WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...
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ω/κ<br />
ω/κ<br />
1.5<br />
1.4<br />
1.3<br />
1.2<br />
1.1<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
B.S. Lazarov, J.S. Jensen / International Journal of Non-Linear Mechanics 42 (2007) 1186–1193 1189<br />
ω/κ<br />
0.6<br />
0.6<br />
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1<br />
1.5<br />
1.4<br />
1.3<br />
1.2<br />
1.1<br />
1<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
1.5<br />
1.4<br />
1.3<br />
1.2<br />
1.1<br />
1<br />
Re(μ u ) Im(μ u )<br />
Fig. 3. Dispersion relation for = 0.1, = 0 and different damping ratios.<br />
0 0.5 1 1.5 2 2.5 3<br />
ω/κ<br />
0.9<br />
0.8<br />
0.7<br />
1.5<br />
1.4<br />
1.3<br />
1.2<br />
1.1<br />
1<br />
Re(μ u ) Im(μ u )<br />
0.9<br />
0.8<br />
0.7<br />
0.6<br />
ζ=1%<br />
ζ=2%<br />
ζ=5%<br />
ζ=10%<br />
ζ=20%<br />
ζ=30%<br />
β=0.1<br />
β=0.3<br />
β=0.5<br />
β=1<br />
0 0.5 1 1.5 2 2.5<br />
Fig. 4. Dispersion relation for = 0.01, A1,j Ā1,j = 0.1 and different mass ratios.<br />
the form<br />
cosh(u) = 1 − 1<br />
2 2<br />
− 1 <br />
2<br />
2(2 + 2i + 32j A1,j Ā1,j )<br />
(2 + 2i + 32j A1,j Ā1,j − 2 . (14)<br />
)<br />
The dispersion relation for the linear case can be obtained from<br />
(14) by setting the non-linear parameter j equal to zero and<br />
removing the damping in the attached oscillators. The influence<br />
of the damping parameter on the dispersion relation in the linear<br />
case can be seen in Fig. 3. Den Hartog has shown in [8] that<br />
the introduction of viscous damping in the vibration absorber<br />
increases the frequency interval in which the device is effective.<br />
Similar behaviour can be observed for wave propagation<br />
problems. The maximal value of the attenuation rate decreases<br />
with increasing the damping in the attached oscillators, and for<br />
high values of the band gap practically disappears. Similar<br />
behaviour is observed for two-dimensional wave propagation<br />
problems in [16]. The value of Im(u) is always different from<br />
0or± in the damped case, and thus oscillatory behaviour in<br />
the spatial domain can always be observed. The influence of the<br />
mass ratio on the band gap is shown in Fig. 4. The width of<br />
the stop band can be increased by increasing the attached mass.<br />
For the non-linear spring–mass chain, u depends on the amplitude<br />
of the attached oscillator at position j. Plots of the dispersion<br />
relation for different values of the non-linear parameter<br />
A1,j Ā1,j are shown in Fig. 5. As can be seen, the position of<br />
the maximal value of the attenuation rate varies with the value<br />
of the non-linear parameter. The maximum is shifted above the<br />
linear natural frequency of the attached oscillators, and the shift<br />
increases for larger values of the amplitude A1,j .Foranegative<br />
value of the non-linear parameter , the shift will occur in<br />
the opposite direction, below the linear natural frequency. As<br />
the wave propagates, the amplitude A1,j decreases, due to the<br />
reflections from the attached oscillators, as well as due to the<br />
damping in the system, and the position of the maximal value<br />
of Re( u) moves toward / = 1. The amplitudes vary along<br />
the chain, and u becomes a function of the spatial coordinate<br />
j. A fixed wave propagation constant cannot be defined in the<br />
non-linear case.<br />
3.3. Transmission properties based on analytical calculations<br />
For given values of the amplitudes B1,j and A1,j of mass<br />
j and the corresponding attached oscillator, the value of u<br />
can be calculated using (14). The amplitude of the mass at<br />
position j +1 is calculated as B1,j+1 =e uB1,j . The amplitude<br />
3