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( a ) The five low frequency vibration modes of example 2, i.e. mode I, II, III, IV and V and ( b ) the dependence of the frequency associated with the five modes on the number of tubes with one end clamped.
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Heterogeneous end constraints are imposed on multiwall carbon nanotubes (MWCNTs) by sequentially clamping one end of their originally simply supported constituent tubes. The finite element method is employed to study the vibration of such MWCNTs with an emphasis on the effect of the mixed boundary conditions. The results show that the clamping proc...
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... their counterparts of thin example 1 the absolute frequency shift of example 2 is 6 to 20 times that of thin example 1. Furthermore, figure 4( b ) shows that for the five modes of example 2, the frequency up-shift obtained by clamping one single layer decreases from the outermost tube to the innermost one with the value of f 1 / f 5 around 9 to 20 for modes I to IV. It is noted that the slopes of the curves in figure 4( b ) are generally greater than those shown in figure 4( a ) for example 1, which means that the outermost few tubes of example 2 generally play a more important role in supporting the whole MWCNT than those of thin example 1. This can be explained by the fact that, for thick example 2 substantial difference in the radius and the aspect ratio exists between the constituent tubes, whereas such difference becomes much smaller in thin example 1. As shown above, clamping the constituent tubes has stronger impacts on mode I than mode II. In figure 2( b ), when N = 0, i.e. the simply supported boundary condition for example 2, the frequency of mode I is 34.6 GHz lower than that of model II; it then catches up with the frequency of model II when N rises to 3, i.e. one end of the outermost three layers is clamped. Further fixing the inner tube or the innermost two tubes the frequency of mode I turns out to be even higher than that of mode II. In other words, for example 2, the vibration mode associated with the fundamental frequency will transform from shell-like vibration with m = 1 and n = 2 to beam-like vibration with m = n = 1 when the one end of the tubes is clamped consecutively. Here, it is reasonable to expect that such a mode transformation would occur earlier, i.e. at smaller N , and result in the model I frequency much higher that of mode II, when both ends of the constituent tubes are clamped. In practical applications, the fundamental mode of shell structures is of major interest, and the alteration of this vibration could either significantly affect the performance of CNT-based nanostructures or have some potential applications directly exploiting this unique feature. The output of this study can thus provide useful guidance to facilitate the design and development of these CNT-based nanotechnology. Here it is worth mentioning that, similar transformation due to the boundary condition change has also been observed for the critical buckling mode of the thick and short MWCNTs [30]. In this section, we shall focus on a (almost) solid and slender five-wall CNT, i.e. example 3 in table 1. It is easy to see that this example consists of five beam-type SWCNTs whose aspect ratio increases from 10 (the outermost tube) to 50 (the innermost tube), much larger than those of examples 1 and 2. On the other hand, example 3 is (almost) solid with a small innermost radius 0.34 nm. It therefore exhibits radial stiffness even higher than that of example 2. As a result, it is seen in figure 3( a ) that, in modes I, II, III and IV, the vibration of example 3 follows the beam-like bending modes where n = 1 remains unchanged (i.e. the cross-section is circular without any deformation) while m increases substantially from 1 to 4. In clamping one end of the constituent tubes, the maximum frequency shift obtained for modes I, II, III and IV is 24.8 GHz, 37 GHz, 41.3 GHz and 43.7 GHz, respectively, (figure 3( b)) and the corresponding relative change is calculated as 52%, 20.6%, 11.3% and 7.6%. These values are significantly larger than those obtained for shell-like vibration of examples 1 and 2, showing that the beam modes of a slender CNT are more sensitive to clamping the constituent tubes in axial direction. This observation can probably be explained by the fact that the energy of the beam modes of slender example 3 is completely determined by its bending in axial direction. However, the energy of the shell-like vibrations of short examples 1 and 2 comes not only from off-plane bending but also from in-plane stretching/compression of the tube walls. On the other hand, it is interesting to see in figure 2( b ) that, in contrast to the case of slender example 3, the beam mode of short example 2 (mode II in figure 2( a)) cannot be significantly affected by the clamped tube ends. Here the major difference is that, ...
