Journal of Sound and Vibration 425 (2018) 41e52
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Modelling vibrating nano-strings by lattice, finite difference and Eringen's nonlocal models H. Zhang a, C.M. Wang a, *, N. Challamel b a
School of Civil Engineering, The University of Queensland, St Lucia, Queensland 4072, Australia ^me, Centre de Recherche, Rue de Saint Maud Universit e de Bretagne Sud, IRDL, UBS, Lorient, Institut de Recherche Dupuy de Lo e e BP, 92116, 56321 Lorient Cedex, France b
a r t i c l e i n f o
a b s t r a c t
Article history: Received 20 December 2017 Received in revised form 10 March 2018 Accepted 2 April 2018 Available online 7 April 2018 Handling Editor: G. Degrande
This paper focuses on the modelling of a tightly stretched nano-string with elastically supported ends by the central finite difference model (FDM), the lattice model and the Eringen's nonlocal model (ENM). The equivalence between the FDM and the lattice model is confirmed provided that the nodal spacing of FDM is equal to the segmental length of lattice model. The analytical vibration solutions for lattice model or FDM can be obtained by solving the linear second-order finite difference equation derived herein. By matching the vibration solutions for lattice model (or FDM) and ENM, the Eringen's small length scale coefficient e0 is calibrated, which is a constant equal to 0.289, regardless of the vibration modes or boundary conditions. The influence of end lateral spring stiffnesses on the vibration solutions for a taut local string is also discussed. For a taut symmetrically restrained local string with sliding ends, the fundamental frequency always approaches to zero and the second frequency always approaches to p. © 2018 Elsevier Ltd. All rights reserved.
Keywords: Elastic boundary conditions String Vibration Finite difference Lattice model Eringen's nonlocal string Small length scale coefficient
1. Introduction The vibration problem of string plays an important role in the historical development of the vibration theory. The characteristics of many engineering systems, such as guy wires, electric transmission lines, ropes and belt used in machinery, and thread manufacturing, can be derived from a study of the dynamics of taut strings [1]. In the recent years, nanoscale strings made of different materials have been produced by chemists. Such nanoscale strings have a wide application in diagnostic tools and nano electro mechanical systems due to its superior heating, electrical, magnetic and mechanical properties. For example, Qin et al. [2] built a microfiber-nanowire hybrid structure for harvesting the energy from vibration or disturbance originating from footsteps, heartbeats, ambient noise and air flow. In particular, the piezoelectric zinc oxide nanowires grown radially around textile fibers are used to convert low-frequency vibration/friction energy into electricity. Further research is done by Kudaibergenov et al. [3] who studied a string with high-contrast in material and geometrical properties for better harvesting of low-frequency vibration energy. For nanoscale structures, the effect of small length scale between atoms or small grain size cannot be neglected anymore. Therefore, the traditional continuum mechanics theory which does not capture size effect is inadequate for analysis of nanostructures. To properly
* Corresponding author. E-mail addresses:
[email protected] (C.M. Wang),
[email protected] (N. Challamel). https://doi.org/10.1016/j.jsv.2018.04.001 0022-460X/© 2018 Elsevier Ltd. All rights reserved.
