10743 FACTA UNIVERSITATIS Series: Mechanical Engineering Vol. 20, No 2, 2022, pp. 445 - 455 https://doi.org/10.22190/FUME220420023M © 2022 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper SIMPLIFIED HAMILTONIAN-BASED FREQUENCY-AMPLITUDE FORMULATION FOR NONLINEAR VIBRATION SYSTEMS Hongjin Ma School of Science, Xi'an University of Architecture and Technology, Xi’an, China Abstract. The Hamiltonian-based frequency formulation has been hailed as an unprecedented success for it gives a straightforward insight into a complex nonlinear vibration system with simple calculation. This paper gives a systematical analysis of the formulation, and two simplified formulations are suggested. The cubic-quintic Duffing oscillator is used as an example to show extremely simple calculation and remarkable accuracy. It can be used as a paradigm for many other applications, and the one-step solving process has cleaned up the road of the nonlinear vibration theory. Key Words: He’s Frequency Formulation, Duffing Equation, Hamilton Principle, One-step Solving 1. INTRODUCTION Nonlinear vibration systems arise everywhere in mechanical engineering, as in thin- walled structures [1], rotor vibration [2,3], gyroscope systems [4,5], micro-electro- mechanical systems [6,7], packaging systems [8], building vibration [9], heavy machinery [10], to name but a few. The most famous nonlinear oscillators include the pendulum systems [11, 12], van der Pol oscillator and Duffing oscillator [13]. The spring-pendulum systems have many applications in satellites, submarines, aircraft, and energy harvesting device [14]. The recent research frontier in the nonlinear vibration theory is the fractal vibration; hence the description of a vibrating system in a fractal space [15-18]. Generally a nonlinear vibration system is complex and it is difficult to solve it exactly. In the last 200 years, engineers had little mathematical tools. The most available one was the perturbation method. However, a small parameter assumption involved in the perturbation method hinders its widely applications for accurate prediction of the periodic properties of a practical problem; hence it is valid only for weakly nonlinear oscillators. Received April 20, 2022 / Accepted May 17, 2022 Corresponding author: Hongjin Ma School of Science, Xi'an University of Architecture and Technology, Xi’an, China E-mail: hjma@xauat.edu.cn 446 H. MA The homotopy perturbation method [19-21] and the variational iteration method [22-24] are much advanced ones for strongly nonlinear oscillators, but both methods require some special mathematics skills for problem-solving; especially the former asks for a suitable homotopy equation, and the latter requires a complex identification of the Lagrange multiplier involved in the iteration algorithm. For engineering problems, as claimed by the Chinese mathematician Ji-Huan He, the simpler is better for a complex problem [18]. Now the condition is completely changed; the Hamiltonian-based frequency formulation [25] sheds a promising light on a fast but reliable insight into the physical properties of a nonlinear oscillator with simple calculation. The Hamiltonian-based frequency formulation is mathematically beautiful and physically penetrating since only a simple integration calculation is needed. It gives a vivid picture of the whole solution properties. The 2021 Hamiltonian-based frequency formulation [25] is a modification