On the behaviour of current-carrying wire-conductors and bucking of a column/Sroves perdavimo laidu ir atramu virpesiu tarpusavio saveika.

Ganji, S.S. ; Barari, A. ; Fereidoon, Abdolhossein 等

1. Introduction

In the last few decades, Single-Degree-Of-Freedom (SDOF) oscillator has been widely used to study the behavior of machines used in pile driving, compacting, rock drilling, impact printing and marine structures [1-4].

Dario Aristizabal-Ochoa analyzed the large-deformation-small strain and postbuckling behavior of Timoshenko beam-columns subjected to conservative as well as non-conservative end loads. He investigated the combined effects of shear, axial and bending deformations in a simplified manner. Later, a one-dimensional composite frame element for nonlinear static and cyclic behavior of concrete-filled steel beam columns is formulated by Valipour and Foster [5].A nonlinear fiber element analysis is presented through the work presented by Liang et al. [6] for predicting the ultimate strengths of thin-walled steel box columns with local buckling behavior.

Apart from the studies mentioned above, many works have been carried out to analyze the nonlinear vibrations, most of which on developing governing equations for Dynamic response of axially loaded Euler-Bernoulli beams [7, 8], inextensible beams [9, 10], transportation [11], cubic-quintic Duffing [12, 13], mass-spring systems [14], and more [15-23].

In addition, the ability to determine the magnetic fields and the resulting parameters (force, impedance, power losses) is very important in the optimization of electric machines and equipment. Gasiorski in 1986 [24] presented a general method which is based on combination of Bubnov-Galerkin methods by means of finite element method. The presented approach was utilized for calculating impedance of polygonal and symmetrical shape conductors carrying current chosen for simulations.

In the present paper, we obtain an approximate expression for the periodic solutions to two practical cases [25, 26] of nonlinear SODF oscillation systems, namely oscillation of current-carrying wire in a magnetic field and the model of bucking of a column by means of iteration perturbation method (IPM), variational approach (VA), and perturbation expansion method (PEM). These techniques yield a very rapid convergence using an iteration and lead to high accuracy of the solution. The results presented in this paper reveal that these methods are very effective and convenient for conservative nonlinear oscillators.

2. The models of nonlinear SODF systems

2.1. Case 1: Motion of a current-carrying conductor

Fig. 1 shows a pair of current-carrying wire-conductors restrained by a wire to a fixed wall by linear elastic springs. Assume x, k, and m as displacement of the wire, stiffness of the springs and mass of wire respectively. The differential equation describing the motion of wire is [25]:

m [d.sup.2] [[??].sup.2] / [dt.sup.2] + k[??] - 2[i.sub.1][i.sub.2] 1 / b - [??] = 0, x (0) = [??], dx / dt (0) = 0, (1)

where k[??] the restoring forces due to is springs and 2[i.sub.1] [i.sub.2] 1 / (b- [??] is the attraction force between the conductors due to magnetic fields produced by the currents. Eq. (1) can be rewritten as a conservative nonlinear oscillatory system with a rational form:

[d.sup.2]x / d[t.sup.2] + x - [DELTA] / 1 - x = 0, x (0) = [??], dx / dt (0) = , (2)

where x = [??]/b, t = [[omega].sub.0][??], [[omega].sup.2.sub.0] = k/m, and [DELTA] = 2[i.sub.1] [i.sub.2]l/[kb.sup.2]. [??] is also the initial condition for x.

[FIGURE 1 OMITTED]

The following four cases should be separately considered: [DELTA]< 0, [DELTA] = 0, 0 <[DELTA]< 1/4 and [DELTA][greater than or equal to] 1/4 [25]. Authors are interested in constructing analytical approximate periodic solutions to Eq. (2). Three different approximate as IPM, VA, and PEM are utilized to construct analytical approximations to periodic oscillation of the current-carrying wire for the cases [DELTA] < 0 and 0 < [DELTA] < 1/4 .

The first-order approximate procedure yields rapid convergence with respect to the "exact" solution obtained by numerical integration. In addition, the results are valid for all permitted oscillation amplitude.

2.2. Case 2: Model of a buckling column

In this section we consider the structure exposed to buckling as shown in Fig. 2. The mass m moves in the horizontal direction only. It is therefore studied the static stability by determining the nature of the singular point at x = 0 of the dynamic equations. The proposed dynamic approach is more convenient and effective to use than the static concept [26].