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H. Zhang et al. / Journal of Sound and Vibration 425 (2018) 41e52
study vibration problems of such a small structure, a discrete model or modified continuum model (which has been referred as nonlocal model) has to be adopted. In this paper, the nano-string is modelled by the first order central finite difference model (FDM), the lattice model and the Eringen's nonlocal model (ENM) in vibration problems. The first two models are naturally discrete and contains the size effect. These two models have been proven to be equivalent for beams [4,5], circular arches [6] and plates [7,8]. However, whether the FDM and the lattice model will achieve an equivalency for string problems is hitherto unknown. Eringen's nonlocal continuum theory [9] which is modified based on the traditional local continuum theory for analyzing small scale structures accommodates the size effect by introducing a small length scale parameter. However, what value should the small length scale coefficient take has yet to be discovered. This parameter may be identified through experiments or molecular dynamics simulations [10]. Recently, it has been shown that there exists an analogy between the lattice model and the ENM [11e13]. The lattice model is a one-dimensional discrete beam comprising rigid segments of equal length. The rigid segments are connected by rotational springs with spring constants equal to the beam flexural rigidity divided by the length of the rigid segment. It is found that when the segmental number increases and approaches to infinity, the vibration frequencies and buckling loads converge to their classical local counterparts from below. When the number of segments is finite, the solution becomes size (i.e., segmental length) dependent. Therefore, the ENM is phenomenologically similar to the lattice model due to their softening effect when comparted to the local model and the small length scale coefficient of ENM may be calibrated with the aid of the lattice model. 2. FDM for nano-string Consider a tightly stretched elastic nano-string under a constant tension T and supported by elastic lateral springs at both ends, as shown in Fig. 1. The stiffnesses of the elastic lateral springs are xA T=L at the left end point A (i.e., at x ¼ 0) and xB T=L at the right end point B (i.e., at x ¼ L), respectively. Note that zero values of xA and xB imply sliding ends and infinite values imply fixed ends. The length of the string is L. The string has a constant density r (mass per unit length). We first model the nano-string (see Fig. 1) using the FDM with evenly distributed nodes with nodal spacing a ¼ L/n along the string as shown in Fig. 2. The nodes from the left end to the right end are numbered from 0 to n and two fictitious nodes 1 and nþ1 are extended from the two ends to accommodate the boundary conditions. The governing equation for the vibrating FDM are given by
T
wjþ1 2wj þ wj1 a2
þ ru2 wj ¼ 0 for j ¼ 0:::n
(1)
which is obtained from the finite difference discretization of the governing equation for a classical string [14]:
T
d2 w þ ru2 w ¼ 0 dx2
(2)
The boundary conditions at the two ends of the string modelled by FDM may be written as
w1 w1
2xA w ¼ 0 at j ¼ 0 n 0
wnþ1 wn1 þ
2xB wn ¼ 0 at j ¼ n n
(3a) (3b)
which are the finite difference form of the continuous boundary conditions for the elastically lateral supported ends [14]:
T
dw xA T w ¼ 0 at x ¼ 0 dx L
Fig. 1. Nano-string under constant tension with elastic lateral supports at two ends.
(4a)
H. Zhang et al. / Journal of Sound and Vibration 425 (2018) 41e52
43
Fig. 2. FDM for the string with elastic end supports.
T
dw xB T þ w ¼ 0 at x ¼ L dx L
(4b)
By assembling the FDM governing equation (1) and boundary conditions (3) into a matrix form and solving the resulting characteristic equation, the vibration frequencies of FDM with elastic end supports can be obtained. 3. Lattice model for nano-string Next, we model the tightly stretched nano-string by a chain of rigid segments with mass concentrated at the segment joints, as shown in Fig. 3. The stiffnesses of the elastic lateral end springs are xA T=L at the left end point A and xB T=L at the right end point B, respectively. The string is subjected to a constant tension force T and is composed of n repetitive cells of length denoted by a ¼ L/n. The cell length a may be related to the interatomic distance for a physical model where the microstructure is directly related to the atomic discreteness of the matter. According to the energy formulation of the lattice model [15,16], the elastic potential energy U of the deformed springs in the lattice model (see Fig. 3) with elastic lateral supports at both ends is given by
U¼
1 xA T 2 1 xB T 2 w þ w 2 L 0 2 L n
(5)
Note that the node numbering follows the same one as in Section 2 where the first node of the string has the number 0, and the last node of the string has the number n (with nþ1 nodes). xA and xB represent the stiffness coefficients of the elastic lateral end springs for the lattice model. The potential energy V due to the constant tension force T in the lattice model can be expressed as
V¼
2 n X 1 wj wj1 T a 2 a j¼1
(6)
The kinetic energy G due to the free vibration of the lattice model is given by
G¼
2 n1 X 1 M vwj 1 M vw0 2 1 M vwn 2 þ þ 2 n vt 2 2n vt 2 2n vt j¼1
Fig. 3. Lattice model for string with elastic end supports.
