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wave domain analysis for axially moving pre-stressed nanobeam by wavelet-based spectral element method
Author’s Accepted Manuscript Time/wave domain analysis for axially moving prestressed nanobeam by wavelet-based spectral element method Ali Mokhtari, Hamid Reza Mirdamadi, Mostafa Ghayour, Vahid Sarvestan www.elsevier.com/locate/ijmecsci
PII: DOI: Reference:
S0020-7403(15)00383-5 http://dx.doi.org/10.1016/j.ijmecsci.2015.11.006 MS3137
To appear in: International Journal of Mechanical Sciences Received date: 7 June 2015 Revised date: 12 October 2015 Accepted date: 5 November 2015 Cite this article as: Ali Mokhtari, Hamid Reza Mirdamadi, Mostafa Ghayour and Vahid Sarvestan, Time/wave domain analysis for axially moving pre-stressed nanobeam by wavelet-based spectral element method, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2015.11.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Time/wave domain analysis for axially moving pre-stressed nanobeam by wavelet-based spectral element method Ali Mokhtari, Hamid Reza Mirdamadi*, Mostafa Ghayour, Vahid Sarvestan Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran *Corresponding author. Tel.: +98 313 391 5248; fax: +98 313 391 2628. E-mail addresses: [email protected] (A. Mokhtari), [email protected] (H.R. Mirdamadi), [email protected] (M.Ghayour), [email protected] (V. Sarvestan).
Abstract In this article, a time/ wave domain analysis is presented for an axially moving pre-stressed nanobeam by wavelet-based spectral element (WSE) method. WSE scheme is constructed as spectral element method (SEM), except that Daubechies wavelet basis functions are used for transforming governing partial differential equation. These basis functions help to rule out some deficiencies of SEM due to periodicity assumption, especially for time domain analysis. Numerical examples are used for validating the accuracy and efficiency of model. The higher accuracy of WSE approach is then evaluated by comparing its results with those of classical finite element and SEM. The effects of moving nanobeam properties, such as velocity, pretention and nonlocal (small-scale) parameter, on vibration and wave characteristics and dispersion curves are investigated. In addition, the instability of moving nanobeam is studied both analytically and numerically considering divergence and flutter.
Abbreviations AD: artificial damping; BVP: boundary-value problem; CNT: carbon nanotube; FEM: finite element model; NP/BVP: non-periodic boundary-value problem; ODE’s: ordinary differential equations; P/BVP: periodic boundary-value problem; PDE: partial differential equation; P/WSE (P/WSFE): periodic wavelet-based spectral (finite) element; SEM (SFEM): spectral (finite) element method; WAD: weak artificial damping; WSE (WSFE): wavelet-based spectral (finite) element.
Keywords Axially moving pre-stressed nanobeam; nonlocal parameter; wavelet-based analysis; wave domain analysis; spectral element model; divergence/ flutter instability; Daubechies wavelet.
Nomenclature c
constant transport speed [ ⁄ ] divergence speed [ ⁄ ] flutter speed [ ⁄ ] nonlocal (small-scale) parameter [ flexural rigidity [
]
]
excitation force [ ] cut-off frequency [ Nyquist frequency [ {̂ }
] ]
global WSE nodal forces [ WSE wavenumber [
] ]
wavenumber for the Euler-Bernoulli beam theory [√ [̂ ]
⁄ ]
global dynamic stiffness matrix span between two end supports [ ] length of an element [ ] bending moment [
]
number of sampling points order of the Daubechies wavelet constant axial pretension [ ] shear force [ ] tensile force-to- flexural rigidity ratio [ ⁄
]
first-order connection coefficient matrix (time domain) second-order connection coefficient matrix (time domain) eigenvalues of
[
⁄ ]
first-order connection coefficient matrix (wave domain) second-order connection coefficient matrix (wave domain) eigenvalues of
[
⁄ ]
mass per length of beam [
⁄ ]
Daubechies scaling function at an arbitrary scale ,
connection coefficients
1. Introduction Axially moving structures and continua can be found in numerous engineering devices in mechanical, civil, electrical, and aerospace applications, e.g., thread-lines in fabric industry, rolled steel beams, chain and belt drives, high-speed sheets, magnetic tapes, band saw blades, aerial cable tramways, and so on. Due to this pervasiveness, the analysis of moving structures and continua has been a motivation for a large amount of publications, mostly on axially moving beams. The lateral vibration of such axially moving systems is commonly modeled as a prestressed beam. It is necessary to forecast the dynamic characteristics of such systems precisely, in order to reach safe, reliable, and successful designs, while hazards and accidents are prevented. There are many articles which analyze axially moving macro-scale (classical) beams. The solutions of equation of motion for moving classical beam models were obtained by several solution techniques, containing the Galerkin's [1-3], assumed modes [4], finite element [5], Green's function [6], transfer function [7], perturbation [8], the Laplace transform [9], artificial parameters [10], and the FFT-based spectral element methods (SEM) [11]. Despite the large number of studies for axially moving classical beams, few analyses have been conducted on similar problems at micro- and nano-scales. Assumed modes method and Galerkin approach [12], higher-order strain gradient solutions [13], and modified couple stress theory [14] are used to investigate moving micro- and nano-scale beam models. As a versatile numerical method, the FEM has an important role in structural analysis. This method may provide accurate dynamic response of a structure if the wavelength is large compared to the mesh size. However, the FEM results become increasingly inaccurate as the frequency bandwidth increases. As a drawback of FEM, it is well known that a large number of
FE’s should be generated to obtain trustworthy and accurate solutions, especially at higher frequencies. Evidently, this provision may increase the computational time and cost. These problems are generally resolved by SEM. SEM [15] transforms the governing partial differential equation (PDE) of motion to a set of ordinary differential equations (ODE’s) by FFT. These space-dependent ODE’s are solved exactly, which are then used as dynamic shape functions for SEM. It is well known that the wavelet-based spectral element (WSE) model is an exact solution method for dynamic analysis of structures [16-18]. WSE formulation is very similar to SEM formulation, except that Daubechies wavelet basis functions are used for transformation of governing PDE. The ensuing coupled ODE’s can be decoupled by performing a waveletdependent eigenvalue problem. The decoupled ODE’s are then solved similarly as in SEM. In SEM, the time window is dependent on the value of damping and the dimensions of structure. It requires to be more wide for weakly damped and shorter dimension structures. A potential remedy to reduce these dependencies is to use artificial damping (AD) [15]. By using nonperiodic boundary condition assumptions [19] for WSE, exactness of results could be free from those deficiencies previously noted such as lack of damping, structures having short dimensions, and small time windows. It should be noted that WSE method could also be used for analyzing undamped structures, where SEM could not work. This model can be used in time domain analysis without any transformations between different domains, something unlike SEM. Periodic boundary condition-based WSE formulation [19] can extract frequency dependent wave characteristics, like wavenumbers, directly. Moreover, [20, 21] show the application of wavelet technique in FE domain. Recent developments in research on axially moving structures were reviewed by Marynowski and Kapitaniak [22]. However, to the best of authors’ knowledge, the WSE model has not yet been introduced in the literature for axially moving nanobeam structures and this should be the most remarkable innovation of this study. Thus, the objectives of this article are: (1) to develop a WSE model for axially moving EB pre-stressed beams based on nonlocal elasticity theory, (2) to highlight higher accuracy of this model as compared with those of classical FEM and SEM, and (3) to investigate the effects of nanobeam properties, such as velocity, pretention and nonlocal parameter, on the vibration and wave characteristics, dispersion curves, and instability.
2. Mathematical model Based on the formulation derived in Appendix A, the governing equation for an element of axially moving EB pre-stressed beam based on nonlocal elasticity theory can be represented as [
] (1)
[
]
The force boundary conditions are given as
(2)
{
where
and
are the shear force and bending moment define as [
] (3)
[
[
]
] (4)
[
]
Fig. 1 shows an element of the nanobeam where
and
are the bending moment
and transverse shear force applied at
and
are the bending moment
and transverse shear force applied at
, and .
Fig. 1. A finite element of axially moving pre-stressed nanobeam.
3. Temporal discretization The scaling transform of functions Daubechies scaling function ⁄
at an arbitrary scale
can be done by a sequence of , as
∑
⁄
(5)
∑
Daubechies scaling function ⁄
where
and
(6)
at an arbitrary scale
is defined as [23]
⁄
and
(7)
, noted by
and
hereafter, are the approximation coefficients at a
definite spatial dimension . Substituting Eqs. (5) and (6) into Eq. (1) gives
⁄
]∑
[ ⁄
(
∑
)
⁄
⁄
∑
(8) ⁄
∑ (
⁄
⁄
∑
) ∑[
]
Taking inner product on both sides of Eq. (8) by ]∑
[
∑
∑
∫
(9)
∫
∑
, gives
∫
∫
∑
, where
∫
∑
∑[
∑
∫
] ∫
where the scaling functions are orthogonal, i.e.,
∫
(10)
By substituting Eq. (10) into Eq. (9), this can be written as
[
simultaneous ODE’s
∑
]
∑ (11)
∑
where,
∑
is the order of the Daubechies wavelet.
and
are the connection coefficients
defined as ∫
(12)
∫
(13)
Beylkin [24] computed the connection coefficients. For compactly supported wavelets, are nonzero only in the interval
.