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... their counterparts of thin example 1 the absolute frequency shift of example 2 is 6 to 20 times that of thin example 1. Furthermore, figure 4( b ) shows that for the five modes of example 2, the frequency up-shift obtained by clamping one single layer decreases from the outermost tube to the innermost one with the value of f 1 / f 5 around 9 to 20 for modes I to IV. It is noted that the slopes of the curves in figure 4( b ) are generally greater than those shown in figure 4( a ) for example 1, which means that the outermost few tubes of example 2 generally play a more important role in supporting the whole MWCNT than those of thin example 1. This can be explained by the fact that, for thick example 2 substantial difference in the radius and the aspect ratio exists between the constituent tubes, whereas such difference becomes much smaller in thin example 1. As shown above, clamping the constituent tubes has stronger impacts on mode I than mode II. In figure 2( b ), when N = 0, i.e. the simply supported boundary condition for example 2, the frequency of mode I is 34.6 GHz lower than that of model II; it then catches up with the frequency of model II when N rises to 3, i.e. one end of the outermost three layers is clamped. Further fixing the inner tube or the innermost two tubes the frequency of mode I turns out to be even higher than that of mode II. In other words, for example 2, the vibration mode associated with the fundamental frequency will transform from shell-like vibration with m = 1 and n = 2 to beam-like vibration with m = n = 1 when the one end of the tubes is clamped consecutively. Here, it is reasonable to expect that such a mode transformation would occur earlier, i.e. at smaller N , and result in the model I frequency much higher that of mode II, when both ends of the constituent tubes are clamped. In practical applications, the fundamental mode of shell structures is of major interest, and the alteration of this vibration could either significantly affect the performance of CNT-based nanostructures or have some potential applications directly exploiting this unique feature. The output of this study can thus provide useful guidance to facilitate the design and development of these CNT-based nanotechnology. Here it is worth mentioning that, similar transformation due to the boundary condition change has also been observed for the critical buckling mode of the thick and short MWCNTs [30]. In this section, we shall focus on a (almost) solid and slender five-wall CNT, i.e. example 3 in table 1. It is easy to see that this example consists of five beam-type SWCNTs whose aspect ratio increases from 10 (the outermost tube) to 50 (the innermost tube), much larger than those of examples 1 and 2. On the other hand, example 3 is (almost) solid with a small innermost radius 0.34 nm. It therefore exhibits radial stiffness even higher than that of example 2. As a result, it is seen in figure 3( a ) that, in modes I, II, III and IV, the vibration of example 3 follows the beam-like bending modes where n = 1 remains unchanged (i.e. the cross-section is circular without any deformation) while m increases substantially from 1 to 4. In clamping one end of the constituent tubes, the maximum frequency shift obtained for modes I, II, III and IV is 24.8 GHz, 37 GHz, 41.3 GHz and 43.7 GHz, respectively, (figure 3( b)) and the corresponding relative change is calculated as 52%, 20.6%, 11.3% and 7.6%. These values are significantly larger than those obtained for shell-like vibration of examples 1 and 2, showing that the beam modes of a slender CNT are more sensitive to clamping the constituent tubes in axial direction. This observation can probably be explained by the fact that the energy of the beam modes of slender example 3 is completely determined by its bending in axial direction. However, the energy of the shell-like vibrations of short examples 1 and 2 comes not only from off-plane bending but also from in-plane stretching/compression of the tube walls. On the other hand, it is interesting to see in figure 2( b ) that, in contrast to the case of slender example 3, the beam mode of short example 2 (mode II in figure 2( a)) cannot be significantly affected by the clamped tube ends. Here the major difference is that, ...
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... their counterparts of thin example 1 the absolute frequency shift of example 2 is 6 to 20 times that of thin example 1. Furthermore, figure 4( b ) shows that for the five modes of example 2, the frequency up-shift obtained by clamping one single layer decreases from the outermost tube to the innermost one with the value of f 1 / f 5 around 9 to 20 for modes I to IV. It is noted that the slopes of the curves in figure 4( b ) are generally greater than those shown in figure 4( a ) for example 1, which means that the outermost few tubes of example 2 generally play a more important role in supporting the whole MWCNT than those of thin example 1. This can be explained by the fact that, for thick example 2 substantial difference in the radius and the aspect ratio exists between the constituent tubes, whereas such difference becomes much smaller in thin example 1. As shown above, clamping the constituent tubes has stronger impacts on mode I than mode II. In figure 2( b ), when N = 0, i.e. the simply supported boundary condition for example 2, the frequency of mode I is 34.6 GHz lower than that of model II; it then catches up with the frequency of model II when N rises to 3, i.e. one end of the outermost three layers is clamped. Further fixing the inner tube or the innermost two tubes the frequency of mode I turns out to be even higher than that of mode II. In other words, for example 2, the vibration mode associated with the fundamental frequency will transform from shell-like vibration with m = 1 and n = 2 to beam-like vibration with m = n = 1 when the one end of the tubes is clamped consecutively. Here, it is reasonable to expect that such a mode transformation would occur earlier, i.e. at smaller N , and result in the model I frequency much higher that of mode II, when both ends of the constituent tubes are clamped. In practical applications, the fundamental mode of shell structures is of major interest, and the alteration of this vibration could either significantly affect the performance of CNT-based nanostructures or have some potential applications directly exploiting this unique feature. The output of this study can thus provide useful guidance to facilitate the design and development of these CNT-based nanotechnology. Here it is worth mentioning that, similar transformation due to the boundary condition change has also been observed for the critical buckling mode of the thick and short MWCNTs [30]. In this section, we shall focus on a (almost) solid and slender five-wall CNT, i.e. example 3 in table 1. It is easy to see that this example consists of five beam-type SWCNTs whose aspect ratio increases from 10 (the outermost tube) to 50 (the innermost tube), much larger than those of examples 1 and 2. On the other hand, example 3 is (almost) solid with a small innermost radius 0.34 nm. It therefore exhibits radial stiffness even higher than that of example 2. As a result, it is seen in figure 3( a ) that, in modes I, II, III and IV, the vibration of example 3 follows the beam-like bending modes where n = 1 remains unchanged (i.e. the cross-section is circular without any deformation) while m increases substantially from 1 to 4. In clamping one end of the constituent tubes, the maximum frequency shift obtained for modes I, II, III and IV is 24.8 GHz, 37 GHz, 41.3 GHz and 43.7 GHz, respectively, (figure 3( b)) and the corresponding relative change is calculated as 52%, 20.6%, 11.3% and 7.6%. These values are significantly larger than those obtained for shell-like vibration of examples 1 and 2, showing that the beam modes of a slender CNT are more sensitive to clamping the constituent tubes in axial direction. This observation can probably be explained by the fact that the energy of the beam modes of slender example 3 is completely determined by its bending in axial direction. However, the energy of the shell-like vibrations of short examples 1 and 2 comes not only from off-plane bending but also from in-plane stretching/compression of the tube walls. On the other hand, it is interesting to see in figure 2( b ) that, in contrast to the case of slender example 3, the beam mode of short example 2 (mode II in figure 2( a)) cannot be significantly affected by the clamped tube ends. Here the major difference is that, ...
