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Linkage analysis: The linkage shown below is the kinematic sketch for the rear suspension of a motorcycle. The dimensions are given in the drawing; consider BC = 2” along the AB direction for the given configuration. For the analysis, assume that the angular velocity of the input link (link 2) is constant and operating in the CW direction (corresponding to the motorcycle going over a bump). a. Calculate the mobility of the linkage. How many loop equations are needed to solve for the dependent joint variables? b. Formulate the loop equations. c. Solve the loop equations and give explicit expressions for the dependent variables as a function of the input angle ?. d. Compute the limits for the input angle ?. Is the linkage going to work as expected? (is the range of motion of ? enough?) e. Write the position vector of point C. Use Maple, GIM or similar software to plot the trajectory of point C over the range of ? calculated in d). f. Use Maple or similar software to plot ? and s as a function of ?. g. Use GIM to create a simulation for the motion of the linkage. Provide a snapshot. h. Compute the velocity vector for point C. Give the value of the velocity for the configuration shown in the kinematic sketch (? = 200o), for an input angular velocity of 200 rpm. i. Compute the velocity s of the slider. Plot this velocity as a function of ? for a constant input angular velocity of 200rpm. j. Plot the acceleration of the slide, s, as a function of ?, for the same constant input angular velocity

Linkage analysis: The linkage shown below is the kinematic sketch for the rear suspension of a motorcycle. The dimensions are given in the drawing; consider BC = 2” along the AB direction for the given configuration. For the analysis, assume that the angular velocity of the input link (link 2) is constant and operating in the CW direction (corresponding to the motorcycle going over a bump). a. Calculate the mobility of the linkage. How many loop equations are needed to solve for the dependent joint variables? b. Formulate the loop equations. c. Solve the loop equations and give explicit expressions for the dependent variables as a function of the input angle ?. d. Compute the limits for the input angle ?. Is the linkage going to work as expected? (is the range of motion of ? enough?) e. Write the position vector of point C. Use Maple, GIM or similar software to plot the trajectory of point C over the range of ? calculated in d). f. Use Maple or similar software to plot ? and s as a function of ?. g. Use GIM to create a simulation for the motion of the linkage. Provide a snapshot. h. Compute the velocity vector for point C. Give the value of the velocity for the configuration shown in the kinematic sketch (? = 200o), for an input angular velocity of 200 rpm. i. Compute the velocity s of the slider. Plot this velocity as a function of ? for a constant input angular velocity of 200rpm. j. Plot the acceleration of the slide, s, as a function of ?, for the same constant input angular velocity

