Overview
This lecture covers the velocity and acceleration analysis of mechanisms, focusing on the instantaneous center (I Center) method, Kennedy’s theorem, relative velocity method, and acceleration analysis including Coriolis acceleration.
Velocity Analysis Overview
- Velocity analysis finds unknown velocities of mechanism links, given input velocity in one degree-of-freedom (DOF) mechanisms.
- Two main approaches: Instantaneous Center (I Center) method and Relative Velocity (Graphical) method.
Instantaneous Center (I Center) Method
- The I Center is a point where a link exhibits pure rotation at a given instant; all velocities are perpendicular to lines from the I Center.
- Number of I Centers in a mechanism with n links is n(n–1)/2.
- For a turning pair, the I Center is at the pin joint.
- For a pure rolling pair, the I Center is at the contact point.
- For a sliding pair on a flat surface, the I Center is at infinity on a line perpendicular to the sliding direction.
- For a slider on a curved surface, the I Center is at the center of curvature.
Kennedy’s Theorem
- For any three links with relative motion, the three associated I Centers lie on a straight line.
Angular Velocity Theorem
- To relate angular velocities of different links: ω₂ × distance (I₁₂ to I₂₅) = ω₅ × distance (I₁₅ to I₂₅).
- If I Centers lie on the same side, angular velocities have the same sense; otherwise, opposite.
Relative Velocity (Graphical) Method
- Velocity diagrams represent velocities as vectors; fixed points have zero velocity.
- For each pin joint, draw velocity vectors perpendicular to the link direction.
- The intersection of these vectors determines velocities of other points.
Acceleration Analysis
- Acceleration of a point on a rotating link has two components: radial (centripetal, toward center, rω²) and tangential (due to angular acceleration, rα, perpendicular to link).
- Resultant acceleration is the vector sum of radial and tangential components.
- The angle of total acceleration: tanθ = tangential/radial acceleration.
Coriolis Acceleration
- Occurs when a slider moves on a rotating link; magnitude: a_C = 2vω, where v is sliding velocity and ω is angular velocity of the link.
- Direction: take sliding velocity vector and rotate it by 90° in the sense of ω.
- Total acceleration for a point on the slider includes radial, tangential, and Coriolis components.
Key Terms & Definitions
- I Center (Instantaneous Center) — The point about which a link rotates instantaneously.
- Kennedy’s Theorem — Three I Centers of three links in relative motion are collinear.
- Radial Acceleration — Acceleration directed toward the center; rω².
- Tangential Acceleration — Acceleration perpendicular to the link due to angular acceleration; rα.
- Coriolis Acceleration — Additional acceleration for sliders on rotating links; 2vω.
Action Items / Next Steps
- Practice identifying I Centers and drawing velocity diagrams for mechanisms.
- Review and memorize formulas for velocity and acceleration components.
- Be cautious about calculating total acceleration, not just individual components, in exam questions.