MECH324 - Homework Hints and Answers

MECH324 - Homework Hints, Answers, and Extra Requirements

Homework 4 (Chapter 5)

TO ENHANCE YOUR LEARNING, PLEASE TRY TO WORK THE PROBLEMS FIRST WITHOUT LOOKING AT THE HINTS AND ANSWERS BELOW.

MAKE SURE YOU SATISFY ALL REQUIREMENTS LISTED BELOW AND IN THE ORIGINAL PROBLEM STATEMENTS (UNLESS INDICATED OTHERWISE BELOW).

NOTE - BECAUSE MOST ANSWERS ARE PROVIDED, IT IS VERY IMPORTANT THAT YOU SHOW ALL STEPS IN YOUR WORK. THE GRADING WILL BE BASED ON THE COMPLETENESS, CLARITY, AND CORRECTNESS OF YOUR WORK, NOT JUST THE ANSWERS YOU PROVIDE.

5-1
[30 pts]

I strongly recommend you use the provided incomplete MathCAD file (PDF version). Make sure you create equations to replace all of the assignment statements in the file where unknowns are assigned constant values. Please show your work for any equations you derive. Do not use numbers (constants) in any of the equations you create. Use the variable names provided and define your own variables where necessary. When referring to variable names already defined in the file, it is easiest to just copy and paste the variable for use in your equations. If you prefer to type them yourself, be sure to select "Greek variables" in the Style pull-down after entering the variable name.
You do not need to find the toggle positions or the minimum transmission angle.
This is a two-position function generation problem.
Use Method 2 (Equations 5.8 and 5.12).
Show your work for all answers provided below.
Use point A₁ as the origin and measure coordinates relative to this point.
Use point A₁ for precision point P₁ and A₂ for precision point P₂.
p₂₁ = 2.76, Δ₂ = -15.79 degrees, s=0
For non-quick-return, α₂ = 0, β₂ = 180 degrees, and φ = Δ₂.
Use z=5 and γ₂ = 56.519 degrees as your free design choices.
w = 1.38, θ = 164.21 degrees
u = 2.915, Σ = 225.95 degrees
Use Equation 5.2 to find v (5) and g (5.621).
Calculate the coordinates of 0₂ and 0₄ using W₁ and Z₁, and U₁ and S₁.

5-2
[30 pts]

I strongly recommend you use the provided incomplete MathCAD file (PDF version). See other info above.
You do not need to find the toggle positions or the minimum transmission angle. You do not need to design the driver dyad.
This is a two-position function generation problem.
Use Method 1 (Equations 5.7 and 5.11).
Show your work for all answers provided below.
Use point A₁ as the origin and measure coordinates relative to this point.
Use point A₁ for precision point P₁ and A₂ for precision point P₂.
p₂₁ = 2.76, Δ₂ = -15.79 degrees, α₂ = 56.52 degrees
Use θ = 94.394 degrees, β₂ = -40.366 degrees, and φ = -45.479 degrees as your free design choices.
w = 4, z = 0
Use Σ = 93.449 degrees, γ₂ = 54.330 degrees, and ψ = 134.521 degrees as your free design choices.
u = 4, s = 2.455
Use Equation 5.2 to find v (2.455) and g (6.447).
Calculate the coordinates of 0₂ and 0₄ using W₁ and Z₁, and U₁ and S₁.

5-4
[40 pts]

I strongly recommend you use the provided incomplete MathCAD file (PDF version). See other info above.
You do not need to find the toggle positions or the minimum transmission angle.
You do not need to design the driver dyad.
This is a three-position motion generation problem with fixed pivots (see Section 5.9).
The points C and D in Figure P3-2 are identified by points A and B in the MathCAD file.
Use point A₁ as the origin and measure coordinates relative to this point.
Start by defining the coordinates of each point (A's and B's) and the fixed pivots.
Use point A₁ for precision point P₁, A₂ for precision point P₂, and A₃ for precision point P₃.
Show your work for all answers provided below.
α₂ (57.58 degrees) is the angle change of the coupler from position 1 to position 2.
α₃ (90.92 degrees) is the angle change of the coupler from position 2 to position 3.
R₁ = 5.182, R₂ = 3.342, R₃ = 5.004
ζ₁ = 101.07 degrees, ζ₂ = 72.16 degrees, ζ₃ = 54.05 degrees
Be sure to read the note in the book concerning Equation 5.34a.
K₁ = 1146, K₂ = 1788, K₃ = 1769
β₃ = 23.77 degrees, β₂ = 16.79 degrees
γ₃ = 11.64 degrees, γ₂ = 11.07 degrees
Use Equation 5.27 to solve for W_{1_x}, W_{1_y}, Z_{1_x}, and Z_{1_y} and calculate w (8.597) and z (5.336).
Use Equation 5.31 to solve for U_{1_x}, U_{1_y}, S_{1_x}, and S_{1_y} and calculate u (7.921) and s (3.803).
Use Equation 5.2 to find v (1.711) and g (4.303).
You are not required to draw the three positions of the linkage.