Neglecting the weight of all but the mass, show that the governing equation for the motion of m is [26]:

mu + ([k.sub.1] - 2P/l) u + ([k.sub.3] - 2P/[l.sup.3])[u.sup.3] + ... = 0 (3)

Where the spring force is given by:

[F.sub.spring] = [k.sub.1] u + [k.sub.3] [u.sup.3] + ... (4)

[FIGURE 2 OMITTED]

3. Solution procedures

3.1. Basic idea of IPM

In this paper, we will consider the second-order differential equation:

[??] + f(u,t) = 0. (5)

We introduce the variable y = du/dt, and then Eq. (5) can be replaced by equivalent system:

[??] (t) = y (t);

[??] (t) = -f (u,y). (7)

Assume that its initial approximate guess can be expressed as:

u (t )= Acos ([omega]t) , (8)

where w is the angular frequency of the oscillation. Then we have:

[??] (t) = -A [omega] sin ([omega]t) = y (t) . (9)

Substituting Eq. (8) and 9 into the Eq. (7), we obtain:

[??](t) = -f( A cos([omega]t),t). (10)

Using Fourier expansion series in the right hand of Eq. (10):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Substituting Eq. (11) into Eq. (10) yields:

[[??](t) = - ([[alpha].sub.1] cos([omega]t) + [alpha].sub.3] cos(3[omega]t) + ...) . (12)

Integrating Eq. (12), yields:

y (t) = - [[alpha].sub.1]/[omega] sin([omega]t) - [[alpha].sub.3]/3[omega] sin (3[omega]t) - . . . (13)

Comparing Eq. (9) and (13), we obtain:

- A [omega] = - [alpha].sub.1] / [omega]; (14)

[omega] = [square root of [[alpha].sub.1]]/A; (15)

T = 2[pi] [square root of A/[[alpha].sub.1]]. (16)

3.2. Basic idea of VA

For explaining the VA procedure, we consider a general nonlinear oscillator in the form of Eq. (5). Its variational principle can be established using the semi-inverse method [27, 28]:

J (u) = [[integral].sup.T/4.sub.0] (- 1/ 2 [[??].sup.2] + F (u)) dt, (17)

where T is period of the nonlinear oscillator, F(u) = [integral] f (u)du. Assume that its solution can be expressed as Eq. (8). Substituting (8) into (17) results in:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (18)

Applying the Ritz method, we require:

[partial derivative]J / [partial derivative]A = 0; (19)

[partial derivative] J / [partial derivative] [omega]. (20)

But using a careful inspection, for most cases we

find:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (21)

Thus, we modify conditions (19) and (20) into a simple form:

d J/d [omega] = 0 . (22)

3.3. Basic idea of PEM

In order to use PEM, we rewrite the general form of Duffing equation in Eq. (5) in the following form [9]:

[??] + [alpha]u + [beta] N (u,t) = 0, (23)

where N(u, t) is the nonlinear term after expending the solution u; [alpha] as a coefficient of u and [beta] as a coefficient of N(u,t) , the series of p introduce as follows:

u = [u.sub.0] + [pu.sub.1] + [p.sup.2] [u.sub.2] +...; (24)

[alpha] = [[omega].sup.2] + [p[gamma].sub.1] + [p.sup.2] [[gamma].sub.2] +...; (25)

[beta] = [p[delta].sub.1] + [p.sup.2] [[delta].sub.2] +... (26)

Substituting Eqs. (24) - (26) into Eq. (23) and equating terms with the identical powers of p, we have:

[p.sup.0]: [??].sub.0] + [[omega].sup.2] [u.sub.0] = 0; (27)

[p.sup.l] : [??].sub.1] + [[omega].sup.2] [u.sub.1] + [[gamma].sub.1][u.sub.0] + [[delta].sub.1]N ([u.sub.0] , t) = 0. (28)

Considering initial conditions [u.sub.0] (0) = A and [??].sub.0](0) = 0, the solution of Eq. (27) is [u.sub.0] = A cos ([[omega]t) . Substituting [u.sub.0] into Eq. (28), we obtain:

[p.sup.l] : [??].sub.1] + [[omega].sup.2][u.sub.1] + [[gamma].sub.1] Acos ([omega]t) + [[delta].sub.1]N (A cos ([omega],t) = 0 . (29)

