mechanism I would like to make it very clear that Inverse Kinematics is different from Kinematic Inversion. Read more about Inverse Kinematics.
Every mechanism is formed of a kinematic chain. When one of the links in the kinematic chain is fixed it becomes a mechanism. To determine the inversions of a mechanism consider the kinematic chain forming the mechanism and obtain the desired inversions by fixing any one of the members as the frame link.
Inversions of a Four-Bar Mechanism
A typical four bar mechanism, as the name denotes, is formed of a kinematic chain of four members connected by revolute joints. This mechanism can have four possible configurations with a different link fixed as frame each time.
Configuration 1
Link 1 is taken as the base link or frame. In this configuration the shortest link is jointed to the base link and this joint can fully rotate and hence called as crank. The other link jointed to the base link oscillates and called as a rocker. This configuration of the four-bar kinematic chain is called as Crank-Rocker mechanism.
Configuration 2
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Link 2 is fixed as the base link. In this configuration shortest link is the base and both joints to the base can rotate completely. It is thus called as Double-Crank or a Drag-Link.
Configuration 3
Link 3 is fixed as the base link. It can be observed that this configuration is same as the Crank-Rocker mechanism.
Configuration 4
Link 4 is fixed as the base link. In this configuration shortest link is the coupler and both the links connected to the base link cannot rotate fully, both oscillate. In this configuration the four-bar kinematic chain is called as Double-Rocker mechanism.
This post is part of the series: Kinematics – Design of Mechanisms
Parallel Kinematic Mechanism
Analysis of Mechanisms
Analysis of mechanisms is the study of motion of different members constituting a mechanism and the mechanism as a whole entity while it is being operated or run. This study of motion involves linear as well as angular position, velocity and acceleration of different points on members of mechanisms. Analysis and synthesis are two different aspects of mechanisms and machine design.
Earlier design engineers used drafting equipments to graphically analyse the mechanisms. The continuous contribution by design engineers for years has lead to development of many methods and techniques for analysis of mechanisms. Recently, the development of computer techniques have offered a number of viable and attractive solutions.
Methods and Techniques of Mechanism Analysis
Mechanism analysis methods are basically of two types, graphical and analytical. Each method has many techniques for analysis of mechanisms, where each technique is suitable for a particular category of mechanisms. With the development of sophisticated computer programs design engineers prefer to concentrate their effort on analytical approach. But still the graphical approach to mechanism analysis has not lost its utility, specially in some cases where graphical technique gives the most efficient solution and physical insight to visualize working of the mechanism.
Graphical Method of Mechanism Analysis
Graphical method starts with position analysis by simply drawing the linkage mechanism to scale. Then the velocity analysis is performed which requires the angular position of the links to be determined beforehand. Similarly it is necessary to know angular velocities of links for acceleration analysis. Thus, the sequence for kinematic analysis of mechanisms is – position analysis, then velocity analysis and then acceleration analysis.
Different Techniques of Graphical Analysis
- Velocity and Acceleration Polygon: Velocity and acceleration are vectors and thus their sum or difference will follow vector polygon laws. If velocity of one point on a link is known then the velocity of other points can be found using the vector polygons. This technique is based on vector polygon laws.
- Velocity and Acceleration Image: This technique is used for graphical analysis of mechanisms with more than one loop. If the velocity and acceleration of two points on a link are known then the velocity and acceleration of third point on that link can be determined using velocity and acceleration image.
- Inversion Technique: When it is not possible to analyse the linkage directly using vector polygon approach then Inversion Technique is used. In this technique the driven and driver cranks are interchanged to perform graphical analysis.
- Relative Velocity and Acceleration: This technique is used to analyse mechanisms with large number of members. In this technique the relationships between relative linear/angular velocities and acceleration of points/members are used to analyse the mechanisms.
- Instant Center of Velocity: For a rigid body moving in a plane, at every instant there exists a point that is instantaneously at rest. This instant center of velocity for the given rigid body is found using standard methods. It is useful for finding input-output velocity relationships of complex mechanisms.
Analytical Method of Mechanism Analysis
Analytical method is used when repetitive and extensive analysis of mechanisms is required, as the analytical equations and solutions obtained can be conveniently programmed on a computer. In this approach vector position, velocity and acceleration equations are formulated based on the fact that there are two different paths connecting the points on a vector loop. The equations thus obtained are simplified and programmed using computers. Desirable solutions are obtained by varying the parameters.