MPE Mathematical Problems in Engineering 1563-5147 1024-123X Hindawi Publishing Corporation 721294 10.1155/2013/721294 721294 Research Article Free Vibration Analysis of an Euler Beam of Variable Width on the Winkler Foundation Using Homotopy Perturbation Method Mutman Utkan Coşkun Safa Bozkurt Department of Civil Engineering Kocaeli University 41380 Kocaeli Turkey kocaeli.edu.tr 2013 18 4 2013 2013 24 02 2013 30 03 2013 2013 Copyright © 2013 Utkan Mutman. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Homotopy Perturbation Method (HPM) is employed to investigate the vibration of an Euler beam resting on an elastic foundation. The beam is assumed to have variable stiffness along its length. HPM is an easy-to-use and very efficient technique for the solution of linear or nonlinear problems. HPM produces analytical approximate expression which is continuous in the solution domain. This work assures that HPM is a promising method for the vibration analysis of the variable stiffness Euler beams on elastic foundation. Different case problems have been solved by using the technique, and solutions have been compared with those available in the literature.

1. Introduction

In geotechnical engineering, it is widely seen in application that pipelines, shallow foundations, and piles are modeled as a beam in the analysis procedure. There are also various types of foundation models such as Winkler, Pasternak, and Vlasov that have been used in the analysis of structures on elastic foundations.

The most frequently used foundation model in the analysis of beam on elastic foundation problems is the Winkler foundation model. In the Winkler model, the soil is modeled as uniformly distributed, mutually independent, and linear elastic vertical springs which produce distributed reactions in the direction of the deflection of the beam. The winkler model requires a single parameter k, representing the soil properties. However, since the model does not take into account either continuity or cohesion of the soil, it may be considered as a rather crude representation of the soil.

As stated before pipelines, shallow foundations and piles can be modeled as a beam. Hence, produced results may find various application areas. There are also different beam types in theory. The well-known is the Euler-Bernoulli beam which is suitable for slender beams. For moderately short and thick beams, the Timoshenko beam model has to be used in the analysis.

Vibration of a constant stiffness Euler beam on elastic foundation was studied previously by Balkaya et al.  and Ozturk and Coskun . Balkaya et al.  used Differential Transform Method while Ozturk and Coskun  used HPM in their studies. Avramidis and Morfidis  studied bending of beams on three-parameter elastic foundation. De Rosa  analyzed free vibration of Timoshenko beams on two-parameter elastic foundation. Matsunaga  studied vibration and buckling of deep beam columns on two-parameter elastic foundations. El-Mously  determined fundamental frequencies of the Timoshenko beams mounted on the Pasternak foundation. Chen [7, 8] analyzed vibration of beam resting on an elastic foundation by the differential quadrature element method (DQEM). Coşkun  investigated the response of a finite beam on a tensionless Pasternak foundation subjected to a harmonic load. Chen et al.  developed a mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation. Maheshwari et al.  studied the response of beams on a tensionless extensible geosynthetic-reinforced earth bed subjected to moving loads. Auciello and De Rosa  developed different approaches to the dynamic analysis of beams on soils subjected to subtangential forces.

2. The Equations of Motion and Boundary Conditions

An Euler beam resting on the Winkler foundation shown in Figure 1 is considered in this study. The equation of motion for this problem is given as follows: (1)2x2(EI(x)2wx2)+k(x)w+ρA(x)(2wt2)=0, where k is the spring constant, w is deflection, ρ is the mass density, A is the cross-sectional area, EI is the beam stiffness, and I is the area moment of inertia about the neutral axis. The deflection is a function of both space and time, that is, w=w(x,t) in which space variable x is measured along the length of the beam and t represents any particular instant of time.

Representation of a beam on the Winkler foundation.

Due to the support conditions at both ends of the beam, different conditions have to be imposed to the obtained solution to determine unknowns included in final approximation produced by HPM. These conditions are given as follows.

For clamped-clamped beam the end conditions are (2)w=wx=0atx=0,L.

For cantilever beam (clamped-free) the end conditions are (3a)w=wx=0atx=0,(3b)2wx2=3wx3=0atx=L.

For simply supported beam (pinned-pinned) the end conditions are (4)w=2wx2=0atx=0,L.