Similar to IPM, for achieving the secular term, we use Fourier expansion series as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

Substituting Eq. (30) into Eq. (29) yields:

[p.sup.l]: [[??].sub.1] + [[omega].sup.2] [u.sub.1] + ([[gamma].sub.1]A +[b.sub.1]) cos ([omega]t) = 0 . (31)

For avoiding secular term, we have:

([[gamma].sub.1] + [b.sub.1]) = 0. (32)

Setting p=1 in Eqs. (25) and (26), and Substituting [[gamma].sub.1] = [alpha] - [[omega].sup.2] and [[delta].sub.1] = [beta] in Eq. (32), we can achieve the frequency and period of Eq. (5).

4. Applications of analytical solutions for Eq. (2)

To show the applicability, accuracy and effectiveness of proposed methods, they are applied to the first practical case presented in Eqs. (2). We use the simple form of Eq. (2) to obtain the approximate solutions based on IPM, VA, and PEM. For this sake, we let x = [x.sub.2] + u in Eq. (2) and expand the resulting equation in a Taylor series about u = 0 . The result is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](33)

[[alpha].sub.1] = 1 [delta] / [(1 - [alpha]).sup.2] , [[alpha].sub.2] = - [delta] / [(1 - [alpha]).sup.3], [[alpha].sup.3] = - [delta] / [(1 - [alpha]).sup.4] (34)

4.1. Implementation of IPM

As it can be seen in the basic idea of IPM, after introducing the variable y = du/dt, and substituting

u = Acos ([omega]t) into the Eq. (33), we obtain:

[??] = [[alpha].sub.1] A cos ([omegat]t) - [[alpha].sub.2] [A.sup.2] [cos.sup.2] ([omega]t) - [[alpha].sub.3] [A.sup.3] [cos.sup.3] ([omega]t) . (35)

By using Fourier series expansion, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (36)

By integrating Eq. (36), and comparing with Eq. (9), we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (37)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (38)

4.2. Implementation of VA

In this section, we will use the VA solution for Eq. (33). The variational principle of this equation can be obtained:

j(u) = [[integral].sup.T/4.sub.0] (- 1/2 [??].sup.2] + [integral] ([alpha].sub.1]u + [[alpha].sub.2] [u.sup.2] + [[alhpa].sub.3][u.sup.3])du)dt. (39)

Using a trial function u = Acos at into (39), the solution of (33) can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (40)

Thus, the stationary condition with respect to A is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (41)

This leads to the result:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](42)

Thus, we obtain the following frequency and period as same as IPM solution:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (43)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (44)

4.3. Implementation of PEM

In order to use PEM procedure, we can rewrite (33) in following form:

[??] + ([[alpha].sub.1])u +l([[alpha].sub.2] [u.sup.2] + [[alpha].sub.3] [u.sup.3] ) = 0 . (45)

This equation is same as the Eq. (23) form where [alpha] = [[alpha].sub.1] and [beta] = 1. According to PEM and Substituting [alpha] = [[alpha].sub.1] and = [beta] into Eqs. (25)-(26), we have:

[[alpha].sub.1] = [[omega].sup.2] + p[[gamma].sub.1] + [p.sup.2] [[gamma].sub.2] +...; (46)

1 = p[[delta].sub.1] + [p.sup.2] [[delta].sub.2] + ... (47)

Substituting Eqs. (24) and (46)-(47) into Eqs. (45) and equating the terms with the identical powers of p, we obtain:

[p.sup.0] : [??].sub.0] + [[omega].sup.2] [u.sub.0] =0; (48)

[p.sup.1] : [??].sub.1] + [[omega].sup.2] [u.sub.1] + [[gamma].sub.1] [u.sub.0] + [[delta].sub.1] [[alpha].sub.2] [u.sup.2.sub.0] = 0. (49)

Considering initial conditions u(0) = A and [??] (0) = 0, the solution of Eq. (48) is [u.sub.0] = A cos ([omega]t) . Substituting u0 into Eq. (49), we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (50)

It is possible to perform the following Fourier series expansion:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (51)

Substituting Eq. (51) into Eq. (50) gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](52)

No secular term in [u.sub.1] requires that:

[[gamma].sub.1] A + / [A.sup.2] [[delta].sub.1](32[[alpha].sub.2] + 9 A [[alpha].sub.3] [pi] = 0. (53)