Similar to Eq. (5), scaling transform of functions ⁄
⁄
and
∑
∑
and
are written as (14)
(15)
Substituting Eqs. (14) and (15) into Eqs. (3) and (4), respectively, give the following differential form [
] ∑
[
∑
(16)
]
[
] ∑
[
(17)
∑
]
While numerical execution deals with a finite interval of time = ⁄
, where
in the above equations could be changed by ⁄
and ⁄
where
[23]. Therefore,
, is . Consequently,
, and
, respectively. One may also
observe from [25] that for extreme efficiency of computation, the optimal length of sequence needs to be an integer power of two, i.e.,
,
.
The ODE’s given by Eqs. (11), (16), and (17) are the wavelet-based spectral element formulation for an axially moving EB pre-stressed beam based on nonlocal elasticity theory. Eq. (11) gives coupled ODE’s, which are to be solved for
, using methods defined afterward. Since numerical
simulations could deal with a small number of sequences, finite interval,
. Likewise,
should only be known in a
should appear in the corresponding finite interval
. In Eq. (11), the ODE’s only relating to interval include coefficients
that lie outside
or . Mitra and Gopalakrishnan in
[19] described the information of two approaches for solving this kind of boundary-value problem (BVP). For a periodic BVP (P/BVP), the function
is assumed to be periodic in
. Having this assumption in mind, the coupled ODE’s given by Eq. (11)
time, with time period
can be written in matrix form as [
]{
}
[
[
]{
{
where
and
]{ }
}
(
){
}
(
){
}
(18)
}
are
circulant connection coefficient matrices obtained from P/BVP
assumption. This periodic WSE (P/WSE) algorithm can be used directly in frequency domain analysis, something similar to SEM. However, this algorithm cannot be proper for time domain analysis because of the periodicity assumption. The second method, i.e., the wavelet extrapolation method of Daubechies compactly supported wavelets could be applied to non-periodic BVP (NP/BVP), here called NP/WSE algorithm, as proposed by Amaratunga and Williams [23, 25, 26]. With this algorithm, the coupled ODE’s, given by Eq. (11), can be written in matrix form as [
]{
}
[
[
]{
{
where
and
]{ }
(
} ){
}
(
){
}
(19)
}
are connection coefficient matrices obtained from NP/BVP assumption.
NP/WSE algorithm can be used in time domain analysis without any transformations between different domains, something unlike SEM. It should be noted that the connection coefficient matrices
,
, and
,
are entirely
-dependent and problem-independent [19]. Where
the order of wavelet used. The second-order connection coefficient matrices calculated from Eqs. (20) and (21), respectively [19].
and
can be
is
(20) (21)
It should be noted that all the formulations of P/WSE are exactly the same as those of NP/WSE, except that
are replaced by
.
4. Spatial discretization and wavelet spectral element formulation WSE method transforms the PDE of motion to a set of ODE’s by using Daubechies wavelet basis functions. These space-dependent ODE’s are solved exactly, which are then used to derive dynamic stiffness matrix. In this section, WSE’s are formulated for an axially moving EB pre-stressed beam based on nonlocal elasticity theory. In WSE model, the resulting ODE’s are coupled. However, diagonalizing the connection coefficient matrix can decouple the set of equations [16]. This can be done by wavelet-dependent eigenvalue analysis of the matrix as (22)
where
is the eigenvector matrix of
eigenvalues
, where
√
and
is the diagonal matrix, containing corresponding
. Similar expression holds for
, where
and
are known
analytically [19], as follows
(23)
[
]
The eigenvalues
are
∑
and the corresponding orthonormal eigenvectors
(24)
,
, are
(25)
√
From Eqs. (20) and (21),
and
can be written respectively, as (26)
where,
is a diagonal matrix with diagonal terms
and (27)
where
is a diagonal matrix including diagonal terms
. P/WSE and NP/WSE connection
coefficient matrices are entirely problem-independent; therefore, this eigenvalue analysis can be done once and stored forever. Then, these eigenvalues can be used directly in a dynamic analysis. This decreases computational time. The ODE’s obtained from decoupling Eq. (18), as well as Eq. (19), can be written as [
]
̂
̂
[
]
̂
̂
̂
(28)
̂
̂
where {̂ }
{
{ ̂}
{ }
}
(29) (30)
Forced boundary conditions given in Eqs. (16) and (17) are similarly transformed as ̂
[
] ̂
̂
̂
̂
(31)
̂
[
]
̂
̂
̂ (32)
̂
Hereafter, the subscript is dropped for establishing easy-to-work notation, noticing all the equations hold for
.
Fig. 2 shows a WSE for an axially moving EB nanobeam with nodes and two DOFs ̂ and ̂
at each node. The nodal transverse forces and moments are ̂ and ̂ ,
̂
respectively.
Fig. 2. A WSE axially moving EB nanobeam with equivalent nodal forces and DOFs.
Solving homogeneous form of decoupled ODE’s, i.e., Eq. (28), gives the exact interpolating function as ̂
(33)
where
is the WSE wavenumber. Substituting interpolating function into homogeneous
equation, resulting from decoupled ODE’s, i.e., Eq. (28), provides a dispersion relation {
[
]}(
) (
(
)
{
}(
)
(34)
)
where
(35)
(
)
is the span between two end supports.
is the tensile force-to-flexural rigidity ratio.
is the
wavenumber for the EB beam theory [16]. is the dimensionless velocity of moving nanobeam. is the dimensionless nanobeam.
. Similarly,
is the dimensionless wavenumber for moving
is the dimensionless nonlocal parameter
. WSE wavenumbers
are calculated from dispersion relation, i.e., Eq. (35). Then the general solution, i.e., Eq. (33), can be written as ̂
(36)
̂
̂
and {
}
(37)
{
} are the constants to be derived from the boundary
conditions at the two nodes. The general solutions can also be written in matrix form as ̂
{
}
(38)
̂
{
}
(39)
where
,
matrix with diagonal terms
. In addition, for
is a
diagonal
Applying the essential boundary conditions
to the two nodes gives ̂
̂
̂
̂
{ {
̂
̂
̂
̂
{ {
} } } }
(40) (41) (42) (43)
where
,
,
, and
.
The nodal displacement vector, as a function of generalized coordinates, can be derived by augmenting Eqs. (40)-(43) as
{̂ }
̂ ̂ ̂ {̂
]{
[
}
{
}
(44)
}
Homogeneous form of differential equations of systems, i.e., regardless of the effect of excitation force, is required for constructing dynamic stiffness matrix. Consequently, For computing force boundary conditions, it is first required to derive the end shear forces, ̂ moments, ̂ ̂
, as given by Eqs. (31) and (32) with ̂
, which can be written in matrix form
}
}
{ { {
̂
and end bending
}
}
{{
} {
{
}
{
}
(45)
(46)
}
where
and
. The boundary values for
the end shear forces and end bending moments, at two nodes can be written as ̂
̂
̂
̂
̂
̂
̂
̂
where and
̂
{ ̂
{
{
{
|
}
(47)
}
(48)
{
|
}
(49)
}
(50)
}
,
,
{
}
,
. The nodal force vector can be written by combining Eqs. (47)-(50) as
̂ {̂ }
̂
{
̂ [
{̂ }
}
{
}
(51)
]
At last, substituting {
} from the nodal displacement vector, i.e., Eq. (44), into nodal force
vector, i.e., Eq. (51), the nodal displacements and nodal forces are interconnected as {̂ }
{̂ }
[ ̂ ]{ ̂ }
(52)
Here, [ ̂ ] is the elemental dynamic stiffness matrix for moving EB pre-stressed nanobeam. In WSE model, the connection between neighboring FE’s (or assembly procedure) and boundary conditions can be processed easily, as done for classical FEM, to obtain the global dynamic stiffness form as {̂ }
[ ̂ ]{ ̂ }
(53)
where [ ̂ ], { ̂ }, and { ̂ } denote the exact global dynamic stiffness matrix for a moving EB pre-stressed nanobeam, the global wavelet-based spectral nodal DOF vector, and the global wavelet-based spectral nodal force vector, respectively. The computations of the above matrices for
are done numerically. The above equations can be solved to derive the
nodal displacement vector, { ̂ }, for recognized nodal forces. Hence, four nodal DOFs of { ̂ } can be substituted into general solutions, i.e., Eq. (36), to derive the constants {
}. These can be
substituted into general solutions, i.e., Eqs. (36) and (37), to obtain the generalized displacements ̂
and ̂
at any arbitrary point of the nanobeam. Then, ̂
can be substituted into
wavelet transform equation, i.e., Eq. (29), to obtain the time domain responses.
5. Wave domain analysis From a wave-domain standpoint, there is a specific relationship between the transformed ODE’s appeared in WSE model and those in SEM. Hence, WSE solutions can also be used in wave domain analysis, the same as SEM, but without any transformations between different domains. The matrix
is a circulant matrix with the following properties [17]
{
[
where { ̃ ⁄ given { eigenvalues,
{̃
] }
̃
}
(54)
⁄
} are FFT of { }
}, and { ̃ } are the FFT coefficients of the first column of }. Thus, the FFT coefficients { ̃ } are the
{ , of the matrix
[
]{
̃
. Operating FFT on both sides of Eq. (18) yields }
[
]{ ̃
{̃
̃
{ ̃
}
}
(
){
]{
̃
}
{
[
]{ ̃
{̃
}
(
){ ̃ }
(55)
}
In SEM, the transformed ODE’s are exactly the same, except that [
̃
̃
}
̃
are replaced by
.