For clamped-simply supported (pinned) beam the end conditions are(5a)w=wx=0atx=0,(5b)w=2wx2=0atx=L.

Now, free vibration analysis of the beams with variable stiffness resting on elastic foundations will be formulated.

A solution is assumed as the following form to formulate the analysis of the presented problem by the separation of variables (6)w(x,t)=W(x)eiωt, where ω is the circular frequency for the vibration. Substituting (6) into (1), equations of motion become as follows: (7)d2dx2(EI(x)d2Wdx2)+kW=ρAω2W.

This equation can be rearranged as (8)4wx4+2EI(x)EI(x)3wx3+EI(x)′′EI(x)2wx2+k(x)EI(x)w+ρA(x)EI(x)(2wt2)=0, where (·) denotes total derivative with respect to x. The governing equation is now rewritten in a nondimensional form. This procedure is provided from  in which a constant stiffness beam was analyzed. The notation is maintained in this study for the comparison purposes. The nondimensional parameters for the Eulerbeam on the Winkler foundation are defined as  (9)x-=xL,W¯=WL,λ=kL4EI,ω¯=ωρAk.

Using these parameters, nondimensional form of the equations and formulation procedures are explained in the following sections.

3. Homotopy Perturbation Method

In recent years, efforts towards application of analytical approximate solution techniques for nonlinear problems have increased. In these studies, He’s Homotopy Perturbation Method  can be considered as one of the most promising methods for nonlinear problems. The HPM provides an analytical approximate expression as the solution for the problems which are continuous in the solution domain. The technique is applied to an equation of the form L(u)+N(u)=f(r),rΩ with boundary conditions B(u,u/n)=0, rΓ, where L is a linear operator, N is a nonlinear operator, B is a boundary operator, Γ is the boundary of the domain Ώ, and f(r) is a known analytic function. HPM defines a homotopy as v(r,p)=Ω×[0,1]R which satisfies the following inequalities: (10)H(v,p)=(1-p)[L(v)-L(u0)]+p[L(v)+N(v)-f(r)]=0

or (11)H(v,p)=L(v)-L(u0)+pL(u0)+p[N(v)-f(r)]=0, where rΩ and p[0,1] are imbedding parameters, u0 is an initial approximation which satisifies the boundary conditions. From (10)-(11), we have (12)H(v,0)=L(v)-L(u0)=0,H(v,1)=L(v)+N(v)-f(r)=0.

The changing process of p from zero to unity is that of v(r,p) from u0 to u(r). In topology, this deformation L(v)-L(u0) and L(v)+N(v)-f(r) are called homotopic. The method expresses the solution of (10)-(11) as a power series in p as follows: (13)v=v0+pv1+p2v2+p3v3+.

The approximate solution of L(u)+N(u)=f(r), rΩ can be obtained as (14)u=limp1v=v0+v1+v2+.

The convergence of the series in (14) has been proven in .

4. HPM Formulation and Solution Procedure for Presented Problem

In this study, a variable stiffness is assumed for the beam considered. The variation is applied by changing the width of the beam along its length. Two different variations in stiffness are assumed. These are linear and exponential variations.

In the linear variation, beam width varies in a linear manner along the beam’s length which is shown in Figure 2.

Beam with a linearly varying width.

The variable width is formulated as (15)b(x)=b0(1-αx), where the dimension of α is [1/L]. For the exponential variation, beam width varies exponentially along the beam’s length as shown in Figure 3.

Beam with an exponentially varying width.

Variable width of the beam in this case is (16)b(x)=b0e-αx, where α has the same dimension as in linear case. By the use of variable width, both cross-sectional area and moment of inertia are variable, (17)A(x)=b(x)h,I(x)=b(x)h312.

In the following sections, formulation procedures are given separately for each case.

4.1. Formulation for Linear Variation

Employing (15) into (17), variable cross-sectional areas and variable stiffness become (18)A(x)=b0h(1-αx)=A0(1-αx),EI(x)=Eb0h312(1-αx)=EI0(1-αx), where A0 and I0 are the cross-sectional area and moment of inertia of the section at the origin, respectively. Inserting (18) into (8) (19)4wx4-2αEI0EI0(1-αx)3wx3+k(x)EI0(1-αx)w+ρA0EI0(2wt2)=0.

