Kinetic Energy

The kinetic energy K of a particle Q in F can be written as

m

к = 2 VQ ■ VQ (2.5)

where m is the mass of the particle and vq is the velocity of Q in F. To use this expression for the kinetic energy in mechanics, F must be an inertial frame.

The kinetic energy of a rigid body B in F can be written as

„ m 1

K = 2 vc ■ vc + 2 mb ■ Ic ■ mb

where m is the mass of the body, Ic is the inertia tensor of B about C, vc is the velocity of C in F, and mb is the angular velocity of B in F. In two-dimensional motion of a balanced body, we may simplify this to

where Ic is the moment of inertia of B about C about z, 0 is the angular velocity of B in F about z, and z is an axis perpendicular to the plane of motion. A similar equation also holds if C is replaced by O, a pivot, such that

K = I0 02 (2.8)

where I0 is the moment of inertia of B about an axis z passing through O. To use these expressions for kinetic energy in mechanics, F must be an inertial frame.

2.1.4 Work

The work W done in a reference frame F by a force F acting at a point Q, which may be either a particle or a point on a rigid body, may be written as

W = f ‘f ■ VQdt (2.9)

Jti

where vq is the velocity of Q in F, and ti and t2 are arbitrary fixed times. When there are contact and distance forces acting on a rigid body, we may express the work done by all such forces in terms of their resultant R, acting at C, and the total torque T of all such forces about C, such that

(2.10)

The most common usage of these formulae in this text is the calculation of virtual work (i. e., the work done by applied forces through a virtual displacement).

2.1.5 Lagrange’s Equations

There are several occasions to make use of Lagrange’s equations when calculating the forced response of structural systems. Lagrange’s equations are derived in the Appendix and can be written as

where L = K – P is called the “Lagrangean”—that is, the difference between the total kinetic energy, K, and the total potential energy, P, of the system. The general­ized coordinates are ft; the term on the right-hand side, Et, is called the “generalized force.” The latter represents the effects of all nonconservative forces, as well as any conservative forces that are not treated in the total potential energy.

Under many circumstances, the kinetic energy can be represented as a function of only the coordinate rates so that

K = K(fb f2, Із, • • •)

The potential energy P consists of contributions from strain energy, discrete springs, gravity, applied loads (conservative only), and so on. The potential energy is a function of only the coordinates themselves; that is

Thus, Lagrange’s equations can be written as

2.2 Modeling the Dynamics of Strings

Among the continuous systems to be considered in other chapters, the string is the simplest. Typically, by this time in their undergraduate studies, most students have had some exposure to the solution of string-vibration problems. Here, we present for future reference a derivation of the governing equation, the potential energy, and the kinetic energy along with the virtual work of an applied distributed transverse force.