(7)
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H. Zhang et al. / Journal of Sound and Vibration 425 (2018) 41e52
where the total mass M ¼ rL of the lattice model is distributed as follows: for internal nodes j ¼ 1, 2, …, n1, the lumped mass is M/n and for the two end nodes j ¼ 0 and n, the lumped mass is M/(2n). To derive the equations of motion, we use Hamilton's principle which requires
Zt2 ðU þ V GÞdt ¼ 0
d
(8)
t1
where t1 andt2 are the initial and final times. By substituting Eqs. (5)e(7) into Eq. (8) and assuming a harmonic motion, i.e., pffiffiffiffiffiffiffi wj ðx; tÞ ¼ wj ðxÞeiut where i ¼ 1 and u is the angular frequency of vibration, one obtains the following governing equation for the vibration problem of the lattice model as
wjþ1 2wj þ wj1 þ where u ¼
u2 n2
wj ¼ 0 for j ¼ 1:::n 1
(9)
qffiffiffiffiffiffi
L2 r T u.
At the boundary points (i.e., j ¼ 0 and n), we have the following equations by minimizing the total energy function in equation (8) with respect to the end point displacements:
w1 w0 þ
!
u2
xA w0 ¼ 0 for j ¼ 0 2n2 !
u2
wn wn1
2n
(10a)
xB wn ¼ 0 for j ¼ n 2
(10b)
It can be seen that the governing equation (9) for lattice model is exactly the same as the governing equation for (1) for FDM. For equivalency between the lattice model and the FDM, equivalent boundary conditions for the lattice model and the FDM have to be derived. By applying the discrete governing equation (1) at the boundary points (i.e., j ¼ 0 and n) in the FDM, we have
w1 2w0 þ w1 þ
u2 n2
w0 ¼ 0
(11a)
u2
(11b)
wnþ1 2wn þ wn1 þ
n2
wn ¼ 0
In order to eliminate the deflection values at the two fictitious nodes (i.e., j ¼ 1 and nþ1), we substitute the boundary conditions (3a) and (3b) respectively into (11a) and (11b) and the resulting boundary conditions are found to be exactly the same as Eqs. (10a) and (10b). Therefore, we can conclude that the lattice model is the equivalent physical structural model to the FDM. 4. Analytical vibration solutions for lattice model (or FDM) Referring to Eq. (1), the general solution for wj can be represented as
wj ¼ A1 sin jq þ A2 cos jq
(12)
where cosq ¼ 1 ra2Tu and A1, A2 are constants which can be determined by the boundary conditions. By substituting the general solution (12) into the boundary conditions (3), one obtain two homogeneous equations which may be expressed as 2
2
2 6 6 4
xA
sin q
xB n
sinðnqÞ þ sin q cosðnqÞ
xB n
n
cosðnqÞ sin q sinðnqÞ
3
7 A1 7 5 A2 ¼ 0
(13)
For a nontrivial solution, the exact characteristic equation for the frequency is given by the vanishing of the determinant of the coefficient matrix in Eq. (13), i.e.
