04.4 - Kinematics Analysis
For the mobility calculation, Gruebler's equation was used:
M = 3*(n-1) - 2 * J_1 - J_2
Links (n) = 4, 1 DOF joints (J_1)=4, 2 DOF joints (J_2) = 0
M = 34 - 1 - 24 - 0 = 1
The mechanism has 1 DOF
We begin by defining a point A as a fixed pivot. Point B moves with the input rotation 2 about A, therefore its coordinates are:
Next, the point C is constrained to slide along a straight guideline inclined at 135°. Thus, its coordinates are parameterized by t:
Similarly, a point D lies on another fixed guide inclined at 105° and is parameterized by s:
To satisfy the loop-closure constraint for link l3= 65, the distance between points B and C must always remain 65:
Expanding the distance equation gives us a quadratic in t:
Solving for t yields the sliding position of point C:
We now apply the same approach to link l4= 42, requiring the distance between C and D to remain constant:
This again produces a quadratic, this time in s:
Where:
and the solution for s becomes:
After determining C and D, we compute the dependent angles 3 and 4using inverse tangent relationships:
Finally, differentiating these angles gives the angular velocities and acceleration for links 3 and 4:
The plotted motion confirms correct kinematic behavior with points C and D sliding smoothly along their guides. The effector velocity curve shows a distinct peak where the output velocity exceeds 5 m/s, indicating that the mechanism can generate sufficient speed for the intended motion. This peak value demonstrates that the achieved velocity is adequate to launch the projectile over a reasonable distance, as verified in the subsequent projectile analysis.
The obtained effector velocity is sufficient to launch the projectile to a reasonable distance, as shown by the trajectories where a 5 m/s launch speed achieves ranges between 1.48 m and 2.55 m, depending on the angle.