Kinematic Analysis of Plane Mechanisms
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Kinematic analysis of plane mechanisms The objective of kinematic analysis is to determine the kinematic quantities such as displacements, velocities, and accelerations of the elements of a mechanism when the input motion is given. It establishes the relationship between the motions of various components of the linkage. Displacement Analysis When the kinematic dimensions and the configurations of the input link of a mechanism are prescribed, the configurations of all the other links are determined by displacement analysis. 1. Graphical Method 2. Analytical Method In a graphical method of displacement analysis, the mechanism is drawn to a convenient scale and the desired unknown quantities are determined through suitable geometrical constructions and calculations. 1. The configurations of a rigid body in plane motion are completely defined by the locations of any two points on it. 2. Two intersecting circles have two points of intersection and one has to be careful, when necessary, to choose the correct point for the purpose in hand. 3. The use of tracing paper, as an overlay, is very convenient and very often provides an unambiguous and quick solution. 4. The graphical method fails if no closed loop with four links exists in the mechanism. An analytical method of displacement analysis, is preferred whenever
Transcript of Kinematic Analysis of Plane Mechanisms
Kinematic analysis of plane mechanismsThe objective of kinematic analysis is to determine the kinematicquantities such as displacements, velocities, and accelerationsof the elements of a mechanism when the input motion is given. Itestablishes the relationship between the motions of variouscomponents of the linkage.
Displacement Analysis
When the kinematic dimensions and the configurations of the inputlink of a mechanism are prescribed, the configurations of all theother links are determined by displacement analysis.
1. Graphical Method
2. Analytical Method
In a graphical method of displacement analysis, the mechanism isdrawn to a convenient scale and the desired unknown quantitiesare determined through suitable geometrical constructions andcalculations.
1. The configurations of a rigid body in plane motion arecompletely defined by the locations of any two points on it.
2. Two intersecting circles have two points of intersection andone has to be careful, when necessary, to choose the correctpoint for the purpose in hand.
3. The use of tracing paper, as an overlay, is very convenientand very often provides an unambiguous and quick solution.
4. The graphical method fails if no closed loop with four linksexists in the mechanism.
An analytical method of displacement analysis, is preferredwhenever
1. high level of accuracy is required
2. a large number of configurations have to be solved
3. The graphical method fails.
Graphical method starts with position analysis by simply drawingthe linkage mechanism to scale. Then the velocity analysis isperformed which requires the angular position of the links to bedetermined beforehand. Similarly it is necessary to know angularvelocities of links for acceleration analysis. Thus, the sequencefor kinematic analysis of mechanisms is - position analysis, thenvelocity analysis and then acceleration analysis.
COORDINATE SYSTEMS
Global or Absolute system is often taken to be attached to MotherEarth.
Local coordinate systems are typically attached to a link at somepoint of interest.
The term inertial reference frame is used to denote a system whichitself has no acceleration.
All angles are measured according to the right-hand rule.That is, counterclockwise angles, angular velocities, andangular accelerations are positive in sign.
POSITION: The position of a point in the plane can bedefined by the use of a position vector as shown in previousfigure. The choice of reference axes is arbitrary and isselected to suit the observer.
The Polar form provides the magnitudes and the angle of thevector. The Cartesian form provides X and Y components ofthe vector.
DISPLACEMENT: Displacement of a point is the change in itsposition and can be defined as the straight line distancebetween the initial and final position of a point which hasmoved in the reference frame.
Translation: All points on the body have the same displacement.
Rotation: Different points in the body undergo differentdisplacements and thus there is a displacement differencebetween any two points chosen.
Complex Motion: The general case of complex motion is thesum of the translation and rotation components.
GRAPHICAL POSITION ANALYSIS OF LINKAGES
For any one-DOF linkage, such as a fourbar, only oneparameter is needed to completely define the positions ofall links.
The parameter usually chosen is the angle of the input link.This is shown as θ2 in figure. We want to find θ3 and θ4. Thelink lengths are known. In this example will consistentlynumber the ground link as 1 and the driver link as 2.
Velocity Analysis of Mechanisms
Important Concepts in Velocity Analysis
1. The absolute velocity of any point on a mechanism is thevelocity of that point with reference to ground.
2. Relative velocity describes how one point on a mechanism movesrelative to another point on the mechanism.
7. The velocity of any point on a link with respect to another point onthe same link is always perpendicular to the line joining these pointson the space diagram.
Rubbing velocity at a pin joint
It is defined as the algebraic sum between the angular velocitiesof the two links which are connected by pin joints, multiplied bythe radius of pin.
Hence the rubbing velocity when two links are moving in oppositedirections is given by
Rubbing velocity = r (ω1 + ω2)
Where r is the radius of the pin
But if the two connected links move in the same direction,
Rubbing velocity = r (ω1 - ω2)
