Characteristics of a Two-Body Holonomic Constraint Mechanical System

Abstract

Two massive blocks are connected with a massless unstretchable line of 2ℓ. One of the masses is placed on a horizontal frictionless table, ℓ distance away from the edge of the table the other one is held horizontally equidistance from the edge along the extension of the line. The latter is released from rest. As it falls under gravity's pull, it drags the one on the table. It is the interest of this investigation to analyze the kinematics of the system. Because of the holonomic constraint of the system, analysis of the problem encounters complicated super nonlinear coupled differential equations. Utilizing Mathematica we solve the equations numerically. Applying the solutions we quantify numerous kinematic quantities; most interestingly we evaluate the run-time, and the trajectory of the falling block. Analysis is robust allowing us to address the "what if" scenarios.

Kinematics of a Holonomic Constraint Rod & Cube System

“…Motivation of suggested investigation stems from the fact that the proposed assembly is composed of a sliding cube that acts as a point-like object with only linear kinematics in contrast to the rod that in addition to the former possesses rotational degrees of freedom. The length of the rod adds additional features to the physics of the problem in contrast to the movement of the pair point-like objects in the two-body problem reported in [3]. The leaning rod against the cube constitutes a holonomic constraint reducing the number of degrees of freedom.…”

“…In other words the rod from its initial position θ(t = 0) = 14.47˚ falls only 4.86˚ and separates. Equation(14)is a cubic trigonometric equation, its cubic term, 3sin3 (θ), can be dropped reducing it to a linear trig equation, 6sin(θ) − 1 = 0, with a root of, θ = 9.59˚. This neat approach underestimating the exact solution only by 0.2%!…”

Modelling Atwood’s Machine with Three Degrees of Freedom

“…Although a general solution of this system can not be found in symbolic form, one can choose some realistic values of the system parameters and solve the equations numerically. Such simulation and real experiments show that the oscillations can modify the system motion significantly and some unexpected kinds of motion such as a quasi-periodic or chaotic motion can arise (see [2][3][4][5][6][7][8][9]). …”