Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics http://pik.sagepub.com/ Dynamic Response of Inextensible Beams by Improved Energy Balance Method M G Sfahani, A Barari, M Omidvar, S S Ganji and G Domairry Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics 2011 225: 66 DOI: 10.1177/2041306810392113 The online version of this article can be found at: http://pik.sagepub.com/content/225/1/66 Published by: http://www.sagepublications.com On behalf of: Institution of Mechanical Engineers Additional services and information for Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics can be found at: Email Alerts: http://pik.sagepub.com/cgi/alerts Subscriptions: http://pik.sagepub.com/subscriptions Reprints: http://www.sagepub.com/journalsReprints.nav Permissions: http://www.sagepub.com/journalsPermissions.nav Citations: http://pik.sagepub.com/content/225/1/66.refs.html >> Version of Record - Mar 1, 2011 What is This? Downloaded from pik.sagepub.com by guest on October 11, 2013 66 Dynamic response of inextensible beams by improved energy balance method M G Sfahani1 , A Barari2∗ , M Omidvar3 , S S Ganji4 , and G Domairry5 1 Department of Civil Engineering, Babol University of Technology, Babol, Iran 2 Department of Civil Engineering, Aalborg University, Aalborg, Denmark 3 Engineering Faculty, University of Golestan, Gorgan, Iran 4 Department of Transportation Engineering, Islamic Azad University, Science and Research Branch, Tehran, Iran 5 Department of Mechanical Engineering, Babol University of Technology, Babol, Iran The manuscript was received on 2 December 2009 and was accepted after revision for publication on 7 October 2010. DOI: 10.1177/2041306810392113 Abstract: An improved He’s energy balance method (EBM) for solving non-linear oscillatory differential equation using a new trial function is presented. The problem considered represents the governing equations of the non-linear, large-amplitude free vibrations of a slender cantilever beam with a rotationally flexible root and carrying a lumped mass at an intermediate position along its span. Based on the simple EBM, the variational integral of the non-linear conservative system is established, and the Fourier series expansion is employed to address the governing algebraic equations. An alternate procedure for a particular value of the initial condition is then used to estimate the constants. This semi-analytical representation gives excellent approximations to the exact solutions for the whole range of the oscillation amplitude, reducing the respective error of angular frequency in comparison with the simple EBM. Two illustrative examples are considered in order to elucidate the methods described, and to reveal the improvements made by the modified method. Keywords: energy balance method, non-linear conservative oscillators, semi-analytical solution 1 INTRODUCTION The study of the oscillating behaviour of a lumped mass supported on a slender cantilever beam element represents the simplified model of many important engineering systems. Large-amplitude non-linear vibration results when the flexibility of the supporting cantilever beam is increased. Many researchers have addressed the non-linear vibration behaviour of beams, both experimentally and theoretically [1–10]. The problem finds applications in multi-body dynamics as well, including in dynamic vibration absorbers [10], while other applications include bridges, aircraft structures, and turbo-machinery blades [11]. Nonlinearity in the oscillation of beams, as reviewed by Hamdan and Shabaneh [12], may be classified into three broad categories, namely geometrical, inertial, ∗ Corresponding author: Department of Civil Engineering, Aalborg University, Sohngårdsholmsvej 57, 9000 Aalborg, Denmark. email: ab@civil.aau.dk; amin78404@yahoo.com Proc. IMechE Vol. 225 Part K: J. Multi-body Dynamics and material. The aforementioned sources of nonlinearity have been the subject of many investigations [9, 13–18]. It is very common to simplify the equations of motion by introducing various assumptions which allow for the derivation of manageable governing equations. Some of the simplifying assumptions include neglecting axial inertia [19–23] and assuming linear curvature [18, 24]. Hamdan and Shabaneh [12] considered the non-linear, large-amplitude free vibrations of a slender, inextensible cantilever beam with a rotationally flexible root and carrying a lumped mass at an intermediate position along