H. Zhang et al. / Journal of Sound and Vibration 425 (2018) 41e52
sin2 q sinðnqÞ
xA þ xB n
sin q cosðnqÞ
xA xB n2
sinðnqÞ ¼ 0
45
(14)
By solving Eq. (14), one obtains the analytical vibration frequencies of the FDM or lattice model with elastic end supports having stiffness constants xA T=L and xB T=L. By analyzing Eq. (14), it is worth noting that when xA and xB are very small, the vibration solutions for lattice model will not be influenced by the segmental number n or the segmental length a. The lattice model and FDM revert back to the continuum string model for xA and xB dropping into the size-independent stiffness zone as shown in Fig. 4. Next, we derive the exact vibration frequency equation of the FDM or lattice model with special boundary conditions (i.e., fixed-fixed, sliding-sliding, fixed-sliding and sliding-fixed). 4.1. Fixed-fixed case For a taut string with fixed-fixed ends (i.e., xA ¼ xB ¼ ∞), Eq. (14) becomes
sinðnqÞ ¼ 0
(15)
The solution to Eq. (15) is
nq ¼ kp for k ¼ 1; 2; 3; :::
(16)
In view of Eq. (12), we evaluate cosq as
cosq ¼ cos
kp rau2 ¼1 n 2T
(17)
By using trigonometric relations, we obtain
rau2 2T
¼ 1 cos
kp kp ¼ 2 sin2 n 2n
(18)
where k should be set equal to unity for the first natural frequency. Since a ¼ L/n, Eq. (18) reduces to the exact vibration frequency equation of a discrete string with the fixed-fixed boundary condition as
uk;n
sffiffiffi 2n kp T sin where k ¼ 1; 2; 3; ::: ¼ L 2n r
(19)
Note that Eq. (19) is exactly the same as that derived by Lagrange [17] for a discrete string with the fixed-fixed boundary qffiffiffi condition. Equation (19) reduces to the local string counterpart when n/∞, i.e., uk;∞ ¼ kLp Tr , where k ¼ 1, 2, 3, …. The reader is referred to the paper by Challamel et al. [18] for a brief discussion on the vibration problem of a discrete string with the fixed-fixed boundary conditions.
Fig. 4. Size-independent stiffness zone and softening stiffness zone for lattice model.
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H. Zhang et al. / Journal of Sound and Vibration 425 (2018) 41e52
4.2. Sliding-sliding case For a taut string with sliding-sliding ends (i.e., xA ¼ xB ¼ 0), Eq. (14) becomes 2
ðsinqÞ sinðnqÞ ¼ 0
(20)
The solution of Eq. (20) may be expressed as
nq ¼ kp for k ¼ 0; 1; 2; 3; :::
(21)
Similar as in the fixed-fixed case, the exact vibration frequency equation for a discrete string with the sliding-sliding boundary condition can be obtained as
uk;n
sffiffiffi 2n kp T sin where k ¼ 0; 1; 2; 3; ::: ¼ L 2n r
Equation (22) reduces to the local string counterpart when n/∞, i.e., uk;∞ ¼ kLp
(22)
qffiffiffi T , where k ¼ 0, 1, 2, 3, …. Note that the r
fundamental vibration frequencies of both local continuous and discrete string with the sliding-sliding boundary condition are zero which indicates a rigid motion. 4.3. Fixed-sliding or sliding-fixed cases For a taut string with fixed-sliding ends (i.e., xA ¼ ∞ and xB ¼ 0) or sliding-fixed ends (i.e., xA ¼ 0 and xB ¼ ∞) ends, Eq. (14) becomes
cosðnqÞ ¼ 0
(23)
The solution of Eq. (23) is
nq ¼
kp for k ¼ 1; 3; 5; ::: 2
(24)
The exact vibration frequency equation of a discrete string with the fixed-sliding or sliding-fixed boundary conditions may be obtained as
uk;n
sffiffiffi 2n kp T sin where k ¼ 1; 3; 5; ::: ¼ L 4n r
Note that Eq. (25) reduces to the local string counterpart whenn/∞, i.e., uk;∞ ¼ k2Lp
(25)
qffiffiffi T , where k ¼ 1, 3, 5, …. r
5. ENM for nano-string Recently, a number of papers pointed out the strain-based integral nonlocal model with an exponential kernel leads to an over-constrained nonlocal model for finite solids (with stress-based boundary conditions which may be in contradiction with the natural boundary conditions of the problem). In other words, strain-based integral nonlocal model with an exponential ndez-Sa ez et al. [19] Romano et al. [20] and Challamel [21]). However, in kernel is generally ill-posed for finite solids (see Ferna this paper, we shall adopt a stress gradient model but with over-constrained stress-based boundary conditions. The stress gradient of Eringen [9] does not lead to ill-posed mathematical problems, with the natural boundary conditions. The difference between both models is due to the fact that the strain-based integral nonlocal model with an exponential kernel has additional stress-based boundary conditions that are not present for the stress gradient model of Eringen [9]. The stress gradient model of Eringen [9] has some pathological features, such as the absence of scale effects for some loading or the possible non-self adjointness of the problem for beam vibrations [22]. However, this model can be supported by some lattice arguments; an advantage that many other nonlocal models do not offer. The governing equation for the vibration problem of an Eringen's stress gradient nonlocal string (see Fig. 5) in a dimensional form is given by Refs. [18,23,24] for the vibration model of the axial chain
H. Zhang et al. / Journal of Sound and Vibration 425 (2018) 41e52
47
Fig. 5. Nonlocal string model.
