Page 110 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 4.3 Laplace Transform Formulation 97

Figure 4.2-4.
Solution: The velocity pulse at r = 0 has a sudden jump from zero to ¿Jq, and its rate of
change (or acceleration) is infinite. Differentiating y{t) and recognizing that
id/dt)u(t) = 8{t), a delta function at the origin, we obtain
y = -///() 0(0 u(t)

By substituting ÿ into Eq. (4.2-5), the result is

z(t) = - ^ sin w„(t - i)
•'O ^0
^/'"sinio„(r - ^)d^
COn J q Jg
^’n^n
-(^e sin - cos (4.2-6)
1 + i^nh))

4.3 LAPLACE TRANSFORM FORMULATION

The Laplace transform method of solving the differential equation provides a
complete solution, yielding both transient and forced vibrations. For those unfamil­
iar with this method, a brief presentation of the Laplace transform theory is given
in Appendix B. In this section, we illustrate its use by some simple examples.
Example 4.3-1
Formulate the Laplace transform solution of a viscously damped spring-mass system
with initial conditions x(0) and i(0).
Solution: The equation of motion of the system excited by an arbitrary force F(t) is
mx cx kx = F(t)