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Lab 5 Math 551 Fall 2015 Goal: In this assignment we will look at two fractals, namely the Sierpinski fractal and the Barnsley Fern. During the lab session, your lab instructor will teach you the necessary MATLAB code to complete the assignment, which will be discussed in the lab on Thursday October 8th or Friday October 9th in the lab (CW 144 or CW 145). What you have to submit: An m- le containing all of the commands necessary to perform all the tasks described below. Submit this le on Canvas. Click: \Assignments”, click \MATLAB Project 5″, click \Submit Assignment”, then upload your .m le and click \Submit Assignment” again. Due date: Friday October 16, 5pm. No late submission will be accepted. TASKS A fractal can be de ned as a self-similar detailed pattern repeating itself. Some of the most well know fractals (the Mandelbrot set and Julia set) can be viewed here: http://classes.yale.edu/fractals/ The Sierpinski Fractal The program srnpnski(m,dist,n) gets its name from the mathematician W. Sierpinski. The only parameter that must be speci ed is m which determines the number of vertices that will be part of a regular polygon. For larger m it produces a graph which is similar to a snow ake. The program starts with a randomly chosen seed position given by the internal variable s. At each stage one of the vertices is chosen at random and a new point is produced which is dist away from the old point to the vertex. The value of dist should be between 0 and 1. The default value is 0.5. This process is repeated n times. The default value of n is 1500. 1. Create a new Matlab function: func t i on s rpns k i (m, di s t , n) %This c r e a t e s a snowf lake from m v e r t i c e s us ing n i t e r a t i o n s . i f nargin <3, n=1500; end i f nargin <2, d i s t =0.5; end c l f a x i s ( [ ?1 ,1 , ?1 ,1] ) p=exp (2 pi  i  ( 1 :m)/m) ; pl o t (p , '  ' ) hold s=rand+i  rand ; f o r j =1:n r=c e i l (m rand ) ; s=d i s t  s+(1?d i s t )p( r ) ; pl o t ( s , ' . ' ) end 2. Try out the following commands s rpns k i ( 3 ) s rpns k i ( 3 , 0 . 5 , 2 5 0 0 ) s rpns k i ( 3 , 0 . 5 , 5 0 0 0 ) s rpns k i ( 3 , 0 . 4 ) 1 s rpns k i ( 3 , 0 . 2 ) s rpns k i ( 5 ) s rpns k i ( 5 , 0 . 4 ) s rpns k i ( 5 , 0 . 3 ) s rpns k i ( 6 , 0 . 3 ) s rpns k i ( 8 , 0 . 3 , 5 0 0 0 ) The Barnsley Fern The following program is the famous Barnsley Fern. The only external parameter is n, the number of iterations. 3. Create a new Matlab function: func t i on f e r n (n) A1=[ 0 . 8 5 , 0 . 0 4 ; ?0 . 0 4 , 0 . 8 5 ] ; A2=[ ?0 . 1 5 , 0 . 2 8 ; 0 . 2 6 , 0 . 2 4 ] ; A3=[ 0 . 2 , ?0 . 2 6 ; 0 . 2 3 , 0 . 2 2 ] ; A4=[ 0 , 0 ; 0 , 0 . 1 6 ] ; T1=[ 0 ; 1 . 6 ] ; T2=[ 0 ; 0 . 4 4 ] ; T3=[ 0 ; 1 . 6 ] ; T4=[ 0 , 0 ] ; P1=0.85; P2=0.07; P3=0.07; P4=0.01; c l f ; s=rand ( 2 , 1 ) ; pl o t ( s ( 1 ) , s ( 2 ) , ' . ' ) hold f o r j =1:n r=rand ; i f r<=P1 , s=A1 s+T1 ; e l s e i f r<=P1+P2 , s=A2 s+T2 ; e l s e i f r<=P1+P2+P3 , s=A3 s+T3 ; e l s e s=A4 s ; end pl o t ( s ( 1 ) , s ( 2 ) , ' . ' ) end 4. Try the following commands: f e r n (100) f e r n (500) f e r n (1000) f e r n (3000) f e r n (5000) f e r n (10000) 2 5. Change the parameters in the fern program: A1=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; A2=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; A3=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; T1=[ 1 ; 1 ] ; T2=[ 1 ; 5 0 ] ; T3=[ 5 0 ; 5 0 ] ; P1=0.33; P2=0.33; P3=0.34; Call the new program srptri.m. Try the command s r p t r i (5000) You should see a familiar looking result. 6. Change the parameters in the fern program: A1=[ 0 , 0 ; 0 , 0 . 5 ] ; A2=[ 0 . 4 2 , ?0 . 4 2 ; 0 . 4 2 , 0 . 4 2 ] ; A3=[ 0 . 4 2 , 0 . 4 2 ; ?0 . 4 2 , 0 . 4 2 ] ; A4=[ 0 . 1 , 0 ; 0 , 0 . 1 ] ; T1=[ 0 ; 0 ] ; T2=[ 0 ; 0 . 2 ] ; T3=[ 0 ; 0 . 2 ] ; T4=[ 0 , 0 . 2 ] ; P1=0.05; P2=0.4; P3=0.4; P4=0.15; Call the new program srptree.m. Try the command s r p t r e e (5000) This is an example of a fractal tree. Some nice animations of fractal trees can be seen here: http://classes.yale.edu/fractals/ MATLAB commands to learn: Cl f , c e i l , imaginary uni t I , i f . . e l s e i f . . e l s e . . end 3