T
d2 w d2 w þ ru2 w ru2 l2c 2 ¼ 0 2 dx dx
(26a)
where lc is the length scale parameter (¼ e0 a ), e0 is the small length scale coefficient and a is the internal characteristic length. By adopting x ¼ x=L, Eq. (26a) becomes
d2 w dx
2
þ
r L 2 u2 w¼0 T ru2 l2c
(26b)
The general solution of Eq. (26b) is given by Refs. [1,14].
w ¼ C1 sin uN x þ C2 cos uN x where u2N ¼
rL2 u2 T ru2 l2c
(27)
The two constants Ci (i ¼ 1 and 2) can be evaluated by the two boundary conditions at the two ends of the string. Note that Rao [1] and Wang and Wang [14] treated only the local string problem. From Eq. (27), the frequency parameter u may be written as
u2 ¼
u2N 1 þ u2N
2 , lc L
T
(28a)
rL2
Based on the definition in Eq. (9), we have
u2 ¼
u2N 1 þ u2N
(28b)
2 lc L
The boundary conditions for nonlocal strings with elastically supported ends are still unknown. As discussed by Wang et al. [12], the traditional continuum nonlocal boundary conditions may have some deficiencies. Therefore, a set of continualized discrete boundary conditions are employed herein and they are given by
1 1 2x w A wð0Þ ¼ 0 at x ¼ 0 w n n n 1 1 2x w 1 þ B wð1Þ ¼ 0 at x ¼ 1 w 1þ n n n
(29a)
(29b)
By substituting the general solution (27) into the two boundary conditions (29), we obtain two homogeneous equations which may be expressed as
2 6 6 4
sin
u N
n u xB sinðuN Þ þ sin N cosðuN Þ n n
xA
3
7 C1 n 7 5 C2 ¼ 0 u xB N cosðuN Þ sin sinðuN Þ n n
For nontrivial C1 and C2, the exact characteristic equation for the frequency is given by
(30)
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H. Zhang et al. / Journal of Sound and Vibration 425 (2018) 41e52
sin2
u x þ x B uN x x N sinðuN Þ A sin cosðuN Þ A 2B sinðuN Þ ¼ 0 n n n n
(31)
By solving the characteristic equation (31), one obtains the solutions of uN and finally one can obtain the dimensionless vibration frequencies u of the nonlocal continuum model through Eq. (28b). 6. Calibration of small length scale coefficient e0 It is interesting to note that the characteristic equation (31) for the nonlocal string has a similar form to the characteristic equation (14) for lattice model. The vibration solutions for the nonlocal string model and lattice model (or FDM) will be exactly the same if we let
q¼
uN
(32)
n
In view of Eqs. (32) and (12), we obtain
cos q ¼ cos
u N
n
¼1
ra2 u2
(33)
2T
By using trigonometric relations, we obtain
ra2 u2 2T
¼ 1 cos
u N
n
¼ 2 sin2
u N
2n
(34)
An asymptotic expansion shows that
u2 ¼
u2 T 2 4T a 2 uN 2 a N ¼ 1 þ /for < < 1 sin u N L 2n ra2 r L2 12L2
(35)
The best calibration of the small length scale coefficient e0 is obtained by comparing Eqs. (28a) and (35), and thereby leading to
u2N
2 a2 lc 1 2 lc ¼ u 0e0 ¼ ¼ pffiffiffiffiffiffiz0:289 N L a 12L2 12
(36)
Note that Eq. (36) applies to all end lateral spring stiffnesses because only the characteristic equations are used in the foregoing derivation. For strings with both ends fixed, Challamel et al. [18] have shown that the same small length scale pffiffiffiffiffiffi coefficient e0 ¼ 1 12 by making use of the analytical vibration mode shape solution. Interestingly, Eq. (36) is different from pffiffiffi the conclusions by Challamel [25], Wang et al. [16] and Zhang et al. [15] who have shown that e0 ¼ 1= 6 for pure vibration problem of an Euler beam or a Timoshenko beam. This indicates that the Eringen's small length scale also depends on the type of