106
K
M
X(T)
M.
F sin (NT)
(1)X
100
-100
(
XImax
80
0
0
4 Dimensional Scaling Analysis
+ KX = F sin (2T), X(0) = Xo,
1
n
2
10
T
3
Fig. 4.1 The elementary problem of a mass on spring with an applied force: (Left) dimensional
parameters, (Right) solutions X(T) for various parameters and (Inset) the amplitude in relation to
the forcing frequency
ada
4.2 The governing equation for the linear mass-spring system shown in Fig. 4.1 is
d²X
dT²
dX
dT\T=0
20
= Vo,
where X(T) [m] is the position of the mass as a function of time with mass M [kg], and
spring coefficient K [N/m], magnitude of the applied force, F [N], forcing frequency,
2 [s], as well as general initial conditions,
Nondimensionalize and select length- and time-scales L, T to normalise the coef-
ficients of the terms on the left side of the ODE and the initial condition for position.
Write and solve the scaled problem, and identify the dimensionless parameter for the
ratio of the forcing frequency to resonant frequency, where the solution's amplitude
grows without bound, as shown for 22 in Fig. 4.1.
4.3 Consider the initial value problem for a damped driven nonlinear oscillator:
