Accepted Manuscript Cubic Quintic Septic Duffing Oscillator: An analytical study S.K. Remmi, M.M. Latha PII: DOI: Reference: S0577-9073(18)30818-9 https://doi.org/10.1016/j.cjph.2018.08.009 CJPH 606 To appear in: Chinese Journal of Physics Received date: Revised date: Accepted date: 13 June 2018 6 August 2018 12 August 2018 Please cite this article as: S.K. Remmi, M.M. Latha, Cubic Quintic Septic Duffing Oscillator: An analytical study, Chinese Journal of Physics (2018), doi: https://doi.org/10.1016/j.cjph.2018.08.009 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. ACCEPTED MANUSCRIPT Highlights • A three-spring structure is considered and its behaviour is analysed. CR IP T • The dynamics is found to be governed by a Cubic Quintic Septic Duffing equation. • Linearised Harmonic Balance Method is used to solve the resulting equation. • Low Natural frequency is observed around the equilibrium point due to low stiffness. AC CE PT ED M AN US • The result is compared with Cubic Quintic Duffing equation. 1 ACCEPTED MANUSCRIPT S. K. Remmia , M. M. Lathab a CR IP T Cubic Quintic Septic Duffing Oscillator: An analytical study AN US Register No: 12585, Department of Physics, Women’s Christian College, Nagercoil 629 001, India (Affiliated to Manonmaniam Sundaranar University, Abishekapatti, Tirunelveli- 627 012, Tamilnadu, India) b Associate Professor, Department of Physics, Women’s Christian College, Nagercoil 629 001, India Abstract PT ED M This paper deals with a particular arrangement of a statically balanced system using 3 springs of prescribed material stiffness and critical geometrical parameter. The dynamics is described by a nonlinear differential equation upto septic power following odd nonlinearity for small disturbance from static equilibrium position. The governing differential equation is solved analytically by the combination of the linearization of the equation with the method of Harmonic Balancing to observe the low natural frequency at fixed point and a finite displacement range in the neighbourhood of the equilibrium point where the dynamic stiffness is low. By this approximation method, the behaviour of the displacement with increase in time as well as the phase-plot of Cubic Quintic Septic Duffing equation for a set of parameter values is studied at the equilibrium position and its neighbourhood. CE 1. Introduction AC The way to interpret an object before performing real experiments is through modelling. It is a mathematical representation which allows simulation and analysing about the system. For physical understanding a simple analytic model is sufficient to describe significant properties of a natural object. In this deem, mechanical systems that has been designing using ∗ Tel:+91 4652 - 231461; Fax:+91 4652 - 228834 Email address: lathaisaac@yahoo.com (M. M. Latha) Preprint submitted to Chinese Journal of Physics August 14, 2018 ACCEPTED MANUSCRIPT AC CE PT ED M AN US CR IP T spring-mass would be the most common in engineering applications. We are recurrently familiar with single and two spring kind of models that have good understanding in their behaviours. But analysing the dynamics of mass with three springs typically arranged nonlinearly is a fascinating model in practice. Recently, this configuration is widely used in vibration isolators under specified condition. The three-spring elastic model has the property to deform with zero stiffness where any change in configuration requires no external work [1]. The configuration is attained by appropriate combination of geometry, stiffness and prestress. The elastic structure typically designed with suitable system parameters can establish a concept of neutrally stable configuration called Quasi zero stiffness (QZS) which is ultimately either stable or unstable for small variations around the equilibrium state. This fascinating phenomenon maintains high static stiffness and low dynamic stiffness at equilibrium and thus yields low natural frequency especially in passive vibration isolators. This QZS behaviour near the operating region can be effectively obtained by means of both geometrical and physical nonlinearity of the springs. This consequently