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... stiffness of the MWCNT. The contribution of the constituent tubes then declines rapidly from the outermost tube to the innermost one. For examples 2 and 3, similar calculation has been done and the results are shown graphically in figures 4( b ) and ( c ), respectively, which will be discussed in detail in the following sections. Example 2 is a thick and short five-wall CNT with R i /H 1 and L/D 0 = 5 (see table 1) which is a mixture of four shell- type tubes with the aspect ratio 5 to 8, i.e. the outer four SWCNTs, and one beam-type tube with the aspect ratio 10, i.e. the innermost SWCNT. The five vibration modes of this example are shown in figure 2( a ) where modes I, III, IV and V are the shell-like vibrations similar to those observed in figure 1( a ) while mode II is a beam-like vibration which is not seen in figure 1( a ). The shell-like vibration of example 2 can be attributed to the four shell-type constituent tubes in the five- wall CNT, while the beam-like vibration is a result of the core beam-type tube and the coupling between adjacent tubes via the interlayer vdW interaction. Here, in contrast to the case of thin example 1, the radii of individual tubes differ considerably in thick example 2, which, in turn, results in noticeable difference in their transverse displacements and the significant changes in the interlayer spacing (see the cross-sections in figure 2( a)) . It is easy to understand that thick example 2 with the innermost radius 1 nm displays higher radial stiffness than that of thin example 1 with the innermost radius 5 nm. Thus, when the vibration mode changes from modes I to V, the circumferential wave number n of example 2 alters between 1 and 3 while the axial half wave number m increases from 1 to 4 (figure 2( a)) . This behaviour is different from that of thin example 1, where from modes I to V, n rises substantially from 1 to 4 but m only varies between 1 and 2 (figure 1( a)) . Figure 2( b ) shows the number N -dependence of the frequencies associated with the five vibration modes illustrated in figure 2( a ). It is seen from figure 2( b ) that all the frequencies tend to increase with the increasing number N . Specifically, clamping one end of all the five layers raises the frequency of model I by 48 GHz. Considering its initial frequency (at N = 0) 116.3 GHz the relative increment is calculated as 41.2%. For modes III ( m = 2 ) and V ( m = 4 ) , respectively, with higher frequencies, the corresponding frequency shift decreases to 23.3 GHz and 30.2 GHz and the relative increase reduces to 10.9% and 5.6%, which, however, are still larger than the frequency shift 7.06 GHz and 9.1 GHz and the relative increase 4.65% and 2% obtained for modes II ( m = 1 ) and IV ( m = 3 ) , respectively. These behaviours are qualitatively similar to those observed for example 1 and thus, can be understood based on the same theories. In addition, it is noted that the relative frequency changes of mode I to mode V obtained for example 2 are 41.2%, 4.7%, 11%, 2% and 5.6%, respectively, close to those of example 1, i.e. 39%, 5%, 12%, 0.77% and 2.7%. However, since the frequencies (or the dynamic stiffness) of example 2 are around 10 times as ...
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... stiffness of the MWCNT. The contribution of the constituent tubes then declines rapidly from the outermost tube to the innermost one. For examples 2 and 3, similar calculation has been done and the results are shown graphically in figures 4( b ) and ( c ), respectively, which will be discussed in detail in the following sections. Example 2 is a thick and short five-wall CNT with R i /H 1 and L/D 0 = 5 (see table 1) which is a mixture of four shell- type tubes with the aspect ratio 5 to 8, i.e. the outer four SWCNTs, and one beam-type tube with the aspect ratio 10, i.e. the innermost SWCNT. The five vibration modes of this example are shown in figure 2( a ) where modes I, III, IV and V are the shell-like vibrations similar to those observed in figure 1( a ) while mode II is a beam-like vibration which is not seen in figure 1( a ). The shell-like vibration of example 2 can be attributed to the four shell-type constituent tubes in the five- wall CNT, while the beam-like vibration is a result of the core beam-type tube and the coupling between adjacent tubes via the interlayer vdW interaction. Here, in contrast to the case of thin example 1, the radii of individual tubes differ considerably in thick example 2, which, in turn, results in noticeable difference in their transverse displacements and the significant changes in the interlayer spacing (see the cross-sections in figure 2( a)) . It is easy to understand that thick example 2 with the innermost radius 1 nm displays higher radial stiffness than that of thin example 1 with the innermost radius 5 nm. Thus, when the vibration mode changes from modes I to V, the circumferential wave number n of example 2 alters between 1 and 3 while the axial half wave number m increases from 1 to 4 (figure 2( a)) . This behaviour is different from that of thin example 1, where from modes I to V, n rises substantially from 1 to 4 but m only varies between 1 and 2 (figure 1( a)) . Figure 2( b ) shows the number N -dependence of the frequencies associated with the five vibration modes illustrated in figure 2( a ). It is seen from figure 2( b ) that all the frequencies tend to increase with the increasing number N . Specifically, clamping one end of all the five layers raises the frequency of model I by 48 GHz. Considering its initial frequency (at N = 0) 116.3 GHz the relative increment is calculated as 41.2%. For modes III ( m = 2 ) and V ( m = 4 ) , respectively, with higher frequencies, the corresponding frequency shift decreases to 23.3 GHz and 30.2 GHz and the relative increase reduces to 10.9% and 5.6%, which, however, are still larger than the frequency shift 7.06 GHz and 9.1 GHz and the relative increase 4.65% and 2% obtained for modes II ( m = 1 ) and IV ( m = 3 ) , respectively. These behaviours are qualitatively similar to those observed for example 1 and thus, can be understood based on the same theories. In addition, it is noted that the relative frequency changes of mode I to mode V obtained for example 2 are 41.2%, 4.7%, 11%, 2% and 5.6%, respectively, close to those of example 1, i.e. 39%, 5%, 12%, 0.77% and 2.7%. However, since the frequencies (or the dynamic stiffness) of example 2 are around 10 times as ...
