Theory of Machines 804319-3 Lecture 4 Position Analysis

Position Analysis Introduction Coordinate Systems Position and Displacement Translation, Rotation, and Complex Motion Graphical Position Analysis of Linkages Algebraic Position Analysis of Linkages Vector Loop Representation of Linkages Complex Numbers as Vectors

Introduction A principal goal of kinematic analysis is to determine the accelerations of all the moving parts in the assembly Design Stresses Static Forces Dynamic Forces Acceleration Graphical Approach Algebraic Approach Velocity Position

Coordinate Systems Global or absolute coordinate system Local coordinate systems

Position and Displacement The Position of a point in the plane can be defined by the use of a Position vector. The attributes of the position vector can be expressed in either Polar or Cartesian coordinates. Each form is directly convertible into the other by: The Pythagorean theorem: And the trigonometry:

Position and Displacement Displacement is defined as “the straight line distance between the initial and final position of a point which has moved in the reference Frame.” Position difference equation: RBA = RB - RA This expression is read: The position of B with respect to A is equal to the (absolute) position of B minus the (absolute) position of A, where absolute means with respect to the origin of the global reference frame.

Position and Displacement CASE 1: One body in two successive positions =>position difference CASE 2: Two bodies simultaneously in separate positions => relative position

Translation, Rotation, and Complex Motion “All points on the body have the same displacement.” Curvilinear Translation Rectilinear Translation RA′A = RB′B.

Translation, Rotation, and Complex Motion “Different points in the body undergo different displacements and thus there is a displacement difference between any two points chosen”. The link now changes its angular orientation in the reference frame, and all points have different displacements. RB′B = RB′A - RBA

Translation, Rotation, and Complex Motion The total complex displacement of point B is defined by the following expression: Total displacement = translation component + rotation component The new absolute position of point B referred to the origin at A is: RB′′B = RB′B + RB′′B′ RB′′A = RA′A + RB′′A′

Translation, Rotation, and Complex Motion Euler's theorem: “The general displacement of a rigid body with one point fixed is a rotation about some axes.” Chasles' theorem: “Any displacement of a rigid body is equivalent to the sum of a translation of any one point on that body and a rotation of the body about an axis through that point.

GRAPHICAL POSITION ANALYSIS OF LINKAGES

GRAPHICAL POSITION ANALYSIS OF LINKAGES

ALGEBRAIC POSITION ANALYSIS OF LINKAGES The coordinates of point A are: The coordinates of point B are found using the equations of circles about A and 04.

ALGEBRAIC POSITION ANALYSIS OF LINKAGES

Vector Loop Representation of linkages An alternate approach to linkage position analysis creates a vector loop (or loops) around the linkage. This loop closes on itself making the sum of the vectors around the loop zero.

Complex Numbers as Vectors There are many ways to represent vectors. They may be defined in Polar coordinates, by their magnitude and angle, or in Cartesian coordinates as x and y components. We can represent vectors by unit vectors or by complex number notation.

Complex Numbers as Vectors Euler identity:

Complex Numbers as Vectors

Complex Numbers as Vectors