transformation (3.264) will not work for that: When hitting two stops, motions of
the mass cannot be adequately approximated by slowly amplitude- and
phase-modulated time-harmonic functions. Instead we rely on the well-proven
general method of transforming weakly nonlinear differential equations into the
first-order form appropriate for standard averaging (cf. ending of Sect. 3.5.5), that
is: First solve the unperturbed system, then consider the free constants of this
solutions as new time-dependent variables, and finally substitute the solution into
the nonlinear equations of motions to obtain the equations governing the slow
evolution of these variables. For classical weakly nonlinear oscillators this approach
results in the van der Pol transformation (3.264). Here, to find a workable transformation, we consider the unperturbed system corresponding to (3.298), which can
be written in terms of a potential V:
€ z 0 þ
dV
dz 0
¼ 0; V z 0
ð Þ ¼
Z z 0
0
M f
ð ÞP f
ð Þdf;
ð3:300Þ
where subscript zero indicates unperturbed variables. Multiplying by _
z 0 and integrating over time gives:
E ¼
1
2
_
z
2
0 þ Vðz 0 Þ;
ð3:301Þ
where the potential energy V(z 0 ) is p-periodic with each period defined by a positive
parabola, and the constant of integration E is the mechanical energy of the
unperturbed system. This energy is limited by two conditions: First, the condition
_
z
2
0 [ 0 with (3.301) gives E > V(z 0 ), which implies E [ maxðVðz 0 ÞÞ ¼ Vðp=2Þ ¼
p
2
=8: Second, we consider only slipping motions, i.e. _
s\v b ; which with (3.297),
(3.301), and M
2
¼ 1 implies that E\
p 2
8 ð1 þ ðv b =DÞ
2 Þ: Hence:
1\
8E
p 2 \1 þ v b =D
ð
Þ
2 :
ð3:302Þ
With E thus restricted, (3.301) gives the velocity of z 0 :
_
z 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 E À Vðz 0 Þ
ð
Þ
p
:
ð3:303Þ
Then we use this solution for the unperturbed system as a basis for a transformation of variables, where E is considered the new dependent variable, i.e. we
let:
_
z ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 E À VðzÞ
ð
Þ
p
;
ð3:304Þ
194
3 Nonlinear Vibrations: Classical Local Theory
the mass cannot be adequately approximated by slowly amplitude- and
phase-modulated time-harmonic functions. Instead we rely on the well-proven
general method of transforming weakly nonlinear differential equations into the
first-order form appropriate for standard averaging (cf. ending of Sect. 3.5.5), that
is: First solve the unperturbed system, then consider the free constants of this
solutions as new time-dependent variables, and finally substitute the solution into
the nonlinear equations of motions to obtain the equations governing the slow
evolution of these variables. For classical weakly nonlinear oscillators this approach
results in the van der Pol transformation (3.264). Here, to find a workable transformation, we consider the unperturbed system corresponding to (3.298), which can
be written in terms of a potential V:
€ z 0 þ
dV
dz 0
¼ 0; V z 0
ð Þ ¼
Z z 0
0
M f
ð ÞP f
ð Þdf;
ð3:300Þ
where subscript zero indicates unperturbed variables. Multiplying by _
z 0 and integrating over time gives:
E ¼
1
2
_
z
2
0 þ Vðz 0 Þ;
ð3:301Þ
where the potential energy V(z 0 ) is p-periodic with each period defined by a positive
parabola, and the constant of integration E is the mechanical energy of the
unperturbed system. This energy is limited by two conditions: First, the condition
_
z
2
0 [ 0 with (3.301) gives E > V(z 0 ), which implies E [ maxðVðz 0 ÞÞ ¼ Vðp=2Þ ¼
p
2
=8: Second, we consider only slipping motions, i.e. _
s\v b ; which with (3.297),
(3.301), and M
2
¼ 1 implies that E\
p 2
8 ð1 þ ðv b =DÞ
2 Þ: Hence:
1\
8E
p 2 \1 þ v b =D
ð
Þ
2 :
ð3:302Þ
With E thus restricted, (3.301) gives the velocity of z 0 :
_
z 0 ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 E À Vðz 0 Þ
ð
Þ
p
:
ð3:303Þ
Then we use this solution for the unperturbed system as a basis for a transformation of variables, where E is considered the new dependent variable, i.e. we
let:
_
z ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 E À VðzÞ
ð
Þ
p
;
ð3:304Þ
194
3 Nonlinear Vibrations: Classical Local Theory