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... stiffness of the MWCNT. The contribution of the constituent tubes then declines rapidly from the outermost tube to the innermost one. For examples 2 and 3, similar calculation has been done and the results are shown graphically in figures 4( b ) and ( c ), respectively, which will be discussed in detail in the following sections. Example 2 is a thick and short five-wall CNT with R i /H 1 and L/D 0 = 5 (see table 1) which is a mixture of four shell- type tubes with the aspect ratio 5 to 8, i.e. the outer four SWCNTs, and one beam-type tube with the aspect ratio 10, i.e. the innermost SWCNT. The five vibration modes of this example are shown in figure 2( a ) where modes I, III, IV and V are the shell-like vibrations similar to those observed in figure 1( a ) while mode II is a beam-like vibration which is not seen in figure 1( a ). The shell-like vibration of example 2 can be attributed to the four shell-type constituent tubes in the five- wall CNT, while the beam-like vibration is a result of the core beam-type tube and the coupling between adjacent tubes via the interlayer vdW interaction. Here, in contrast to the case of thin example 1, the radii of individual tubes differ considerably in thick example 2, which, in turn, results in noticeable difference in their transverse displacements and the significant changes in the interlayer spacing (see the cross-sections in figure 2( a)) . It is easy to understand that thick example 2 with the innermost radius 1 nm displays higher radial stiffness than that of thin example 1 with the innermost radius 5 nm. Thus, when the vibration mode changes from modes I to V, the circumferential wave number n of example 2 alters between 1 and 3 while the axial half wave number m increases from 1 to 4 (figure 2( a)) . This behaviour is different from that of thin example 1, where from modes I to V, n rises substantially from 1 to 4 but m only varies between 1 and 2 (figure 1( a)) . Figure 2( b ) shows the number N -dependence of the frequencies associated with the five vibration modes illustrated in figure 2( a ). It is seen from figure 2( b ) that all the frequencies tend to increase with the increasing number N . Specifically, clamping one end of all the five layers raises the frequency of model I by 48 GHz. Considering its initial frequency (at N = 0) 116.3 GHz the relative increment is calculated as 41.2%. For modes III ( m = 2 ) and V ( m = 4 ) , respectively, with higher frequencies, the corresponding frequency shift decreases to 23.3 GHz and 30.2 GHz and the relative increase reduces to 10.9% and 5.6%, which, however, are still larger than the frequency shift 7.06 GHz and 9.1 GHz and the relative increase 4.65% and 2% obtained for modes II ( m = 1 ) and IV ( m = 3 ) , respectively. These behaviours are qualitatively similar to those observed for example 1 and thus, can be understood based on the same theories. In addition, it is noted that the relative frequency changes of mode I to mode V obtained for example 2 are 41.2%, 4.7%, 11%, 2% and 5.6%, respectively, close to those of example 1, i.e. 39%, 5%, 12%, 0.77% and 2.7%. However, since the frequencies (or the dynamic stiffness) of example 2 are around 10 times as ...
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... stiffness of the MWCNT. The contribution of the constituent tubes then declines rapidly from the outermost tube to the innermost one. For examples 2 and 3, similar calculation has been done and the results are shown graphically in figures 4( b ) and ( c ), respectively, which will be discussed in detail in the following sections. Example 2 is a thick and short five-wall CNT with R i /H 1 and L/D 0 = 5 (see table 1) which is a mixture of four shell- type tubes with the aspect ratio 5 to 8, i.e. the outer four SWCNTs, and one beam-type tube with the aspect ratio 10, i.e. the innermost SWCNT. The five vibration modes of this example are shown in figure 2( a ) where modes I, III, IV and V are the shell-like vibrations similar to those observed in figure 1( a ) while mode II is a beam-like vibration which is not seen in figure 1( a ). The shell-like vibration of example 2 can be attributed to the four shell-type constituent tubes in the five- wall CNT, while the beam-like vibration is a result of the core beam-type tube and the coupling between adjacent tubes via the interlayer vdW interaction. Here, in contrast to the case of thin example 1, the radii of individual tubes differ considerably in thick example 2, which, in turn, results in noticeable difference in their transverse displacements and the significant changes in the interlayer spacing (see the cross-sections in figure 2( a)) . It is easy to understand that thick example 2 with the innermost radius 1 nm displays higher radial stiffness than that of thin example 1 with the innermost radius 5 nm. Thus, when the vibration mode changes from modes I to V, the circumferential wave number n of example 2 alters between 1 and 3 while the axial half wave number m increases from 1 to 4 (figure 2( a)) . This behaviour is different from that of thin example 1, where from modes I to V, n rises substantially from 1 to 4 but m only varies between 1 and 2 (figure 1( a)) . Figure 2( b ) shows the number N -dependence of the frequencies associated with the five vibration modes illustrated in figure 2( a ). It is seen from figure 2( b ) that all the frequencies tend to increase with the increasing number N . Specifically, clamping one end of all the five layers raises the frequency of model I by 48 GHz. Considering its initial frequency (at N = 0) 116.3 GHz the relative increment is calculated as 41.2%. For modes III ( m = 2 ) and V ( m = 4 ) , respectively, with higher frequencies, the corresponding frequency shift decreases to 23.3 GHz and 30.2 GHz and the relative increase reduces to 10.9% and 5.6%, which, however, are still larger than the frequency shift 7.06 GHz and 9.1 GHz and the relative increase 4.65% and 2% obtained for modes II ( m = 1 ) and IV ( m = 3 ) , respectively. These behaviours are qualitatively similar to those observed for example 1 and thus, can be understood based on the same theories. In addition, it is noted that the relative frequency changes of mode I to mode V obtained for example 2 are 41.2%, 4.7%, 11%, 2% and 5.6%, respectively, close to those of example 1, i.e. 39%, 5%, 12%, 0.77% and 2.7%. However, since the frequencies (or the dynamic stiffness) of example 2 are around 10 times as ...