admits the system to exhibit nonlinear behaviour. Carrella [2] studied the geometrical nonlinear three spring structure designed using negative stiffness element which turned out to be the prominent feature of QZS mechanism and in addition there is an optimum geometry and stiffness relationship to achieve stable QZS property in the structure. As well a small change in stiffness around the equilibrium point is also studied by introducing physical nonlinearity to the system by means of the initial geometry, pre-stress and the stiffness of the springs [3]. The dynamics of QZS system under small or large displacement from the static equilibrium position may be linear or nonlinear and its strength of nonlinearity depends on the nonlinear spring force acting on the mass. The quest for exact solution of such nonlinear systems supports to evolve number of analytical techniques. In general, it is impossible to obtain an accurate solution for a strong nonlinear differential equation [4]. However, a well-known Perturbation method [5,6] has been used in the past few decades for solving the nonlinear systems which hold only small parameters in its equation of motion. But at present state of knowledge an approximate analytical approach is applicable to solve the nonlinear differential equation contains large as well as small parameters. Subsequently many other methods emerge including Lindstedt-Poincare method [7,8], Krylov Bogoliubov averaging Method (KBM) [9], Power-series method [10] which allow for the creation of an approximation solution valid for large amplitudes of oscillation. Though 3 ACCEPTED MANUSCRIPT AN US CR IP T these methods involve minimum algebraic manipulations, it is usually difficult to achieve higher-order analytic approximation to solutions. Recent study reports based on higher order approximation [11-13] present the solution for nonlinear differential equation with odd and even nonlinearity that immensely involves perplex calculations, computation, and consumption of time. One among these is the Newtons’s Harmonic Balance Method (HBM) [14-18]. An improvised version of standard Harmonic Balance Method is the Linearized Harmonic Balancing approach [19,20] used for finding solution for strong nonlinear differential equations very near to accuracy by more than one approximation procedure. This method is significant over other methods in simplicity, non-requirement of initial condition from the outset and numerical integration [4]. Also this approach holds good for small as well as large amplitudes of oscillation. ED M Duffing oscillators are extremely nonlinear that exhibit stunning behaviour even if it is in unforced condition. Hopefully, no researches have done a detailed dynamics about a system under a specific range of displacement which is described by a Cubic Quintic Septic Duffing equation. This paper presents the dynamics of three springs carrying equal stiffness constructed geometrically with negative and positive stiffness elements which are kept at a particular oblique angle of Quasi Zero Stiffness condition. AC CE PT The paper is systemised as follows: Details of the negative stiffness property of two springs and the configration of three-spring conservative system for optimised QZS condition are given in Section 2. In section 3 we discuss about the Force-Displacement phenomenon to analyse the system for the displacement upto the maximum exertion range from equilibrium position and arrive at second order nonlinear differential Duffing equation. Section 4 describes the Linearised Harmonic Balance Method (LHBM) to solve the Duffing equation analytically upto third order approximation to observe the natural frequency about the operating point. The exact solution by direct integration of governing equation is included in Section 5. Section 6 provides the frequency comparision results obtained from the proposed technique and the direct integration method and further investigation of the prominent features of the Frequency-Amplitude graph. In addition, the behaviour of the system with increase in time and the phase plot for a particular set of parameter values are examined . Section 7 compares the