structures, e.g., nano-string or nano-beam. In order to illustrate the validity of the approach for calibrating the small length scale coefficient e0 of the nonlocal continuum string models, two cases of a vibrating taut string supported at two ends by elastic springs are considered. The first case is to consider a string supported at two ends by elastic lateral springs having stiffness coefficients xA ¼ 104 and xB ¼ p/2 (same results for xA ¼ p/2 and xB ¼ 104 because of symmetry). The second case is to consider a string supported at two ends by elastic lateral springs having stiffness coefficients xA ¼ xB ¼ 3.6. Fig. 6 presents the fundamental vibration frequencies of the lattice model (or FDM), the ENM and the local string model for the first considered string (xA ¼ 104 and xB ¼ p/2). It can be seen that the fundamental vibration frequencies of lattice model and ENM agree well with each other with e0 ¼ 0.289 and they finally converge to the local continuum result. The local continuum result is obtained from the calculations that are presented in the Appendix. Fig. 7 shows the good agreement of second vibration frequency of the lattice model (or FDM) with that of the ENM for the second considered string (xA ¼ xB ¼ 3.6). =
7. Concluding remarks This paper presents the vibration solutions of a tightly stretched nano-string with elastically supported ends which are derived from (a) the first order central finite difference string model, (b) the lattice model and (c) the Eringen's nonlocal string model.
H. Zhang et al. / Journal of Sound and Vibration 425 (2018) 41e52
49
Fig. 6. Fundamental frequencies of lattice model (or FDM), ENM (with e0 ¼ 0.289) and local string for various n.
Fig. 7. Second frequencies of lattice model (or FDM), ENM (with e0 ¼ 0.289) and local string for various n.
It is shown that the FDM is equivalent to the lattice model provided that the nodal spacing of FDM is equal to the segmental length of lattice model. The analytical solution of lattice model (or FDM) is obtained by solving the associated linear secondorder finite difference equation. It is found that when the stiffness coefficients xA and xB of the end lateral springs fall into a certain region (size-independent stiffness zone in Fig. 4), the fundamental vibration frequency of the lattice model or FDM becomes independent of n. Also in this study, we calibrate the Eringen's small length scale coefficient e0 by matching the vibration solutions for ENM and lattice model. It is found that the value of e0 remains a constant equal to 0.289 regardless of the vibration modes or boundary conditions. The results in this paper also apply to the problems of longitudinal vibration of bars and torsional vibration of shafts, since their equations of motion have the same form as that of the transverse vibration of strings [1]. Future studies may examine the vibration problems of strings using higher-order finite difference schemes. Challamel et al. [26] have shown that higher-order finite difference schemes lead to higher-order enriched constitutive laws with a higher convergence rate. Appendix A. Vibration solutions of taut local string supported by elastic lateral springs at two ends The exact characteristic equation of the vibration frequency u of a taut local string supported by elastic lateral springs at two ends can be obtained by solving Eqs. (2) and (4) as [14].
