position analysis - cairo university displacement • displacement of a point: is the change in its...
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Position Analysis
2015-03-02
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Position • The position of a point in the plane can be
defined by the use of a position vector – Cartesian coordinates – Polar coordinates
Each form is directly convertible into the other by:
REVISION
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Position Coordinate Transformation • lf the position of point A is expressed in the local xy
system, and it is desired to transform its coordinates to the global XY system, the equations are:
REVISION
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Displacement • Displacement of a point: is the change in its
position and can be defined as the straight-line distance between the initial and final position of a point which has moved in the reference frame.
REVISION
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Displacement • Position difference equation: The position of B
with respect to A is equal to the (absolute) position of B minus the (absolute) position of A
OR
REVISION
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Translation, Rotation, and Complex motion
• Translation
• Rotation
REVISION
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Translation, Rotation, and Complex motion
• Complex motion – Total displacement =translation component +
rotation component
REVISION
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Coordinate Systems
Coordinate Systems: GCS = Global Coordinate System, (X, Y) LNCS = Local Non-Rotating Coordinate System , (x, y) LRCS = Local Rotating Coordinate System , (x’, y’)
x’ y’
REVISION
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Graphical Position Analysis • The graphical analysis of this problem is
trivial and can be done using only high school geometry.
Position at θ3
Position at θ3, θ3’
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Algebraic Position Analysis • Complex Numbers as Vectors
– Remember the Euler identity:
REVISION
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Algebraic Position Analysis • Complex Numbers as Vectors
We use complex number notation for vectors to develop and derive the equations for position, velocity, and acceleration of linkages.
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Algebraic Position Analysis The Vector loop Equation for a Slider-Crank 1. Write the vector loop equation:
2. We substitute the complex number notation for each
position vector:
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Algebraic Position Analysis The Fourbar Slider-Crank Position Solution
√ √ √ Input ? √ √ ?
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Algebraic Position Analysis The Fourbar Slider-Crank Position Solution
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Algebraic Position Analysis The Fourbar Slider-Crank Position Solution From and
? ?
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Algebraic Position Analysis The Vector loop Equation for a Fourbar linkage 1. Write the vector loop equation:
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Algebraic Position Analysis The Vector loop Equation for a Fourbar linkage 1. Write the vector loop equation:
2. We substitute the complex number notation for each
position vector:
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Algebraic Position Analysis The Vector loop Equation for a Fourbar linkage We have two unknowns θ2, θ3. We need two equations.
√ √ √ √
Input √
=0 √
? ?
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Algebraic Position Analysis The Vector loop Equation for a Fourbar linkage Subsituting Euler equivalents for the into we get:
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Algebraic Position Analysis The Vector loop Equation for a Fourbar linkage rhgf
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Algebraic Position Analysis The Vector loop Equation for a Fourbar linkage rhgf
=1
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Algebraic Position Analysis The Vector loop Equation for a Fourbar linkage rhgf
Freudenstein's equation
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Algebraic Position Analysis The Vector loop Equation for a Fourbar linkage Using the trigonometric rule We get
The solution is:
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Algebraic Position Analysis The Vector loop Equation for a Fourbar linkage Back to: Squaring and adding:
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Algebraic Position Analysis The Vector loop Equation for a Fourbar linkage Rhgf similarly
The solution is:
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Algebraic Position Analysis Inverted Slider-Crank Position Solution
• This is inversion #3 of the common fourbar slider-crank linkage in which the sliding joint is between links 3 and 4 at point B. This is shown as an offset slider-crank mechanism. The slider block has pure rotation with its center offset from the slide
• axis.
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Algebraic Position Analysis Inverted Slider-Crank Position Solution
RB=R2-R3
√ ? √ √ Input
? ? √
We need 3rd equation
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Algebraic Position Analysis Inverted Slider-Crank Position Solution
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Algebraic Position Analysis Inverted Slider-Crank Position Solution
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Position of a Point on a Link Once the angles of all the links are found, it is simple and straightforward to define and calculate the position of any point on any link for any input position of the linkage. Example: 4-bar linkage
RA
x’ y’
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Transmission Angles • We will expand that definition here to
represent the angle between any two links in a linkage, as a linkage can have many transmission angles.
• It is easy to define the transmission angle algebraically. It is the difference between the angles of the two joined links.
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Transmission Angles Extreme Values of the Transmission Angle
• For a Grashof crank-rocker fourbar linkage the
minimum value of the transmission angle occurs when the crank is colinear with the ground link
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Transmission Angles Extreme Values of the Transmission Angle
θ2=0, π µ=θ4-θ3
Grashof
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Transmission Angles Extreme Values of the Transmission Angle
We label the links
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Transmission Angles Extreme Values of the Transmission Angle • For a Grashof double-rocker linkage the transmission angle can vary
from 0 to 90 degrees because the coupler can make a full revolution
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Transmission Angles Extreme Values of the Transmission Angle
θ2=θ3 µ=θ4-θ3
Non-Grashof
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Transmission Angles Extreme Values of the Transmission Angle • For a non-Grashof triple-rocker linkage the transmission angle will be
zero degrees in the toggle positions which occur when the output rocker c and the coupler b are colinear.
Not minimum µ