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... stiffness of the MWCNT. The contribution of the constituent tubes then declines rapidly from the outermost tube to the innermost one. For examples 2 and 3, similar calculation has been done and the results are shown graphically in figures 4( b ) and ( c ), respectively, which will be discussed in detail in the following sections. Example 2 is a thick and short five-wall CNT with R i /H 1 and L/D 0 = 5 (see table 1) which is a mixture of four shell- type tubes with the aspect ratio 5 to 8, i.e. the outer four SWCNTs, and one beam-type tube with the aspect ratio 10, i.e. the innermost SWCNT. The five vibration modes of this example are shown in figure 2( a ) where modes I, III, IV and V are the shell-like vibrations similar to those observed in figure 1( a ) while mode II is a beam-like vibration which is not seen in figure 1( a ). The shell-like vibration of example 2 can be attributed to the four shell-type constituent tubes in the five- wall CNT, while the beam-like vibration is a result of the core beam-type tube and the coupling between adjacent tubes via the interlayer vdW interaction. Here, in contrast to the case of thin example 1, the radii of individual tubes differ considerably in thick example 2, which, in turn, results in noticeable difference in their transverse displacements and the significant changes in the interlayer spacing (see the cross-sections in figure 2( a)) . It is easy to understand that thick example 2 with the innermost radius 1 nm displays higher radial stiffness than that of thin example 1 with the innermost radius 5 nm. Thus, when the vibration mode changes from modes I to V, the circumferential wave number n of example 2 alters between 1 and 3 while the axial half wave number m increases from 1 to 4 (figure 2( a)) . This behaviour is different from that of thin example 1, where from modes I to V, n rises substantially from 1 to 4 but m only varies between 1 and 2 (figure 1( a)) . Figure 2( b ) shows the number N -dependence of the frequencies associated with the five vibration modes illustrated in figure 2( a ). It is seen from figure 2( b ) that all the frequencies tend to increase with the increasing number N . Specifically, clamping one end of all the five layers raises the frequency of model I by 48 GHz. Considering its initial frequency (at N = 0) 116.3 GHz the relative increment is calculated as 41.2%. For modes III ( m = 2 ) and V ( m = 4 ) , respectively, with higher frequencies, the corresponding frequency shift decreases to 23.3 GHz and 30.2 GHz and the relative increase reduces to 10.9% and 5.6%, which, however, are still larger than the frequency shift 7.06 GHz and 9.1 GHz and the relative increase 4.65% and 2% obtained for modes II ( m = 1 ) and IV ( m = 3 ) , respectively. These behaviours are qualitatively similar to those observed for example 1 and thus, can be understood based on the same theories. In addition, it is noted that the relative frequency changes of mode I to mode V obtained for example 2 are 41.2%, 4.7%, 11%, 2% and 5.6%, respectively, close to those of example 1, i.e. 39%, 5%, 12%, 0.77% and 2.7%. However, since the frequencies (or the dynamic stiffness) of example 2 are around 10 times as ...
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... stiffness of the MWCNT. The contribution of the constituent tubes then declines rapidly from the outermost tube to the innermost one. For examples 2 and 3, similar calculation has been done and the results are shown graphically in figures 4( b ) and ( c ), respectively, which will be discussed in detail in the following sections. Example 2 is a thick and short five-wall CNT with R i /H 1 and L/D 0 = 5 (see table 1) which is a mixture of four shell- type tubes with the aspect ratio 5 to 8, i.e. the outer four SWCNTs, and one beam-type tube with the aspect ratio 10, i.e. the innermost SWCNT. The five vibration modes of this example are shown in figure 2( a ) where modes I, III, IV and V are the shell-like vibrations similar to those observed in figure 1( a ) while mode II is a beam-like vibration which is not seen in figure 1( a ). The shell-like vibration of example 2 can be attributed to the four shell-type constituent tubes in the five- wall CNT, while the beam-like vibration is a result of the core beam-type tube and the coupling between adjacent tubes via the interlayer vdW interaction. Here, in contrast to the case of thin example 1, the radii of individual tubes differ considerably in thick example 2, which, in turn, results in noticeable difference in their transverse displacements and the significant changes in the interlayer spacing (see the cross-sections in figure 2( a)) . It is easy to understand that thick example 2 with the innermost radius 1 nm displays higher radial stiffness than that of thin example 1 with the innermost radius 5 nm. Thus, when the vibration mode changes from modes I to V, the circumferential wave number n of example 2 alters between 1 and 3 while the axial half wave number m increases from 1 to 4 (figure 2( a)) . This behaviour is different from that of thin example 1, where from modes I to V, n rises substantially from 1 to 4 but m only varies between 1 and 2 (figure 1( a)) . Figure 2( b ) shows the number N -dependence of the frequencies associated with the five vibration modes illustrated in figure 2( a ). It is seen from figure 2( b ) that all the frequencies tend to increase with the increasing number N . Specifically, clamping one end of all the five layers raises the frequency of model I by 48 GHz. Considering its initial frequency (at N = 0) 116.3 GHz the relative increment is calculated as 41.2%. For modes III ( m = 2 ) and V ( m = 4 ) , respectively, with higher frequencies, the corresponding frequency shift decreases to 23.3 GHz and 30.2 GHz and the relative increase reduces to 10.9% and 5.6%, which, however, are still larger than the frequency shift 7.06 GHz and 9.1 GHz and the relative increase 4.65% and 2% obtained for modes II ( m = 1 ) and IV ( m = 3 ) , respectively. These behaviours are qualitatively similar to those observed for example 1 and thus, can be understood based on the same theories. In addition, it is noted that the relative frequency changes of mode I to mode V obtained for example 2 are 41.2%, 4.7%, 11%, 2% and 5.6%, respectively, close to those of example 1, i.e. 39%, 5%, 12%, 0.77% and 2.7%. However, since the frequencies (or the dynamic stiffness) of example 2 are around 10 times as ...