}
(
){
̃
}
(
){ ̃ }
(56)
}
where (57)
Fig. 3(a) shows that for a same interval are equal to fraction, i.e.,
, for both WSE method and SEM,
up to a definite fraction of Nyquist frequency, ⁄
, is
and real part of
⁄
⁄
. This
-dependent and is greater for higher order basis functions, for
which it is problem-independent. Where
for
and it is close to
for
. Therefore, WSE model can be used in wave domain analysis, similar to SEM, just up to a definite fraction of
. In addition, the eigenvalues
in NP/WSE algorithm are complex-
valued and the imaginary part of those are calculated and plotted for different bases in Fig. 3(b). From a wave-domain standpoint, P/WSE algorithm, as compared with NP/WSE, has lower computational cost and time. Therefore, P/WSE is implemented for wave domain analysis in this article.
(a)
(b) Fig. 3. (a) Examination of resemblances between ( ) imaginary part of
,
, and real part of
, for different orders of basis,
, and
.
6. Stability analysis The eigenfrequencies,
, of an assembled system can be calculated from the constraint
that the determinant of P/WSE dynamic stiffness matrix, [ ̂ ], as a function of
(or ), vanishes
at
. That is [̂ ]
(58)
Stability analysis requires presuming the free vibration responses of the moving nanobeam, as follows [11] (59)
Substituting free vibration responses, i.e., Eq. (59), into the free vibration equation of wave differential equation of system, i.e., Eq. (1), gives an eigenvalue problem. ODE’s obtained from the formula of free vibration response are precisely the same as those from P/WSE algorithm, i.e., Eq. (55), except that
are exchanged by . Eventually, this eigenvalue problem can be
written as [̂
]
(60)
For each class of instability, the dynamic behavior of these moving structures is described by the sign of real and imaginary parts of , as follows [11]
(61)
As it is mentioned in [11], considering the existence of non-trivial dynamic equilibrium positions by setting
√
(or
) and
{ {
yields the divergence velocity, }
}
, in closed form as
(62)
7. Numerical simulations A uniform carbon nanotube (CNT) over two simple supports is considered that is axially moving. CNTs would be an excellent selection for high speed nano-scale tools. Moving CNTs would be used in spacecraft antennas, space elevator cables, high speed vehicles, and high speed
magnetic nanotapes. Here, a moving CNT is modeled by a moving EB nanobeam. The structural and material properties of the nanobeam are span between two end supports Young’s modulus ⁄
[12], radius of cross-section ⁄
,
, mass density
[12]. A transverse excitation load is applied at the mid-span where the nanobeam is
subjected to an axial tensile force
. The load is an impulsive function that has
amplitude and the duration of 0 to
.
and
are
and
, respectively, unless
otherwise mentioned. P/WSE algorithm is used to study stability analysis, the natural frequencies, and the dispersion relation curves. NP/WSE algorithm is used to simulate wave propagation in the same structure. In order to demonstrate the validity of proposed model, several numerical examples are carried out in this section. The results are compared with FEM and SEM to validate and show the vantages of WSE model for time-frequency domain analysis of wave propagation. The order of Daubechies scaling functions used is
.
7.1 Wave domain and stability analyses
The natural frequencies for the structure are calculated using P/WSE algorithm with . Consequently, wave domain characteristics could be acquired for a frequency range of
resulting from SEM but for a frequency range of from P/WSE with
. The P/WSE algorithm is valued by comparing the
natural frequencies calculated from this algorithm, those obtained from the analytical approach [15, 27], those from the SEM, and those from classical FEM. Table 1 demonstrates that the P/WSE and SEM results using one single element are equal to those from exact analytical results. This means that, P/WSE algorithm can be used to calculate the eigenfrequencies,
, like SEM. On the other hand, for FEM dynamic analysis, the total
number of FE’s used in the analysis is varying in order to improve the numerical accuracy. Contrary to the classical FEM, both the present P/WSE model and SEM provide highly accurate results by using only one element. Table 1 shows that when the axial speed divergence speed
raises to the
, the fundamental frequency disappears for the first time. Once again, for a
constant axial velocity and nonlocal parameter, as the axial pretension increases, the natural frequencies would generally increase. One may also mention from Table 1 that for a fixed pretension, as the axial velocity of the nanobeam increases, all the natural frequencies would
decrease. It is observed from Table 1 that the natural frequencies decrease with increasing nonlocal parameter for a fixed pretension and axial velocity. Therefore, a suitable determination of interval
would be an essential decision making for an accurate wave domain analysis at
higher frequencies by P/WSE model and SEM.
Table 1. Natural frequencies of the axially moving EB pre-stressed nanobeam Moving velocity
Investigated
Method (number of
(m/s)
parameters
elements)
0
Natural frequencies (GHz)
Theory [15]
34.91
139.63
2234.02
WSE (1)
34.91
139.63
2234.02
SEM (1)
34.91
139.63
2234.02
FEM (30)
34.91
139.63
2234.77
FEM (5)
34.91
139.86
2891.26
Theory [27]
34.17
128.78
1144.92
WSE (1)
34.17
128.78
1144.92
SEM (1)
34.17
128.78
1144.92
FEM (30)
34.17
128.78
1145.30
FEM (5)
34.17
128.99
1435.95
Theory [15]
34.97
139.69
2234.08
WSE (1)
34.97
139.69
2234.08
SEM (1)
34.97
139.69
2234.08
FEM (30)
34.97
139.69
2234.83
FEM (5)
34.97
139.92
2891.32
2053.80
WSE (1)
0.00
110.61
1091.70
(Divergence
SEM (1)
0.00
110.61
1091.70
FEM (30)
0.01
110.61
1092.28
FEM (5)
0.43
111.09
1433.80
0
0
velocity,
)
3936.55
WSE (1)
-
28.67
949.30
(Flutter velocity,
SEM (1)
-
28.67
949.30
FEM (30)
-
28.83
950.38
FEM (5)
-
34.99
1423.62
)
Fig. 4 shows the stability diagram of the classical beam and nanobeam. There are four distinct regions seen in the figure for nanobeam, as explained below: (1) Since all the eigenvalues pure imaginary for
are
, the moving beam would be stable. The meaning of ⁄ . (2) For
is first divergence velocity for nanobeam and its value is
, there exist purely real positive eigenvalues , indicating the occurrence of divergence instability. The numerical value of For the velocity interval, since all the eigenvalues
is
⁄ . (3)
, the moving beam is stable again are pure imaginary.
flutter velocity for nanobeam. (4) For
⁄ is the lowest , the complex-valued eigenvalues
with
real positive parts indicate the flutter instability. Similar to nanobeam, there are four separate zones as seen in this figure for classical beam. A closer scrutiny of Fig. 4 shows that the divergence velocity as well as flutter one would appear at lower levels of speed as nonlocal parameter grows up from zero.
Fig. 4. Stability diagram for eigenvalues for
and solid line for
against moving speed of nanobeam c, for different values of the nonlocal parameters, (dash line ).
Fig. 5 demonstrates the effects of axial pretension, nanobeam,
,
, and
when
, on three critical axial velocities of
. In addition, the effects of nonlocal parameter,
, on those velocities are presented in Fig. 6. Furthermore, the stable and unstable regions are evident in Fig. 5 and Fig. 6. Four different zones appear in these figures from bottom to top, as expressed below: (1, 2) The zone under the curve and and
and the narrower zone between two curves
are the first and second stable regions, respectively. (3) The region among two curves points to the first divergence instability region. (4) The zone just up the curve
shows the flutter instability region. It is apparent from Fig. 5 that these three critical axial velocities are uniformly increasing as the axial pretension increases. Fig. 6 reveals that these three critical velocities reduce as nonlocal parameter becomes more highlighted. Both Figs. 4 and 6 explain that small-scale parameter leads to instability of moving nanobeam at lower levels of velocity.
Fig. 5. The critical moving velocities against the axial pretension at
.
Fig. 6. The critical moving velocities against the nonlocal parameter at
In Fig. 7, the percentage of relative error is plotted versus the functions where
.
⁄
for different orders of basis
is wavenumber resulting from P/WSE algorithm, while
is the
wavenumber obtained from SEM, respectively. It can be shown that the error is negligible up to a certain
⁄
, and then it increases very suddenly. The fraction,
from Fig.6 for different
.
, can be derived numerically
Fig. 7. Examination of resemblances between wavenumbers of basis,
(SEM) and
(WSE model), for different orders
.
In Figs. 8-10, the wavenumber against frequency is plotted using P/WSE algorithm at several moving velocities and the plots are compared with those of values from SEM. It can be shown that
should be identical to
up to a definite value for
structure are computed using P/WSE algorithm with
. The wavenumber for the .
As shown in Fig. 8(a) and Fig. 8(b), when the nanobeam is in a stationary state (i.e., the wavenumber
⁄ ),
is purely real-valued and negative, which means there exist damping
waves within the nanobeam. When the nanobeam is moving, the wavenumber
gets complex-valued, as shown in Fig.
9(a) and Fig. 9(b). In addition, these figures show that as far as the axial velocity of nanobeam raises to a critical value
, all the wavenumbers at zero frequency join together at zero-value.
(a)
(b) Fig. 8. (a) The dispersion relation curve for wave propagation in moving EB nanobeam at and
,
,
, by using SEM and WSE model, and (b) Zoom-in on the dispersion relation curve in subfigure (a).
(a)
(b) Fig. 9. (a) The dispersion relation curve for wave propagation in moving EB nanobeam at
,
, and
, by using SEM and WSE model, and (b) Zoom-in on the dispersion relation curve in subfigure (a).
The critical axial velocity,
, can be derived from Eq. (34) by putting
detecting the state that all the roots of √
(or
) and
become zero, as shown below
⁄
(63)
Fig. 10(a) and Fig. 10(b) show that if the axial velocity of nanobeam remains to grow beyond the wavenumber
becomes purely imaginary-valued at the frequency range
,
,
while
becomes again complex-valued at the range
velocity is larger than given by
. This means that when the axial
, there could exit traveling waves within a narrow low-frequency band
. Therefore, the frequency
could be termed as the cut-off frequency.
