The condition under (3.258), that f and g should be bounded and
Lipschitz-continuous in (A, h) 2 D & R
2 , is seen to be fulfilled when D/A is
bounded and Lipschitz-continuous in A; this holds true under the already stated
assumption A |D|.
With the conditions fulfilled, application of (3.259) with (3.256) to (3.266) gives
the averaged system
_
A 1 ¼ À ~
bA 1 ;
_
h 1 ¼ À
2D
pA 1
:
ð3:269Þ
where
~
b ¼ b þ
1 À R
p
;
ð3:270Þ
and where also here the subscript 1 indicates a first-order approximate solution,
asymptotically valid under assumptions (3.261). The effective damping constant ~
b
is seen to integrate the two dissipative effects present: inelastic restitution during
impacts (parameter R), and viscous damping in-between impacts (parameter b). The
linear system (3.269) can readily be integrated to give:
A 1 ¼ C 1 e
À ~
bt
; C 1 [ 0;
h 1 ¼ À
2D
pC 1 ~
b
e
~
bt
þ C 2 ;
ð3:271Þ
where the constants C 1 and C 2 are determined by the initial conditions.
Back-substituting this into (3.264) and (3.262) then gives the approximate solution
s 1 for the original variable s:
s 1 ¼ C 1 e
À ~
bt sin t À
2D
pC 1 ~
b
e
~
bt
þ C 2
À D; 0\t\t à ;
t à ¼ O min ~
b
À1
; D
j j
À1 ; ~
b
À1 ln C 1 D
j j
À1
n
o
;
ð3:272Þ
where the time-horizon t * ensures the error in s 1 to be of the same magnitude order
as the small parameters (cf. the error estimate under (3.259)), and that A 1 > |D| (to
ensure boundedness and Lipschitz continuity, cf. the remark under (3.268)).
Fig. 3.28 compares this approximate solution with results of numerical simulation of the original equation of motion (3.260) (using the MATLAB-function
ode23, with impacts handled by the event function feature), for parameters as
indicated in the figure legend. The equilibrium mass-to-stop distance D is positive
3.9 Vibro-Impact Analysis Using Discontinuous Transformations
183
Lipschitz-continuous in (A, h) 2 D & R
2 , is seen to be fulfilled when D/A is
bounded and Lipschitz-continuous in A; this holds true under the already stated
assumption A |D|.
With the conditions fulfilled, application of (3.259) with (3.256) to (3.266) gives
the averaged system
_
A 1 ¼ À ~
bA 1 ;
_
h 1 ¼ À
2D
pA 1
:
ð3:269Þ
where
~
b ¼ b þ
1 À R
p
;
ð3:270Þ
and where also here the subscript 1 indicates a first-order approximate solution,
asymptotically valid under assumptions (3.261). The effective damping constant ~
b
is seen to integrate the two dissipative effects present: inelastic restitution during
impacts (parameter R), and viscous damping in-between impacts (parameter b). The
linear system (3.269) can readily be integrated to give:
A 1 ¼ C 1 e
À ~
bt
; C 1 [ 0;
h 1 ¼ À
2D
pC 1 ~
b
e
~
bt
þ C 2 ;
ð3:271Þ
where the constants C 1 and C 2 are determined by the initial conditions.
Back-substituting this into (3.264) and (3.262) then gives the approximate solution
s 1 for the original variable s:
s 1 ¼ C 1 e
À ~
bt sin t À
2D
pC 1 ~
b
e
~
bt
þ C 2
À D; 0\t\t à ;
t à ¼ O min ~
b
À1
; D
j j
À1 ; ~
b
À1 ln C 1 D
j j
À1
n
o
;
ð3:272Þ
where the time-horizon t * ensures the error in s 1 to be of the same magnitude order
as the small parameters (cf. the error estimate under (3.259)), and that A 1 > |D| (to
ensure boundedness and Lipschitz continuity, cf. the remark under (3.268)).
Fig. 3.28 compares this approximate solution with results of numerical simulation of the original equation of motion (3.260) (using the MATLAB-function
ode23, with impacts handled by the event function feature), for parameters as
indicated in the figure legend. The equilibrium mass-to-stop distance D is positive
3.9 Vibro-Impact Analysis Using Discontinuous Transformations
183