386
9 Topologic Optimization of Vibrations of Size-Dependent Beams
Fig. 9.16 Functions “load-deflection” q(w) (a) and “fundamental frequency-deflection” ω(w) (b)
for θ = 0 [reprinted with permission from International Journal of Non-Linear Mechanics publishers]
the size-dependent effect (k 4 = 0) for the topologically optimal microstructure; 2—
with the account of the size-dependent effect (k 4 = 0.3) of the homogeneous beam;
2*—taking into account the size-dependent effect for the topologically optimal
microstructure.
Figure 9.16 shows the dependencies “load-deflection” q(w) and “fundamental
frequency-deflection” ω(w) for θ = 0 for the nonlinear problem. The analysis of the
obtained results shows that for the optimal beam, the deflection w is smaller than the
counterpart homogeneous beam for the same load magnitudes, taking or not taking
into account the size-dependent behaviour. In the case of presence of the temperature
field, for large deflections, there is a tendency to bring closer the results for optimal
and non-optimal beams. This benefit, i.e. a decrease in the maximum deflection of the
topologically optimal beam, comparing with the homogeneous beam, equals 12.7%
(for θ = −100 and q = 500) and for the case without the size-dependent behaviour,
it reaches 10.4%. For the case θ = +100 and q = 500, the decrease is 13.25/11.5%
for the problem without/with the account of the size-dependent parameter. For the
lack of the temperature field, the results are 12.4% and 13.8%, respectively.
The analysis of changes in the fundamental frequencies for different temperature magnitudes and the scale-size length parameters yields the following results.
For θ = 100, the difference between the frequencies of the homogeneous and nonhomogeneous (optimal) beam for the deflection 0.00001 achieves 5.7 and 3.76%,
without and with the account of the size-dependent parameter, respectively. In the
case of a unit (9.71) deflection of the beam, the corresponding values are 3.9 and
3.15%.
For the temperature θ = −100 and for deflection 0.00001, we have 7.4% and
5.1%, respectively, and for the deflection of the magnitude 1, we have 4.4% and
4.47%, respectively. Finally, for the deflection 0.00001, we get 6.5 and 6.1%, and
for the deflection 1, we obtain 4% and 3.12%, respectively.
9 Topologic Optimization of Vibrations of Size-Dependent Beams
Fig. 9.16 Functions “load-deflection” q(w) (a) and “fundamental frequency-deflection” ω(w) (b)
for θ = 0 [reprinted with permission from International Journal of Non-Linear Mechanics publishers]
the size-dependent effect (k 4 = 0) for the topologically optimal microstructure; 2—
with the account of the size-dependent effect (k 4 = 0.3) of the homogeneous beam;
2*—taking into account the size-dependent effect for the topologically optimal
microstructure.
Figure 9.16 shows the dependencies “load-deflection” q(w) and “fundamental
frequency-deflection” ω(w) for θ = 0 for the nonlinear problem. The analysis of the
obtained results shows that for the optimal beam, the deflection w is smaller than the
counterpart homogeneous beam for the same load magnitudes, taking or not taking
into account the size-dependent behaviour. In the case of presence of the temperature
field, for large deflections, there is a tendency to bring closer the results for optimal
and non-optimal beams. This benefit, i.e. a decrease in the maximum deflection of the
topologically optimal beam, comparing with the homogeneous beam, equals 12.7%
(for θ = −100 and q = 500) and for the case without the size-dependent behaviour,
it reaches 10.4%. For the case θ = +100 and q = 500, the decrease is 13.25/11.5%
for the problem without/with the account of the size-dependent parameter. For the
lack of the temperature field, the results are 12.4% and 13.8%, respectively.
The analysis of changes in the fundamental frequencies for different temperature magnitudes and the scale-size length parameters yields the following results.
For θ = 100, the difference between the frequencies of the homogeneous and nonhomogeneous (optimal) beam for the deflection 0.00001 achieves 5.7 and 3.76%,
without and with the account of the size-dependent parameter, respectively. In the
case of a unit (9.71) deflection of the beam, the corresponding values are 3.9 and
3.15%.
For the temperature θ = −100 and for deflection 0.00001, we have 7.4% and
5.1%, respectively, and for the deflection of the magnitude 1, we have 4.4% and
4.47%, respectively. Finally, for the deflection 0.00001, we get 6.5 and 6.1%, and
for the deflection 1, we obtain 4% and 3.12%, respectively.