This equation can be rewritten as (20)4wx4-2α(11-αx)3wx3+k(x)EI0(11-αx)w+ρA0EI0(2wt2)=0.

Employing (6) (21)Wiv-2αξ(x)W′′′+k(x)EI0ξ(x)W-ρA0EI0ω2W=0.

Equation (21) can be made nondimensional in view of (9) as follows: (22)W¯iv-2α¯ξ(x¯)W¯′′′+λ(ξ(x¯)-ω¯2)W=0, where (23)x-=xL,W¯=WL,λ=kL4EI0,ω¯=ωρA0k,ξ(x-)=11-α-x-,α-=αL.

By the application of HPM, the following iteration algorithm is obtained:(24a)W-0iv-u0iv=0,(24b)W-1iv+u0iv-2α-ξ(x-)W-0′′′+λ(ξ(x-)-ω-2)W-0=0,(24c)W-niv-2α-ξ(x-)W-n-1′′′+λ(ξ(x-)-ω-2)W-n-1=0,n2.

4.2. Formulation for Exponential Variation

Employing (16) into (17), variable cross-sectional areas and variable stiffness become (25)A(x)=b0he-αx=A0e-αx,EI(x)=Eh312b0e-αx=EI0e-αx.

Inserting (25) into (8) (26)4wx4-2α3wx3+α22wx2+k(x)EI0e-αxw+ρA0EI0(2wt2)=0.

Employing (6) (27)Wiv-2αW′′′+α2W′′+k(x)EI0eαxW-ρA0EI0ω2W=0.

Equation (21) can be made nondimensional in view of (9) as follows: (28)W¯iv-2α¯W¯′′′+α¯2W¯′′+λ(eα¯x¯-ω¯2)W=0.

By the application of HPM, the following iteration algorithm is obtained:(29a)W-0iv-u0iv=0,(29b)W-1iv+u0iv-2α-W-0′′′+α-2W-0′′+λ(eα-x--ω-2)W-0=0,(29c)W-niv-2α-W-n-1′′′+α-2W-n-1′′+λ(eα-x--ω-2)W-n-1=0,n2.

4.3. Solution Procedure

An initial approximation may be chosen as a cubic polynomial with four unknown coefficients. There exist four boundary conditions, that is, two at each end of the column, in the presented problem. Hence, an initial approximation of the following choice may be employed: (30)W-0=Ax-3+Bx-2+Cx-+D.

In the computations, twenty iterations are conducted and four boundary conditions for each case are rewritten by using the final approximation of iteration. Each boundary condition produces an equation containing four unknowns spread from the initial approximation. These nondimensional boundary conditions are as follows.

Clamped-clamped beam: (31)W¯=dW¯dx-=0atx-=0,1.

Clamped-free (cantilever) beam:(32a)W¯=dW¯dx-=0atx-=0,(32b)d2W¯dx-2=d3W¯dx-3=0atx-=1.

Pinned-pinned (simply supported) beam: (33)W¯=d2W¯dx-2=0atx-=0,1.

Clamped-pinned beam:(34a)W¯=dW¯dx-=0atx-=0,(34b)W¯=d2W¯dx-2=0atx-=1.

Hence, four equations in four unknowns may be written with respect to the boundary conditions of the problem. These equations can be represented in matrix form as follows: (35)[M(ω-)]{A}={0}, where {A}=ABCDT. For a nontrivial solution, determinant of coefficient matrix must be zero. Determinant of coefficient matrix yields a characteristic equation in terms of ω-. Positive real roots of this equation are the normalized free vibration frequencies for the case considered.

5. Numerical Results 5.1. Constant Stiffness Case

As the first example, Euler beam of constant stiffness (i.e., EI is constant) with different boundary conditions is investigated. For the sake of comparison, all the values are set to unity such as I=E=A=ρ=1; hence, λ=1, according to previous studies . Both algorithms given for linear and exponential variations lead to constant stiffness when α=0.

In Table 1, first three normalized free vibration frequencies for simply supported (pinned-pinned) beam are compared with previous available results in the literature and the exact solution. Excellent agreement is observed with the exact solution.

Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation.