its span. The problem is schematically depicted in Fig. 1 with the parameters introduced by Hamdan and Shabaneh [12]. Inextensibility refers to the quality whereby the beam preserves a constant neutral axis length during vibration. Aside from the assumption of inextensibility, Hamdan and Shabaneh [12] also accepted the simplifying assumption that the shear and rotary inertial effects were negligible, and that the beam was assumed to undergo planar flexural vibrations. They then employed the Hamilton’s principle and Dynamic response of inextensible beams by improved energy balance method Fig. 1 Schematic representation of the free vibration of a beam carrying a lumped mass along its span [12] the single-mode Lagrangian method to construct two formulations for the governing equations of the nonlinear, large-amplitude free vibrations of a slender cantilever beam with a rotationally flexible root and carrying a lumped mass at an intermediate position along its span. One of the formulations pertains to extensible beams, while the other considers inextensibility. The time transformation approach [25] was then used to obtain approximate solutions to the period of oscillation for the two formulations. Due to the central role which non-linear oscillators play in many fields of engineering, much attention has been devoted in the past towards developing analytical and numerical solutions for such nonlinear problems. Some of the methods that have been introduced through the literature over the years include the Lindstedt–Poincare method [26], the parameter-expansion method [27–31], the variational iteration method [32–34], the homotopy perturbation method [35–39], the max–min method [40, 41], non-perturbative methods [42], and differential transform method [43], among others. He [44, 45] gave a comprehensive review of the recently developed techniques for solving non-linear oscillation problems, which comprise the relatively newer family of solutions which lie within the framework of periodic solutions. Other methods have also been developed in recent years which seem to be just as promising in obtaining accurate solutions to the generally more difficult non-linear oscillation problems. He’s energy balance method (EBM) [46–49] is one such method, which is actually a heuristic approach valid not only for weakly non-linear systems, but also for strongly non-linear ones [46]. In this article, He’s EBM is first employed to solve the governing equations of the nonlinear, large-amplitude free vibrations of a slender cantilever beam with a rotationally flexible root and carrying a lumped mass at an intermediate position along its span. Then, by applying a new functional 67 form derived from the generalized harmonic balance method [50], the already highly accurate results obtained by the simple EBM are further improved. Two examples are considered in order to demonstrate the application of the method to actual oscillation problems, and to assess the capability of the methods in arriving at accurate solutions to the problem. Felix and Balthazar [51] introduced a device of nonlinear electromechanical vibration absorber from the model described by Yamapi and Woafo [52]. They did a modification in the non-linear component of the voltage of the condenser, taking a non-linear friction of type cubic–quintic Duffing oscillator. Ganji et al. [53] exerted the EBM in a cubic–quintic Duffing oscillator. Finally, in order to show the capability, effectiveness, convenience, and high accuracy of the EBM, the semianalytical results and the exact solution proposed by Ganji et al. [53] and Lai et al. [54], respectively, are presented in Appendix 2. 2 BASIC IDEA OF THE ENERGY BALANCE METHOD He [46] gave a preliminary report on the EBM for non-linear oscillators in which he elucidated the basic idea of the method through several illustrative examples. The EBM may be described shortly as follows. First, a variational principle for the non-linear oscillation under consideration is obtained; then a Hamiltonian (H ) of the relation is constructed. It is held that throughout the oscillation, the sum of the kinetic energy (E) and potential energy (T ) remains constant, or H = E + T = constant. A trial function is used to determine the angular frequency, and the method of collocation along with other methods such as the method of least squares and Galerkin method is employed, as required, to obtain the actual angular frequency. The method is straightforward and robust, and is especially advantageous for strongly non-linear problems where analytical solutions are harder to establish. The aforementioned EBM is improved herein by employing the above-mentioned harmonic balance method [50] and by implementing it into the steps described above. The governing equation of the problem under consideration in the present study is introduced below, followed by implementation of the simple EBM as well as the proposed improved EBM for obtaining the angular frequency. 