Kinematics and Mechanism Fundamentals for Engineers

Posted on Aug 26, 2025 in Technology

Kinematics and Mechanism Fundamentals

This document outlines fundamental concepts in kinematics, focusing on methods for calculating point velocities in mechanisms, key definitions of machine elements, and classifications of kinematic pairs.

Velocity Analysis in Mechanisms

Relative Velocity Method

The velocity of a point (V) can be determined relative to another point (A) on the same rigid body using the formula: V = V_A + V_B/A, where V_A is the absolute velocity of point A, and V_B/A is the velocity of point B relative to point A.

The relative velocity of a point on a rigid body with respect to any other point on that body has a magnitude equal to the product of the angular velocity and the distance between the points. Its direction is perpendicular to the line joining the two points.

Instantaneous Center of Rotation (CIR)

The Instantaneous Center of Rotation (CIR) is a point on a rigid body (or its extension) that has zero velocity at a particular instant. It is located at the intersection of lines drawn perpendicular to the velocity vectors of any two points on the rigid body.

Determining the CIR

The CIR can be determined by:

  • Direct inspection (for simple cases).
  • Using the Law of Three Centers (Aronhold-Kennedy Theorem).
CIR for Specific Motions
  • If a member undergoes rectilinear motion, its CIR is located at infinity, along any line perpendicular to the direction of movement.
  • To determine the CIR for a mechanism at a particular time:
    1. Calculate the total number of Instantaneous Centers of Rotation (CIRs) that exist in the mechanism.
    2. List all possible combinations of CIRs.
    3. Locate the CIRs on the mechanism diagram. Immediate CIRs are found at physical joints; others are determined using Kennedy’s Law.

Aronhold-Kennedy Theorem (Law of Three Centers)

When three rigid bodies are in relative motion, their three instantaneous centers of rotation (CIRs) are collinear.

Grashof’s Law for Four-Bar Mechanisms

Grashof’s Law predicts the type of motion (crank or rocker) for a four-bar linkage based on the lengths of its links. Let L_s be the shortest link, L_l the longest link, and L_p1, L_p2 be the two intermediate links.

  • If L_s + L_l ≤ L_p1 + L_p2 (Grashof Condition):
    • If the shortest link (L_s) is fixed: The mechanism is a Double Crank (both adjacent links can make full rotations).
    • If a link adjacent to the shortest (L_p1 or L_p2) is fixed: The mechanism is a Crank-Rocker (one link makes full rotations, the other oscillates).
    • If the longest link (L_l) is fixed: The mechanism is a Double Rocker (neither link can make a full rotation).
  • If L_s + L_l > L_p1 + L_p2 (Non-Grashof Condition):
    • All inversions of the mechanism will result in a Double Rocker (no link can make a full rotation).

Definitions of Machine Elements and Mechanisms

  • Machine: A system formed by one or more mobile parts, mechanical assemblies, and possibly other components, designed to perform a particular task, typically involving the realization of work or processing power.
  • Apparatus: Similar to a machine, but its basic principle of operation is not purely mechanical; it can be electrical, electronic, computational, or other.
  • Mechanism: An assembly of mechanical elements that perform functions related to guidance and transmission of movements and forces within a machine.
  • Member: An element of a machine or mechanism that may be liquid or solid, rigid or nearly rigid. It refers to parts that are rigidly connected together.
  • Piece: Each of the elementary parts that constitute a member.
  • Kinematic Chain: An assembly or sub-assembly of members within a mechanism linked to each other, such as the piston-crank-connecting rod system, or the transmission chain of a bicycle.
  • Structure: An assembly of mechanical elements forming a machine without relative movement. Primarily, it supports the forces acting on the machine.

Classification of Mechanism Members and Movement Types

  • Members of a Mechanism: Can be classified as binary, ternary, quaternary, etc., based on the number of joints they connect.
  • Types of Movement:
    • Continuous
    • Intermittent
    • Reciprocating (alternate)
    This includes pure rotation/reciprocating rotation, and pure translation/reciprocating translation.

Kinematic Pairs and Degrees of Freedom

Kinematic Pair Definition

A Kinematic Pair is the type of joint that connects two links or members, allowing relative motion between them.

Types of Kinematic Pairs

  • Lower Kinematic Pair: Contact between members occurs over a surface (e.g., revolute joint, prismatic/sliding joint).
  • Higher Kinematic Pair: Contact between members occurs at a point or along a line (e.g., rolling contact, cam-follower).

Degrees of Freedom (DoF)

The Degrees of Freedom (DoF) represent the number of independent relative movements that can exist between the members of a kinematic pair or a mechanism.

Types of Lower Kinematic Pairs

  • Revolute Pair (or Articulation): Allows pure rotation. The contact surface is a surface of revolution, guided laterally. DoF = 1.
  • Prismatic Pair: Allows independent relative translational movement between two members. DoF = 1.
  • Cylindrical Pair: Allows two independent relative movements between members (one rotation and one translation). DoF = 2.
  • Spherical Pair: Allows three independent rotational movements (e.g., ball-and-socket joint). DoF = 3.
  • Planar Pair: Allows three relative movements (two translations and one rotation) on a planar contact surface. DoF = 3.
  • Helical Pair: Allows a translational movement that is dependent on rotational movement, with a helical contact surface (e.g., screw joint). DoF = 1.