Rigid Body Guidance Design

Rigid Body Guidance Design 901508820 AME40423 Mechanisms and Machines December 3, 2010 Abstract The purpose of this project was to design a four bar mechanism to guide bicycle and rack to a given overhead position. The links could not collide with the wall or floor as the bicycle is lifted. It was ideal to have the ground pivots located on the footboard, but not necessary. The first and third positions were given in the problem statement and are shown in Figure 1. The third position angle, a3x, and a3y coordinates were determined to be 90o, -39 inches, and 56 inches, respectively. The second position was chosen to have a position angle, a2x, and a2y coordinates with values of 45o, -3 inches, and 28 inches, respectively. The first ground position was determined to have x and y coordinates of -46.5 inches and 9.2 inches, respectively, using a circling point with x and y coordinates both with a value of 0 inches. This corresponded to a link length of 47.4 inches. The second ground position was determined to have x and y coordinates of -35.3 inches and 10.4 inches, respectively, using a circling point with x and y coordinates with values of 0 inches and -26.3 inches, respectively. This corresponded to a link length of 50.9 inches.

An animation of this design is presented in the Results section.

1 Engineering Analysis The initial and final positions of the bike were specified and are shown in Figure 1. The intermediate position was left as a design choice.

Figure 1: Dormitory Bicycle Storage Mechanism Relative to position 1, the second and third positions were determined as, Position 2: ̅​̅​̅

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Position 3: ]

The bike rack needs to pivot to the storage rack so the top of the bike rack in position 1 was chosen as the location for the first moving pivot with coordinates, ̅​̅​̅ using equation 5.8,

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using equation 5.10 the fixed pivot was determined to have coordinates, ̅

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Then from equation 5.4 the link length, R, was calculated to be 47.4 inches. The same procedure was used with the moving pivot at the bottom of the bike rack which has coordinates, ̅​̅​̅

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from equation 5.8,

using equation 5.10 the fixed pivot was determined to have coordinates, ̅

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Then from equation 5.4 the link length, R, was calculated to be 50.9 inches.

2 Results The results from the engineering analysis were used to create an animation of the four-bar mechanism. The links do not hit the wall or floor and therefore satisfy the constraints. In Figures 2-4, the progression of the four-bar from position 1 to position 3 is shown. Since a crossed fourbar mechanism was designed a barrier would have to be made so the user is protected when the links cross.

Figure 2: Four-Bar in Position 1

Figure 3: Four-Bar in Position 2

Figure 3: Four-Bar in Position 3

Appendix A: MATLAB Code %Charles Hunter %Bike Rack Design Problem %11/28/10 close all clear all clc %Design constraints a2x = -3; a2y = 28; T2 = pi/4; a3x = -39; a3y = 56; T3 = pi/2; %Circling Points x1 = 0; y1 = -26.3; %Finding Center Points U2 = x1*(cos(T2)-1) - y1*sin(T2) + a2x U3 = x1*(cos(T3)-1) - y1*sin(T3) + a3x V2 = x1*sin(T2) + y1*(cos(T2) - 1) + a2y V3 = x1*sin(T3) + y1*(cos(T3) - 1) + a3y W2 = ((a2x^2+a2y^2)/2) + x1*(a2x*cos(T2)+a2y*sin(T2)) + y1*(a2y*cos(T2) a2x*sin(T2)) W3 = ((a3x^2+a3y^2)/2) + x1*(a3x*cos(T3)+a3y*sin(T3)) + y1*(a3y*cos(T3) a3x*sin(T3)) xstar = ((W2*V3)-(V2*W3))/((U2*V3)-(V2*U3)) ystar = ((U2*W3)-(W2*U3))/((U2*V3)-(V2*U3)) x1bar = [x1;y1]; xstarbar = [xstar;ystar]; R = sqrt((x1bar-xstarbar).*(x1bar-xstarbar)); linklength = sqrt((R(1)^2+R(2)^2))