9.4 - Kinematic Analysis
Plots/description of the ideal motion/for profiles for the problem statement:
The animation shown in Figure 1 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 plots shown in Figure 2 show that the resulting output demonstrates non-uniform acceleration, velocity, and position motion. The pink dotted line represents the velocity vector while the black represents acceleration. The output is the node connecting links 3 and 4 in the upper most loop that the velocity and acceleration vectors are tied to. the remaining three dotted lines represent the L1 links for each loop. They are imaginary and not physically connected.
Mobility Calculation:
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
Linkage design L = 10 J1 = 12 J2 = 0
M = 3(10-1) - 2(12) - 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 = 8 J1 = 10 J2 = 0
M = 3(8-1) - 2(10) - 0 = 1
This adjustment gives us the correct 1 degree of freedom.
Grashof's Law:
We will solve for the Grashof condition of each four-bar mechanism separately in Table 1. Due to the fact that on each mechanism, aside from the lowest four bar mechanism, the value of link 1 (“ground”) is in flux, we will solve for the Grashof condition of each 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
Sub Mechanism | Config. | Link 1 [mm] | Link 2 [mm] | Link 3 [mm] | Link 4 [mm] | Grashof Condition |
|---|---|---|---|---|---|---|
Crank & Rocker w/ coupler | Min | 73 [P] | 45 [S] | 120 [L] | 120 [Q] | < (Class 1) |
Max | 175 [L] | 45 [S] | 120 [P] | 120 [Q] | < (Class 1) | |
Middle Crank & Rocker | Min | 84 [P] | 35 [S] | 120 [L] | 100 [Q] | < (Class 1) |
Max | 144 [L] | 35 [S] | 120 [P] | 100 [Q] | < (Class 1) | |
Lowest Crank & Rocker | Static | 100 [P] | 25 [S] | 120 [L] | 60 [Q] | < (Class 1) |
Table 1: Grashoff calculations for all sub-mechanisms under all configurations
As is shown in the table all three 4 Bar mechanisms experience Grashoff class 1 motion under all configurations confirming that this linkage assembly will not run into any problems with impossible geometries or toggle positions.
Kinematic Analysis (position, velocity, and accel. for the mouse):
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.
Animation of linkage operation in full range:
The proposed linkage mechanism was created in Motiongen to get a better understanding of the path that each node on the linkage will take. This motion simulation is shown below in Figure 3. The output motion is illustrated by the purple motion profile.