(a)
(b) Fig. 10. (a) The dispersion relation curve for wave propagation in moving EB nanobeam at and
,
,
, by using SEM and WSE model, and (b) Zoom-in on the dispersion relation curve in subfigure (a).
7.2 Time domain analysis
Unlike the use of SEM, NP/BVP assumption leads to a direct use of WSE model for time domain analysis. In Fig. 11, the results of WSE formulations and SEM are compared in order to highlight vantages of the former, especially for time domain analysis. It is noticed that weak
artificial damping (WAD) model is used for SEM, i.e., a model of damping having values lower than that recommended by [15]. It should be noted that the SEM could not work for this structure without attributing any AD, while WSE method could work for this case study without any limitation. For WSE results, the time window of
and
appropriate, while for WAD formulation of SEM,
should be , and
are used. Moreover, WSE model, similar to SEM one, needs two FE’s for calculating the results adequately accurate. Fig. 11 shows that the length of time window,
, as used in the
SEM including a WAD mechanism, is extended to increase its accuracy.
(a)
(b) Fig. 11. (a) Examination of resemblances between WSE and WAD-(i.e., weak artificial damping) based SEM results at
⁄ ,
, and
, and ( ) Zoom-in on deflection of mid-span in subfigure (a).
In this part, the AD that recommended by [15] is used for SEM. Figs. 12-15 show the time responses of moving nanobeam at several moving velocities. By increasing the value of AD, the SEM results reach more accuracy than those of WAD based SEM. However, as the previous case, as much as the width of time window,
, used in the SEM, is extended its results
improves. As could be anticipated from Fig. 4 and Figs. 12-15, the moving nanobeam is stable at ⁄ (Figs. 12(a) and 12(b)), unstable by divergence at and 13(b)), stable again at
⁄ (Figs. 13(a)
⁄ (Figs. 14(a) and 14(b)), and unstable by flutter at
⁄ (Figs. 15(a) and 15(b)).
(a)
(b) Fig. 12. (a) Examination of resemblances between WSE and SEM time responses at stable zone,
, and
means weak artificial damping.
, i.e., the first
, and ( ) Zoom-in on deflection of mid-span in subfigure (a). WAD
Fig. 13. WSE time responses at moving speed happening divergence,
,
, and
.
Fig. 14. WSE time responses at
, i.e., the second stable zone,
Fig. 15. WSE time responses at moving speed happening flutter, .
, and
,
.
, and
The WSE and FEM results are compared to validate the higher accuracy of WSE results for time domain analysis in Fig. 16. WSE results are obtained by using sampling rate and just two FE’s. FEM results are obtained by using 10 and 100 beam FE’s and Newmark’s time integration scheme with a sampling rate of
. Fig. 16 shows that the results
predicted by FEM would converge to those predicted by the present WSE model provided that the total number of FE’s used in FEM could increase beyond 100.
Fig. 16. Examination of resemblances between WSE and FEM time responses of EB nanobeam at moving velocity happening flutter,
, and
.
Generally, elastic structures should have more stiffness and fundamental frequency for lower moving velocities. Both the shift-in-time and shift-in-magnitude for response amplitudes in Fig. 17 emphasize this fact of the moving nanobeam.
Fig. 17. The effects of moving speed on time responses at
and
.
Again, structures should have more stiffness and fundamental frequency for higher pretensions. Therefore, narrower time periods and lower amplitudes of time responses in Fig. 18 would demonstrate this fact of the moving nanobeam.
Fig. 18. The effects of axial pretension on time responses at
Fig. 19 shows the effects of nonlocal parameter, i.e.,
and
.
, on the nanobeam time responses.
Again, this structure would have lower natural frequencies for higher nonlocal parameter. Therefore, narrower time periods of time response would appear for lower nonlocal parameter. The horizontal shifts toward the time scales would demonstrate the fact for the mid-point deflection of nanobeam.
Fig. 19. The effects of nonlocal parameter, i.e., .
, on the nanobeam time responses at
and
8. Conclusions The objective in this article was to present a WSE model, formulated in terms of Daubechies scaling functions and exact dynamic shape functions, for a moving EB pre-stressed nanobeam. The wavelet transform was applied for transforming the temporal/spatial-dependent PDE of the system to a set of spatial-dependent ODE’s, as a replacement for using Fourier transformation. Both P/WSE and NP/WSE algorithms were used to process wave and time domains, respectively. From a wave-domain standpoint, wave properties could also be extracted, but just up to a certain fraction of the Nyquist frequency. From a time-domain standpoint, the present method proved to be more efficient as it could eliminate the deficiencies associated with SEM for a time domain analysis. Numerical simulations were done to study the wave propagation due to the vibration of an axially moving pre-stressed nanobeam. These numerical tests demonstrated the feasibility and capability of using WSE model in moving structures. The effects of axial pretension, velocity and nonlocal parameter on the stiffness, deflection, and instability of moving system were also studied. Comparisons with SEM results were also presented to focus the vantages and limitations of WSE model. It was shown that this formulation could provide extremely accurate solutions even by using fewer number of FE’s (one or two WSE’s), as compared with those obtained from exact analytical solutions and SEM, as well as FEM procedure.
Appendix A Imagine an EB pre-stressed beam that moves in its axial direction at a constant velocity c, as shown in Fig. A.1.
Fig. A.1. Axially moving EB pre-stressed beam and it’s Cartesian coordinate system.
The governing wave differential equation of this system is given as [15]
(A.1)
here
is the transverse deflection,
moment,
is constant axial pretension, and
moment,
is the excitation force,
is bending
is the mass per length of beam. Bending
, can be expressed as ∫
(A.2)
Based on the Eringen’s nonlocal continuum theory, the constitutive relation for beam model is [12, 13, 28, 29] (A.3)
where
is axial stress, is the axial strain, and
is nonlocal parameter (or small-scale
parameter). Multiplying Eq. (A.3) by and taking integration of results over cross-sectional area of the beam, lead to (A.4)
where
is flexural rigidity. Plugging the second derivative of bending moment,
, from
Eq. (A.1) into Eq. (A.4), gives
(A.5)
[
]
The governing equation of axially moving pre-stressed nanobeam, is found by substituting nonlocal bending moment, i.e. Eq. (A.5), into Eq. (A.1), as
[
] (A.6)
[
]
Eq. (A.6) governs the vibration and axially rigid body translation of a nanobeam under arbitrary transient excitation.
References [1] S.J. Hwang, N.C. Perkins, "Super-critical stability of an axially moving beam", Journal of Sound and Vibration, vol. 154, no. 3, pp. 381-409, 1992. [2] A.A.N. AI-Jawi, C. Pierre, A.G. Ulsoy, "Vibration localization in dual-span axially moving beams, Part I: Formulation and results", Journal of Sound and Vibration, vol. 179, no. 2, pp. 243-266, 1995. [3] F. Pellicano, F. Vestroni, "Non linear dynamics and bifurcations of an axially moving beam", Journal of Vibration and Acoustics, vol. 22, pp. 21-30, 2001. [4] H.P. Lee, "Dynamics of a beam moving over multiple supports", International Journal of Solids and Structures, vol. 30, no. 2, pp. 199-209, 1993. [5] M. Stylianou, B. Tabarrok, "Finite Element analysis of an axially moving beam, Part I : Time integration", Journal of Sound and Vibration, vol. 178, no. 4, pp. 433-453, 1994. [6] J.A. Wickert, C.D. Mote, "Classical vibration analysis of axially moving continua", Journal of Applied Mechanics, vol. 57, pp. 738-744, 1990. [7] C.H. Riedel, C.A. Tan, "Dynamic characteristics and mode localization of elastically constrained axially moving strings and beams", Journal of Sound and Vibration, vol. 215, no. 3, pp. 455-473, 1998. [8] H.R. Oz, "On the vibrations of an axially traveling beam on fixed supports with variable velocity", Journal of Sound and Vibration, vol. 239, no. 3, pp. 556-564, 2001. [9] S. Chonan, "Steady state response of an axially moving strip subjected to a stationary lateral load", Journal of Sound and Vibration, vol. 107, no. 1, pp. 155-165, 1986. [10] T. Yang, B. Fang, X.D. Yang, Y. Li, "Closed-form approximate solution for natural frequency of axially moving beams", International Journal of Mechanical Science, vol. 74, pp 154–160, 2013. [11] H. Oh, U. Lee, D. Park, "Dynamics of an axially moving Euler-Bernoulli beam: Spectral Element modeling and analysis", KSME International Journal, vol. 18, no. 3, pp. 395-406, 2004. [12] K. Kiani, "Longitudinal, transverse, and torsional vibrations and stabilities of axially moving singlewalled carbon nanotubes", Current Applied Physics, vol. 13, pp. 1651-1660, 2013.
