184
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
6.7 Construction of the Charts of Vibration Characters
Verusus Amplitude and Frequency of the Exciting Load
and Their Analysis (First-, Second- and Third-Order
Kinematic Hypotheses)
The use of the charts of vibration characters in order to study system evolution in time
recalls Poincaré’s statement that nonlinear dynamics should be viewed from different
perspectives. It also allows to obtain full picture of the studied process development.
Investigation of dynamic process of nonlinear mathematical models of Euler–
Bernoulli, Timoshenko and Sheremetev–Pelekh with a help of the vibration regime
(character) charts constructed with the help of the harmonic excitation parameters,
i.e. amplitude and frequency {q 0 , ω p }, allows to analyse the wide palette of problems
which gives hints to analyse behaviour of real constructions.
One of the important problems while constructing and analysing charts of vibrations is the accuracy with which we get information about the dynamic processes.
The second important problem deals with getting minimum of computational time
needed for the chart construction.
We have investigated suitability of the information mapping in vibration regimes
for the charts of the vibration characters with the resolution of 100 × 100, 200 ×
200, 300 × 300, 600 × 600 points. The charts with the resolutions 300 × 300 and
600 × 600 practically coincide. In order to obtain the charts with vibration regimes
of 300 × 300, there is a need to solve 9 · 10
4 case studies, and to analyse the obtained
results on the basis of frequency spectrum, autocorrelation functions and spectrum of
Lyapunov exponents. In order to construct a chart with resolution of 600 × 600, we
need to analyse 12 · 10
4 case studies, which requires long computational time. This
is why, we have chosen the resolution 300 × 300. Table 6.18 presents the charts of
the vibration regimes of the various resolutions obtained while solving problems of
the Bernoulli–Euler beam model vibrations without an account of the size-dependent
behaviour for the boundary clamping conditions.
w(0, t) = w(1, t) = 0,
∂w(0, t)
∂ x
=
∂w(1, t)
∂ x
= 0,
u(0, t) = u(1, t) = 0.
The same notation is used in the remaining charts in this section.
6.7.1 Analysis of the Charts of the Vibration Regimes for the
Euler–Bernoulli, Timoshenko and Sheremetev–Pelekh
Modes Versus λ
We consider the charts of the vibration regimes of the mentioned models with and
account of different values of the beam relative thickness λ for two different values
of the size-dependent parameter l/ h = 0 and l/ h = 0.3.
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
6.7 Construction of the Charts of Vibration Characters
Verusus Amplitude and Frequency of the Exciting Load
and Their Analysis (First-, Second- and Third-Order
Kinematic Hypotheses)
The use of the charts of vibration characters in order to study system evolution in time
recalls Poincaré’s statement that nonlinear dynamics should be viewed from different
perspectives. It also allows to obtain full picture of the studied process development.
Investigation of dynamic process of nonlinear mathematical models of Euler–
Bernoulli, Timoshenko and Sheremetev–Pelekh with a help of the vibration regime
(character) charts constructed with the help of the harmonic excitation parameters,
i.e. amplitude and frequency {q 0 , ω p }, allows to analyse the wide palette of problems
which gives hints to analyse behaviour of real constructions.
One of the important problems while constructing and analysing charts of vibrations is the accuracy with which we get information about the dynamic processes.
The second important problem deals with getting minimum of computational time
needed for the chart construction.
We have investigated suitability of the information mapping in vibration regimes
for the charts of the vibration characters with the resolution of 100 × 100, 200 ×
200, 300 × 300, 600 × 600 points. The charts with the resolutions 300 × 300 and
600 × 600 practically coincide. In order to obtain the charts with vibration regimes
of 300 × 300, there is a need to solve 9 · 10
4 case studies, and to analyse the obtained
results on the basis of frequency spectrum, autocorrelation functions and spectrum of
Lyapunov exponents. In order to construct a chart with resolution of 600 × 600, we
need to analyse 12 · 10
4 case studies, which requires long computational time. This
is why, we have chosen the resolution 300 × 300. Table 6.18 presents the charts of
the vibration regimes of the various resolutions obtained while solving problems of
the Bernoulli–Euler beam model vibrations without an account of the size-dependent
behaviour for the boundary clamping conditions.
w(0, t) = w(1, t) = 0,
∂w(0, t)
∂ x
=
∂w(1, t)
∂ x
= 0,
u(0, t) = u(1, t) = 0.
The same notation is used in the remaining charts in this section.
6.7.1 Analysis of the Charts of the Vibration Regimes for the
Euler–Bernoulli, Timoshenko and Sheremetev–Pelekh
Modes Versus λ
We consider the charts of the vibration regimes of the mentioned models with and
account of different values of the beam relative thickness λ for two different values
of the size-dependent parameter l/ h = 0 and l/ h = 0.3.