Page 461 - Dynamics of Mechanical Systems

0593_C13_fm Page 442 Monday, May 6, 2002 3:21 PM

442 Dynamics of Mechanical Systems

By comparing the solution procedures between the initial and boundary value problems,
we see that the solution of the initial value problem is simpler and more direct than that
of the boundary value problem. Therefore, for simplicity, in the sequel we will consider
primarily initial value problems.
Consider next the damped linear oscillator equation:

+
+
˙
˙˙
mx cx kx = 0 (13.2.18)
where m, c, and k are constants. This is the classical second-order homogeneous linear
ordinary differential equation with constant coefficients. From References 13.9 to 13.13,
we see that the solution depends upon the relative magnitudes of m, c, and k. If the product
2
4km is less than c , we can write the solution in the form:
x = e [ cos ω t Bsin ω t] (13.2.19)
+
−µ
t
A
where µ and ω are defined as:
/
D
D

2
µ = cm and ω =  k − c 2 2   12 (13.2.20)
 m 4 m 
2
If, in Eq. (13.2.18), the product c exceeds 4km, the solution takes the form:
x = Ae −(µ + vt ) + Be −(µ − vt ) (13.2.21)

where µ and ν are defined as:

/
D
2
µ = cm and ν =  c 2 − k   12 (13.2.22)

 m4 2 m 
Finally, if in Eq. (13.2.18), the product 4km is exactly equal to c , the solution takes the
2
form:
x = e ( A Bt) (13.2.23)
−µ
+
t
where µ is:

2
µ= cm = k m (13.2.24)

2
Next, consider the forced linear oscillator described by the equation (with 4km > c ):
mx cx kx = () (13.2.25)
+
+
˙˙
˙
f t
where f(t) is the forcing function. The forcing function typically has the form:
F
ft () = cos pt (13.2.26)