Setting p = 1 in Eqs. (25) and (26), we have:

[[alpha].sub.1] = [omega].sup.2] + [[gamma].sub.1] (54)

1 = [[delta].sub.1] (55)

From Eqs. (53)-(55), we obtain:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (56)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (57)

5. Applications of analytical solutions for Eq. (3)

Similar to prior section, in this part, we applied the approximate methods for another practical case presented in Eq. (3). This equation can be put in the following general form:

[??] + [[alpha].sub.1] u + [[alpha].sub.3] [u.sup.3] = 0 (58)

where [[alpha].sub.1] = ([k.sub.1] / m - 2P / 1m), [[alpha].sub.3] = ([k.sub.3] / m - 2P [1.sup.3]m).

5.1. Applying the IPM

After introducing the variable y = du/dt, and Substituting u = A cos ([omega]t) into the Eq. (58), we obtain:

[??] = -[[alpha].sub.1]A cos ([omega]t) - [[alpha].sub.3]A [cos.sup.3] ([omega]t). (59)

By using Fourier series expansion, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](60)

By integrating Eq. (60), and comparing with Eq. (9), we obtain:

[omega] = [square root of [[alpha].sub.1] + 3/4 [[alpha].sub.3] [A.sup.2]]; (61)

T = 4[pi] [square root of 4 [[alpha].sub.1] + 3 [[alpha].sub.3] [A.sup.2]] (60)

5.2. Applying the VA

In this section, we will use the Variational Approach solution for Eq. (58). The variational principle of Eq. (58), can be obtained:

j(u) = [[integral].sup.T/4.sub.0] ( - 1/2 [??].sup.2] + [integral] ([[alpha].sub.1]u + [[alpha].sub.2] [u.sup.3]) du)dt. (63)

Using a trial function u = A cos [omega]t into Eq. (63), the solution of Eq. (58) can be expressed as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (64)

Thus, the stationary condition with respect to A is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (65)

This leads to the result:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (66)

Thus, we obtain the following frequency and period same as the IPM solution:

[omega] = [square root of [[alpha].sub.1] + 3/4 [[alpha].sub.3] [A.sup.2]]; (67)

T = 4 [pi] [square root of 4 [alpha].sub.1] + 3[alpha].sub.3] [A.sup.2]] (68)

5.3. Applying the PEM

We assume that in Eq. (58), [alpha] = [[alpha].sub.1] and [beta] = [[alpha].sub.3]. Similar to sections 3.3 and 4.3, we expand the solution u, and its coefficients [[alpha].sub.1] and 1 (Eqs. (24)-(26)), and substituting into Eq. (58), we can obtain:

[p.sup.0] : [[??].sub.0] + [[omega].sub.2][u.sub.0] =0 ; (69)

[p.sub.1] : [[??].sub.1] + [[omega].sub.2][u.sub.1] + [gamma].sub.1][u.sub.0] + [[delta].sub.1] [[alpha].sub.2] [u.sup.3.sub.0] = 0. (70)

Considering Eq. (69) with initial conditions u(0)=A and [??](0) = 0 gives [u.sub.0] = A cos ([omega]t). Substituting [u.sub.0] into Eq. (70), we obtain:

[p.sup.1] : [??].sub.1] + [[omega].sup.2] [u.sub.1] + [[gamma].sub.1] A cos ([omega]t) + [[delta].sub.1] [[alpha].sub.2][A.sup.3] [cos.sup.3] ([omega]t) = 0 . (71)

Using the following Fourier series expansion, we have:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (72)

Substituting Eq. (72) into Eq. (71) gives:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (73)

No secular term in [u.sub.1] requires that:

[[gamma].sub.1]A + 3[A.sup.3][[alpha].sub.2] [[delta].sub.1] / 4 = 0 (74)

Setting p = 1 in Eqs. (25) and (26), we have:

[[alpha].sub.1] = [[omega].sub.2] + [[gamma].sub.1]; (75)

1 = [[delta].sub.1]. (76)

From Eqs. (75) and (76), we obtain:

[omega] = [square root of [[alpha].sub.1] + 3/4 [[alpha].sub.3] [A.sup.2]]; (77)

T = 4[pi] [square root of 4[[alpha].sub.1] + 3[[alpha].sub.3] [A.sup.2]] (78)