[13] C. W. Lim, C. Li, J-L. Yu, "Dynamic behaviour of axially moving nanobeams based on nonlocal elasticity approach", Acta Mechanica Sinica, vol. 26, pp. 755-765, 2010. [14] K. Marynowski, "Axially moving microscale panel model based on modified couple stress theory", Journal of Nanomechanics and Micromechanic ASCE. DOI: http://dx.doi.org/10.1061/(ASCE)NM.21535477.0000071 , A4014002. [15] U. Lee, Spectral Element Method in Structural Dynamics, first ed., John Wiley & Sons (Asia), 2009. [16] M. Mitra, S. Gopalakrishnan, "Spectrally formulated wavelet finite element for wave propagation and impact force identification in connected 1-D waveguides", International Journal of Solids and Structures, vol. 42, pp. 4695-4721, 2005. [17] M. Mitra, S. Gopalakrishnan, "Extraction of wave characteristics from wavelet-based spectral finite element formulation", Mechanical Systems and Signal Processing, vol. 20, pp. 2046–2079, 2006. [18] M. Mitra, S. Gopalakrishnan, "Wavelet-based spectral finite element for analysis of coupled wave propagation in higher order composite beams", Composite Structures, vol. 73, pp. 263–277, 2006. [19] S. Gopalakrishnan, M. Mitra, Wavelet Methods for Dynamical Problems, first ed., Taylor & Francis Group, 2010. [20] F. Jin, T.Q. Ye, "Instability analysis of prismatic members by wavelet-Galerkin method", Advances in Engineering Software, vol. 30, pp 361–367, 1999. [21] X. Chen, J. Xiang, B. Li, Zh. He, "A study of multiscale wavelet-based elements for adaptive finite element analysis", Advances in Engineering Software, vol. 41, pp 196–205, 2010. [22] K. Marynowski, T. Kapitaniak, "Dynamics of axially moving continua (review) ", International Journal of Mechanical Science. DOI: http://dx.doi.org/10.1016/j.ijmecsci.2014.01.017 [23] K. Amaratunga, J.R. Williams, "Time integration using wavelets", Proceedings of SPIE, Wavelet Application for Dual Use, vol. 2491, pp 894–902, 1995. [24] G. Beylkin, "On the representation of operators in bases of compactly supported wavelets", SIAM Journal of Numerical Analysis, vol. 6, no. 6, pp. 1716-1740, 1992. [25] J. R. Williams, K. Amaratunga, "A discrete wavelet transform without edge effects using wavelet extrapolation", Journal of Fourier Analysis and Applications, vol. 3, no. 4, pp. 435–449, 1997. [26] K. Amaratunga, J. R. Williams, "Wavelet-Galerkin solution of boundary value problems", Archives of Computational Methods in Engineering, vol. 4, no. 3, pp. 243–285, 1997. [27] C. M, Wang, Y. Y. Zhang, X. Q. He, "Vibration of nonlocal Timoshenko beams", Nanotechnology, vol. 18, 2007. [28] J. Peddison, G. R. Buchanan, R. P. McNitt, "Application of nonlocal continuum models to nanotechnology", International Journal of Engineering Science, vol. 41, pp 305-312, 2003. [29] J. N. Reddy, "Nonlocal theories for bending, buckling and vibration of beams", International Journal of Engineering Science, vol. 45, pp 288-307, 2007.
Highlights
Develop wavelet-based spectral element model for moving pre-stressed nanobeam.
Process time/wave domain analysis for the moving nanostructure.
Highlight deficiencies of classical finite element and spectral element methods.
Study effects of nanobeam moving velocity and pre-stress on vibration and stability.
Investigate effects of small-scale parameter on vibration and stability.
PII: DOI: Reference:
S0020-7403(15)00383-5 http://dx.doi.org/10.1016/j.ijmecsci.2015.11.006 MS3137
To appear in: International Journal of Mechanical Sciences Received date: 7 June 2015 Revised date: 12 October 2015 Accepted date: 5 November 2015 Cite this article as: Ali Mokhtari, Hamid Reza Mirdamadi, Mostafa Ghayour and Vahid Sarvestan, Time/wave domain analysis for axially moving pre-stressed nanobeam by wavelet-based spectral element method, International Journal of Mechanical Sciences, http://dx.doi.org/10.1016/j.ijmecsci.2015.11.006 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Time/wave domain analysis for axially moving pre-stressed nanobeam by wavelet-based spectral element method Ali Mokhtari, Hamid Reza Mirdamadi*, Mostafa Ghayour, Vahid Sarvestan Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iran *Corresponding author. Tel.: +98 313 391 5248; fax: +98 313 391 2628. E-mail addresses: [email protected] (A. Mokhtari), [email protected] (H.R. Mirdamadi), [email protected] (M.Ghayour), [email protected] (V. Sarvestan).
Abstract In this article, a time/ wave domain analysis is presented for an axially moving pre-stressed nanobeam by wavelet-based spectral element (WSE) method. WSE scheme is constructed as spectral element method (SEM), except that Daubechies wavelet basis functions are used for transforming governing partial differential equation. These basis functions help to rule out some deficiencies of SEM due to periodicity assumption, especially for time domain analysis. Numerical examples are used for validating the accuracy and efficiency of model. The higher accuracy of WSE approach is then evaluated by comparing its results with those of classical finite element and SEM. The effects of moving nanobeam properties, such as velocity, pretention and nonlocal (small-scale) parameter, on vibration and wave characteristics and dispersion curves are investigated. In addition, the instability of moving nanobeam is studied both analytically and numerically considering divergence and flutter.
Abbreviations AD: artificial damping; BVP: boundary-value problem; CNT: carbon nanotube; FEM: finite element model; NP/BVP: non-periodic boundary-value problem; ODE’s: ordinary differential equations; P/BVP: periodic boundary-value problem; PDE: partial differential equation; P/WSE (P/WSFE): periodic wavelet-based spectral (finite) element; SEM (SFEM): spectral (finite) element method; WAD: weak artificial damping; WSE (WSFE): wavelet-based spectral (finite) element.
Keywords Axially moving pre-stressed nanobeam; nonlocal parameter; wavelet-based analysis; wave domain analysis; spectral element model; divergence/ flutter instability; Daubechies wavelet.
Nomenclature c
constant transport speed [ ⁄ ] divergence speed [ ⁄ ] flutter speed [ ⁄ ] nonlocal (small-scale) parameter [ flexural rigidity [
]
]
excitation force [ ] cut-off frequency [ Nyquist frequency [ {̂ }
] ]
global WSE nodal forces [ WSE wavenumber [
] ]
wavenumber for the Euler-Bernoulli beam theory [√ [̂ ]
⁄ ]
global dynamic stiffness matrix span between two end supports [ ] length of an element [ ] bending moment [
]
number of sampling points order of the Daubechies wavelet constant axial pretension [ ] shear force [ ] tensile force-to- flexural rigidity ratio [ ⁄
]
first-order connection coefficient matrix (time domain) second-order connection coefficient matrix (time domain) eigenvalues of
[
⁄ ]
first-order connection coefficient matrix (wave domain) second-order connection coefficient matrix (wave domain) eigenvalues of
[
⁄ ]
mass per length of beam [
⁄ ]
Daubechies scaling function at an arbitrary scale ,
connection coefficients
1. Introduction Axially moving structures and continua can be found in numerous engineering devices in mechanical, civil, electrical, and aerospace applications, e.g., thread-lines in fabric industry, rolled steel beams, chain and belt drives, high-speed sheets, magnetic tapes, band saw blades, aerial cable tramways, and so on. Due to this pervasiveness, the analysis of moving structures and continua has been a motivation for a large amount of publications, mostly on axially moving beams. The lateral vibration of such axially moving systems is commonly modeled as a prestressed beam. It is necessary to forecast the dynamic characteristics of such systems precisely, in order to reach safe, reliable, and successful designs, while hazards and accidents are prevented. There are many articles which analyze axially moving macro-scale (classical) beams. The solutions of equation of motion for moving classical beam models were obtained by several solution techniques, containing the Galerkin's [1-3], assumed modes [4], finite element [5], Green's function [6], transfer function [7], perturbation [8], the Laplace transform [9], artificial parameters [10], and the FFT-based spectral element methods (SEM) [11]. Despite the large number of studies for axially moving classical beams, few analyses have been conducted on similar problems at micro- and nano-scales. Assumed modes method and Galerkin approach [12], higher-order strain gradient solutions [13], and modified couple stress theory [14] are used to investigate moving micro- and nano-scale beam models. As a versatile numerical method, the FEM has an important role in structural analysis. This method may provide accurate dynamic response of a structure if the wavelength is large compared to the mesh size. However, the FEM results become increasingly inaccurate as the frequency bandwidth increases. As a drawback of FEM, it is well known that a large number of
FE’s should be generated to obtain trustworthy and accurate solutions, especially at higher frequencies. Evidently, this provision may increase the computational time and cost. These problems are generally resolved by SEM. SEM [15] transforms the governing partial differential equation (PDE) of motion to a set of ordinary differential equations (ODE’s) by FFT. These space-dependent ODE’s are solved exactly, which are then used as dynamic shape functions for SEM. It is well known that the wavelet-based spectral element (WSE) model is an exact solution method for dynamic analysis of structures [16-18]. WSE formulation is very similar to SEM formulation, except that Daubechies wavelet basis functions are used for transformation of governing PDE. The ensuing coupled ODE’s can be decoupled by performing a waveletdependent eigenvalue problem. The decoupled ODE’s are then solved similarly as in SEM. In SEM, the time window is dependent on the value of damping and the dimensions of structure. It requires to be more wide for weakly damped and shorter dimension structures. A potential remedy to reduce these dependencies is to use artificial damping (AD) [15]. By using nonperiodic boundary condition assumptions [19] for WSE, exactness of results could be free from those deficiencies previously noted such as lack of damping, structures having short dimensions, and small time windows. It should be noted that WSE method could also be used for analyzing undamped structures, where SEM could not work. This model can be used in time domain analysis without any transformations between different domains, something unlike SEM. Periodic boundary condition-based WSE formulation [19] can extract frequency dependent wave characteristics, like wavenumbers, directly. Moreover, [20, 21] show the application of wavelet technique in FE domain. Recent developments in research on axially moving structures were reviewed by Marynowski and Kapitaniak [22]. However, to the best of authors’ knowledge, the WSE model has not yet been introduced in the literature for axially moving nanobeam structures and this should be the most remarkable innovation of this study. Thus, the objectives of this article are: (1) to develop a WSE model for axially moving EB pre-stressed beams based on nonlocal elasticity theory, (2) to highlight higher accuracy of this model as compared with those of classical FEM and SEM, and (3) to investigate the effects of nanobeam properties, such as velocity, pretention and nonlocal parameter, on the vibration and wave characteristics, dispersion curves, and instability.
