Page 112 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 4.3 Laplace Transform Formulation 99

g

Figure 4.3-3.
the floor, the differential equation of motion for m applicable as long as the spring
remains in contact with the floor is
mx kx = mg (a)
Taking the Laplace transform of this equation with the initial conditions x(0) = 0 and
i( 0) = ^J2gh, we can write the subsidiary equation as

x{s) = ^ ------T + (b)
+ Oil +

where u>„ = ^k/m is the natural frequency of the system. From the inverse transfer-
mation of the displacement equation becomes
g
-
x(t) = --------sin (oj H------ t (1 - cos o)j)
^ ^ ojI
(c)
2gh g Ÿ . ,

t —</)) + —2 -^(0 ^ ^
0)n
where the relationship is shown in Fig. 4.3-3. By differentiation, the velocity and
acceleration are

+ cos{o)„t - 4>)

jc(t) = - i o l J + l - ^ \ sin(&>„i - (f))
V V /
We recognize here that g/(o^ = 8^^ and that the maximum displacement and acceler­
ation occur at sin - (/>) = 1.0. Thus, the maximum acceleration in terms of
gravity is found to depend only on the ratio of the distance dropped to the statical
deflection as given by the equation
X 2h .
(d)

A plot of this equation is shown in Fig. 4.3-4.