This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
This note was uploaded on 10/28/2009 for the course MECH 320 taught by Professor D.a during the Spring '09 term at American University of Beirut.
 Spring '09
 D.A

Click to edit the document details