6. Results and discussions

As it is apparent in section 4, the periodic solutions of IPM, VA, and PEM for a current-carrying conductor with cubic non-linearity are equal. In order to, substitute [[alpha].sub.3] = 1 -[DELTA]/[(1 -[alpha]).sup.2] , [[alpha].sub.2] = -[DELTA][(1 - [alpha]).sup.3] and [[alpha].sub.3] = -[DELTA] / [(1-[alpha]).sup.4] into results of periodic solutions for example Eqs. (56) and (57) , the frequency and period values of Eq. (33) can be written as follow:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (79)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (80)

Similarly, substituting [[alpha].sub.1] =([k.sub.1]/m - 2P/lm) and [[alpha].sub.2] =([k.sub.3] / m - 2p / [1.sup.3] m) into Eqs. (77) and (78) gives the following frequency and period values for Eq. (58):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (81)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (82)

6.1. Analytical solutions of current-carrying wire conductor equation

In this section, we compare the analytical approximate periods of Eq. (33) with the exact ones. Considering [25], the exact solution of Eq. (33) is expressed in appendix A.

Using [DELTA] = -3/4, [alpha] = -1/2 and [DELTA] = 1/8, [alpha] = (2 -[square root of 2]/4, the exact period [T.sub.e] [25] and approximate periods [T.sub.IPM] , [T.sub.VA] , and [T.sub.EPM] are listed in Table 1. From Table 1 we can obtain that the presented approximate solutions are excellent for all permitted oscillation amplitudes. In general, the first approximate periods of IPM, VA, and PEM are acceptable.

Comparisons of the approximate analytical solution with the exact solutions for given [DELTA] = -3/4, [alpha] = -1/2 and [DELTA] = 1/8, [alpha] = (2 -[square root of 2]/4 and different amplitudes of oscillation A are shown in Figs. 3-6, respectively.

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

6.2. Analytical solutions of bucking of a column equation

The exact frequency [[omega].sub.e] for a dynamic system governed by Eq. (58) is presented in Appendix B. As we know Eq. (58) is similar to a type of Duffing equation. So, the maximum amplitude A of the oscillation satisfies [[alpha].sub.a][A.sup.2] = -[[alpha].sub.1] ; the Duffing equation has a heteroclinic orbit with period +[infinity] [26]. Hence, in order to avoid the hetero clinic orbit with period + [infinity], the value of [k.sub.3] in the bucking of a column equation should satisfy the following equation:

[k.sub.3] > [k.sub.1]/[A.sup.2] + 2p/l (1/[A.sup.2] + 1[l.sup.2]), (83)

where [k.sub.1], l [member of] [R.sup.+] and A, p [member of] R.

To further illustrate and verify the accuracy of the proposed analytical approaches for Eq. (58), the corresponding comparisons of analytical solutions with exact results for specific parameters and initial values consisting m, p, l, [k.sub.1] , [k.sub.2] and A are tabulated in Table 2.

[FIGURE 7 OMITTED]

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

Figs. 7-10 indicate the comparison of these analytical methods for different parameters with initial values which are in an excellent agreement with the exact solutions.

Of course the accuracy of these methods can be improved upon using higher-order approximate solutions for approximations methods. Hence, it is concluded for providing an excellent agreement with exact solutions for the nonlinear Duffing equation.

7. Conclusions

In summary, three analytical approximations to the periodic solution of SDOF systems including current carrying conductor and bucking of a column are constructed using IPM, PEM, and VA approaches. According to the results (Tables. 1-2, and Figs. 3-10), we can see that the presented approximate results are absolutely equal and differences between analytical and exact solutions are negligible. In other words, the first-order approximate solutions of IPM, EBM, and VA benefit a high accuracy and the percentage error improves significantly from lowerorder to higher-order analytical approximations for different parameters and initial amplitudes.

Appendix A

For achieving the exact period [T.sub.e] of Eq. (1), substituting a new independent variable u = x - [alpha] into Eq. (1) leads to [25]:

[??] + [alpha] + u - [DELTA] / 1 - [alpha] - u = 0, u(0) = A, [??] (0)=0 (A.1)

where [alpha] is one of the stable equilibrium points and A = [??] - [alpha]. The corresponding potential energy function is:

V (u) = 1/2 [(u + [alpha]).sup.2] +[DELTA]ln [absolute value of 1 - [alpha]]. (A.2)

And it reaches its minimum at u = 0. Thus, the system will oscillate between asymmetric limits [-B, A] where both -B (B > 0) and A have the same energy level, i.e.:

V(-B) = V(A) . (A.3)

The exact period [T.sup.e] (A) is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (A.4)

where B is given by Eqs. (A.2) and (A.3).

