where 〈〉 denotes averaging over u during a single period of f:
g u
ð Þ
h
i u ¼
1
2p
Z 2p
0
g u
ð Þdu:
ð3:256Þ
For application of the method it is important to recall that a function f(x, u), x(u)
2 D & R
n , is bounded and Lipschitz-continuous in x on D if, and only if,
non-negative constants M f and k f exist such that, for all (x 1 , x 2 ) 2 D:
fðx; uÞ
k
k M f and fðx 2 ; uÞ À fðx 1 ; uÞ
k
k k x x 2 À x 1
k
k :
ð3:257Þ
In other words, a Lipschitz-continuous function f(x, u) is limited in how sudden
it can change with x: The slope of a line (in R
n ) joining any two points on the graph
(in R
n
) of the function will never exceed its Lipschitz constant k.
Extended First-order Averaging – for Discontinuous Systems For
vibro-impact problems formulated with kinematic impact conditions, the inherent
discontinuities in velocity preclude using standard averaging. However, a special
form of the averaging theorem can be proved (Zhuravlev and Klimov 1988), that
holds for systems with small (i.e. O(e)) discontinuities in the state variables, that is,
instead of (3.254):
dx
du
¼ efðx; uÞ for u 6 ¼ jp; j ¼ 0; 1; . . .; xðuÞ 2 D & R
n
; e ( 1;
x þ À x À ¼ egðx À Þ for u ¼ jp;
ð3:258Þ
where f is 2p-periodic (not necessarily continuous) in u, f and g are bounded and
Lipschitz-continuous in x on x 2 D & R
n , and x – and x + are the states x immediately before and after the j’th impact, corresponding to the passage of u through the
value jp. For that case the averaged system becomes, instead of (3.255) (Zhuravlev
1988; Fidlin 2005, 2006):
dx 1
du
¼ e fðx 1 ; uÞ
h
i u þ p
À1
gðx 1 Þ
;
ð3:259Þ
where again the subscript 1 indicates the first level of approximation to x, i.e.
x = x 1 + ex 2 + O(e
2 ), so that the error ||x 1 (t) – x(t)|| is O(e) on the time-scale 1/e.
The averaged motions x 1 may themselves be determined at different levels of
accuracy, i.e. x 1 = x 11 + ex 12 + O(e
2 ), of which (3.259) gives the first order
approximation x 11 , while the second-order approximation(s) x 12 (and x 2 ) are
derived in Thomsen and Fidlin 2008.
180
3 Nonlinear Vibrations: Classical Local Theory
g u
ð Þ
h
i u ¼
1
2p
Z 2p
0
g u
ð Þdu:
ð3:256Þ
For application of the method it is important to recall that a function f(x, u), x(u)
2 D & R
n , is bounded and Lipschitz-continuous in x on D if, and only if,
non-negative constants M f and k f exist such that, for all (x 1 , x 2 ) 2 D:
fðx; uÞ
k
k M f and fðx 2 ; uÞ À fðx 1 ; uÞ
k
k k x x 2 À x 1
k
k :
ð3:257Þ
In other words, a Lipschitz-continuous function f(x, u) is limited in how sudden
it can change with x: The slope of a line (in R
n ) joining any two points on the graph
(in R
n
) of the function will never exceed its Lipschitz constant k.
Extended First-order Averaging – for Discontinuous Systems For
vibro-impact problems formulated with kinematic impact conditions, the inherent
discontinuities in velocity preclude using standard averaging. However, a special
form of the averaging theorem can be proved (Zhuravlev and Klimov 1988), that
holds for systems with small (i.e. O(e)) discontinuities in the state variables, that is,
instead of (3.254):
dx
du
¼ efðx; uÞ for u 6 ¼ jp; j ¼ 0; 1; . . .; xðuÞ 2 D & R
n
; e ( 1;
x þ À x À ¼ egðx À Þ for u ¼ jp;
ð3:258Þ
where f is 2p-periodic (not necessarily continuous) in u, f and g are bounded and
Lipschitz-continuous in x on x 2 D & R
n , and x – and x + are the states x immediately before and after the j’th impact, corresponding to the passage of u through the
value jp. For that case the averaged system becomes, instead of (3.255) (Zhuravlev
1988; Fidlin 2005, 2006):
dx 1
du
¼ e fðx 1 ; uÞ
h
i u þ p
À1
gðx 1 Þ
;
ð3:259Þ
where again the subscript 1 indicates the first level of approximation to x, i.e.
x = x 1 + ex 2 + O(e
2 ), so that the error ||x 1 (t) – x(t)|| is O(e) on the time-scale 1/e.
The averaged motions x 1 may themselves be determined at different levels of
accuracy, i.e. x 1 = x 11 + ex 12 + O(e
2 ), of which (3.259) gives the first order
approximation x 11 , while the second-order approximation(s) x 12 (and x 2 ) are
derived in Thomsen and Fidlin 2008.
180
3 Nonlinear Vibrations: Classical Local Theory