11.3 - Kinematic Analysis

Kinematic Analysis

For this report, link 1 is ground, link 2 is the crank, link 3 is the connecting rod, and link 4 connects link 3 to the ground. Point A is the connection point between links 2 and 3 and point B is the connection point between links 3 and 4. Point P is the location of water bottle’s cap, and the end-effector as a whole lies along the line connecting points B and P.

The system parameters are as follows:

L₁ = 4.5048 inches
L₂ = 2.5224 inches
L₃ = 4.5336 inches
L₄ = 2.631 inches
AP = 6.993 inches
θ₁ = 110°
∠BAP = 6.971°
ω₂ = -7.1 rad/s

Our linkage system is composed of a singular fourbar loop. A kinematic diagram is shown below in Figure 11.3.1.

Figure 11.3.1. Kinematic diagram of our mechanism.

Gruebler Equation

Number of Links (L) = 4
Number of Full Joints (J₁) = 4
Number of Half Joins (J₂) = 0

M = 3(L - 1) - 2J₁ - J₂ = 3(4 - 1) - 2(4) - 0 = 1

Our mechanism will have 1 degree of freedom, which is all a mechanism like this needs.

Grahof Condition

Shortest Link (S) = 2.5224 in
Longest Link (L) = 4.5336 in
Other Links (P, Q) = 4.5048 in, 2.631 in

S + L = 7.058 in

P + Q = 7.1358 in

Because S + L ≤ P + Q, this linkage meets the Grashof condition, meaning at least one link can make a full rotation. This is ideal because we intend to control it with a single motor without sensors.

Position, Velocity, and Acceleration Analysis

Shown below in Figures 11.3.2 - 11.3.6 are the position, angle, velocity, acceleration, and mechanical advantage profiles of our linkage configuration (for the bottle cap specifically, as that is the main point of interest). For the position, angle, and velocity the ideal profiles are overlaid onto the theoretical profiles for the purpose of comparison. Across the board, these ideal profiles align with the theoretical ones quite nicely. We’ve also included an animation of the linkage, shown in Figure 11.3.7.

Figure 11.3.2. The ideal and actual position profiles of the end-effector’s tip and the bottle cap.

Figure 11.3.3. The ideal and actual angular profiles of the end-effector.

Figure 11.3.4. A comparison of the magnitude (left) and angle (right) of the cap’s velocity with respect to the input angle (θ₂) between the linkage’s actual profile and the ideal profile. Link 2 is assumed to rotate at a constant angular velocity of 7.1 radians per second clockwise.

Figure 11.3.5. The magnitude (left) and angle (right) of the cap’s acceleration with respect to θ2. This assumes a constant inputs speed of ω2 = 7.1 radians per second clockwise

Figure 11.3.6. The mechanical advantage with respect to the crank’s angle. There appear to be asymptotes near 120 and 310 degrees.

Figure 11.3.7. Animation of linkage with velocity (orange arrow) and acceleration (pink arrow) of cap included as well as the ideal and theoretical position profiles of the bottle cap and end-effector tip.

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