186
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
Table 6.19 The Bernoulli–Euler model—charts of vibration regimes: (a) without the sizedependent behaviour l/ h = 0; (b) with the size-dependent behaviour l/ h = 0.3 [reprinted with
permission from International Journal of Non-Linear Mechanics publishers]
is not observed. Comparison of the charts of vibration regimes shows a shift of the
zone of damped vibrations from vicinity of the frequency ω p = 6.9 and for l/ h = 0
into the zone of high frequencies for l/ h = 0.3.
As in the previous case, here zones of periodic vibrations are larger on the chart
of vibration regime for l/ h = 0.3, though rather marginally. On contrary to the
Bernoulli–Euler model, the area of the chaotic zones of vibrations is larger in the chart
of the vibration regimes for l/ h = 0.3. The most lengthy zone along the parameter
of the load, amplitude q 0 , is localized on the frequency ω p = 7.3. Another difference
with regard to the Bernoulli–Euler model relies on occurrence on two charts of the
vibration regimes in the case of Timoshenko model of the chaotic islands surrounded
by periodic vibrations. An account of the size-dependent behaviour transmits that
zone into larger frequencies. It should be mentioned that for the chart l/ h = 0.3,
there is a shift of the periodic vibrations over the frequency ω p = 6.9, in comparison
to the chart of the vibration regimes without account of the size-dependent behaviour.
Zones of chaos for the chart of vibration regimes for l/ h = 0.3 begin earlier with
regard to q 0 then for the chart with (l/ h = 0). For the chart l/ h = 0.3 in the frequency
window ω p ∈ [7.5; 9.3] for q 0 ∈ [2000; 5000] in zone of vibrations spent on nondependent frequency, there is a zone of subharmonic vibrations ω p /3. On contrary to
the chart of vibration regimes l/ h = 0 in the chart l/ h = 0.3 in the subinterval of high
frequencies ω > ω p = 9.5, there is a zone of periodic vibration located in chaotic
zone and being surrounded by a zone spanned on the non-dependent frequency.
Zones of periodic vibrations in the chart of vibration regimes for l/ h = 0.3 possesses the enlarged, and they are located lower than in the case of the chart for
l/ h = 0 with regard to the parameter q 0 as it was happened earlier while compared
with the vibration regimes of the Bernoulli–Euler model. The latter behaviour is
implied by the occurrence of the biharmonic operator governing equation of deflection zones of periodic vibrations that have the shape of strips. In contrast to the chart
6 Mathematical Models of Micro- and Nano-cylindrical Panels in Temperature Field
Table 6.19 The Bernoulli–Euler model—charts of vibration regimes: (a) without the sizedependent behaviour l/ h = 0; (b) with the size-dependent behaviour l/ h = 0.3 [reprinted with
permission from International Journal of Non-Linear Mechanics publishers]
is not observed. Comparison of the charts of vibration regimes shows a shift of the
zone of damped vibrations from vicinity of the frequency ω p = 6.9 and for l/ h = 0
into the zone of high frequencies for l/ h = 0.3.
As in the previous case, here zones of periodic vibrations are larger on the chart
of vibration regime for l/ h = 0.3, though rather marginally. On contrary to the
Bernoulli–Euler model, the area of the chaotic zones of vibrations is larger in the chart
of the vibration regimes for l/ h = 0.3. The most lengthy zone along the parameter
of the load, amplitude q 0 , is localized on the frequency ω p = 7.3. Another difference
with regard to the Bernoulli–Euler model relies on occurrence on two charts of the
vibration regimes in the case of Timoshenko model of the chaotic islands surrounded
by periodic vibrations. An account of the size-dependent behaviour transmits that
zone into larger frequencies. It should be mentioned that for the chart l/ h = 0.3,
there is a shift of the periodic vibrations over the frequency ω p = 6.9, in comparison
to the chart of the vibration regimes without account of the size-dependent behaviour.
Zones of chaos for the chart of vibration regimes for l/ h = 0.3 begin earlier with
regard to q 0 then for the chart with (l/ h = 0). For the chart l/ h = 0.3 in the frequency
window ω p ∈ [7.5; 9.3] for q 0 ∈ [2000; 5000] in zone of vibrations spent on nondependent frequency, there is a zone of subharmonic vibrations ω p /3. On contrary to
the chart of vibration regimes l/ h = 0 in the chart l/ h = 0.3 in the subinterval of high
frequencies ω > ω p = 9.5, there is a zone of periodic vibration located in chaotic
zone and being surrounded by a zone spanned on the non-dependent frequency.
Zones of periodic vibrations in the chart of vibration regimes for l/ h = 0.3 possesses the enlarged, and they are located lower than in the case of the chart for
l/ h = 0 with regard to the parameter q 0 as it was happened earlier while compared
with the vibration regimes of the Bernoulli–Euler model. The latter behaviour is
implied by the occurrence of the biharmonic operator governing equation of deflection zones of periodic vibrations that have the shape of strips. In contrast to the chart