result of the governing equation with the Cubic-Quintic Duffing equation. Finally, Section 8 details 4 ACCEPTED MANUSCRIPT the brief results of the present work and ends with concluding comments and future scope. CR IP T 2. Construction of three spring model Two identical linear elastic springs each of length L0 having equal stiffness kh are joined at one end say B to hold the mass and other two ends are separately hinged at two different supports A and C as shown in the Figure 1 (a). The springs are initially at an angle say θ0 which is at the height h0 from the horizontal. ED M AN US When a force is applied by placing a mass at B, the system behaves as a negative stiffness mechanism. The free dynamics of this single degree of freedom body is purely nonlinear when the mass reaches the horizontal position. This is because of the change of force in the springs exerted on the mass along the vertical direction. This nonlinear force causes change in stiffness which results in the variation of vibration response in the system. For instance, a sudden increase in dynamic stiffness causes reduction in vibration resulting in high static stiffness which leads to undesirable deflection of mass. This limitation can be succeeded by including another linear spring of stiffness kv parallel to the vertical component of the inclined springs as shown in Figure 1 (b). The added vertical spring acts as a positive stiffness element. CE PT Thus the combined effect of negative and positive stiffness springs can control the change in stiffness of the system. Hence, we obtain a smooth dynamics over a certain range of displacement around the horizontal position. This phenomena of low dynamic stiffness around the equilibrium region is known as Quasi Zero Stiffness (QZS). The QZS property allows the system to respond with low natural frequency when it is disturbed from the equilibrium point. Also, this mechanism explicitly relies on the system parameters. AC However, there are infinite number of possible combinations of geometrical and stiffness system parameters to give QZS character in which the optimised system parameter falls under the obligue angle ranging from 480 to 570 which would be determined from the relation cos θ0 = Ld00 [2]. To understand the free vibration of the three spring system under QZS condition, we make an attempt to construct the system by a particular set of geometric 5 CR IP T ACCEPTED MANUSCRIPT AN US Figure 1: (a) Arrangement of the Three-Spring system, (b) Three-spring system leading to Cubic, Quintic and Septic oscillations parameters say d0 =0.3m, L0 =0.5m which corresponds to the inclined angle θ0 =530 . 3. Formulation of Duffing equation PT ED M Let the system in Figure 1 is compressed to an angle say θ by placing an arbitrary mass m at B so that the oblique springs (kh ) reaches the horizontal position in such a way that it is at a height h from the equilibrium point. The mass is supported rigidly upright by the vertical spring (kv ). Now disturb the mass to some distance x from the horizontal position so that the resulting force on the spring system is F = 2kh ∆L sin θ + kv x, (1) CE where ∆L = |L − Ld | is the change in length of the spring. L and Ld are the length of the p compressed and p displaced springs respectively. They take the form L = d20 + h2 , Ld = d20 + (h + x)2 and sin θ = h+x . Ld AC Hence Eq. (1) becomes F = 2kh (h + x)[ √ L d20 +(h+x)2 − 1] + kv x. (2) Binomial expansion of [d20 +(h+x)2 ]1/2 yields a polynomial equation with infinite number of terms raised to higher powers in which the terms upto seventh 6 ACCEPTED MANUSCRIPT power is considered here to describe the dynamics of the three-spring system. Now Eq. (2) is written as Lkh Lkh (h + x) − 2kh (h + x) − 3 (h + x)3 + d0 d0 3 Lkh 15 Lkh (h + x)5 − (h + x)7 + kv x. 5 7 4 d0 24 d0 2 CR IP T F = (3) F = AN US The system is stable for QZS condition at the equilibrium point, h = 0. Thus the above equation is reformed into kh 3 3 kh 5 15 kh 7 x + x − x + kv x. d20 4 d40 24 d60 (4) The equation of motion of the system having equal stiffness (kh = kv = k) for all the three springs is obtained from F − (−mẍ) = 0 as M d2 x + αx(t) − βx3 (t) + γx5 (t) − δx7 (t) = 0, dt2 15 k 24 md60 are ED 3 k k k where α = m is the linear coefficient and β = md 2 , γ = 4 md4 , δ = 0 0 the cubic, quintic, septic power coefficients respectively. (5) AC CE PT This equation is a second order ordinary differential equation of mass holding odd nonlinear forces and is termed as Cubic Quintic septic conservative Duffing equation. This equation normally gives the real dynamics of the model undergoing moderately with small as well as large amplitudes of motion. But, our interest is to find the system dynamics about the equilibrium position where the dynamic stiffness is found to be low that results in low natural frequency. For this, we try to calculate numerically the change in frequency of the system for different initial displacements by assigning a set of values for the geometric parameters k = 200N/m, d0 = 0.3m, m = 2.8kg, L = 0.5m. Generally, the low stiffness region lies in a certain displacement range about the equilibrium point. In this region, the dynamic stiffness is zero at the equilibrium point which corresponds to low natural frequency and near the neighbourhood of the equilibrium point for short length the stiffness of 7 ACCEPTED MANUSCRIPT the the the CR IP T the system is low. The expressions for the static equilibrium point and maximum excursion distance as a function of maximum stiffness from equilibrium point are calculated from the stiffness equation explored in Force-Displacement characteristic theory [2] and are given as √ xeq = L0 1 − d2 , q dp = L0 d [1 − K(1 − d)]−2/3 − 1, (6) (7) AN US where d = Ld00 . It is determined that the value of d for which dp is a maximum, changes depending on the value of the prescribed stiffness K. It is clear that the above two relations (6) and (7) only varies with the geometrical parameters. M For the chosen set of geometrical parameters the above expressions yield the values for the equilibrium point xeq and the maximum exertion distance dp as xeq = 0.4, dp = 0.12 for K = 0.5. By using these values we can define a range xeq − dp < x < xeq + dp , that is from x = 0.28 to 0.52, where the dynamic stiffness is found to be low and the overall performance of the system under this range leads to negative stiffness mechanism. ED It is possible to determine the abstract ideas that have mentioned above from the solution of the equation of motion of the three-spring sytem. To examine this we substitute the values of k, d0 , m in α, β, γ, δ to obtain AC CE PT d2 x + 71.43x − 793.65x3 + 6613.75x5 − 30619.24x7 = 0. (8) 2 dt This Duffing equation involves large system parameters with strong fifth and seventh power nonlinearities. Though Eq. (8) looks simple in apperance, it is rather difficult to solve exactly due to the presence of Cubic, Quintic and Septic nonlinear force terms. However, for large nonlinear parameters, the Higher order Approximation Method finds suitable to provide more accurate higher-order approximate frequencies and the corresponding periodic solutions for various initial amplitudes compared with the exact solution. 4. Proposed Analytical Methodology For the higher order approximation technique, linearisation of governing equation is performed earlier to the Harmonic Balance Method to estabilish approximate expressions for frequencies and periodic displacements. 8 ACCEPTED MANUSCRIPT CR IP T 4.1. Linerization of governing equation The presence of nonlinear restoring force function of x in Eq. (8) limits the occurance of a periodic solution in t. By the action of linearization, one can assume a periodic solution satisfying the initial condition x(0) = (0) = 0. A; dx dt AN US This is done by introducing a new independent variable τ = ωt such that the solution of Eq. (8) i) is a periodic function of τ ii) satisfies the above initial condition and iii) whose amplitude A depend on frequency. Thus Eq (5) is rewritten in terms of angular frequency ω as ω2 d2 x(τ ) + αx(τ ) − βx3 (τ ) + γx5 (τ ) − δx7 (τ ) = 0. 