2. Mathematical model Based on the formulation derived in Appendix A, the governing equation for an element of axially moving EB pre-stressed beam based on nonlocal elasticity theory can be represented as [
] (1)
[
]
The force boundary conditions are given as
(2)
{
where
and
are the shear force and bending moment define as [
] (3)
[
[
]
] (4)
[
]
Fig. 1 shows an element of the nanobeam where
and
are the bending moment
and transverse shear force applied at
and
are the bending moment
and transverse shear force applied at
, and .
Fig. 1. A finite element of axially moving pre-stressed nanobeam.
3. Temporal discretization The scaling transform of functions Daubechies scaling function ⁄
at an arbitrary scale
can be done by a sequence of , as
∑
⁄
(5)
∑
Daubechies scaling function ⁄
where
and
(6)
at an arbitrary scale
is defined as [23]
⁄
and
(7)
, noted by
and
hereafter, are the approximation coefficients at a
definite spatial dimension . Substituting Eqs. (5) and (6) into Eq. (1) gives
⁄
]∑
[ ⁄
(
∑
)
⁄
⁄
∑
(8) ⁄
∑ (
⁄
⁄
∑
) ∑[
]
Taking inner product on both sides of Eq. (8) by ]∑
[
∑
∑
∫
(9)
∫
∑
, gives
∫
∫
∑
, where
∫
∑
∑[
∑
∫
] ∫
where the scaling functions are orthogonal, i.e.,
∫
(10)
By substituting Eq. (10) into Eq. (9), this can be written as
[
simultaneous ODE’s
∑
]
∑ (11)
∑
where,
∑
is the order of the Daubechies wavelet.
and
are the connection coefficients
defined as ∫
(12)
∫
(13)
Beylkin [24] computed the connection coefficients. For compactly supported wavelets, are nonzero only in the interval
.
Similar to Eq. (5), scaling transform of functions ⁄
⁄
and
∑
∑
and
are written as (14)
(15)
Substituting Eqs. (14) and (15) into Eqs. (3) and (4), respectively, give the following differential form [
] ∑
[
∑
(16)
]
[
] ∑
[
(17)
∑
]
While numerical execution deals with a finite interval of time = ⁄
, where
in the above equations could be changed by ⁄
and ⁄
where
[23]. Therefore,
, is . Consequently,
, and
, respectively. One may also
observe from [25] that for extreme efficiency of computation, the optimal length of sequence needs to be an integer power of two, i.e.,
,
.
The ODE’s given by Eqs. (11), (16), and (17) are the wavelet-based spectral element formulation for an axially moving EB pre-stressed beam based on nonlocal elasticity theory. Eq. (11) gives coupled ODE’s, which are to be solved for
, using methods defined afterward. Since numerical
simulations could deal with a small number of sequences, finite interval,
. Likewise,
should only be known in a
should appear in the corresponding finite interval
. In Eq. (11), the ODE’s only relating to interval include coefficients
that lie outside
or . Mitra and Gopalakrishnan in
[19] described the information of two approaches for solving this kind of boundary-value problem (BVP). For a periodic BVP (P/BVP), the function
is assumed to be periodic in
. Having this assumption in mind, the coupled ODE’s given by Eq. (11)
time, with time period
can be written in matrix form as [
]{
}
[
[
]{
{
where
and
]{ }
}
(
){
}
(
){
}
(18)
}
are
circulant connection coefficient matrices obtained from P/BVP
assumption. This periodic WSE (P/WSE) algorithm can be used directly in frequency domain analysis, something similar to SEM. However, this algorithm cannot be proper for time domain analysis because of the periodicity assumption. The second method, i.e., the wavelet extrapolation method of Daubechies compactly supported wavelets could be applied to non-periodic BVP (NP/BVP), here called NP/WSE algorithm, as proposed by Amaratunga and Williams [23, 25, 26]. With this algorithm, the coupled ODE’s, given by Eq. (11), can be written in matrix form as [
]{
}
[
[
]{
{
where
and
]{ }
(
} ){
}
(
){
}
(19)
}
are connection coefficient matrices obtained from NP/BVP assumption.
NP/WSE algorithm can be used in time domain analysis without any transformations between different domains, something unlike SEM. It should be noted that the connection coefficient matrices
,
, and
,
are entirely
-dependent and problem-independent [19]. Where
the order of wavelet used. The second-order connection coefficient matrices calculated from Eqs. (20) and (21), respectively [19].
and
can be
is
(20) (21)
It should be noted that all the formulations of P/WSE are exactly the same as those of NP/WSE, except that
are replaced by
.
4. Spatial discretization and wavelet spectral element formulation WSE method transforms the PDE of motion to a set of ODE’s by using Daubechies wavelet basis functions. These space-dependent ODE’s are solved exactly, which are then used to derive dynamic stiffness matrix. In this section, WSE’s are formulated for an axially moving EB pre-stressed beam based on nonlocal elasticity theory. In WSE model, the resulting ODE’s are coupled. However, diagonalizing the connection coefficient matrix can decouple the set of equations [16]. This can be done by wavelet-dependent eigenvalue analysis of the matrix as (22)
where
is the eigenvector matrix of
eigenvalues
, where
√
and
is the diagonal matrix, containing corresponding
. Similar expression holds for
, where
and
are known
analytically [19], as follows
(23)
[
]
The eigenvalues
are
∑
and the corresponding orthonormal eigenvectors
(24)
,
, are
(25)
√
From Eqs. (20) and (21),
and
can be written respectively, as (26)
where,
is a diagonal matrix with diagonal terms
and (27)
where
is a diagonal matrix including diagonal terms
. P/WSE and NP/WSE connection
coefficient matrices are entirely problem-independent; therefore, this eigenvalue analysis can be done once and stored forever. Then, these eigenvalues can be used directly in a dynamic analysis. This decreases computational time. The ODE’s obtained from decoupling Eq. (18), as well as Eq. (19), can be written as [
]
̂
̂
[
]
̂
̂
̂
(28)
̂
̂
where {̂ }
{
{ ̂}
{ }
}
(29) (30)
Forced boundary conditions given in Eqs. (16) and (17) are similarly transformed as ̂
[
] ̂
̂
̂
̂
(31)
̂
[
]
̂
̂
̂ (32)
̂
Hereafter, the subscript is dropped for establishing easy-to-work notation, noticing all the equations hold for
.
Fig. 2 shows a WSE for an axially moving EB nanobeam with nodes and two DOFs ̂ and ̂
at each node. The nodal transverse forces and moments are ̂ and ̂ ,
̂
respectively.
Fig. 2. A WSE axially moving EB nanobeam with equivalent nodal forces and DOFs.
Solving homogeneous form of decoupled ODE’s, i.e., Eq. (28), gives the exact interpolating function as ̂
(33)
where
is the WSE wavenumber. Substituting interpolating function into homogeneous
equation, resulting from decoupled ODE’s, i.e., Eq. (28), provides a dispersion relation {
[
]}(
) (
(
)
{
}(
)
(34)
)
where
(35)
(
)
is the span between two end supports.
is the tensile force-to-flexural rigidity ratio.
is the
wavenumber for the EB beam theory [16]. is the dimensionless velocity of moving nanobeam. is the dimensionless nanobeam.
. Similarly,
is the dimensionless wavenumber for moving
is the dimensionless nonlocal parameter
. WSE wavenumbers
are calculated from dispersion relation, i.e., Eq. (35). Then the general solution, i.e., Eq. (33), can be written as ̂
(36)
̂
̂
and {
}
(37)
{
} are the constants to be derived from the boundary
conditions at the two nodes. The general solutions can also be written in matrix form as ̂
{
}
(38)
̂
{
}
(39)
where
,
matrix with diagonal terms
. In addition, for
is a
diagonal
Applying the essential boundary conditions
to the two nodes gives ̂
̂
̂
̂
{ {
̂
̂
̂
̂
{ {
} } } }
(40) (41) (42) (43)
where
,
,
, and
.