of He’s 2006 frequency formulation [26]. The formulation observes a nonlinear vibration system in an energy view by establishing a Hamilton principle, which can elucidate clearly the energy conservation of the vibrating process, that is the total energy of kinetic and potential energy keeps unchanged during oscillation. As the formulation is derived from an energy form of the nonlinear oscillator, its approximate solution is valid for the whole solution domain without limitations such as those in the traditional perturbation method. The frequency formulation begins with two arbitrary guesses of the frequency, and two residual integrals have to be calculated to estimate an accurate frequency. It is amazing that the accurate result does not depend upon the initial choice of two arbitrary frequencies even for strong nonlinearity. Due to these unprecedented advantages, the formulation caught immediate attention after publication due to its extremely simplicity and remarkable reliability. It has been successfully applied to Kundu-Mukherjee-Naskar oscillator [27], nonlinear vibration of embedded carbon nanotube [28], fractal vibration systems [29,30], packaging systems [31] and MEMS oscillators [32]. It has been hailed as an unprecedented success. Though the achievement has contributed to the fast development of the nonlinear vibration theory, the solving process is relative complex, so a modification is needed. 2. PROBLEM STATEMENT Consider the following conservative oscillator [25]: ( ) 0, (0) 0 (0)w p w w w B + = = = (1) where p(w) is an odd function of the displacement w and B is the displacement amplitude. Its Hamilton principle can be easily established by the semi-inverse method [33] dtwPwwJ        −= )( 2 1 )( 2 (2) where P(w) is the potential energy defined so that ( ) ( ) d P w p dw w= (3) Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems 447 According to Hamilton principle, we have the conversation law 2 1 ( ) 2 w P Hw + = (4) where H is the Hamiltonian constant, which depends only upon the initial conditions, so the following first-order differential equation is obtained: 0)()( 2 1 2 =−+ BPwPw (5) The Hamiltonian-based frequency formulation is to find an approximate solution in the form tBtw ωcos)( = (6) where ω is the frequency to be solved later. The Hamiltonian-based frequency formulation begins with two arbitrary tries ω1 and ω2, and the following residual equations are obtained： 2 2 2 1 1 1 1 1 ( ) sin ( cos ) ( ) 2 R t B t P B t P B= + −ω ω ω (7) 2 2 2 2 2 2 2 1 ( ) sin ( cos ) ( ) 2 R t B t P B t P B= + −ω ω ω (8) The residual integrals are defined as tdtR T R T 1 4/ 0 1 1 1 cos 4~ 1 = (9) tdtR T R T 2 4/ 0 2 2 2 cos 4~ 2 = (10) where T1=2π/ω1, T2=2π/ω2. The frequency formulation is [25] 2 2 2 1 2 2 1 1 2 R R R R = − − ω ω ω (11) We can see that the frequency formulation requires only a simple calculation of two integrals, making it much attractive and promising in the nonlinear vibration theory. 3. MODIFIED HE’S FREQUENCY FORMULATIONS The present author shows an interest in two arbitrary choices of ω1 and ω2, e.g., ω1=1 and ω2=2, or ω1=100 and ω2=0.1, all cases leading to the same result. This two-trial assumption goes back to an ancient Chinese algorithm [26], which was developed to He’s frequency formulation to solve nonlinear oscillators by Ji-Huan He [26], and Chun-Hui He’s iteration algorithm for numerical simulation [34,35]. 