Method ω 1 ω 2 ω 3 ω 4 ω 5
HPM 9.92014 39.4911 88.8321 157.9168 246.7421
DTM  9.92014 39.4911 88.8321
DQEM  9.92014 39.4913 89.4002
Exact solution  9.92014 39.4911 88.8321

The first five natural frequencies for clamped-clamped beam and cantilever (clamped-free) beam are presented in Tables 2 and 3, respectively. They are compared with the available solutions.

Normalized free vibration frequencies of clamped-clamped beam resting on the Winkler foundation.

Method ω 1 ω 2 ω 3 ω 4 ω 5
HPM 22.3956 61.6809 120.908 199.862 298.557
DTM  22.3733 61.6728 120.903 199.859 298.556
DQEM  22.3956 61.6811 120.910 199.885 298.675

Normalized free vibration frequencies of cantilever beam resting on the Winkler foundation.

Method ω 1 ω 2 ω 3 ω 4 ω 5
HPM 3.65546 22.0572 61.7053 120.906 199.862
DTM  3.65546 22.0572 61.7053 120.906 199.862
DQEM  3.65544 22.0572 61.7057 120.911 199.894

Excellent agreement is observed with previous available results for both cantilever and clamped-clamped beams. Clamped-pinned beam was not included in the studies used for comparison. Hence, only the result obtained from HPM is tabulated for this case in Table 4.

Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation.

Method ω 1 ω 2 ω 3 ω 4 ω 5
HPM 15.451 49.975 104.253 178.273 272.033

As one can see, perfect agreement is obtained for constant stiffness case. This issue is mainly due to constant coefficient governing equation. In the following sections, variable stiffness cases are investigated.

5.2. Linearly Varying Stiffness

Linearly varying beam width results in a linearly varying flexural stiffness. The variation is based on parameter α. A number of case studies are conducted with respect to parameter α, and the results are given in Tables 5, 6, 7, and 8.

Normalized free vibration frequencies of cantilever beam resting on Winkler foundation with linearly varying flexural stiffness.

α 0.00 0.10 0.20 0.30 0.40 0.50
ω 1 3.65546 3.77785 3.91814 4.08113 4.27363 4.50571
ω 2 22.0572 22.2779 22.5271 22.8129 23.1475 23.5506
ω 3 61.7053 61.9182 62.1616 62.4458 62.7867 63.2104
ω 4 120.9061 121.1194 121.3645 121.6528 122.0024 122.4434
ω 5 199.8620 200.0755 200.3213 200.6117 200.9661 201.4172

Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation with linearly varying flexural stiffness.

α 0.00 0.10 0.20 0.30 0.40 0.50
ω 1 15.4506 15.5615 15.6801 15.8069 15.9427 16.0879
ω 2 49.9749 50.0781 50.1878 50.3052 50.4316 50.5685
ω 3 104.2525 104.3556 104.4655 104.5836 104.7121 104.8544
ω 4 178.2725 178.3756 178.4853 178.6036 178.7333 178.8785
ω 5 272.0328 272.1358 272.2455 272.3640 272.4944 272.6418

Normalized free vibration frequencies of clamped-clamped beam resting on the Winkler foundation with linearly varying flexural stiffness.

α 0.00 0.10 0.20 0.30 0.40 0.50
ω 1 22.3956 22.3922 22.3777 22.3477 22.2958 22.2120
ω 2 61.6809 61.6752 61.6541 61.6114 61.5380 61.4183
ω 3 120.9075 120.9009 120.8775 120.8302 120.7485 120.6144
ω 4 199.8620 199.8549 199.8302 199.7803 199.6939 199.5512
ω 5 298.5572 298.5499 298.5243 298.4729 298.3834 298.2351

Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation with linearly varying flexural stiffness.

α 0.00 0.10 0.20 0.30 0.40 0.50
ω 1 9.9201 9.9217 9.9210 9.9170 9.9084 9.8932
ω 2 39.4911 39.4928 39.4970 39.5047 39.5166 39.5340
ω 3 88.8321 88.8340 88.8398 88.8511 88.8697 88.8986
ω 4 157.9168 157.9189 157.9257 157.9389 157.9612 157.9966
ω 5 246.7421 246.7443 246.7516 246.7661 246.7907 246.8302

Variation of normalized free vibration frequencies (ω-) with respect to nondimensional variation parameter α- for each beam are also given in Figures 4, 5, 6, 7, and 8.