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H. Zhang et al. / Journal of Sound and Vibration 425 (2018) 41e52
u2 sinu ðxA þ xB Þucosu xA xB sinu ¼ 0
(A.1)
By solving the characteristic equation (A.1), we obtain the vibration frequencies of the local string with elastic end supports modelled by lateral springs having stiffness constants xA T=L and xB T=L. Note that by letting T ¼ r ¼ L ¼ 1 in Eq. (A.1), we recover Wang and Wang [14] results because u ¼ u. For a special case of a string with the fixed-sliding or sliding-fixed (i.e., xA ¼ ∞ and xB ¼ 0, or xA ¼ 0 and xB ¼ ∞) boundary conditions, the vibration frequencies as computed from Eq. (A.1) are given by
uk ¼
kp where k ¼ 1; 3; 5; ::: 2
(A.2)
If we let
xkA ¼ xkB ¼
kp where k ¼ 1; 3; 5; ::: 2
(A.3)
the vibration frequencies of a fixed-sliding or sliding-fixed string will be exactly the same as the vibration frequencies of a string with elastic lateral springs of stiffness kp/2 at the two ends and the solutions are given by Eq. (A.2). To summarize the influence of the stiffnesses of end lateral springs on the vibration frequencies of the string, three special cases are considered. xA ¼ 0, Eq. (A.1) is simplified to usinu xB cosu ¼ 0. xA ¼ ∞, Eq. (A.1) is simplified to ucosu þ xB sinu ¼ 0. 2 xA ¼ xB , Eq. (A.1) is simplified to u2 sinu 2xB ucosu xB sinu ¼ 0. Fig. A1 shows the variations of the first and second vibration frequencies of the local string with respect to the lateral springs stiffness coefficient xB for the three specific lateral constraints.
Fig. A1. Variations of first and second vibration frequencies with respect to lateral spring stiffness coefficients xA and xB .
As shown in Fig. A1, it can be seen that the fundamental vibration frequency vanishes and the second vibration frequency approaches to p for the sliding-sliding boundary conditions (i.e., xA ¼ xB ¼ 0). The zero fundamental frequency u1 ¼ 0 indicates a rigid body motion. The findings here can also be confirmed from an asymptotic expansion for small values of x. Consider the frequency equation in the specific case of symmetrical lateral restraints
u2 sinu 2xucosu x2 sinu ¼ 0 with xA ¼ xB ¼ x
(A.4)
By assuming a half-power asymptotic expansion (Puiseux series), one finds for the fundamental frequency
u1 ¼
pffiffiffiffiffi 2x þ oðxÞ
By assuming a Taylor series expansion, one finds that the second frequency is given by
(A.5)
H. Zhang et al. / Journal of Sound and Vibration 425 (2018) 41e52
u2 ¼ p þ
2x
p
2 þo x
51
(A.6)
Eqs. (A.5) and (A.6) clearly show that the fundamental frequency becomes zero and the second frequency becomes p when
x tends to zero. By observing Fig. A1, it is also found that the fundamental vibration frequency approaches to p when x is very large. Dividing Eq. (A.4) byx2 and it can be transformed to
x2 u2 1 tanu ¼ 2xu where x ¼ 1=x
(A.7)
For x [ 1, we have x / 0 and an asymptotic expansion of Eq. (A.7) leads to
u1 ¼ p
2p
x
þo
1
!
x2
(A.8)
Equation (A.8) clearly shows that the fundamental frequency becomes p when x [ 1. Fig. A2 shows the comparisons between the exact solutions solved from Eq. (A.4) and the approximate solutions solved from Eq. (A.5), (A.6) and (A.8).
Fig. A2. Comparisons between exact solution solved from Eq. (A.4) and approximate solutions obtained from Eqs. (A.5), (A.6) and (A.8).
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