The nodal displacement vector, as a function of generalized coordinates, can be derived by augmenting Eqs. (40)-(43) as
{̂ }
̂ ̂ ̂ {̂
]{
[
}
{
}
(44)
}
Homogeneous form of differential equations of systems, i.e., regardless of the effect of excitation force, is required for constructing dynamic stiffness matrix. Consequently, For computing force boundary conditions, it is first required to derive the end shear forces, ̂ moments, ̂ ̂
, as given by Eqs. (31) and (32) with ̂
, which can be written in matrix form
}
}
{ { {
̂
and end bending
}
}
{{
} {
{
}
{
}
(45)
(46)
}
where
and
. The boundary values for
the end shear forces and end bending moments, at two nodes can be written as ̂
̂
̂
̂
̂
̂
̂
̂
where and
̂
{ ̂
{
{
{
|
}
(47)
}
(48)
{
|
}
(49)
}
(50)
}
,
,
{
}
,
. The nodal force vector can be written by combining Eqs. (47)-(50) as
̂ {̂ }
̂
{
̂ [
{̂ }
}
{
}
(51)
]
At last, substituting {
} from the nodal displacement vector, i.e., Eq. (44), into nodal force
vector, i.e., Eq. (51), the nodal displacements and nodal forces are interconnected as {̂ }
{̂ }
[ ̂ ]{ ̂ }
(52)
Here, [ ̂ ] is the elemental dynamic stiffness matrix for moving EB pre-stressed nanobeam. In WSE model, the connection between neighboring FE’s (or assembly procedure) and boundary conditions can be processed easily, as done for classical FEM, to obtain the global dynamic stiffness form as {̂ }
[ ̂ ]{ ̂ }
(53)
where [ ̂ ], { ̂ }, and { ̂ } denote the exact global dynamic stiffness matrix for a moving EB pre-stressed nanobeam, the global wavelet-based spectral nodal DOF vector, and the global wavelet-based spectral nodal force vector, respectively. The computations of the above matrices for
are done numerically. The above equations can be solved to derive the
nodal displacement vector, { ̂ }, for recognized nodal forces. Hence, four nodal DOFs of { ̂ } can be substituted into general solutions, i.e., Eq. (36), to derive the constants {
}. These can be
substituted into general solutions, i.e., Eqs. (36) and (37), to obtain the generalized displacements ̂
and ̂
at any arbitrary point of the nanobeam. Then, ̂
can be substituted into
wavelet transform equation, i.e., Eq. (29), to obtain the time domain responses.
5. Wave domain analysis From a wave-domain standpoint, there is a specific relationship between the transformed ODE’s appeared in WSE model and those in SEM. Hence, WSE solutions can also be used in wave domain analysis, the same as SEM, but without any transformations between different domains. The matrix
is a circulant matrix with the following properties [17]
{
[
where { ̃ ⁄ given { eigenvalues,
{̃
] }
̃
}
(54)
⁄
} are FFT of { }
}, and { ̃ } are the FFT coefficients of the first column of }. Thus, the FFT coefficients { ̃ } are the
{ , of the matrix
[
]{
̃
. Operating FFT on both sides of Eq. (18) yields }
[
]{ ̃
{̃
̃
{ ̃
}
}
(
){
]{
̃
}
{
[
]{ ̃
{̃
}
(
){ ̃ }
(55)
}
In SEM, the transformed ODE’s are exactly the same, except that [
̃
̃
}
̃
are replaced by
.
}
(
){
̃
}
(
){ ̃ }
(56)
}
where (57)
Fig. 3(a) shows that for a same interval are equal to fraction, i.e.,
, for both WSE method and SEM,
up to a definite fraction of Nyquist frequency, ⁄
, is
and real part of
⁄
⁄
. This
-dependent and is greater for higher order basis functions, for
which it is problem-independent. Where
for
and it is close to
for
. Therefore, WSE model can be used in wave domain analysis, similar to SEM, just up to a definite fraction of
. In addition, the eigenvalues
in NP/WSE algorithm are complex-
valued and the imaginary part of those are calculated and plotted for different bases in Fig. 3(b). From a wave-domain standpoint, P/WSE algorithm, as compared with NP/WSE, has lower computational cost and time. Therefore, P/WSE is implemented for wave domain analysis in this article.
(a)
(b) Fig. 3. (a) Examination of resemblances between ( ) imaginary part of
,
, and real part of
, for different orders of basis,
, and
.
6. Stability analysis The eigenfrequencies,
, of an assembled system can be calculated from the constraint
that the determinant of P/WSE dynamic stiffness matrix, [ ̂ ], as a function of
(or ), vanishes
at
. That is [̂ ]
(58)
Stability analysis requires presuming the free vibration responses of the moving nanobeam, as follows [11] (59)
Substituting free vibration responses, i.e., Eq. (59), into the free vibration equation of wave differential equation of system, i.e., Eq. (1), gives an eigenvalue problem. ODE’s obtained from the formula of free vibration response are precisely the same as those from P/WSE algorithm, i.e., Eq. (55), except that
are exchanged by . Eventually, this eigenvalue problem can be
written as [̂
]
(60)
For each class of instability, the dynamic behavior of these moving structures is described by the sign of real and imaginary parts of , as follows [11]
(61)
As it is mentioned in [11], considering the existence of non-trivial dynamic equilibrium positions by setting
√
(or
) and
{ {
yields the divergence velocity, }
}
, in closed form as
(62)
7. Numerical simulations A uniform carbon nanotube (CNT) over two simple supports is considered that is axially moving. CNTs would be an excellent selection for high speed nano-scale tools. Moving CNTs would be used in spacecraft antennas, space elevator cables, high speed vehicles, and high speed
magnetic nanotapes. Here, a moving CNT is modeled by a moving EB nanobeam. The structural and material properties of the nanobeam are span between two end supports Young’s modulus ⁄
[12], radius of cross-section ⁄
,
, mass density
[12]. A transverse excitation load is applied at the mid-span where the nanobeam is
subjected to an axial tensile force
. The load is an impulsive function that has
amplitude and the duration of 0 to
.
and
are
and
, respectively, unless
otherwise mentioned. P/WSE algorithm is used to study stability analysis, the natural frequencies, and the dispersion relation curves. NP/WSE algorithm is used to simulate wave propagation in the same structure. In order to demonstrate the validity of proposed model, several numerical examples are carried out in this section. The results are compared with FEM and SEM to validate and show the vantages of WSE model for time-frequency domain analysis of wave propagation. The order of Daubechies scaling functions used is
.
7.1 Wave domain and stability analyses
The natural frequencies for the structure are calculated using P/WSE algorithm with . Consequently, wave domain characteristics could be acquired for a frequency range of
resulting from SEM but for a frequency range of from P/WSE with
. The P/WSE algorithm is valued by comparing the
natural frequencies calculated from this algorithm, those obtained from the analytical approach [15, 27], those from the SEM, and those from classical FEM. Table 1 demonstrates that the P/WSE and SEM results using one single element are equal to those from exact analytical results. This means that, P/WSE algorithm can be used to calculate the eigenfrequencies,
, like SEM. On the other hand, for FEM dynamic analysis, the total
number of FE’s used in the analysis is varying in order to improve the numerical accuracy. Contrary to the classical FEM, both the present P/WSE model and SEM provide highly accurate results by using only one element. Table 1 shows that when the axial speed divergence speed
raises to the
, the fundamental frequency disappears for the first time. Once again, for a
constant axial velocity and nonlocal parameter, as the axial pretension increases, the natural frequencies would generally increase. One may also mention from Table 1 that for a fixed pretension, as the axial velocity of the nanobeam increases, all the natural frequencies would
decrease. It is observed from Table 1 that the natural frequencies decrease with increasing nonlocal parameter for a fixed pretension and axial velocity. Therefore, a suitable determination of interval
would be an essential decision making for an accurate wave domain analysis at
higher frequencies by P/WSE model and SEM.
Table 1. Natural frequencies of the axially moving EB pre-stressed nanobeam Moving velocity
Investigated
Method (number of
(m/s)
parameters
elements)
0
Natural frequencies (GHz)
Theory [15]
34.91
139.63
2234.02
WSE (1)
34.91
139.63
2234.02
SEM (1)
34.91
139.63
2234.02
FEM (30)
34.91
139.63
2234.77
FEM (5)
34.91
139.86
2891.26
Theory [27]
34.17
128.78
1144.92
WSE (1)
34.17
128.78
1144.92
SEM (1)
34.17
128.78
1144.92
FEM (30)
34.17
128.78
1145.30
FEM (5)
34.17
128.99
1435.95
Theory [15]
34.97
139.69
2234.08
WSE (1)
34.97
139.69
2234.08
SEM (1)
34.97
139.69
2234.08
FEM (30)
34.97
139.69
2234.83
FEM (5)
34.97
139.92
2891.32
2053.80
WSE (1)
0.00
110.61
1091.70
(Divergence
SEM (1)
0.00
110.61
1091.70
FEM (30)
0.01
110.61
1092.28
FEM (5)
0.43
111.09
1433.80
0
0
velocity,
)
3936.55
WSE (1)
-
28.67
949.30
(Flutter velocity,
SEM (1)
-
28.67
949.30
FEM (30)
-
28.83
950.38
FEM (5)
-
34.99
1423.62
)
Fig. 4 shows the stability diagram of the classical beam and nanobeam. There are four distinct regions seen in the figure for nanobeam, as explained below: (1) Since all the eigenvalues pure imaginary for
are
, the moving beam would be stable. The meaning of ⁄ . (2) For
is first divergence velocity for nanobeam and its value is
, there exist purely real positive eigenvalues , indicating the occurrence of divergence instability. The numerical value of For the velocity interval, since all the eigenvalues
is
⁄ . (3)
, the moving beam is stable again are pure imaginary.
flutter velocity for nanobeam. (4) For
⁄ is the lowest , the complex-valued eigenvalues
with
real positive parts indicate the flutter instability. Similar to nanobeam, there are four separate zones as seen in this figure for classical beam. A closer scrutiny of Fig. 4 shows that the divergence velocity as well as flutter one would appear at lower levels of speed as nonlocal parameter grows up from zero.
Fig. 4. Stability diagram for eigenvalues for
and solid line for
against moving speed of nanobeam c, for different values of the nonlocal parameters, (dash line ).
