1549312215-Complex_Analysis_for_Mathematics_and_Engineering_5th_edition__Mathews

{

1,

6. U(O) = 0,
   
   
   • 1,
   
   
   
   

12.3 • VIBRATIONS IN MECHANICAL SYSTEMS 529

for ~ < 8 < 1Ti

for - 2 " < 0 < ~;

for - 1T<0 < -;.

Approximations for U7 (8) and U1 (rcos8,rsin8) are shown in Figure 12.16.

Figure 12.16

'· u (8) = l

•. u (8) = l

0,

.. - 6

-2- ·

ill 2 ,

0,

0,

• 1,
  1,
  0,

s

for ~ $ 8 $ 1Ti

0 $ 0 < ~;

for - 2 " $ 8 < O;

for -1T < 8 < - 2 ".

for~ < 0 < 1T;

for0<0<~;

for-;< 8 < O;

for -1T < 8 < - 2 ".

12 .3 Vibrations in Mechanical Systems

Consider a spring that resists compression as well as extension, is suspended
vertically from a fixed support, and has a body of mass m attached to its lower
end. We make the assumption that m is much larger than the mass of the spring
so that we can neglect the mass of the spring. If there is no motion, then the
system is in static equilibrium, as illustrated in Figure 12.17(a). If the mass is
pulled down farther and released, then it will undergo an oscillatory motion.
If there is no friction to slow the motion of the mass, then we say that the
system is 11ndamped. We determine the motion of this mechanical system by
considering the forces acting on the mass during the motion. Doing so leads to
a differential equation relating the displacement as a function of time. The most
obvious force is that of gravitational attraction acting on the mass m and given