448 H. MA It is amazing that the Hamiltonian-based frequency formulation does not depend upon the choice of ω1 and ω2. Here a strict proof is given. From Eqs. (7) and (8), we have 2 2 21 1 1 1 2 2 2 2 21 1 1 1 ( ) sin ( cos ) ( ) 2 2 ( ) cos ( ) 2 2 cos B R t t P B t P B B P B B t P B t = + − − = − + ω ω ω ω ω ω ω (12) 2 2 22 2 2 2 2 2 2 2 22 2 2 2 ( ) sin ( cos ) ( ) 2 2 ( ) cos ( ) 2 2 cos B R t t P B t P B B P B B t P B t = + − − = − + ω ω ω ω ω ω ω (13) The residual integrals become 1 2 2 / 4 1 21 1 1 1 0 0 1 6 ( )4 2 cos ( 1 ) 3 T B P B R R tdt P B x dx T   − = = + −  ω ω (14) 2 2 2 / 4 1 22 2 2 0 0 2 6 ( )4 2 cos ( 1 ) 3 T B P B R R tdt P B x dx T   − = = + − 2 ω ω (15) So a modified frequency formulation is obtained, which is 2 2 1 2 2 2 2 1 2 2 1 1 2 2 2 2 2 1 1 2 2 2 21 2 2 1 0 0 2 2 2 2 1 2 1 2 2 2 1 2 0 2 2 2 1 2 2 + 6 ( ) 6 ( )2 2 ( 1 ) ( 1 ) 3 3 6 ( ) 6 ( ) 3 3 6 ( ) 2 ( 1 ) ( ) 3 3 6 ( ) 6 ( 1 ) = = ( ) ( ) R R R R B P B B P B P B x dx P B x dx B P B B P B P B P B x dx B P B P B x d          − − = − − − + − + − − − − − − − − − − =               ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω 1 0 2 x B  (16) This modification seems much simpler than the original one because no calculation of integrals is needed, so it is easily applicable. Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems 449 Alternatively, we locate at [26] 1 2 cos cost t k= =ω ω (17) Eqs. (7) and (8) become 2 2 2 1 1 1 (1 ) ( ) ( ) 2 R B k P kB P B= − + −ω (18) 2 2 2 2 2 1 (1 ) ( ) ( ) 2 R B k P kB P B= − + −ω (19) Another modification of the Hamiltonian-based frequency formulation is 2 2 2 2 1 1 2 1 2 2 2 2 2 2 2 2 2 2 1 1 2 2 2 2 2 2 2 1 2 2 2 2 2 2 1 2 1 2 2 2 2 1 2 1 1 ( (1 ) ( ) ( )) ( (1 ) ( ) ( )) 2 2 1 1 ( (1 ) ( ) ( )) ( (1 ) ( ) ( )) 2 2 ( ) ( ) ( ) ( ) 1 (1 ) ( ) 2 ( ) ( ) 1 (1 2 R R R R B k P kB P B B k P kB P B B k P kB P B B k P kB P B P kB P B k B P B P kB − = − − + − − − + − = − + − − − + − − − − = − − − = − ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω 2 2 )k B (20) We recommend k=1/√2 , the simplified Hamiltonian-based frequency formulation is 2 2 4( ( ) ( / 2))P B P B B − =ω (21) It is astonishing that all intermediate steps involved in the original Hamiltonian-based frequency formulation have disappeared, making the two modified formulations unprecedentedly simple. 4. EXAMPLES Consider the following well-known Duffing equation [25] 3 0, (0) 0 (0)w w w w w B + + = = =ε (22) For this example, we have 2 4 1 1 ( ) 2 4 P w w w= + ε (23) In order to compare the two modifications with the original one, we illustrate first the solving process of the original Hamiltonian-based frequency formulation, which begins with two arbitrary choices ω1 and ω2, and the following residual equations are obtained: 450 H. MA 2 2 2 1 1 1 1 2 2 2 2 2 4 2 41 1 1 1 1 ( ) sin ( cos ) ( ) 2 2 ( ) cos cos 2 2 4 R t B t P B t P B B P B B B B t t = + − − − = + + ω ω ω ω ω ε ω ω (24) 2 2 2 2 2 2 2 2 2 2 2 2 4 2 42 2 2 2 1 ( ) sin ( cos ) ( ) 2 2 ( ) cos cos 2 2 4 R t B t P B t P B B P B B B B t t = + − − − = + + ω ω ω ω ω ε ω ω (25) The residual integrals are: 1 2 4 2 / 4 2 1 1 1 1 0 1 4 4 10 30 ( ) cos 3 15 T B B B P B R R tdt T   + − = = + ε ω ω (26) 2 4 4 2 / 4 2 2 2 2 2 0 2 4 4 10 30 ( ) cos 3 15 T B B B P B R R tdt T   + − = = + ε ω ω (27) The Hamiltonian-based frequency formulation leads to the result 2 2 2 21 2 2 1 1 2 7 1 10 R R B R R = = − + − ω ω ω ε (28) Now the modified formulation of Eq. (16) gives a one-step solving process: 1 2 2 0 2 1 2 4 2 2 4 4 2 0 2 2 4 2 4 2 2 4 2 2 6 ( ) 6 ( 1 ) 1 1 1 1 6 6 (1 ) ( 2 1) 2 4 2 4 3 4 3 2 2 5 7 10 7 1 10 P B P B x dx B B B B x B x x dx B B B B B B B B B B − − = + − − + − + = + − + = + = = +                           ω ε ε ε ε ε ε (29) which is the same as Eq. (28). Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems 451 The modified formulation of Eq. (21) is also simple, which leads to the following result 2 2 2 4 2 4 2 2 4 2 2 4( ( ) ( / 2 )) 1 1 1 1 4 2 4 4 16 3 4 3 1 4 P B P B B B B B B B B B B B − =   + − −    = + = = + ω ε ε ε ε (30) This result is the same as those obtained by the variational iteration method or the homotopy perturbation method [26]. Consider another example of the cubic-quintic Duffing oscillator: 3 5 0, (0) 0 (0)w w aw bw w w B + + + = = = (31) The original Hamiltonian-based frequency formulation is to calculate the residuals: 2 2 2 1 1 1 1 2 2 2 2 2 4 6 2 4 61 1 1 1 1 1 ( ) sin ( cos ) ( ) 2 2 ( ) cos cos cos 2 2 4 6 R t B t P B t P B B P B B B aB bB t t t = + − − − = + + + ω ω ω ω ω ω ω ω (32) 2 2 2 2 2 2 2 2 2 2 2 2 4 6 2 4 62 2 2 2 2 1 ( ) sin ( cos ) ( ) 2 2 ( ) cos cos cos 2 2 4 6 R t B t P B t P B B P B B B aB bB t t t = + − − − = + + + ω ω ω ω ω ω ω ω (33) and the residual integrals: 1 2 6 4 2 / 4 2 1 1 1 1 0 1 4 16 28 70 210 ( ) cos 3 105 T B bB aB B P B R R tdt T   + + − = = + ω ω (34) 2 2 6 4 2 / 4 2 2 2 2 2 0 2 4 16 28 70 210 ( ) cos 3 105 T B bB aB B P B R R tdt T   + + − = = + ω ω (35) The original Hamiltonian-based frequency formulation gives the following result 2 2 2 41 2 2 1 1 2 2 7 19 1 10 35 R R aB bB R R = = − + + − ω ω ω (36) 452 H. MA Eq. (16) gives immediately the following results 2 4 61 1 1 ( ) + 2 4 6 P w w aw bw= + (37) and 1 2 2 0 2 1 2 4 6 2 2 4 2 2 6 2 3 0 2 2 4 6 2 4 6 2 2 4 6 2 2 4 6 ( ) 6 ( 1 ) 1 1 1 1 1 1 6 6 (1 ) (1 ) (1 ) 2 4 6 2 4 6 3 4 16 3 2 2 5 35 7 19 10 35 7 19 1 10 35 P B P B x dx B B aB bB B x B x bB x dx B B aB bB B aB bB B B aB bB B aB bB − − = + + − − + − + − = + + − + + = + + = = + +                           ω (38) This is the same with Eq. (36). The one-step solving process by Eq. (21) leads to the following result 2 2 2 4 6 2 4 6 2 2 4 6 2 2 4 4( ( ) ( / 2)) 1 1 1 1 1 1 4 2 4 6 4 16 48 3 7 4 12 3 7 1 4 12 P B P B B B aB bB B aB bB B B aB bB B aB bB − =   + + − − −    = + + = = + + ω (39) Fig. 1 shows the comparison of approximate solution w(t)=Bcosωt, where the frequency is determined by Eq. (38) or Eq. (39), with the exact one when a=b=1 for various values of amplitudes. 4. CONCLUSIONS This paper simplifies Hamiltonian-based frequency formulation for nonlinear oscillators. All the intermediate steps involved in the original formulation become totally unnecessary, making the formulation unprecedentedly simple. Both the simplified frequency formulations will play an important role in the nonlinear vibration theory for a fast and reliable insight into the periodic properties of a complex nonlinear oscillator, and it will open up a flood of opportunities for various applications. The one-step solving process given in this paper confirms again Ji-Huan He’s famous saying: “The simpler is the better for the nonlinear vibration theory”. Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems 453 Fig. 1 Comparison of the exact solutions with the approximate solutions based on Eqs. (38) and (39) for amplitude B ranging from 0.01 to 10000 454 H. MA REFERENCES 1. Rama G., Marinkovic D., Zehn M., 2018, High performance 3-node shell element for linear and geometrically nonlinear analysis of composite laminates, Composites Part B: Engineering, 151, pp. 118-126. 