3 APPLICATION OF THE ENERGY BALANCE METHOD TO NON-LINEAR BEAM VIBRATION 3.1 Application to inextensible beams – case I The unimodal temporal equation of the largeamplitude free vibrations of a restrained uniform Proc. IMechE Vol. 225 Part K: J. Multi-body Dynamics 68 M G Sfahani, A Barari, M Omidvar, S S Ganji, and G Domairry beam carrying an intermediate lumped mass system is represented by the following equation as described by Hamdan and Shabaneh [12] rewritten as     1 1 1 1 2 4 2 1 2 4 2 1 + ε1 u + ε2 u + u λ + ε3 u + ε 4 u u̇ 2 2 2 2 4 6 ü + λu + ε1 u2 ü + ε1 uu̇2 + ε2 u4 ü + 2ε2 u3 u̇2 + ε 3 u 3 + ε4 u 5 = 0 (1) λA2 ε3 A 4 ε4 A 6 + + 2(1 + B)2 4(1 + B)4 6(1 + B)6 as u̇(0) = 0 (2) in which λ is an integer which may take values of λ = 0, 1 or, −1; dots represent derivates; and ε1 , ε2 , ε3 , ε4 are positive parameters that were defined by Hamdan and Shabaneh [12] and need not be small in value. The first step in obtaining the simple energy balance approximation of the periodic solution to equation (1) is to establish the variational principle as t  1 1 1 1 J (u) = − u̇2 + λu2 + ε1 u2 u̇2 + ε2 u4 u̇2 2 2 2 2 0  1 1 + ε3 u4 + ε4 u6 dt 4 6 u̇(t) = −Aω 1 2 1 2 1 1 u̇ + λu + ε1 u2 u̇2 + ε2 u4 u̇2 2 2 2 2 1 1 + ε3 u 4 + ε4 u 6 4 6 1 2 1 1 = λu0 + ε3 u04 + ε4 u06 2 4 6 ω2 6 Fk cos(2k + 1)θ m=0 Dm cos(2m + 1)θ + 8 (1 + B cos 2θ) (1 + B cos 2θ)6 ε3 A 4 ε4 A 6 λA2 + + (10) = 2 4 2(1 + B) 4(1 + B) 6(1 + B)6 8 k=0 where Fk and Dm are functions stated in terms of A, B and the set of known parameters ε1 , ε2 , ε3 , ε4 and λ. Multiplying both sides of equation (10) by (1 + B cos 2θ)8 , the following relation is obtained 8 (4) 8 ω2 k=0 Following He [46], the initial trial function is chosen as u(t) = u0 cos ωt and, after substituting into equation (4) and collocating at ωt = π/4, the simple energy balance approximation is obtained as u(t) = u0 cos 0.5  t A cos(ωt) 1 + B cos(2ωt) where A, B, functions of equation (2). function, the m=0 Dm cos(2m + 1)θ = 0 Further manipulation of equation (11) by introducing new parameters yields (ω2 F0 + E0 ) cos θ + (ω2 F1 + E1 ) cos 3θ + HOH = 0 (12) (6) and ω are to be determined as the initial conditions expressed by By implementing the updated trial Hamiltonian of equation (3) may be Proc. IMechE Vol. 225 Part K: J. Multi-body Dynamics Fk cos(2k + 1)θ − (11) (5) Next, on the basis of the generalized harmonic balance method, a new trial function for the already obtained approximate solution is constructed following Mickens [50] in the form u(t) = (8) in which θ = ωt and (C1 , C2 , C3 ) are given functions of B only. Thus, the Hamiltonian is rewritten by collocation equations (9) and (6) in equation (7) to give (3) H= 9ε3 u02 + 12λ + 7ε4 u04 6u02 ε1 + 3u04 ε2 + 12 sin ωt(1 + B cos 2ωt + 4B cos2 ωt) (1 + B cos 2ωt)2 Using Fourier series expansion equation (8) may be rewritten as   C1 cos θ + C2 cos 3θ + C3 cos 5θ u̇(t) = −Aω (9) (1 + B cos 2θ)2 Its Hamiltonian may therefore be written in the form  (7) Next, the first derivative of equation (6) is obtained with the following initial conditions u(0) = u0 , = By applying energy balancing for the cos θ and cos 3θ terms, the following two equations are obtained ω2 F0 + E0 = 0, ω 2 F1 + E 1 = 0 (13) or  F0 F1   2   ω 0 E0 × = 1 E1 0 (14) From the initial conditions and the assumed solution given by equation (6), A can be calculated as a Dynamic response of inextensible beams by improved energy balance method function of u0 and B as achieved terms as stated above, the following approximate solution is obtained for equation (1) as A = u0 (1 + B) (15) F 0 E1 − F 1 E0 = 0 3 3 (16) 7ε4 u04 + 9ε3 u02 + 12λ ε2 u04 + 2ε1 u02 + 4 (17) For u0 = 1, the angular frequency, ω, was determined by numerically integrating equation (1). A comparison of the numerically derived value and that obtained by means of equation (17) for the set of parameters taken from reference [12] as ε1 = 0.326 845, ε2 = 0.129 579, ε3 = 0.232 598, ε4 = 0.087 