whereas the method of multiple scales and similar are better suited for capturing
slowly modulated fast motions (Fig. 7.1(b)).
7.3 Simple Examples
Next we use the MDSM to illustrate the stiffening, the biasing, and the smoothening
effect of HF excitation in simple physical settings, which will also introduce much
of the terminology to be used subsequently.
7.3.1 Pendulum on a Vibrating Support (Stiffening
and Biasing)
Studies of the pendulum with a rapidly vibrating support (Fig. 7.2) have a long
history (e.g., Stephenson 1908a,b, 1909; Hirsch 1930; Lowenstern 1932; Kapitza
1951; Panovko and Gubanova 1965; Bogdanoff and Citron 1965), which is still
continuing (e.g., Acheson 1993, 1995; Acheson and Mullin 1993; Anderson and
Tadjbakhsh 1989; Fenn et al. 1998; Guran and Plaut 1993; Jensen et al. 2004; Sah
et al. 2013; Schmitt and Bayly 1998; Sudor and Bishop 1999; Weibel et al. 1997).
Here it serves to illustrate the stiffening and biasing effects of HF excitation, which
in turn influence the effective natural frequency and the existence and stability of
equilibriums. The equation of motion is:
Fig. 7.1 (a) Large, slow motions o verlaid by small, fast oscillations, u(t) = sin(et) + e sin(xt),
e ( 1; (b) Slowly modulated fast motion, u(t) = sin(ext)sin(xt)
Fig. 7.2 Pendulum with a vibrating support
7.2 The Method of Direct Separation of Motions (MDSM)
393