![]() q + and dq +/dt (the positions and velocities of the system after impact) are defined as dummy variables symbolically to make them distinct from q and dq/dt (the positions and velocities of the system before impact). Note that only the solution that generates a non-zero λ value will be used, since a λ of zero will generate a trivial solution that results in the jack going right through the wall of the box. Derive 16 sets of impact update equations (one corresponding to each impact condition), which change the velocities of the system ( dq/dt) post impact to make the jack and the box bounce off each other.This results in 16 different impact conditions that the simulation continuously checks for at each time step ( dt). two corners of the jack hitting at once) is exceedingly low, an “impact” is defined as any time any of the 4 corners of the jack hits any of the four walls of the box (with some tolerance built in to account for numerical integration making the jack “miss”). Since the probability of having the face of the jack hit the wall of the box (e.g. A correct simulation should always conserve the Hamiltonian regardless of whether it’s an open or closed system. The Hamiltonian will be used to derive the impact update equations and can also be plotted with respect to time to examine if the simulation is correct. Calculate the Hamiltonian of the system, which is a conserved value in any dynamic system.The equation is shown below where F is a 6-vector consisting of F x, F y, and zeros: Forces in the x b and y b direction are added here to “shake up” the box. Derive the forced Euler-Lagrange (EL) equations to simulate the trajectory of the jack and the box when it is not actively impacting.m is simply the mass of the object and g is the gravity scalar 9.8 m/s 2. M here is defined as the 6圆 three dimensional inertia matrix of the object (note that only inertia about z is being considered here since the simulation is two dimensional) and V b= where the superscript v represents taking the skew-symmetric form of the vector and G represents the transformation matrix (could be G wb or G wj depending on whether one is determining the KE of the jack or box.). Define the total kinetic energy ( KE) and potential energy ( PE) of the system (jack + box).A diagram of the system can be seen in the image below: The jack and box frames have origins at the center of mass (CoM) of both these objects. First, define the coordinate system ( q = ), frames (jack frame, box frame, and a fixed world frame), and the frames’ relative transformation matrices G wj (transforms jack coordinates to world coordinate) and G wb (transforms box coordinate to world coordinate).The overall process of generating the simulation is as the following: The final simulation looks like this: Demo.mp4 numeric calculations were also explored in this project. Other topics such as multiple impacts, rotational inertia, and symbolic vs. The physics of the dice and the cup was derived and simulated from scratch using the SymPy package in Python, and both Lagrangian dynamics and rigid body transformations were used to generate the trajectory of these two objects. The Dice in Cup project was done for the final assignment of Northwestern University’s Theory of Machines - Dynamics (MECH_ENG 314) course to simulate the dynamics of a 2-dimensional square dice (or jack) bouncing around inside a cup (or box) that is being shaken up by external forces under gravity.
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