584, and λ = 1 gives ωnum = 1.0101 and ωsimple EBM = 1.0123, with a respective error of 0.22 per cent. Thus, the simple energy balance estimate for ω is sufficiently accurate. However, it is difficult to estimate A and B in practice, and an alternate procedure may be used to estimate these parameters. A particular value for u0 , that is, u0 = 1 as well as the assumption of t = t̄ is adopted. The EBM as described above is once again provoked and the total energy is balanced at ωt = π/4, and the value of t̄ = T /8 is therefore adopted. Thus cos(ωt) = cos √ π 2 = , 4 2 π =0 2 (18) (22) For the assumed value of u0 = 1 the modified ω becomes ω = 1.0096, which contains a respective error of 0.05 per cent. Therefore, it can be deduced that the modification imposed on the method reduces the error and thus improves the obtained solution for the non-linear oscillation problem under consideration. 3.2 Application to inextensible beams – case II The second example considered for the purpose of further elucidation is taken from reference [55] and involves the planar, flexural large-amplitude free vibration of a slender inextensible cantilever beam carrying a lumped mass with rotary inertia at an intermediate position along its span. The governing equation of the problem was found by Hamdan and Dado [55] as  (1 + αu2 )ü + αuu̇2 + u + βu3 = 0 (23) with initial conditions same as that of equation (1). In order to solve this problem using the energy balance concept, the variational form of the equation is written in the form (19) Substitution of equation (15) into equation (19) and solving for B gives B= √ 2u(t̄) − 1 (20) Using Runge–Kutta fourth-order numerical method, u(t̄) can be determined as u(t̄) = 0.7280. Hence, from equations (19) and (20) the following value for B may be obtained as B = 0.029. Putting together the t  0  1 1 1 (−1 + αu2 )u̇2 + u2 + βu4 dt 2 2 4  (24) By using the trial function u(t) = u0 cos ωt, the Hamiltonian is obtained as As a result, equation (5) takes the form √ A 2 u(t̄) = 2 14.1936ε4 u04 + 17.9928ε3 u02 + 23.5288λ 6.5838ε2 u04 + 12.4358ε1 u02 + 23.4894 ω(u0 ) = J (u) = cos(2ωt) = cos (21) where which introduces a polynomial expression for parameter B. In principle, the smallest magnitude root of equation (16) provides the required value of B. The estimate for the angular frequency may now be determined from equations (13) and (14). For the simple EBM, ω(u0 ) was obtained as ω(u0 ) = (1.029)u0 cos ωt 1 + (0.029) cos 2ωt u(t) = Also, from equation (14) it can be written √ 69  1 1 1 1 1 (1 + αu2 )u̇2 + u2 + βu4 = u02 + βu04 2 2 4 2 4 (25) Collocating at ωt = π/4 and solving for ω the following expression is obtained √  2 (2 + αu02 )(4 + 3βu02 ) ω(u0 ) = (26) 2(2 + αu02 ) or u(t) = u0 cos  4 + 3βu02 4 + 2αu02  t (27) As in the first example, the initial condition u0 = 1 is considered for α = 0.1 and β = 1.0, for which Hamdan Proc. IMechE Vol. 225 Part K: J. Multi-body Dynamics 70 M G Sfahani, A Barari, M Omidvar, S S Ganji, and G Domairry and Dado [55] have provided physical interpretations. The resulting angular frequencies from the solution of Hamdan and Dado [55], ωnum , and that of the simple EBM, ωsimple EBM are obtained as 1.2875 and 1.2910, respectively. The result of the simple EBM thus yields an error of 0.27 per cent. Equation (23) is once again considered and the corresponding energy balance solution of the equation is improved using the initial trial function given in equation (6). The Hamiltonian will therefore be written as u(t) = (0.9758)u0 cos ωt 1 − (0.024) cos 2ωt A2 βA4 = + (1 + B)2 2(1 + B)4 (28) Substituting equations (6) and (9) in equation (28) and using Fourier expansion yields 6 4 Ei cos(2i + 1)θ Pn cos(2n + 1)θ ω + n=0 6 (1 + B cos 2θ ) (1 + B cos 2θ )4 4 2 βA A + (29) = 2 (1 + B) 2(1 + B)4 (31) which gives ω as ω(u0 ) =   1 (1 + αu2 )u̇2 + 1 + βu2 u2 2 2 Following the same procedure employed for obtaining equations (12) to (20), and taking into consideration that here, u(t̄) = 0.69, B may be obtained as B = −0.024. Finally, substituting into equation (6), the following approximate solution is obtained 26.1949 + 25u02 − 5.6668ε2 u02 6.541 + 3.1142ε1 u02 1 2 (32) For the considered case, u0 = 1, the angular frequency is obtained as ω = 1.2888, producing a respective error of about 0.1 per cent, which is an improvement in the solution obtained by the simple EBM. 4 RESULTS i=0 in which Ei and Pn are functions defined in terms of A, B and positive parameters α