Читать книгу Introduction to Mechanical Vibrations - Ronald J. Anderson - Страница 13

First courses on the subject of Dynamics, whether for particles or rigid bodies, are primarily concerned with teaching the basics of kinematics, free body diagrams, and applications of Newton's Laws of Motion. Applying these three concepts sequentially will lead to a set of simultaneous force and moment balance equations that take account of kinematic constraints.

There are different ways of approaching these problems. One can use a formal vector‐based approach and we will start with that here because it gives a complete set of governing equations including solutions for all constraint forces that are required to enforce kinematic constraints on the motion. A shorthand version of this approach which may be called an “informal vector approach” is often used in practice and that will be the second method addressed here. It typically works with two‐dimensional views and leads to the governing equations of motion without necessarily solving for all constraint forces. The third approach will see the equations of motion derived using Lagrange's Equations. This is a work/energy approach that leads to the nonlinear differential equation of motion with minimal effort on the part of the analyst. The kinematic constraint forces are automatically eliminated as the governing equations are derived, leaving a designer with no information about forces acting on elements of the system unless extra work is done to find them. Lagrange's Equations are not typically introduced to undergraduate engineers as often as Newton's Laws are, so extra effort is made in this chapter to introduce the procedures for applying Lagrange's Equations to mechanical systems.

As an example, consider Figure 1.1. The figure shows a small bead with mass, , sliding on a frictionless semicircular wire that rotates about a vertical axis with a constant angular velocity, . The wire has radius . Gravity acts to pull the mass to the bottom of the semicircle while centripetal effects try to move it to the top. The single angular degree of freedom, , is sufficient to describe the motion of the bead on the wire.

Figure 1.1 A bead on a wire.