MathCAD advice:

instead of typing in complex variable names (e.g., with Greek letters, subscripts, and primes), just copy and paste from other places in the file.
all subscripts are literal, created by pressing the period on the keyboard (not by selecting the X_n in the Matrix palette for an iterative or array index subscript).
the primes after variable names can be created by pressing the accent/tilde key to the left of the 1-key on the keyboard.

5-1 [30 pts]	I strongly recommend you use the provided incomplete MathCAD file (PDF version). Make sure you create equations to replace all of the assignment statements in the file where unknowns are assigned constant values. Please show your work for any equations you derive. Do not use numbers (constants) in any of the equations you create. Use the variable names provided and define your own variables where necessary. When referring to variable names already defined in the file, it is easiest to just copy and paste the variable for use in your equations. If you prefer to type them yourself, be sure to select "Greek variables" in the Style pull-down after entering the variable name. You do not need to find the toggle positions or the minimum transmission angle. This is a two-position function generation problem. Use Method 2 (Equations 5.8 and 5.12). Show your work for all answers provided below. Use point A₁ as the origin and measure coordinates relative to this point. Use point A₁ for precision point P₁ and A₂ for precision point P₂. p₂₁ = 2.76, Δ₂ = -15.79 degrees, s=0 For non-quick-return, α₂ = 0, β₂ = 180 degrees, and φ = Δ₂. Use z=5 and γ₂ = 56.519 degrees as your free design choices. w = 1.38, θ = 164.21 degrees u = 2.915, Σ = 225.95 degrees Use Equation 5.2 to find v (5) and g (5.621). Calculate the coordinates of 0₂ and 0₄ using W₁ and Z₁, and U₁ and S₁.
5-2 [30 pts]	I strongly recommend you use the provided incomplete MathCAD file (PDF version). See other info above. You do not need to find the toggle positions or the minimum transmission angle. You do not need to design the driver dyad. This is a two-position function generation problem. Use Method 1 (Equations 5.7 and 5.11). Show your work for all answers provided below. Use point A₁ as the origin and measure coordinates relative to this point. Use point A₁ for precision point P₁ and A₂ for precision point P₂. p₂₁ = 2.76, Δ₂ = -15.79 degrees, α₂ = 56.52 degrees Use θ = 94.394 degrees, β₂ = -40.366 degrees, and φ = -45.479 degrees as your free design choices. w = 4, z = 0 Use Σ = 93.449 degrees, γ₂ = 54.330 degrees, and ψ = 134.521 degrees as your free design choices. u = 4, s = 2.455 Use Equation 5.2 to find v (2.455) and g (6.447). Calculate the coordinates of 0₂ and 0₄ using W₁ and Z₁, and U₁ and S₁.
5-4 [40 pts]	I strongly recommend you use the provided incomplete MathCAD file (PDF version). See other info above. You do not need to find the toggle positions or the minimum transmission angle. You do not need to design the driver dyad. This is a three-position motion generation problem with fixed pivots (see Section 5.9). The points C and D in Figure P3-2 are identified by points A and B in the MathCAD file. Use point A₁ as the origin and measure coordinates relative to this point. Start by defining the coordinates of each point (A's and B's) and the fixed pivots. Use point A₁ for precision point P₁, A₂ for precision point P₂, and A₃ for precision point P₃. Show your work for all answers provided below. α₂ (57.58 degrees) is the angle change of the coupler from position 1 to position 2. α₃ (90.92 degrees) is the angle change of the coupler from position 2 to position 3. R₁ = 5.182, R₂ = 3.342, R₃ = 5.004 ζ₁ = 101.07 degrees, ζ₂ = 72.16 degrees, ζ₃ = 54.05 degrees Be sure to read the note in the book concerning Equation 5.34a. K₁ = 1146, K₂ = 1788, K₃ = 1769 β₃ = 23.77 degrees, β₂ = 16.79 degrees γ₃ = 11.64 degrees, γ₂ = 11.07 degrees Use Equation 5.27 to solve for W_{1_x}, W_{1_y}, Z_{1_x}, and Z_{1_y} and calculate w (8.597) and z (5.336). Use Equation 5.31 to solve for U_{1_x}, U_{1_y}, S_{1_x}, and S_{1_y} and calculate u (7.921) and s (3.803). Use Equation 5.2 to find v (1.711) and g (4.303). You are not required to draw the three positions of the linkage.