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Citations
... The wave dispersion, which is obtained from the molecular dynamics simulations, has good agreement with the data obtained from the elastic shell theory. The finite element technique is applied to analyze the oscillations of CNT in [25]. The elastic shell model is used to analyze the buckling of the multi-walled CNTs by He et al. [26]. ...
System of nonlinear partial differential equations, which describes the multi-walled carbon nanotube nonlinear oscillations, is derived. The Sanders–Koiter nonlinear shell theory and the nonlocal anisotropic Hooke’s law are used. Three kinds of nonlinearities are accounted. First of all, the van der Waals forces are considered as nonlinear functions of the radial displacements. Secondly, the nanotube walls displacements have moderate values, which are described by the geometrically nonlinear shell theory. Thirdly, as the stress resultants are the nonlinear functions of the displacements, the additional nonlinear terms are accounted in the equations of motions. These terms are derived from the natural boundary conditions, which are used in the weighted residual method. The finite degrees of freedom nonlinear dynamical system is derived to describe the oscillations of nanostructure. The multi-mode invariant manifolds are used to describe the free nonlinear oscillations, as the dynamical systems have the internal resonances 1:1. The motions on the invariant manifolds are described by two degrees of freedom nonlinear dynamical systems, which are studied by the multiple scales method. The backbone curves of the nonlinear modes are analyzed. As follows from the results of the numerical simulations, the eigenmode of low eigenfrequency has commensurable longitudinal, transversal and circumference displacements. In this case, the nonlinear parts of the van der Waals forces harden essentially the backbone curve of the oscillations.
... The wave dispersion, which is obtained from the molecular dynamics simulations, shows good agreement with the data obtained using the elastic shell theory. The finite element technique is applied to analyze the oscillations of CNT in [25]. The elastic shell model is used to analyze the buckling of the multi-walled CNTs by X. ...
... The vdW forces are assumed linear. Therefore, the nonlinear terms of the equation (25) are not accounted. The nonlinear dynamical system (34) is analyzed with 42 and 60 DOFs to analyze the convergence of the solutions. ...
... The calculations of the NNMs backbone curve with account of the vdW forces nonlinearity, which are described by the equations (25), are carried out. The results of the CNT calculations with the aspect ratio 3 ⁄ = 9 are shown on Fig.14. ...
System of nonlinear partial differential equations, which describes the multi-walled carbon nanotube nonlinear oscillations, is derived. The Sanders-Koiter nonlinear shell theory and the nonlocal anisotropic Hooke’s law are used in this model. Three kinds of nonlinearities are accounted. First of all, the van der Waals forces are nonlinear functions of the radial displacements. Secondly, the nanotube walls displacements have moderate values, which are described by the geometrically nonlinear shell theory. Thirdly, as the stress resultants are the nonlinear functions of the displacements, the additional nonlinear terms in the equations of motions are obtained. These terms are derived from the natural boundary conditions, which are used in the weighted residual method.
The finite degrees of freedom nonlinear dynamical system is derived to describe the oscillations of nanostructure. The Shaw-Pierre nonlinear normal modes in the form of the multi-mode invariant manifolds are used to describe the free nonlinear oscillations, as the dynamical systems contains the internal resonances 1:1. The motions on the invariant manifolds are described by two degrees of freedom nonlinear dynamical systems, which are analyzed by the multiple scales method. The backbone curves of the nonlinear modes are analyzed.
As follows from the results of the numerical simulations, the eigenmode of low eigenfrequency has commensurable longitudinal, transversal and circumference displacements. The nonlinear parts of the van der Waals forces harden essentially the backbone curve of the oscillations close to this eigenmode.
... Euler-Bernoulli theory is used to model SWCNT as resonators via continuum mechanics [31][32][33]. e motion of free vibration equation can be expressed as ...