Lab 5 Math 551 Fall 2015 Goal: In this assignment we will look at two fractals, namely the Sierpinski fractal and the Barnsley Fern. During the lab session, your lab instructor will teach you the necessary MATLAB code to complete the assignment, which will be discussed in the lab on Thursday October 8th or Friday October 9th in the lab (CW 144 or CW 145). What you have to submit: An m- le containing all of the commands necessary to perform all the tasks described below. Submit this le on Canvas. Click: \Assignments”, click \MATLAB Project 5″, click \Submit Assignment”, then upload your .m le and click \Submit Assignment” again. Due date: Friday October 16, 5pm. No late submission will be accepted. TASKS A fractal can be de ned as a self-similar detailed pattern repeating itself. Some of the most well know fractals (the Mandelbrot set and Julia set) can be viewed here: http://classes.yale.edu/fractals/ The Sierpinski Fractal The program srnpnski(m,dist,n) gets its name from the mathematician W. Sierpinski. The only parameter that must be speci ed is m which determines the number of vertices that will be part of a regular polygon. For larger m it produces a graph which is similar to a snow ake. The program starts with a randomly chosen seed position given by the internal variable s. At each stage one of the vertices is chosen at random and a new point is produced which is dist away from the old point to the vertex. The value of dist should be between 0 and 1. The default value is 0.5. This process is repeated n times. The default value of n is 1500. 1. Create a new Matlab function: func t i on s rpns k i (m, di s t , n) %This c r e a t e s a snowf lake from m v e r t i c e s us ing n i t e r a t i o n s . i f nargin <3, n=1500; end i f nargin <2, d i s t =0.5; end c l f a x i s ( [ ?1 ,1 , ?1 ,1] ) p=exp (2 pi  i  ( 1 :m)/m) ; pl o t (p , '  ' ) hold s=rand+i  rand ; f o r j =1:n r=c e i l (m rand ) ; s=d i s t  s+(1?d i s t )p( r ) ; pl o t ( s , ' . ' ) end 2. Try out the following commands s rpns k i ( 3 ) s rpns k i ( 3 , 0 . 5 , 2 5 0 0 ) s rpns k i ( 3 , 0 . 5 , 5 0 0 0 ) s rpns k i ( 3 , 0 . 4 ) 1 s rpns k i ( 3 , 0 . 2 ) s rpns k i ( 5 ) s rpns k i ( 5 , 0 . 4 ) s rpns k i ( 5 , 0 . 3 ) s rpns k i ( 6 , 0 . 3 ) s rpns k i ( 8 , 0 . 3 , 5 0 0 0 ) The Barnsley Fern The following program is the famous Barnsley Fern. The only external parameter is n, the number of iterations. 3. Create a new Matlab function: func t i on f e r n (n) A1=[ 0 . 8 5 , 0 . 0 4 ; ?0 . 0 4 , 0 . 8 5 ] ; A2=[ ?0 . 1 5 , 0 . 2 8 ; 0 . 2 6 , 0 . 2 4 ] ; A3=[ 0 . 2 , ?0 . 2 6 ; 0 . 2 3 , 0 . 2 2 ] ; A4=[ 0 , 0 ; 0 , 0 . 1 6 ] ; T1=[ 0 ; 1 . 6 ] ; T2=[ 0 ; 0 . 4 4 ] ; T3=[ 0 ; 1 . 6 ] ; T4=[ 0 , 0 ] ; P1=0.85; P2=0.07; P3=0.07; P4=0.01; c l f ; s=rand ( 2 , 1 ) ; pl o t ( s ( 1 ) , s ( 2 ) , ' . ' ) hold f o r j =1:n r=rand ; i f r<=P1 , s=A1 s+T1 ; e l s e i f r<=P1+P2 , s=A2 s+T2 ; e l s e i f r<=P1+P2+P3 , s=A3 s+T3 ; e l s e s=A4 s ; end pl o t ( s ( 1 ) , s ( 2 ) , ' . ' ) end 4. Try the following commands: f e r n (100) f e r n (500) f e r n (1000) f e r n (3000) f e r n (5000) f e r n (10000) 2 5. Change the parameters in the fern program: A1=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; A2=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; A3=[ 0 . 5 , 0 ; 0 , 0 . 5 ] ; T1=[ 1 ; 1 ] ; T2=[ 1 ; 5 0 ] ; T3=[ 5 0 ; 5 0 ] ; P1=0.33; P2=0.33; P3=0.34; Call the new program srptri.m. Try the command s r p t r i (5000) You should see a familiar looking result. 6. Change the parameters in the fern program: A1=[ 0 , 0 ; 0 , 0 . 5 ] ; A2=[ 0 . 4 2 , ?0 . 4 2 ; 0 . 4 2 , 0 . 4 2 ] ; A3=[ 0 . 4 2 , 0 . 4 2 ; ?0 . 4 2 , 0 . 4 2 ] ; A4=[ 0 . 1 , 0 ; 0 , 0 . 1 ] ; T1=[ 0 ; 0 ] ; T2=[ 0 ; 0 . 2 ] ; T3=[ 0 ; 0 . 2 ] ; T4=[ 0 , 0 . 2 ] ; P1=0.05; P2=0.4; P3=0.4; P4=0.15; Call the new program srptree.m. Try the command s r p t r e e (5000) This is an example of a fractal tree. Some nice animations of fractal trees can be seen here: http://classes.yale.edu/fractals/ MATLAB commands to learn: Cl f , c e i l , imaginary uni t I , i f . . e l s e i f . . e l s e . . end 3

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Problem 3 (25 Points) A bracket is subjected to a 610 N force as shown in the figure. (a) Express the force in Cartesian vector form. (8 points) (b) Determine the moment of the force about point C. (10 points) (c) Determine the moment of the force about line CD. (7 points)

Problem 3 (25 Points) A bracket is subjected to a 610 N force as shown in the figure. (a) Express the force in Cartesian vector form. (8 points) (b) Determine the moment of the force about point C. (10 points) (c) Determine the moment of the force about line CD. (7 points)

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