Читать книгу Introduction to Mechanical Vibrations - Ronald J. Anderson - Страница 19

The generalized force, , associated with the generalized coordinate, , accounts for the effect of externally applied forces that are not included in the potential energy. We normally include elastic (i.e. spring) forces and gravitational forces in the potential energy and all others enter through the use of generalized forces.

Given a three‐dimensional applied force

(1.24)

with a position vector

(1.25)

relative to a fixed point, we define the right‐hand side of Lagrange's Equation for generalized coordinate to be the generalized force, , where

(1.26)

The two most common methods for finding the generalized forces are as follows.

1 The formal methodThe most formal approach, and one that always works, starts with the vector expression of the absolute position of the point of application of the forceand then writes the generalized force as(1.27) Equation 1.27 can be written for each of applied forces and the resulting scalar generalized forces can be added together to give the total generalized force for generalized coordinate as(1.28)

2 The intuitive approachLet there be generalized coordinates specifying the position of a force acting on a dynamic system in Cartesian Coordinates. The force will be acting at the point where the coordinates , and are functions of the generalized coordinates through and of time, , as follows(1.29) Variations in the position of the force as the generalized coordinates are varied while time is held constant can be written as(1.30) If we are trying to find the generalized force corresponding to only one of the generalized coordinates, say , we rewrite Equation 1.30 with and , giving(1.31) Now consider Equation 1.26 with each side multiplied by (1.32) The terms from Equation 1.31 can be substituted into the right‐hand side of Equation 1.32 to yield(1.33) The right‐hand side of Equation 1.33 can be seen to be the work done by the applied force as its position varies due to changes in the generalized coordinate while all other generalized coordinates and time are held constant.Using the intuitive approach to finding generalized forces, the analyst will consider, in sequence, the variation of individual generalized coordinates and will write expressions for the total work done during each variation. The generalized force associated with each generalized coordinate will be the work done during the variation of that coordinate, , divided by the variation in the coordinate. That is,(1.34)