7.4 Kinematic Analysis
Ideal Motion of Fourbar linkage:
Mobility Calculations:
Gruebler’s Equation:
Number of Links: 7
Number of 1 DOF Joints: 7
Number of 2 DOF Joints: 1
M = 3 * (7 - 1) - (2 * 7) - 1 = 3 DOF
Grashof Condition:
Link 1 = 6.375 inches (longest link), Link 2 = 2.5 inches (shortest link), Link 3 = 6 inches, Link 4 = 3 inches
S + L = 8.875 inches
P + Q = 9 inches
As S + L < P + Q, the Grashof Condition is satisfied, indicating that the four-bar linkage is able to make a full 360º rotation if needed. However, given the nature of the dice shaking motion, we predict that the motion will be about 40º during the shaking motion and 70º during the throwing motion.
Kinematic Analysis
As the expected motion is 3 shakes followed by a throw, each of the above plots has a clear pattern of 3 periods of repetitive motion followed by similar but different motion at the very end. Additionally, the exact range of motion shown in the ideal motion gif at the top of this page was chosen to maximize m_v, as can be seen in the m_v vs time plot above. We experimentally determined the spring constant k to be approximately 400 N/m based on observed displacement of the spring during motion. Using this value, we then iterated over possible input speeds, tuning them gradually until we achieved a motion profile where the peak linear acceleration exceeded the acceleration due to gravity. This ensured that the dice would be thrown upward with enough force to become airborne, creating a moment of flight that allowed for truly random tumbling. Since we wanted the dice to be flung into the air during each shake rather than simply sliding or rocking in place, this parameter tuning was critical to achieving our desired chaotic behavior. It is shown in the two graphs above depicting linear acceleration accounting for the spring and without accounting for it.