Fig. 5 demonstrates the effects of axial pretension, nanobeam,
,
, and
when
, on three critical axial velocities of
. In addition, the effects of nonlocal parameter,
, on those velocities are presented in Fig. 6. Furthermore, the stable and unstable regions are evident in Fig. 5 and Fig. 6. Four different zones appear in these figures from bottom to top, as expressed below: (1, 2) The zone under the curve and and
and the narrower zone between two curves
are the first and second stable regions, respectively. (3) The region among two curves points to the first divergence instability region. (4) The zone just up the curve
shows the flutter instability region. It is apparent from Fig. 5 that these three critical axial velocities are uniformly increasing as the axial pretension increases. Fig. 6 reveals that these three critical velocities reduce as nonlocal parameter becomes more highlighted. Both Figs. 4 and 6 explain that small-scale parameter leads to instability of moving nanobeam at lower levels of velocity.
Fig. 5. The critical moving velocities against the axial pretension at
.
Fig. 6. The critical moving velocities against the nonlocal parameter at
In Fig. 7, the percentage of relative error is plotted versus the functions where
.
⁄
for different orders of basis
is wavenumber resulting from P/WSE algorithm, while
is the
wavenumber obtained from SEM, respectively. It can be shown that the error is negligible up to a certain
⁄
, and then it increases very suddenly. The fraction,
from Fig.6 for different
.
, can be derived numerically
Fig. 7. Examination of resemblances between wavenumbers of basis,
(SEM) and
(WSE model), for different orders
.
In Figs. 8-10, the wavenumber against frequency is plotted using P/WSE algorithm at several moving velocities and the plots are compared with those of values from SEM. It can be shown that
should be identical to
up to a definite value for
structure are computed using P/WSE algorithm with
. The wavenumber for the .
As shown in Fig. 8(a) and Fig. 8(b), when the nanobeam is in a stationary state (i.e., the wavenumber
⁄ ),
is purely real-valued and negative, which means there exist damping
waves within the nanobeam. When the nanobeam is moving, the wavenumber
gets complex-valued, as shown in Fig.
9(a) and Fig. 9(b). In addition, these figures show that as far as the axial velocity of nanobeam raises to a critical value
, all the wavenumbers at zero frequency join together at zero-value.
(a)
(b) Fig. 8. (a) The dispersion relation curve for wave propagation in moving EB nanobeam at and
,
,
, by using SEM and WSE model, and (b) Zoom-in on the dispersion relation curve in subfigure (a).
(a)
(b) Fig. 9. (a) The dispersion relation curve for wave propagation in moving EB nanobeam at
,
, and
, by using SEM and WSE model, and (b) Zoom-in on the dispersion relation curve in subfigure (a).
The critical axial velocity,
, can be derived from Eq. (34) by putting
detecting the state that all the roots of √
(or
) and
become zero, as shown below
⁄
(63)
Fig. 10(a) and Fig. 10(b) show that if the axial velocity of nanobeam remains to grow beyond the wavenumber
becomes purely imaginary-valued at the frequency range
,
,
while
becomes again complex-valued at the range
velocity is larger than given by
. This means that when the axial
, there could exit traveling waves within a narrow low-frequency band
. Therefore, the frequency
could be termed as the cut-off frequency.
(a)
(b) Fig. 10. (a) The dispersion relation curve for wave propagation in moving EB nanobeam at and
,
,
, by using SEM and WSE model, and (b) Zoom-in on the dispersion relation curve in subfigure (a).
7.2 Time domain analysis
Unlike the use of SEM, NP/BVP assumption leads to a direct use of WSE model for time domain analysis. In Fig. 11, the results of WSE formulations and SEM are compared in order to highlight vantages of the former, especially for time domain analysis. It is noticed that weak
artificial damping (WAD) model is used for SEM, i.e., a model of damping having values lower than that recommended by [15]. It should be noted that the SEM could not work for this structure without attributing any AD, while WSE method could work for this case study without any limitation. For WSE results, the time window of
and
appropriate, while for WAD formulation of SEM,
should be , and
are used. Moreover, WSE model, similar to SEM one, needs two FE’s for calculating the results adequately accurate. Fig. 11 shows that the length of time window,
, as used in the
SEM including a WAD mechanism, is extended to increase its accuracy.
(a)
(b) Fig. 11. (a) Examination of resemblances between WSE and WAD-(i.e., weak artificial damping) based SEM results at
⁄ ,
, and
, and ( ) Zoom-in on deflection of mid-span in subfigure (a).
In this part, the AD that recommended by [15] is used for SEM. Figs. 12-15 show the time responses of moving nanobeam at several moving velocities. By increasing the value of AD, the SEM results reach more accuracy than those of WAD based SEM. However, as the previous case, as much as the width of time window,
, used in the SEM, is extended its results
improves. As could be anticipated from Fig. 4 and Figs. 12-15, the moving nanobeam is stable at ⁄ (Figs. 12(a) and 12(b)), unstable by divergence at and 13(b)), stable again at
⁄ (Figs. 13(a)
⁄ (Figs. 14(a) and 14(b)), and unstable by flutter at
⁄ (Figs. 15(a) and 15(b)).
(a)
(b) Fig. 12. (a) Examination of resemblances between WSE and SEM time responses at stable zone,
, and
means weak artificial damping.
, i.e., the first
, and ( ) Zoom-in on deflection of mid-span in subfigure (a). WAD
Fig. 13. WSE time responses at moving speed happening divergence,
,
, and
.
Fig. 14. WSE time responses at
, i.e., the second stable zone,
Fig. 15. WSE time responses at moving speed happening flutter, .
, and
,
.
, and
The WSE and FEM results are compared to validate the higher accuracy of WSE results for time domain analysis in Fig. 16. WSE results are obtained by using sampling rate and just two FE’s. FEM results are obtained by using 10 and 100 beam FE’s and Newmark’s time integration scheme with a sampling rate of
. Fig. 16 shows that the results
predicted by FEM would converge to those predicted by the present WSE model provided that the total number of FE’s used in FEM could increase beyond 100.
Fig. 16. Examination of resemblances between WSE and FEM time responses of EB nanobeam at moving velocity happening flutter,
, and
.
Generally, elastic structures should have more stiffness and fundamental frequency for lower moving velocities. Both the shift-in-time and shift-in-magnitude for response amplitudes in Fig. 17 emphasize this fact of the moving nanobeam.
Fig. 17. The effects of moving speed on time responses at
and
.
Again, structures should have more stiffness and fundamental frequency for higher pretensions. Therefore, narrower time periods and lower amplitudes of time responses in Fig. 18 would demonstrate this fact of the moving nanobeam.
Fig. 18. The effects of axial pretension on time responses at
Fig. 19 shows the effects of nonlocal parameter, i.e.,
and
.
, on the nanobeam time responses.
Again, this structure would have lower natural frequencies for higher nonlocal parameter. Therefore, narrower time periods of time response would appear for lower nonlocal parameter. The horizontal shifts toward the time scales would demonstrate the fact for the mid-point deflection of nanobeam.
Fig. 19. The effects of nonlocal parameter, i.e., .
, on the nanobeam time responses at
and
8. Conclusions The objective in this article was to present a WSE model, formulated in terms of Daubechies scaling functions and exact dynamic shape functions, for a moving EB pre-stressed nanobeam. The wavelet transform was applied for transforming the temporal/spatial-dependent PDE of the system to a set of spatial-dependent ODE’s, as a replacement for using Fourier transformation. Both P/WSE and NP/WSE algorithms were used to process wave and time domains, respectively. From a wave-domain standpoint, wave properties could also be extracted, but just up to a certain fraction of the Nyquist frequency. From a time-domain standpoint, the present method proved to be more efficient as it could eliminate the deficiencies associated with SEM for a time domain analysis. Numerical simulations were done to study the wave propagation due to the vibration of an axially moving pre-stressed nanobeam. These numerical tests demonstrated the feasibility and capability of using WSE model in moving structures. The effects of axial pretension, velocity and nonlocal parameter on the stiffness, deflection, and instability of moving system were also studied. Comparisons with SEM results were also presented to focus the vantages and limitations of WSE model. It was shown that this formulation could provide extremely accurate solutions even by using fewer number of FE’s (one or two WSE’s), as compared with those obtained from exact analytical solutions and SEM, as well as FEM procedure.
Appendix A Imagine an EB pre-stressed beam that moves in its axial direction at a constant velocity c, as shown in Fig. A.1.
Fig. A.1. Axially moving EB pre-stressed beam and it’s Cartesian coordinate system.
The governing wave differential equation of this system is given as [15]
(A.1)
here
is the transverse deflection,
moment,
is constant axial pretension, and
moment,
is the excitation force,
is bending
is the mass per length of beam. Bending
, can be expressed as ∫
(A.2)
Based on the Eringen’s nonlocal continuum theory, the constitutive relation for beam model is [12, 13, 28, 29] (A.3)
where
is axial stress, is the axial strain, and
is nonlocal parameter (or small-scale
parameter). Multiplying Eq. (A.3) by and taking integration of results over cross-sectional area of the beam, lead to (A.4)
where
is flexural rigidity. Plugging the second derivative of bending moment,
, from
Eq. (A.1) into Eq. (A.4), gives
(A.5)
[
]
The governing equation of axially moving pre-stressed nanobeam, is found by substituting nonlocal bending moment, i.e. Eq. (A.5), into Eq. (A.1), as
[
] (A.6)
[
]
Eq. (A.6) governs the vibration and axially rigid body translation of a nanobeam under arbitrary transient excitation.
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Highlights
Develop wavelet-based spectral element model for moving pre-stressed nanobeam.
Process time/wave domain analysis for the moving nanostructure.
Highlight deficiencies of classical finite element and spectral element methods.
Study effects of nanobeam moving velocity and pre-stress on vibration and stability.
Investigate effects of small-scale parameter on vibration and stability.