Variation of normalized first mode frequency with respect to normalized variation coefficient (linear variation in stiffness).

Variation of normalized second mode frequency with respect to normalized variation coefficient (linear variation in stiffness).

Variation of normalized third mode frequency with respect to normalized variation coefficient (linear variation in stiffness).

Variation of normalized fourth mode frequency with respect to normalized variation coefficient (linear variation in stiffness).

Variation of normalized fifth mode frequency with respect to normalized variation coefficient (linear variation in stiffness).

5.3. Exponentially Varying Stiffness

Exponentially varying beam width results in an exponentially varying flexural stiffness. The variation is again based on parameter α, and a number of case studies are also conducted with respect to parameter α. Results are given in Tables 9, 10, 11, and 12.

Normalized free vibration frequencies of cantilever beam resting on the Winkler foundation with exponentially varying flexural stiffness.

α 0.00 0.25 0.50 0.75 1.00
ω 1 3.65546 3.99564 4.46148 5.08383 5.89198
ω 2 22.0572 22.6410 23.4038 24.4544 25.9407
ω 3 61.7053 62.2689 63.0286 64.1348 65.8187
ω 4 120.9061 121.4706 122.2407 123.3890 125.1956
ω 5 199.8620 200.4266 201.2020 202.3738 204.2517

Normalized free vibration frequencies of clamped-pinned beam resting on the Winkler foundation with exponentially varying flexural stiffness.

α 0.00 0.25 0.50 0.75 1.00
ω 1 15.4506 15.7228 16.0355 16.4684 17.1543
ω 2 49.9749 50.2233 50.5095 50.9401 51.7103
ω 3 104.2525 104.4999 104.7874 105.2357 106.0716
ω 4 178.2725 178.5190 178.8070 179.2648 180.1380
ω 5 272.0328 272.2787 272.5671 273.0313 273.9289

Normalized free vibration frequencies of clamped-clamped beam resting on the Winkler foundation with exponentially varying flexural stiffness.

α 0.00 0.25 0.50 0.75 1.00
ω 1 22.3956 22.3562 22.2409 22.0853 21.9787
ω 2 61.6809 61.6245 61.4646 61.2498 61.1004
ω 3 120.9075 120.8450 120.6698 120.4372 120.2857
ω 4 199.8620 199.7961 199.6124 199.3705 199.2211
ω 5 298.5572 298.4893 298.3002 298.0528 297.9060

Normalized free vibration frequencies of simply supported beam resting on the Winkler foundation with exponentially varying flexural stiffness.

α 0.00 0.25 0.50 0.75 1.00
ω 1 9.9201 9.8928 9.8179 9.7493 9.7849
ω 2 39.4911 39.4736 39.4584 39.5456 39.9202
ω 3 88.8321 88.8182 88.8204 88.9564 89.4478
ω 4 157.9168 157.9048 157.9164 158.0782 158.6327
ω 5 246.7421 246.7312 246.7485 246.9262 247.5200

Variation of normalized free vibration frequencies (ω-) with respect to nondimensional variation parameter α- for each beam are also given in Figures 9, 10, 11, 12, and 13.

Variation of normalized first mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).

Variation of normalized second mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).

Variation of normalized third mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).

Variation of normalized fourth mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).

Variation of normalized fifth mode frequency with respect to normalized variation coefficient (exponential variation in stiffness).

6. Conclusions

In this study, HPM is introduced for the free vibration analysis of variable stiffness Euler beams on elastic foundations. As a demonstration of application of the method, firstly the case studies for which previous results were available are chosen. In these studies, constant stiffness Euler beams were considered, and first analyses are conducted for constant stiffness case for comparison with the available results. HPM produced results which are in excellent agreement with the previously available solutions that encourage the application of the method for variable stiffness beams. To represent a variation in stiffness, a rectangular beam with varying width is considered, and two types of variation are taken into consideration. These are, namely, linear and exponential changes. Such varying widths lead to linearly and exponentially varying stiffnesses. The analyses are expanded for variable stiffness cases. HPM also produced reasonable results for the vibration of variable stiffness beams which show the efficiency of the method. For the variable stiffness problems, the governing equation is a differential equation with variable coefficients, and it is not easy to obtain analytical solutions for these types of problems. However, it is easy to put those variable parameters into the iteration algorithm of HPM, and the results can be obtained after performing some iterations with the method. The results obtained in this study point out that the proposed method is a powerful and reliable method in the analysis of the presented problem.