Using carbon nanotubes for sensing the mass in a biosensor is recently proven as an emerging technology in healthcare industry. This study investigates relative frequency shifts and sensitivity studies of various biological objects such as insulin hormone, immunoglobulin G (IgG), the most abundant type of antibody, and low-density lipoproteins (LDL) masses using the single-wall carbon nanotubes as a biomass sensor via continuum mechanics. Uniform distributed mass is applied to the single-wall carbon nanotube mass sensor. In this study, fixed-free and fixed-fixed type single-wall carbon nanotubes with various lengths of relative frequency shifts are studied. Additionally, the sensitivity analysis of fixed-free and fixed-fixed type CNT biological mass sensors is carried out. Moreover, mode shapes studies are performed. The sensitivity results show better, if the length of the single-wall carbon nanotube is reduced.
... Euler-Bernoulli theory is used to model SWCNT as resonators via continuum mechanics [31][32][33]. e motion of free vibration equation can be expressed as ...
Using carbon nanotubes for sensing the mass in a biosensor is recently proven as an emerging technology in healthcare industry. is study investigates relative frequency shifts and sensitivity studies of various biological objects such as insulin hormone, immunoglobulin G (IgG), the most abundant type of antibody, and low-density lipoproteins (LDL) masses using the single-wall carbon nanotubes as a biomass sensor via continuum mechanics. Uniform distributed mass is applied to the single-wall carbon nanotube mass sensor. In this study, fixed-free and fixed-fixed type single-wall carbon nanotubes with various lengths of relative frequency shifts are studied. Additionally, the sensitivity analysis of fixed-free and fixed-fixed type CNT biological mass sensors is carried out. Moreover, mode shapes studies are performed. e sensitivity results show better, if the length of the single-wall carbon nanotube is reduced.
... Such approaches lead to achieving computational eciency up to certain extent, suitable for carrying out a purely deterministic analysis involving only a few simulations. Examples include vibration of single and multilayer carbon nanotubes [17], graphene sheets along with other 2D materials [18,19,20,21,22,23], family of fullerene molecules [24] to mention a few. Analysis of nano-scale biological systems including dierent viruses have also seen the application of elastic continuum and structural mechanics based approaches including nite element method (FEM). ...
A machine learning assisted efficient, yet comprehensive characterization of the dynamics of coronaviruses, in conjunction with finite element (FE) approach, is presented. Without affecting the accuracy of prediction in low‐frequency vibration analysis, an equivalent model for the FE analysis is proposed, based on which the natural frequencies corresponding to first three non‐rigid modes are analyzed. To quantify the inherent system‐uncertainty efficiently, Monte Carlo simulation is proposed in conjunction with the machine learning based FE computational framework for obtaining complete probabilistic descriptions considering both individual and compound effect of stochasticity. A variance based sensitivity analysis is carried out to enumerate the relative significance of different material parameters corresponding to various constituting parts of the coronavirus structure. Using the modal characteristics like natural frequencies and mode shapes of the virus structure including their stochastic bounds, it is possible to readily identify coronaviruses by comparing the experimentally measured dynamic responses in terms of the peaks of frequency response function. Results from this first of its kind study on coronaviruses along with the proposed generic machine learning based approach will accelerate the detection of viruses and create efficient pathways toward future inventions leading to cure and containment in the field of virology.
... For such applications, the effects of surface irregularity on free torsional vibration of SWCNTs should be realized. During the last three decades, there have been numerous studies on the vibration of SWCNTs [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. ...
This study is an attempt to show the impacts of surface irregularity and compressive initial stresses on the torsional vibration of a single-walled carbon nanotube (SWCNT). The governing equation and corresponding closed-form solutions were derived with the aid of Hamilton's principle. Then, the natural frequencies were obtained analytically and the influences of surface irregularity and compressive initial stresses on the torsional vibration were studied in detail. Numerical results analyzing the torsional vibration incorporating compressive initial stress effects were discussed and presented graphically. The effects of surface irregularity on the natural frequency of torsional vibrations of nanomaterials, especially for SWCNTs, have not been investigated before, and most of the previous research works have been carried for a regular carbon nanotube. Therefore, it must be emphasized that the torsional vibrations of irregular SWCNTs are novel and applicable for the design of nano-oscillators and nanodevices, in which SWCNTs act as the most prevalent nanocomposite structural element. The analytical solutions and numerical results revealed that the surface irregularity and compressive initial stress have notable effects on the natural frequency of torsional vibrations. It has been observed that, as the surface irregularity and compressive initial stress parameters increase, the torsional natural frequency of vibrations of SWCNTs also increases. Since SWCNTs have very small size, they are always subject to initial stresses from different resources; therefore, understanding the influences of compressive initial stresses on the torsional frequency of nanotubes helps the engineers and researchers to design proper nanodevices for different applications with irregular shapes.
... The vibrations of CNTs have a considerable importance in nanomechanical applications such as the design of nano oscillators and nanodevices. During the last decade, numerous studies on vibrations of SWCNTs using continuum shell models were performed, as it can be seen from the examples of ref. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. Based on Euler-Bernoulli beam, Tang and Yang investigated a novel model of fluid-conveying nanotubes made of bidirectional functionally graded materials (FGMs) and presented the dynamic behavior and stability of nanotubes [20]. ...
