which can be rewritten as:
€ z ¼ sðz; _
z; tÞ þ vðz; _
z; tÞ;
ð7:16Þ
where v defines the vibrational forces:
vðz; _
z; tÞ s 1 ðz; _
z; u
Ã
; _
u
Ã
; tÞ
h
i þ fðz þ u
Ã
; _
z þ _
u
Ã
; t; sÞ
h
i :
ð7:17Þ
Vibrational forces do not necessarily have the physical unit of force, but should
be understood in a generalized sense to represent certain forcing terms in the
equation of motion. The physical unit may correspond to acceleration of state
variables such as translation, rotation, temperature, or electric charge.
Note that Eq. (7.16) for the slow motions z is quite similar to the original
Eq. (7.2) for the full motions x, though with the fast forces f replaced by (slow)
vibrational forces v, that have no explicit dependence on the fast time s. Note also
that the fast motions φ
* are determined as approximate solutions to (7.14); however,
φ
* could also be calculated by purely numerical methods, or even measured
experimentally.
To an observer filtering out the small HF components of the motions – as with
measurement instruments, human senses, or dedicated lowpass filters – it will
appear as if the system is influenced by the slow vibrational forces v, while the fast
forces f are not apparent. For example, one out of many ways to explain the
well-known inversion of a pendulum on a vibrating support (Sect. 7.3.1) is to show
that the effect of the HF excitation, on the average, corresponds to a slow vibrational force pulling the pendulum towards the upside-down equilibrium.
7.2.3 The MDSM Compared to Classic Perturbation
Approaches
The MDSM is typically used with excitations well away from system resonances
(Thomsen (2003b) represents an exception), and there should be components of the
excitation that are rapidly oscillating.
Several perturbation approaches also work by splitting motions into slow and fast
components. For example, as explained in Chap. 3, the method of multiple scales
explicitly introduces independent time scales for capturing motions at different scales
of time. Solutions are sought in the form of uniformly valid expansions, e.g.,
u(t;e) = u 0 (T 0 , T 1 , …, T n ) + eu 1 (T 0 , T 1 , …, T n ) + ÁÁÁ + e
n
u n (T 0 , T 1 , …, T n ), where
e ( 1 and T i = e
i
t. Here T 0 = t is the fast time, whereas T 1 , T 2 ,… are slow times.
Each term of the expansion usually becomes a product of slow and fast terms, e.g.,
u = a(et)b(t) + O(e).
With the MDSM, by contrast, solutions are obtained in the form of sums of slow
and fast components, e.g., u = z(et) + eφ(t). Hence, the MDSM lends itself naturally at describing slow motions overlaid by small, fast oscillations (Fig. 7.1(a)),
392
7 Special Effects of High-Frequency Excitation
€ z ¼ sðz; _
z; tÞ þ vðz; _
z; tÞ;
ð7:16Þ
where v defines the vibrational forces:
vðz; _
z; tÞ s 1 ðz; _
z; u
Ã
; _
u
Ã
; tÞ
h
i þ fðz þ u
Ã
; _
z þ _
u
Ã
; t; sÞ
h
i :
ð7:17Þ
Vibrational forces do not necessarily have the physical unit of force, but should
be understood in a generalized sense to represent certain forcing terms in the
equation of motion. The physical unit may correspond to acceleration of state
variables such as translation, rotation, temperature, or electric charge.
Note that Eq. (7.16) for the slow motions z is quite similar to the original
Eq. (7.2) for the full motions x, though with the fast forces f replaced by (slow)
vibrational forces v, that have no explicit dependence on the fast time s. Note also
that the fast motions φ
* are determined as approximate solutions to (7.14); however,
φ
* could also be calculated by purely numerical methods, or even measured
experimentally.
To an observer filtering out the small HF components of the motions – as with
measurement instruments, human senses, or dedicated lowpass filters – it will
appear as if the system is influenced by the slow vibrational forces v, while the fast
forces f are not apparent. For example, one out of many ways to explain the
well-known inversion of a pendulum on a vibrating support (Sect. 7.3.1) is to show
that the effect of the HF excitation, on the average, corresponds to a slow vibrational force pulling the pendulum towards the upside-down equilibrium.
7.2.3 The MDSM Compared to Classic Perturbation
Approaches
The MDSM is typically used with excitations well away from system resonances
(Thomsen (2003b) represents an exception), and there should be components of the
excitation that are rapidly oscillating.
Several perturbation approaches also work by splitting motions into slow and fast
components. For example, as explained in Chap. 3, the method of multiple scales
explicitly introduces independent time scales for capturing motions at different scales
of time. Solutions are sought in the form of uniformly valid expansions, e.g.,
u(t;e) = u 0 (T 0 , T 1 , …, T n ) + eu 1 (T 0 , T 1 , …, T n ) + ÁÁÁ + e
n
u n (T 0 , T 1 , …, T n ), where
e ( 1 and T i = e
i
t. Here T 0 = t is the fast time, whereas T 1 , T 2 ,… are slow times.
Each term of the expansion usually becomes a product of slow and fast terms, e.g.,
u = a(et)b(t) + O(e).
With the MDSM, by contrast, solutions are obtained in the form of sums of slow
and fast components, e.g., u = z(et) + eφ(t). Hence, the MDSM lends itself naturally at describing slow motions overlaid by small, fast oscillations (Fig. 7.1(a)),
392
7 Special Effects of High-Frequency Excitation