Position Analysis
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1 Position Analysis
2 Position REVISION 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:
3 Position REVISION 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:
4 Displacement REVISION 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.
5 Displacement REVISION 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
6 Translation, Rotation, and Complex motion Translation REVISION Rotation
7 Translation, Rotation, and Complex motion Complex motion REVISION Total displacement =translation component + rotation component
8 Coordinate Systems REVISION y x Coordinate Systems: GCS = Global Coordinate System, (X, Y) LNCS = Local Non-Rotating Coordinate System, (x, y) LRCS = Local Rotating Coordinate System, (x, y )
9 Graphical Position Analysis REVISION The graphical analysis of this problem is trivial and can be done using only high school geometry. Position at θ3 Position at θ3, θ3
10 Complex Numbers as Vectors REVISION Remember the Euler identity:
11 Complex Numbers as Vectors We use complex number notation for vectors to develop and derive the equations for position, velocity, and acceleration of linkages.
12 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:
13 The Fourbar Slider-Crank Position Solution? Input?
14 The Fourbar Slider-Crank Position Solution
15 The Fourbar Slider-Crank Position Solution From and??
16 The Vector loop Equation for a Fourbar linkage 1. Write the vector loop equation:
17 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:
18 The Vector loop Equation for a Fourbar linkage?? Input =0 We have two unknowns θ2, θ3. We need two equations.
19 The Vector loop Equation for a Fourbar linkage Subsituting Euler equivalents for the into we get:
20 The Vector loop Equation for a Fourbar linkage rhgf
21 The Vector loop Equation for a Fourbar linkage rhgf =1
22 The Vector loop Equation for a Fourbar linkage rhgf Freudenstein's equation
23 The Vector loop Equation for a Fourbar linkage Using the trigonometric rule We get The solution is:
24 The Vector loop Equation for a Fourbar linkage Back to: Squaring and adding:
25 The Vector loop Equation for a Fourbar linkage Rhgf similarly The solution is:
26 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.
27 Inverted Slider-Crank Position Solution R B =R 2 -R 3??? Input We need 3 rd equation
28 Inverted Slider-Crank Position Solution
29 Inverted Slider-Crank Position Solution
30 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 y x R A
31 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.
32 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
33 Transmission Angles Extreme Values of the Transmission Angle Grashof θ2=0, π µ=θ4-θ3
34 Transmission Angles Extreme Values of the Transmission Angle We label the links
35 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
36 Transmission Angles Extreme Values of the Transmission Angle Non-Grashof θ2=θ3 µ=θ4-θ3
37 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
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