2. Wang, C.C., Yau, H.T., Jang, M.J., Yeh, Y.L., 2007, Theoretical analysis of the non-linear behavior of a flexible rotor supported by herringbone grooved gas journal bearings, Tribology International, 40(3), pp. 533–541. 3. Wang, C.C., Yau, H.T., 2008, Application of a hybrid numerical method to the bifurcation analysis of a rigid rotor supported by a spherical gas journal bearing system, Nonlinear Dynamics, 51(4), pp. 515–528. 4. Yau, H.T., 2008, Generalized projective chaos synchronization of gyroscope systems subjected to dead-zone nonlinear inputs, Physics Letters A, 372(14), pp. 2380-2385. 5. Yau, H.T., 2007, Nonlinear rule-based controller for chaos synchronization of two gyros with linear-plus-cubic damping, Chaos, Solitons and Fractals, 34(4), pp. 1357-1365. 6. Skrzypacz, P., Ellis, G., He, J.H., He, C.H., 2022, Dynamic pull-in and oscillations of current-carrying filaments in magnetic micro-electro-mechanical system, Communications in Nonlinear Science and Numerical Simulation, 109, 106350. 7. Tian, D., He, C.H., 2021, A fractal micro-electromechanical system and its pull-in stability, Journal of Low Frequency Noise, Vibration & Active Control, 40(3), pp. 1380-1386. 8. Ji, Q.P., Wang, J., Lu, L.X., Ge, C.F., 2021, Li-He's modified homotopy perturbation method coupled with the energy method for the dropping shock response of a tangent nonlinear packaging system, Journal of Low Frequency Noise, Vibration & Active Control, 40(2), pp. 675-682. 9. Shen, Y., El-Dib, Y.O., 2021, A periodic solution of the fractional Sine-Gordon equation arising in architectural engineering, Journal of Low Frequency Noise, Vibration & Active Control, 40(2), pp. 683-691. 10. Mitrev, R., Janošević, D., Marinković, D., 2017, Dynamical modelling of hydraulic excavator considered as a multibody system, Tehnicki Vjesnik, 24, pp. 327-338. 11. He, J.H., Amer, T.S., Elnaggar, S., Galal, A.A., 2021, Periodic Property and Instability of a Rotating Pendulum System, Axioms, 10(3), 191. 12. He, J.H., Amer, T.S., El-Kafly, H.F., Galal, A.A., 2022, Modelling of the rotational motion of 6-DOF rigid body according to the Bobylev-Steklov conditions, Results in Physics, 35(2022), 105391. 13. He, C.H., Tian, D., Moatimid, G.M., Salman, H.F., Zekry, M.H., 2022, Hybrid Rayleigh -Van der Pol-Duffing Oscillator: Stability Analysis and Controller, Journal of Low Frequency Noise, Vibration & Active Control, 41(1), pp. 244-268. 14. He, C.H., Amer, T.S., Tian, D., Abolila, A.F., Galal, A.A., 2022, Controlling the kinematics of a spring-pendulum system using an energy harvesting device, Journal of Low Frequency Noise, Vibration & Active Control, doi: 10.1177/14613484221077474. 15. Wang, K.L., Wei, C.F., 2021, A powerful and simple frequency formula to nonlinear fractal oscillators, Journal of Low Frequency Noise, Vibration & Active Control, 40(3), pp. 1373-1379. 16. Feng, G.Q., 2021, He's frequency formula to fractal undamped Duffing equation, Journal of Low Frequency Noise, Vibration & Active Control, 40(4), pp. 1671-1676. 17. Tian, Y., 2022, Frequency formula for a class of fractal vibration system, Reports in Mechanical Engineering, 3(1), pp. 55-61. 18. He, J.H., 2019, The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators, Journal of Low Frequency Noise, Vibration & Active Control, 38(3–4), pp. 1252–1260. 