Page 265 - Dynamics of Mechanical Systems

0593_C08_fm Page 246 Monday, May 6, 2002 2:45 PM

246 Dynamics of Mechanical Systems

FIGURE 8.4.4
A free-body diagram of the
pendulum mass.

and

˙˙
mlθ + mg sinθ = 0 (8.4.2)
or, alternatively,

T = mlθ ˙ 2 + mg cosθ (8.4.3)

and

˙˙ g sinθ = 0 (8.4.4)
θ +( ) l
Equation (8.4.4) is the classic pendulum equation. It is the governing equation for the
orientation angle θ. Observe that it does not involve the pendulum mass m, but simply
the length . This means that the pendulum motion is independent of its mass.
We will explore the solution of Eq. (8.4.4) in Chapter 13, where we will see that it is a
nonlinear ordinary differential equation requiring approximate and numerical methods
to obtain the solution. The nonlinearity occurs in the sinθ term. If it happens that θ is
“small” so that sinθ may be closely approximated by θ, the equation takes the linear form:

˙˙ θ +( ) = 0 (8.4.5)
θ
l
g
Equation (8.4.5) is called the linear oscillator equation. It usually forms the starting point
for a study of vibrations (see Chapter 13). Once Eq. (8.4.4) is solved for θ, the result may
be substituted into Eq. (8.4.3) to obtain the rod tension T.
Finally, it should be noted that dynamics principles such as d’Alembert’s principle or
Newton’s laws simply lead to the governing equation. They do not lead to solutions of
the equations, although some principles may produce equations that are in a form more
suitable for easy solution than others.

8.5 A Smooth Particle Moving Inside a Vertical Rotating Tube
For a second example illustrating the use of d’Alembert’s principle consider a circular tube
T with radius r rotating with angular speed Ω about a vertical axis as depicted in Figure
8.5.1. Let T contain a smooth particle P with mass m which is free to slide within T. Let
the position of P within T be defined by the angle θ as shown. Let n and n be radial and
r θ