6.7 Charts of Vibration Characters Verusus Parameters
187
of vibration regimes for l/ h = 0, there is zone of periodic vibrations, with inclusion
of chaotic zone (lower band of frequencies). In the analysed model, there is also
coincidence of the obtained charts of vibration regimes with those associated with
the Bernoulli–Euler model.
Generalizing the obtained results, one may conclude that zones of vibrations with
ω p ∈ [3.45; 6.9] differ marginally for all studied models. An account of the sizedependent behaviour for all mathematical models changes vibrations character, in
particular, in zone of ω p ∈ [6.9; 10.35].
6.7.1.2 Charts of Vibration Regimes for λ = 50
Areas of periodic and chaotic zones are increased in the chart of vibration regimes
for l/ h = 0. Increase of the periodic zones takes place in the interval q 0 ∈ [30000;
60000].
Zones of periodic vibrations increase in the chart of the vibration regimes for
l/ h = 0.3. On contrary to other charts, zone corresponding to superposition of the
independent frequencies occurs for ω p = 7.1. Chaotic zones are increased in the
chart of vibration regimes without an account of the size-dependent behaviour.
Zones of periodic and chaotic vibrations are slightly larger on the chart of vibration
regimes for l/ h = 0 than for l/ h = 0.3. In both cases, the beginning of chaos with
regard to the loading parameter is the same. Inside the zone of periodic vibrations for
ω p ∈ [4.3; 5.2], there is zone of vibrations spanned on the non-dependent frequency
by chaotic zone.
Comparison of the charts associated with various models yields the following
conclusions. For l/ h = 0 in the interval ω p ∈ [3.45; 6.9], the characters of the picture
is similar. In the chart of the Sheremetev–Pelekh model, there are zone characteristics
only for the Bernoulli–Euler model (periodic vibrations) and only for the Timoshenko
model (bifurcation zones). The vibration character for the interval ω p ∈ [6.9; 10.35]
for all models is different. For l/ h = 0.3, in the window ω p ∈ [3.45; 6.9], the charts
of the vibration regimes of the Timoshenko and the Sheremetev–Pelekh models
exhibit similar dynamic regimes (Tables 6.20, 6.21, 6.22, 6.23, 6.24).
6.7.1.3 Charts of Vibration Regimes for λ = 100
Zone of periodic vibrations shown in Table 6.25b is decreased due to increase of
zones of vibration spanned on the independent frequency. Zones of damped vibrations
are increased in the region corresponding to high frequencies ω p ∈ [7.5; 10.1]; the
same holds for zone of periodic vibrations for q 0 < 2000. On contrary to the chart of
vibration regimes for l/ h = 0, zone of superposition of the independent frequencies
has more sense.
The essential difference in comparison to the Bernoulli–Euler model is observed
in the periodic larger amount of vibrations zones in the size-dependent case.
187
of vibration regimes for l/ h = 0, there is zone of periodic vibrations, with inclusion
of chaotic zone (lower band of frequencies). In the analysed model, there is also
coincidence of the obtained charts of vibration regimes with those associated with
the Bernoulli–Euler model.
Generalizing the obtained results, one may conclude that zones of vibrations with
ω p ∈ [3.45; 6.9] differ marginally for all studied models. An account of the sizedependent behaviour for all mathematical models changes vibrations character, in
particular, in zone of ω p ∈ [6.9; 10.35].
6.7.1.2 Charts of Vibration Regimes for λ = 50
Areas of periodic and chaotic zones are increased in the chart of vibration regimes
for l/ h = 0. Increase of the periodic zones takes place in the interval q 0 ∈ [30000;
60000].
Zones of periodic vibrations increase in the chart of the vibration regimes for
l/ h = 0.3. On contrary to other charts, zone corresponding to superposition of the
independent frequencies occurs for ω p = 7.1. Chaotic zones are increased in the
chart of vibration regimes without an account of the size-dependent behaviour.
Zones of periodic and chaotic vibrations are slightly larger on the chart of vibration
regimes for l/ h = 0 than for l/ h = 0.3. In both cases, the beginning of chaos with
regard to the loading parameter is the same. Inside the zone of periodic vibrations for
ω p ∈ [4.3; 5.2], there is zone of vibrations spanned on the non-dependent frequency
by chaotic zone.
Comparison of the charts associated with various models yields the following
conclusions. For l/ h = 0 in the interval ω p ∈ [3.45; 6.9], the characters of the picture
is similar. In the chart of the Sheremetev–Pelekh model, there are zone characteristics
only for the Bernoulli–Euler model (periodic vibrations) and only for the Timoshenko
model (bifurcation zones). The vibration character for the interval ω p ∈ [6.9; 10.35]
for all models is different. For l/ h = 0.3, in the window ω p ∈ [3.45; 6.9], the charts
of the vibration regimes of the Timoshenko and the Sheremetev–Pelekh models
exhibit similar dynamic regimes (Tables 6.20, 6.21, 6.22, 6.23, 6.24).
6.7.1.3 Charts of Vibration Regimes for λ = 100
Zone of periodic vibrations shown in Table 6.25b is decreased due to increase of
zones of vibration spanned on the independent frequency. Zones of damped vibrations
are increased in the region corresponding to high frequencies ω p ∈ [7.5; 10.1]; the
same holds for zone of periodic vibrations for q 0 < 2000. On contrary to the chart of
vibration regimes for l/ h = 0, zone of superposition of the independent frequencies
has more sense.
The essential difference in comparison to the Bernoulli–Euler model is observed
in the periodic larger amount of vibrations zones in the size-dependent case.