and β. Multiplying both sides of equation (29) by (1 + B cos 2θ )6 results 6 ω2 i=0 6 Ei cos(2i + 1)θ − Table 1 n=0 Pn cos(2n + 1)θ = 0 (30) The results obtained by the simple EBM as well as the modified method are tabulated in Tables 1 and 2 along with the results from the exact solution, revealing the improvements achieved by modifying the simple EBM as described in previous sections of this article. Also, Figs 2 and 3 show the respective error for the time history diagrams. As can be seen, the plots of the improved solution show a smoother variation of the error function as compared to the simple EBM. The discontinuities found in the plots refer to zero points in ωexact . Comparison between exact solution and solutions obtained with equations (18) and (25) Approximate solutions A ωSimple EBM ωModified EBM Exact solution ωex 0.1 0.3 0.5 0.7 0.9 1.0 1.0000 1.0005 1.0019 1.0045 1.0091 1.0123 1.0008 1.0011 1.0019 1.0037 1.0071 1.0096 1.0005 1.0021 1.0018 1.0039 1.0075 1.0101 Table 2 |ω − ωex | × 100/ωex ω = ωSimple EBM ω = ωModified EBM 0.05 0.16 0.01 0.06 0.16 0.22 0.03 0.10 0.01 0.02 0.04 0.05 Comparison between exact solution and solutions obtained with equations (29) and (37) Approximate solutions A ωSimple EBM ωModified EBM Exact solution ωex 0.2 0.4 0.6 0.8 1.0 1.0139 1.0540 1.1169 1.1975 1.2910 1.0143 1.0540 1.1161 1.1960 1.2888 1.0150 1.0540 1.1162 1.1956 1.2875 Proc. IMechE Vol. 225 Part K: J. Multi-body Dynamics |ω − ωex | × 100/ωex ω = ωSimple EBM ω = ωModified EBM 0.11 4.14 × 10−3 0.06 0.16 0.27 0.07 3.8 × 10−3 8.96 × 10−4 0.03 0.10 Error Percentage Dynamic response of inextensible beams by improved energy balance method 10 9 8 7 6 5 4 3 2 1 0 REFERENCES 0 1 2 3 Time (second) Simple EBM Error Percentage 10 9 8 7 6 5 4 3 2 1 0 4 5 6 Modified EBM Respective error of time history diagram of u(t) for the mode; ε1 = 0.326 845, ε2 = 0.129 579, ε3 = 0.232 598, ε4 = 0.087 584, and λ = 1 and u0 = 1.0 Fig. 2 0 1 2 3 4 5 Time (second) Simple EBM Fig. 3 5 71 Modified EBM Respective error of time history diagram of u(t) for the mode; α = 0.1, β = 1.0, and u0 = 1.0 CONCLUDING REMARKS In this article, the simple EBM was employed to solve the governing equations of a system of non-linear autonomous conservative oscillators, describing the large-amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass along its span. The solution of the EBM was found to be highly accurate, with respective errors of 0.22 per cent and 0.27 per cent for the two examples considered. The simple EBM was further improved by employing a numerical solution for a particular value of the initial conditions. The semi-analytical approximations constructed via the above-described modification were found to improve the solutions for all the amplitudes and vibration times within the domain of the problem, as compared to the simple EBM. The successful implementation of the EBM for the large-amplitude non-linear oscillation problem considered in this article further confirms the capability of the EBM in solving non-linear oscillation problems. © Authors 2011 1 Quintana, V. and Grossi, R. Eigenfrequencies of generally restrained Timoshenko beams. Proc. IMechE, Part K: J. Multi-body Dynamics, 2010, 224(1), 117–125. DOI: 10.1243/14644193JMBD189. 2 Ziaei-Rad, S. and Imregun, M. 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Notation x0 kinetic energy energy balance method Hamiltonian time variable potential energy vibration of a restrained uniform beam generalized dimensionless displacement variable (function of t) initial condition ε1 ε2 ε3 ε4 λ ω ωe positive integer positive integer positive integer positive integer 0, 1 or, −1 angular frequency exact frequency x(0) = X0 , x ′ (0) = 0 (34) ωe (X0 ) = k1 = 2 πk1 π/2 (1 + k2 sin t + k3 sin4 t)−1/2 dt α+ γ X04 βX02 + 2 4 0 2 (35) (36) k2 = 3βX02 + 2γ X04 6α + 3βX02 + 2γ X04 (37) k3 = 2γ X04 6α + 3βX02 + 2γ X04 (38) Assuming the α = β = γ = 1 and X0 = 3, the EBM gives [53] x = 3 cos(7.417 768 993t) APPENDIX 2 A cubic–quintic Duffing oscillator has the general form of x ′′ + f (x) = 0 With initial conditions of where f (x) = αx + βx 3 + γ x 5 . The exact frequency ωe by imposing the initial conditions is [54] APPENDIX 1 E EBM H t T u x 73 (33) (39) The comparisons between results obtained by EBM [53] and exact solution [54] are presented in literature [53] and it is found that EBM is very powerful to solve highly non-linear oscillators. Proc. IMechE Vol. 225 Part K: J. Multi-body Dynamics