Balkaya M. Kaya M. O. Saǧlamer A. Analysis of the vibration of an elastic beam supported on elastic soil using the differential transform method Archive of Applied Mechanics 2009 79 2 135 146 2-s2.0-58149523136 10.1007/s00419-008-0214-9 Ozturk B. Coskun S. B. The Homotopy Perturbation Method for free vibration analysis of beam on elastic foundation Structural Engineering and Mechanics 2011 37 4 415 425 2-s2.0-79551707446 Avramidis I. E. Morfidis K. Bending of beams on three-parameter elastic foundation International Journal of Solids and Structures 2006 43 2 357 375 2-s2.0-28044449134 10.1016/j.ijsolstr.2005.03.033 De Rosa M. A. Free vibrations of Timoshenko beams on two-parameter elastic foundation Computers and Structures 1995 57 1 151 156 2-s2.0-0029634766 Matsunaga H. VIbration and buckling of deep beam-columns on two-parameter elastic foundations Journal of Sound and Vibration 1999 228 2 359 376 2-s2.0-0000489957 El-Mously M. Fundamental frequencies of Timoshenko beams mounted on Pasternak foundation Journal of Sound and Vibration 1999 228 2 452 457 2-s2.0-0000983675 Chen C. N. Vibration of prismatic beam on an elastic foundation by the differential quadrature element method Computers and Structures 2000 77 1 1 9 2-s2.0-0033745567 10.1016/S0045-7949(99)00216-3 Chen C. N. DQEM vibration analyses of non-prismatic shear deformable beams resting on elastic foundations Journal of Sound and Vibration 2002 255 5 989 999 2-s2.0-0037194567 10.1006/jsvi.2001.4176 Coşkun I. The response of a finite beam on a tensionless Pasternak foundation subjected to a harmonic load European Journal of Mechanics, A/Solids 2003 22 1 151 161 2-s2.0-2942603414 10.1016/S0997-7538(03)00011-1 Chen W. Q. C. F. Bian Z. G. A mixed method for bending and free vibration of beams resting on a Pasternak elastic foundation Applied Mathematical Modelling 2004 28 10 877 890 2-s2.0-4143092668 10.1016/j.apm.2004.04.001 Maheshwari P. Chandra S. Basudhar P. K. Response of beams on a tensionless extensible geosynthetic-reinforced earth bed subjected to moving loads Computers and Geotechnics 2004 31 7 537 548 2-s2.0-11344270554 10.1016/j.compgeo.2004.07.005 Auciello N. M. De Rosa M. A. Two approaches to the dynamic analysis of foundation beams subjected to subtangential forces Computers and Structures 2004 82 6 519 524 2-s2.0-0742307248 10.1016/j.compstruc.2003.10.020 He J. H. Coupling method of a homotopy technique and a perturbation technique for non-linear problems International Journal of Non-Linear Mechanics 2000 35 1 37 43 2-s2.0-0033702384 10.1016/S0020-7462(98)00085-7 MR1723761 He J. H. The homotopy perturbation method for nonlinear oscillators with discontinuities Applied Mathematics and Computation 2004 151 1 287 292 2-s2.0-1242287587 10.1016/S0096-3003(03)00341-2 He J. H. Application of homotopy perturbation method to nonlinear wave equations Chaos, Solitons & Fractals 2005 26 3 695 700 2-s2.0-18844426016 10.1016/j.chaos.2005.03.006 He J. H. Asymptotology by homotopy perturbation method Applied Mathematics and Computation 2004 156 3 591 596 2-s2.0-4344696077 10.1016/j.amc.2003.08.011 MR2088125 ZBL1061.65040 He J. H. Homotopy perturbation method for solving boundary value problems Physics Letters A 2006 350 1-2 87 88 2-s2.0-30644460357 10.1016/j.physleta.2005.10.005 MR2199322 ZBL1195.65207 He J. H. Limit cycle and bifurcation of nonlinear problems Chaos, Solitons & Fractals 2005 26 3 827 833 2-s2.0-18844391045 10.1016/j.chaos.2005.03.007