19. He, J.H., El-Dib, Y.O., 2021, The enhanced homotopy perturbation method for axial vibration of strings, Facta Universitatis-Series Mechanical Engineering, 19(4), pp. 735-750. 20. He, J.H., El‐Dib, Y.O., Mady, A.A., 2021, Homotopy Perturbation Method for the Fractal Toda Oscillator, Fractal and Fractional, 5(2021), 93. 21. He, C.H., El-Dib, Y.O., 2022, A heuristic review on the homotopy perturbation method for non-conservative oscillators, Journal of Low Frequency Noise, Vibration & Active Control, doi: 10.1177/14613484211059264. 22. Nadeem, M., He, J.H., 2021, He-Laplace variational iteration method for solving the nonlinear equations arising in chemical kinetics and population dynamics, Journal of Mathematical Chemistry, 59(5), pp. 1234-1245. 23. Anjum, N., He, J.H., 2020, Analysis of nonlinear vibration of nano/microelectromechanical system switch induced by electromagnetic force under zero initial conditions, Alexandria Engineering Journal, 59(6), pp. 4343-4352. 24. Skrzypacz, P., He, J.H., Ellis, G., Kuanyshbay, M., 2020, A simple approximation of periodic solutions to microelectromechanical system model of oscillating parallel plate capacitor, Mathematical Methods in the Applied Science, DOI: 10.1002/mma.6898. Simplified Hamiltonian-Based Frequency-Amplitude Formulation for Nonlinear Vibration Systems 455 25. He, J.H., Hou W.F., Qie, N., Gepreel, K.A., Shirazi, A.H., Sedighi, H.M., 2021, Hamiltonian-based frequency-amplitude formulation for nonlinear oscillators, Facta Universitatis-Series Mechanical Engineering, 19(2), pp. 199-208. 26. He, J.H., 2006, Some asymptotic methods for strongly nonlinear equations, International Journal of Modern Physics b, 20(10), pp. 1141–1199. 27. Wang, K.J., Zhu, H.W., 2021, Periodic wave solution of the Kundu-Mukherjee-Naskar equation in birefringent fibers via the Hamiltonian-based algorithm, EPL, doi: 10.1209/0295-5075/ac3d6b. 28. Wang, K.J., Wang, G.D., 2022, Study on the nonlinear vibration of embedded carbon nanotube via the Hamiltonian-based method, Journal of Low Frequency Noise, Vibration & Active Control, 41(1), pp. 112-117. 29. He, C.H., Liu, C., 2022, A modified frequency-amplitude formulation for fractal vibration systems, Fractals, doi: 10.1142/S0218348X22500463. 30. He, J.H., El-Dib, Y.O., 2021, A tutorial introduction to the two-scale fractal calculus and its application to the fractal Zhiber-Shabat Oscillator, Fractals, 29(08), 2150268. 31. He, J.H., Yang, Q., He, C.H., Khan, Y., 2021, A Simple Frequency Formulation for the Tangent Oscillator, Axioms, 10, 320. 32. He, J.H., Qie, N., He, C.H., Gepreel, K., 2022, Fast identification of the pull-in voltage of a nano/micro-electromechanical system, Journal of Low Frequency Noise, Vibration & Active Control, 2022, doi: 10.1177/14613484211068252. 33. He, J.H., Hou, W.F., He, C.H., Saeed, T., Hayat, T., 2021, Variational approach to fractal solitary waves, Fractals, 29(07), 2150199. 34. He, C.H., 2016, An introduction to an ancient Chinese algorithm and its modification, International Journal of Numerical Methods for Heat & Fluid Flow, 26(8), pp. 2486-2491. 35. Khan, W.A., 2022, Numerical simulation of Chun-Hui He's iteration method with applications in engineering, International Journal of Numerical Methods for Heat & Fluid Flow, 32(3), pp. 944-955. https://doi.org/10.1209/0295-5075/ac3d6b https://doi.org/10.1142/S0218348X22500463