9.2 - Project Prototype

The animation below is a 5x speed, accurate representation of our system design. It was created using a 12 RPM on the bottom gear, with an angular acceleration of 0, and the provided gear ratios and lengths. The GIF and plot show that the resulting output demonstrates non-uniform acceleration, velocity, and position motion.

Gruebler's Calculations:

M = 3(L-1) - 2J₁ - J₂

M - Mobility of the system (Degrees of freedom)
L - Number of links in the mechanism
J₁ - Number of 1 DOF joints

J₂ - Number of 2 DOF joints

Proposed design:

L = 7

J₁ = 8

J₂ = 0

M = 3(7-1) - 2(8) - 0 = 2

This gives us 2 degrees of freedom. 

The reason for this is that Gruebler’s equation doesn’t account for the fact that the motion of the two input links is directly related. The best way to adjust for this is to combine both links into 1. This would reduce the number of links by 1 and the number of 1 DOF joints by 1 conforming to the prerequisites for Gruebler's equation.

Adjusted design:

L = 6

J₁ = 7

J₂ = 0

M = 3(6-1) - 2(7) - 0 = 1

This adjustment gives us the correct 1 degree of freedom.

Grashof's Law:

We will solve for the Grashof condition of both four-bar mechanism separately. Due to the fact that the top four bar mechanism has the value of link 1 (“ground”) in flux, we will solve for the grashof condition of this sub-mechanism twice, once at the maximum value of link 1 and once at the minimum.

Grashof Equation:

S + L  <,=,>  P + Q

S: Shortest Link

L: Longest Link

P: First Remaining Link

Q: Second Remaining Link

Table 1: All sub-mechanisms under all configurations undergo Grashof class 1 motion

Our system is composed of three stacked four-bar linkages. To solve the position of a four-bar mechanism, we require the lengths of all links, along with the angles θ₁ and θ₂. For the bottom loop, these values are known: all link lengths are constant, θ₁ is 0 (as link 1 is grounded), and θ₂ is directly controlled by the motor.

However, analyzing the second and third loops is more complex, as link 1 in these loops has a variable length. To address this, we solve the system from the bottom up—first determining the length of link 1 and the corresponding angle for each upper loop. The value of θ₂ for each subsequent loop is derived using the gear ratios between the loops.

For velocity analysis, the angular velocity of each loop is calculated starting from the bottom loop’s known RPM, then scaling it using the gear ratios for the loops above. Similarly, for acceleration analysis, we require the angular acceleration of each loop. But since all gears rotate at a constant speed, the angular acceleration for each is 0.