In this work, an attempt is done to apply the eigenvalue approach as well as Donnell thin-shell theory to find out the vibrational analyses of an irregular single-walled carbon (ISWCNT) incorporating initial stress effects. The effects of surface irregularity and initial stresses on natural frequency of vibration of nano materials, especially for single-walled carbon nanotubes (SWCNTs), have not been investigated before, and most of the previous research have been carried for a regular and initial stress-free CNTs. Therefore, it must be emphasized that the vibrations of prestressed irregular SWCNT are novel and applicable for the design of nano oscillators and nanodevices, in which SWCNTs act as the most prevalent nanocomposite structural element. The surface irregularity is assumed in the parabolic form at the surface of SWCNT. A novel equation of motion and frequency equation is derived. The obtained numerical results provide a better representation of the vibration behavior of prestressed ISWCNTs. It has been observed that the presence of either surface irregularity or initial stress has notable effects on the natural frequency of vibration, particularly in the short-length SWCNTs. Numerical results show that the natural frequency of SWCNT decreases with increase in surface irregularity and initial stress parameters. To the authors’ best knowledge, the effect of surface irregularity and initial stresses on the vibration behavior of SWCNTs has not yet been studied, and the present work is an attempt to find out this effectiveness. In addition, the results of the present analysis may serve as useful references for the application and the design of nano oscillators and nanodevices, in which SWCNTs act as the most prevalent nanocomposite structural element.
... These are the general equations which completely relate the added mass and rotary inertia and the frequency shifts. Note from equation (45) that the value of the identified mass is affected by rotary inertia. In the special case, when the rotary inertia is neglected, substituting β ¼ 0, from equation (45) we have ...
... are available from experiment. These quantities can then be used as an 'input' to equations (44) and (45) to identify the added mass and rotary inertia. In the absence of experimental results, in this work we validate the approximate sensor equation against numerical results obtained from an independent and detailed finite element model. ...
... The Finite element method (FEM) is a powerful numerical technique for solving partial differential equations with general boundary conditions [44]. There are broadly two approaches to model carbon nanotubes using the FEM, namely the continuum approach (see for example [45]) and the discrete approach (see for example [46]). We refer to a recent state-of-the-art review paper for comprehensive discussions on FEM for nanomechanical systems [47]. ...
Nano and micromechanical mass sensing using cantilever oscillators of different length-scales has been an established approach. The main principle underpinning this technique is the shift in the resonance frequency caused by the additional mass in the dynamic system. While the mass of an object to be sensed is useful information, some idea about the shape of the object would be an additional benefit. The shape information may be used to make a distinction between two different objects of the same mass. This paper establishes the conceptual framework for simultaneous sensing of the mass as well as the rotary inertia of an object attached to a vibrating cantilever beam. The rotary inertia of an object gives additional insight into its shape, which is a key motivation of this work. It is shown that by using two modes it is possible to formulate two coupled nonlinear equations, which in turn can be solved to obtain the mass and the rotary inertia simultaneously from the frequency shifts of first two vibration modes. Euler-Bernoulli beam theory and an energy approach are used to derive closed-form expressions for the identified mass and rotary inertia from the measured frequency shifts. Analytical expressions are validated using high fidelity finite element simulation results.
... The natural frequencies versus the change of the geometry of the cantilever SWCNTs were investigated using the FE method with the beam elements by Ke and Li [49]. Chowdhury et al. [51][52] used the FE method to simulate the sliding vibration behavior of the MWCNTs, in which the vdW interaction between adjacent tubes was described by linear spring elements. In their research, the use of linear springs to describe van der Waals forces may not be accurate enough because van der Waals forces are non-linear forces that do not really reflect the interactions between tubes. ...
A nonlinear spring model is proposed to investigate the oscillation behavior of oscillators based on double-walled carbon nanotubes (DWCNTs) with open end by using the finite element (FE) method, where non-linear spring elements are used to represent the van der Waals (vdW) interaction between tubes. Compared to the linear spring FE model, the proposed non-linear springs can more accurately describe the interaction between nanotubes because the vdW interaction is a kind of strongly non-linear force. The influence of boundary conditions, geometric parameters of the DWCNTs, and the layer spacing of tubes on the natural frequencies is especially studied. Various oscillation modes and the corresponding natural frequencies are obtained. Compared to the results obtained by using the linear spring model, the natural frequencies of oscillators based on DWCNTs are in qualitatively better agreement with those obtained from the analytical method and the molecular dynamics (MD) method. From the FE results, it also can be seen that, DWCNTs is expected to be a nanoscale oscillatory device, and its oscillation mode and natural frequency can be adjusted by changing the geometric parameters and boundary condition of the tubes. The proposed nonlinear spring model is helpful for the design of the nano-oscillators under various conditions.
... There are numerous studies on the free vibration of SWCNT [42,43], DWCNT [44], and MWCNT [45] using continuum shell models during the last decade. In addition researchers investigated boundary effect [46,47], coupling between flexural modes [48], nonlinear vibration [49][50][51], wave propagation [52,53], and interlayer degree of freedom [54]. However, these models could not be applied for predicting vibrational behavior of CNS due to technical limitations. ...
Carbon nanoscroll (CNS) is a graphene sheet rolled into a spiral structure with great potential for different applications in nanotechnology. In this paper, an equivalent open shell model is presented to study the vibration behavior of a CNS with arbitrary boundary conditions. The equivalent parameters used for modeling the carbon nanotubes are implemented to simulate the CNS. The interactions between the layers of CNS due to van der Waals forces are included in the model. The uniformly distributed translational and torsional springs along the boundaries are considered to achieve a unified solution for different boundary conditions. To study the vibration characteristics of CNS, total energy including strain energy, kinetic energy, and van der Waals energy are minimized using the Rayleigh-Ritz technique. The first-order shear deformation theory has been utilized to model the shell. Chebyshev polynomials of first kind are used to obtain the eigenvalue matrices. The natural frequencies and corresponding mode shapes of CNS in different boundary conditions are evaluated. The effect of electric field in axial direction on the natural frequencies and mode shapes of CNS is investigated. The results indicate that, as the electric field increases, the natural frequencies decrease.
