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Unformatted text preview: Chapter 2 Position Analysis 2.1 Absolute Cartesian Method The position analysis of a kinematic chain requires the determination of the joint positions, the position of the centers of gravity, and the angles of the links with the horizontal axis. A planar link with the end nodes A and B is considered in Fig. 2.1. Let ( x A , y A ) be the coordinates of the joint A with respect to the reference frame xOy , and ( x B , y B ) be the coordinates of the joint B with the same reference frame. Using Pythagoras the following relation can be written ( x B − x A ) 2 +( y B − y A ) 2 = AB 2 = L 2 AB , (2.1) where L AB is the length of the link AB . Let φ be the angle of the link AB with the horizontal axis Ox . Then, the slope m of the link AB is defined as m = tan φ = y B − y A x B − x A . (2.2) Let n be the intercept of AB with the vertical axis Oy . Using the slope m and the intercept n , the equation of the straight link, in the plane, is y = mx + n , (2.3) where x and y are the coordinates of any point on this link. Fig. 2.1 Planar rigid link with two nodes φ L AB A B x y O ( x B , y B ) ( x A , y A ) 15 16 2 Position Analysis 2.2 SliderCrank (RRRT) Mechanism Exercise The RRRT (slidercrank) mechanism shown in Fig. 2.2a has the dimensions: AB = . 5 m and BC = 1 m. The driver link 1 makes an angle φ = φ 1 = 45 ◦ with the horizontal axis. Find the positions of the joints and the angles of the links with the horizontal axis. 1 A B C 2 3 x y φ 1 A B C 2 x φ 1 C 2 (b) (a) Circle of radius BC y Fig. 2.2 (a) Slidercrank (RRRT) mechanism and (b) two solutions for joint C : C 1 and C 2 Solution The MATLAB R program starts with the statements: clear all % clears all variables and functions clc % clears the command window and homes the cursor close all % closes all the open figure windows The MATLAB commands for the input data are: AB=0.5; BC=1.; The angle of the driver link 1 with the horizontal axis φ = 45 ◦ . The MATLAB com mand for the input angle is: 2.2 SliderCrank (RRRT) Mechanism 17 phi=pi/4; where pi has a numerical value approximately equal to 3.14159. Position of Joint A A Cartesian reference frame xOy is selected. The joint A is in the origin of the reference frame, that is, A ≡ O , x A = , y A = , or in MATLAB: xA=0; yA=0; Position of Joint B The unknowns are the coordinates of the joint B , x B and y B . Because the joint A is fixed and the angle φ is known, the coordinates of the joint B are computed from the following expressions: x B = AB cos φ = ( . 5 ) cos45 ◦ = . 353553 m , y B = AB sin φ = ( . 5 ) sin45 ◦ = . 353553 m . (2.4) The MATLAB commands for Eq. 2.4 are: xB=AB * cos(phi); yB=AB * Sin(phi); where phi is the angle φ in radians....
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 Spring '09
 D.A
 Numerical Analysis, Cartesian Coordinate System, René Descartes

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