Krysko
Vadim A.
Mathematical Sciences
Department of Mathematics and Modeling, Saratov State Technical University, Politekhnicheskaya 77, 410054, Saratov, Russia
Department of Mathematics and Modeling, Saratov State Technical University, Politekhnicheskaya 77, 410054, Saratov, Russia
Chaotic Vibrations of Conical and Spherical Shells and Their Control
Polar
importance
Novel non-linear dynamical phenomena
vibration type charts
result reliability
charts
applications
chaotic vibrations
symmetric deformation
plate/shell surface
shallow rotational shells
type charts
control parameters
types
deformation
non-linear ordinary differential
set
59-80
regime
validity
https://scigraph.springernature.com/explorer/license/
particular importance
closed shallow rotational shells
plate
dynamical phenomena
plates/shells
spherical shell
stability loss phenomena
parameters
thickness
order
initial imperfection/deflection
studied continuous systems
loss phenomenon
shell
account
non-constant thickness
reliability
members
boundary conditions
en
shell surface
rotational shells
deflection
shell slopes
imperfection/deflection
2022-01-01T19:17
2014-09-10
control
classical approach
transition
different shell slopes
engineering applications
structural members
vibration
This chapter is aimed on investigation of non-linear dynamics of conical and spherical shells. The variational equations are derived, and then the problem is reduced to a set of non-linear ordinary differential and algebraic equations. Since further axially symmetric deformation of closed shallow rotational shells and circled plates subjected to uniformly distributed periodic load being normal to the middle plate/shell surface are studied, the polar co-ordinates are used and four types of boundary conditions are investigated. The obtained equations are solved numerically, and the results reliability and validity are discussed in either regular, bifurcation or chaotic regimes including constant and non-constant thickness of the mentioned structural members, taking into account an initial imperfection/deflection. The classical approaches (time histories and frequency power spectra) are used to monitor different transitions from periodic to chaotic vibrations. Novel non-linear dynamical phenomena exhibited by the studied plates/shells are detected and discussed versus the chosen control parameters. In particular, the so called vibration type charts (amplitude—frequency of excitation planes) versus the different shell slopes are reported, which are of a particular importance for direct engineering applications. Finally, it is demonstrated how one may control non-linear dynamics of the studied continuous systems by using their thickness, in order to avoid buckling and stability loss phenomena.
chaotic regime
surface
load
different transitions
false
chapters
problem
https://doi.org/10.1007/978-3-319-02535-3_3
conical
ordinary differential
equations
system
2014-09-10
conditions
differential
non-linear dynamics
algebraic equations
variational equations
investigation
continuous system
chapter
direct engineering applications
periodic load
middle plate/shell surface
phenomenon
bifurcation
slope
dynamics
non-linear dynamical phenomena
chapter
approach
doi
10.1007/978-3-319-02535-3_3
Holm
Altenbach
Pure Mathematics
Jan
Awrejcewicz
Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski Str., 90-924, Łódź, Poland
Department of Automation, Biomechanics and Mechatronics, Lodz University of Technology, 1/15 Stefanowski Str., 90-924, Łódź, Poland
Springer Nature - SN SciGraph project
Gennadi I.
Mikhasev
978-3-319-02534-6
978-3-319-02535-3
Shell and Membrane Theories in Mechanics and Biology
dimensions_id
pub.1007477928
Springer Nature