100
4 Reliability of Chaotic Vibrations of Euler-Bernoulli Beams with Clearance
L 2,c (w i, j , u i, j ) =
1
2
u i, j+1 − 2u i, j + u i, j−1
+
+
1
2c
(w i, j+1 − w i, j−1 )
1
c 2 (w i, j+1 − 2w i, j + w i, j−1 ); j = 0, ..., n.
In Eq. (4.3), for j = 1, j = n − 1, we need to employ the values of the contour
points, which are defined by the boundary conditions.
The obtained Cauchy problem is solved by the fourth- and the second-order
Runge-Kutta, the fourth-order Runge-Kutta-Fehlberg, the fourth-order Cash-Karp,
the eighth-order Runge-Kutta-Prince-Dormand and the implicit second- and fourthorder Runge-Kutta methods. On the basis of the described algorithms, the program
package has been developed, which allows one to solve the given problem with
respect to the control parameters
q 0 , ω p
. The main attention has been paid to control and avoid the occurrence of penetration of the structural elements. As it has
been already pointed out, the studied problems are strongly nonlinear, and hence an
important question regarding the reliability of the obtained results arises.
While studying the problems of contact interaction, one needs to detect the fundamental frequencies of the vibrational processes first, because, as it will be shown
later, a transition into chaotic vibrations takes place after the beam contact. One of
the methods devoted to studying the localization of the fundamental frequencies is
the principal component analysis.
4.4 Principal Component Analysis (PCA)
It should be emphasized that, formally, the principal component analysis (PCA),
aimed at the estimation of principal components with respect to the dissipative problems, can be always employed. However, in the case when the “signal” cannot be distinguished from “noise”, any earlier given accuracy is helpful. Typically, the “signal”
exhibits a relatively small dimension while keeping the relatively large amplitude.
On the other hand, “noise” exhibits a large dimension for a relatively small amplitude. This observations allow us to understand the main features of PCA. Namely,
the method works as a filter, i.e. the signal is mainly kept in projections onto the first
principal components, whereas the remaining components include mainly noise (for
more details see [3–32]).
As a result of solving PDEs by FDM, a matrix W composed of the values of the
time-dependent beam deflections measured in the nodes are obtained. Let us present a
matrix ˜
W in the form of its linear splitting ˜
W = T P
t
+ E. The matrix W contains all
elements, whereas the matrix ˜
W is yielded by the series with the account of k principal
components. In order to compute the score matrix T and the matrix of loading P, the
singular series development of the matrix W is carried out through the so-called autoscaling process. The auto-scaled matrix ¯
W = U SV
t
, where T = U S and P = V t, is
4 Reliability of Chaotic Vibrations of Euler-Bernoulli Beams with Clearance
L 2,c (w i, j , u i, j ) =
1
2
u i, j+1 − 2u i, j + u i, j−1
+
+
1
2c
(w i, j+1 − w i, j−1 )
1
c 2 (w i, j+1 − 2w i, j + w i, j−1 ); j = 0, ..., n.
In Eq. (4.3), for j = 1, j = n − 1, we need to employ the values of the contour
points, which are defined by the boundary conditions.
The obtained Cauchy problem is solved by the fourth- and the second-order
Runge-Kutta, the fourth-order Runge-Kutta-Fehlberg, the fourth-order Cash-Karp,
the eighth-order Runge-Kutta-Prince-Dormand and the implicit second- and fourthorder Runge-Kutta methods. On the basis of the described algorithms, the program
package has been developed, which allows one to solve the given problem with
respect to the control parameters
q 0 , ω p
. The main attention has been paid to control and avoid the occurrence of penetration of the structural elements. As it has
been already pointed out, the studied problems are strongly nonlinear, and hence an
important question regarding the reliability of the obtained results arises.
While studying the problems of contact interaction, one needs to detect the fundamental frequencies of the vibrational processes first, because, as it will be shown
later, a transition into chaotic vibrations takes place after the beam contact. One of
the methods devoted to studying the localization of the fundamental frequencies is
the principal component analysis.
4.4 Principal Component Analysis (PCA)
It should be emphasized that, formally, the principal component analysis (PCA),
aimed at the estimation of principal components with respect to the dissipative problems, can be always employed. However, in the case when the “signal” cannot be distinguished from “noise”, any earlier given accuracy is helpful. Typically, the “signal”
exhibits a relatively small dimension while keeping the relatively large amplitude.
On the other hand, “noise” exhibits a large dimension for a relatively small amplitude. This observations allow us to understand the main features of PCA. Namely,
the method works as a filter, i.e. the signal is mainly kept in projections onto the first
principal components, whereas the remaining components include mainly noise (for
more details see [3–32]).
As a result of solving PDEs by FDM, a matrix W composed of the values of the
time-dependent beam deflections measured in the nodes are obtained. Let us present a
matrix ˜
W in the form of its linear splitting ˜
W = T P
t
+ E. The matrix W contains all
elements, whereas the matrix ˜
W is yielded by the series with the account of k principal
components. In order to compute the score matrix T and the matrix of loading P, the
singular series development of the matrix W is carried out through the so-called autoscaling process. The auto-scaled matrix ¯
W = U SV
t
, where T = U S and P = V t, is