Appendix B

The exact solution of Eq. (58) can be obtained by integrating the governing differential equation and imposing the initial conditions in Eq. (58) as follows:

1/2 [[??].sup.2] + [alpha]/2 [v.sup.2] + [beta]/2 [v.sup.4] = C, [for all]t, (B.1)

which C is a constant. Imposing initial conditions in

Eq. (58) yields:

C = [alpha] / 2 [A.sup.2] + [beta] /4 [A.sup.4.] (B.2)

Equating Eqs. (B.1) and (B.2) yields:

1/2[v.sup.2] + [alpha]/2 [v.sup.2] + [beta]/4 [v.sup.4] = [alpha]/2 [a.sup.2] + [beta] /4 [A.sup.4] (B.3)

or equivalently

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B.4)

Integrating Eq. (B.4), the period of oscillation [T.sub.e] is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B.5)

Substituting v = Acost into Eq. (B.5) and integrating:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B.6)

which

[delta] = [beta] [A.sup.2] / 2([alpha] + [beta] [A.sup.2]) (B.7)

The exact frequency [[omega].sub.e] is also a function of A and can be obtained from the period of the oscillation as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B.8)

10.5755/j01.mech.19.3.4659

Received December 06,2011 Accepted May 15,13

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S.S. Ganji *, A. Barari **, Abdolhossein Fereidoon ***, S. Karimpour ****

* Young Researchers and Elites Club, Science and Research Branch, Islamic Azad University, Tehran, Iran, E-mail: r.alizadehganji@gmail.com

** Department of Civil Engineering, Aalborg University, Sohngardsholmsvej 57, DK-9000 Aalborg, Aalborg, Denmark, E-mail: ab@civil.aau.dk.

*** Department of Mechanical Engineering, Faculty of Engineering, Semnan University, Semnan, Iran

**** Young Researchers and Elites Club, Science and Research Branch, Islamic Azad University, Tehran, Iran

Table 1
Comparison of approximate and "exact" periods for
current-carrying conductor

                                                  [T.sub.app]/
A      B[26]     [T.sub.e] [26]   [T.sub.app]     [T.sub.e]

a) [DELTA] = -3/4                 [T.sub.IPM] =
[alpha] = -1/2                    [T.sub.EPM] =
                                  [T.sub.VA]

0.1    0.10112   5.43974          5.40109         0.99289
0.4    0.41917   5.41309          5.26059         0.97183
0.7    0.76545   5.34444          5.09664         0.95363
1.0    1.16058   5.20343          4.91674         0.94490
1.2    1.48388   5.02518          4.79132         0.95346
1.4    1.97647   4.65034          4.66357         1.00284
1.43   2.10317   4.54583          4.64431         1.02166

b) [DELTA] = 1/8
[alpha] = (2 -
[square root]22)/4

0.1    0.09839   6.91230          6.98305         1.01023
0.3    0.28465   6.99470          7.20094         1.02948
0.5    0.44943   7.25765          7.51543         1.03552
0.6    0.51485   7.63026          7.71964         1.01171
0.63   0.53007   7.85127          7.78834         0.99198

Table 2
Comparison of approximate and "exact" periods
for the bucking of a column

Constant parameters

m       L     P     [k.sub.1]   [k.sub.3]    A

1       1     1        10           5        1
5      1.5    5         5           6        3
10     10     10       10          50       10
50     25     40       30          100      20
70     20    -30       50          100      10
100    50    150       70          20       100
500    150   220       120         500      0.5
1000   500   1000      500         500       1

[T.sub.e]         [T.sub.app]          [T.sub.app]/[T.sub.e]

                 [T.sub.IPM] =
            [T.sub.EBM] = [T.sub.VA]

1.96451             1.96254                   1.00101
3.32368             3.23744                   1.02664
0.33143             0.32426                   1.02212
0.26208             0.25640                   1.02216
0.30993             0.30323                   1.02212
0.16580             0.162206                  1.02218
9.71672             9.676370                  1.00417
6.75871             6.73241                   1.00391