Problem 1: Closed form solution of a four bar linkage

Consider the 4-bar linkage shown below (not drawn to scale).

Assume l₁ = 8m, l₂ = 3m, l₃ = 6m, h = 3m, h_n = 3m, h_n = 3m. The linkage l₂ is the crank and rotates at a constant angular velocity of 60rpm.

a) Plot θ₃ vs. θ₂ for one complete revolution of the crank. (figure 1)

b) Plot θ₄ vs. θ₂ for one complete revolution of the crank. (figure 2)

c) (x_p, x_p,) is the coordinate of point P. Plot x_p vs. θ₂ for one complete revolution of the crank. (figure 3)

d) Plot y_p vs. θ₂ for one complete revolution of the crank (figure 4)

e) Plot the path of point P for one complete revolution of the crank (figure 5)

Hint: Refer to examples in section 11.7.3 and section 11.7.4 on the course reader

Problem 2: Numerical solution of a four bar linkage

The dimension of each linkage is provided in the figure above. The handle (crank) is given a constant angular velocity of 45 rpm in counter-clockwise direction.

Note: question f) and g) in this problem are extra credit questions. No point will be deducted if you do not answer this two questions. If you answer this two questions correctly, you will get some extra credit for this homework.

a) Use Newton-Raphson method to calculate θ₃ and θ₄ for one complete revolution of the crank and plot them as a function of θ₂ in the same figure. (Figure 1)

b) Plot the path of point P for one revolution of the crank; assume the origin of your coordinate system is at the crank location. (Figure 2)

c) Plot the distance of separation (in cm) between the point P and Q as a function of θ₂ during a full rotation of the crank/handle. What is the maximum separation? At what crank angle does it occur? (Figure 3)

d) Plot the angular velocity (in rad/s) of the link DQ. (Figure 4)

e) Plot the velocity (cm/s) of point P vs. θ₂ for one complete revolution. (Figure 5)

f) Extra credit question: Derive the expression of θ₄^k+1 which can be used in Freudenstein method. (Hint: Derivation of Eq. 11.13 on the course reader is a good example. The equation you should obtain is:

θ₄^k+1 = (arccos [ l₃² - l₁x² - l_1y² - l₂² - l_CD² + 2l_1xl₂(cos(θ₂) - 2l_1yl₂sin(θ₄^k) - 2I₂I_CDcos(θ₂-θ₄^k))/2l_1xl_CD

g) Extra credit question: Use Freudenstein method to calculate θ₃ and θ₄ for one complete revolution of the crank and plot them as a function of θ₂ in the same figure. (Figure 6)

Problem 3: SolidWorks Assignment

Complete the SolidWorks tutorial lesson 8 posted on TED. Submit screen shots of trace path of the tip of the triangle, displacement and velocity graphs (3 Figures). Do not forget to include time stamps.

Checklist of what should be submitted for homework # 7

1. Computer printed hardcopy with all work to be graded.
• Use a cover page (Name, PID, Class Name, HW#)
• Show all your work: Include all plots, explain your work and answer all questions
• Label your plots (otherwise points will be deducted)
• Attach a printout of your MATLAB code used to generate the results at the end of your hard copy.

2. Submit the electronic copy of your m-files. The codes are downloaded from the class gmail account using an automated procedure. Please read the following carefully to be sure your file is submitted correctly.
• Subject of the email: write your full name and the assignment
• Include only m-files (no Word document or other file types)
• Do not zip files
• Your files should be named as follows: "lastname_firstname_hw7prob1.m"
• It is OK to include additional function files, but only if they're prefixed with "lastname_firstname_hw# "
• Make sure you are finished before submitting your code to avoid having multiple copies submitted.

Your MATLAB files will be reviewed electronically for uniqueness using the MOSS (measure of software similarity) script developed at Stanford. This script compares your answer to any other answer in the class. No credit will be given if the MOSS test indicates high similarities with another student.