Kinematic Analysis of Mechanisms Using Graphical Methods 1

Overview

• Focus on instantaneous center method, relative velocity method, and relative acceleration method.
• Today's topic: Relative acceleration method for four bar mechanism.

Key Steps in Analysis of Four Bar Mechanism

1. Construct the configuration diagram with known dimensions of links.
2. Draw the velocity polygon to determine unknown velocities.
3. Draw the acceleration polygon to find unknown accelerations of various links.

Assumptions

• Dimensions of all links are known.
• Angular velocity and acceleration of one link are given.
• Example: Four bar chain (links ABCD) with link AD as fixed.

Concepts in Circular Motion

• Linear Tangential Velocity:
  • Magnitude = r × ω
  • Direction is perpendicular to the radius of rotation.
• Components of Acceleration:
  • Centripetal Component:
    • Magnitude = v²/r
    • Direction towards the center of rotation.
  • Tangential Component:
    • Magnitude = r × α
    • Direction perpendicular to the radius of rotation.
• Total Acceleration: Vector sum of centripetal and tangential components.

Velocity Polygon Construction

Example: Link AB

• Velocity of Point B relative to A (VBA):
  • Direction is perpendicular to link AB, represented as VBA.
• Centripetal Component:
  • Magnitude = v²/r (directed towards A).
• Tangential Component:
  • Magnitude = r × α (perpendicular to AB).

Constructing for Other Links

• For links like BC and CD, determine velocities relative to adjacent links similarly.
• Plot directions based on physical rotations and known points in the configuration.

Determining Angular Velocities

1. Link BC:
   • Angular velocity (ωCB) = VCB / length of link BC.
   • Direction determined from velocity vector direction (downwards = clockwise).
2. Link CD:
   • Angular velocity (ωCD) = VCD / length of link CD.
   • Direction determined similarly (counterclockwise).

Acceleration Polygon Construction

• Each link has centripetal and tangential components.

For Link AB

• Plot Centripetal Acceleration (FCBA): Directed from B to A.
• Plot Tangential Acceleration (FTBA): Perpendicular to FCBA.
• Resultant acceleration (FBA) is represented by vector AB.

For Link BC and CD

• Follow similar steps to plot centripetal and tangential components for links BC and CD.
• Use intersection points to determine unknown magnitudes of accelerations.

Summary of Results

• From the acceleration polygon, determine:
  • Magnitude and direction of tangential accelerations for all links.
  • Use relationships to find angular accelerations:
    • Link BC: αCB = FTCB / length of BC (clockwise).
    • Link CD: αCD = FTCD / length of CD (counterclockwise).

Conclusion

• The method illustrates how to derive angular velocities and accelerations using graphical methods in kinematic analysis, specifically for a four bar mechanism.