2 dt (9) M Expanding Eq. (9) with respect to small increments ∆x1 (τ ) and ∆ω12 at x(τ ) = x1 (τ ) and neglecting the powers of x1 greater than seven and linearizing with respect to x1 (τ ) yields 2 d2 x1 2 d x1 + ω ∆ + αx1 − βx31 + γx51 − δx71 + dt2 dt2 (α − 3βx21 + 5γx41 − 7δx61 )∆x1 = 0. ED ω2 (10) PT Let x1 (τ ) = A cos τ is a periodic function of period 2π which is considered as the initial approximation to x(τ ). Simplifing Eq. (10) gives d 2 x1 + αA cos τ + α∆x1 − βA3 cos3 τ dt2 −3βA2 cos2 τ ∆x1 + γA5 cos5 τ + 5γA4 cos4 ∆x1 − δA7 cos7 τ − 7δA6 cos6 τ ∆x1 = 0. CE −ω12 A cos τ + ω12 ∆ (11) AC This is a linear equation in ∆x1 and also a periodic function of τ of period 2π. 4.2. Harmonic Balancing Method After linearization, Newton’s Harmonic Balance procedure is used to solve for approximate angular frequency and periodic displacement. Before applying this procedure the above linear Eq. (11) should undergo the Fourier series 9 ACCEPTED MANUSCRIPT expansion of odd and even powers of cos τ . The Fourier coefficients for Eq. (11) are found to be CR IP T 3 5 35 a1 = αA − βA3 + γA5 − δA7 , 4 8 64 1 3 5 21 a3 = − βA + γA5 − γA7 , 4 16 64 1 7 1 a5 = γA5 − δA7 , a7 = − δA7 16 64 64 15 4 35 6 γA − δA , 4 8 3 2 5 4 105 6 b2 = − βA + γA − δA , 2 2 32 5 21 7 b4 = γA4 − δA6 , b6 = − δA6 , 8 16 32 b8 = 0. AN US b0 = 2α − 3βA2 + (12) (13) M 4.2.1. First order approximation For the first order approximation, we set ∆x1 (τ ) = 0, and ∆ω12 = 0. PT ED The initial approximation to the periodic function becomes x(τ ) = x1 (τ ) = Acosτ. Substituting Eqs. (12), (13) and the above approximation in Eq. (11) gives −Aω12 cos τ + a1 cos τ + a3 cos 3τ + a5 cos 5τ + a7 cos 7τ = 0. (14) AC CE Setting the coefficient of cos τ to zero consequences first order analytical approximate frequency ω1 and periodic solution x1 (t) as q 35 δA6 , ω1 (A) = α − 43 βA2 + 58 γA4 − 64 x1 (t) = A cos(ω1 t). (15) Here the angular frequency ω1 as well as the periodic solution x1 (t) depend on the initial amplitude A. 10 ACCEPTED MANUSCRIPT CR IP T 4.2.2. Second order approximation To construct second order analytical approximate solution, the correction term ∆x1 is extended to ∆x1 (τ ) = k1 (cos τ − cos 3τ ) and ω2 , x2 takes the form ω2 = ω1 + ∆ω1 , x2 (τ ) = x1 (τ ) + ∆x1 (τ ). Substituting the above correction term and Eqs. (12), (13) in Eq. (11) and applying trignometric series expansion, we obtain −(ω12 + ∆ω12 )A cos τ + ω12 ∆ d2 x1 + (a1 cos τ + a3 cos 3τ dt2 b0 + b2 cos 2τ + b4 cos 4τ + 2 b6 cos 6τ + b8 cos 8τ )∆x1 = 0. AN US +a5 cos 5τ + a7 cos 7τ ) + (16) Setting zero coefficients to cos τ and cos 3τ results in the production of quadratic equations which on solving simultaneously yields the expression for k1 and ∆ω12 as 2Aa3 , −18a1 + a(b0 − b2 − b4 + b6 ) a3 (−2a1 + (b0 − b4 )A) ∆ω12 = . −18Aa1 + A2 (b0 − b2 − b4 + b6 ) (17) ED M k1 = The second order approximated angular frequency ω2 is solved as CE PT ω2 = 3 5 35 [α − βA2 + γA4 − δA6 ]1/2 4 8 64 a3 [−2a1 + (b0 − b4 )A] +[ ]1/2 . 2 −18Aa1 + A (b0 − b2 − b4 + b6 ) (18) AC It has more accuracy than ω1 which is acquired from first order approximation technique. The corresponding approximate periodic solution x2 (t) is obtained as x2 (t) = a cos(ω1 t) + k1 [cos(ω2 t) − cos 3(ω2 t)]. (19) Here also, both the frequency and periodic solution of second order approximation depends on the amplitude A. 11 ACCEPTED MANUSCRIPT 4.2.3. Third order approximation For the third order analytical approximation, the terms x1 (τ ) and ∆x1 (τ ) are replaced by x2 (τ ) and ∆x2 (τ ) respectively. CR IP T The approximation is set as ∆x2 (τ ) = k2 (cos τ − cos 3τ ) + k3 (cos 3τ − cos 5τ ) and ω3 , x3 are assigned as ω3 = ω2 + ∆ω2 , x3 (τ ) = x2 (τ ) + ∆x2 (τ ). Substituting Eqs. (12), (13) in Eq. (11) and setting the coefficients of cos τ , cos 3τ and cos 5τ to zero in the trignometric series expansion gives the following expressions for k2 , k3 and ∆ω32 : (20) k3 = p8 (−p2 p9 + p1 p10 ) + p4 (p6 p9 − p5 p10 ) , p8 (−p3 p10 + p2 p11 ) + p4 (p7 p10 − p6 p11 ) (21) AN US p8 (p3 p9 − p1 p11 ) + p4 (−p7 p9 + p5 p11 ) , p8 (−p3 p10 + p2 p11 ) + p4 (p7 p10 − p6 p11 ) p3 (−p6 p9 + p5 p10 ) + p2 (−p7 p10 + p6 p11 ) + p2 (p7 p9 − p5 p11 ) , p8 (−p3 p10 + p2 p11 ) + p4 (p7 p10 − p6 p11 ) M ∆ω22 = k2 = ED where (22) CE PT p1 = −2Aω22 + 2c1 − 2k1 ω22 , p2 = d0 − d4 − 2ω22 , p3 = d2 − d6 , p4 = −(2A + 2k1 ), p5 = 9k1 ω22 + c3 , d0 d6 d0 d2 d6 + − , p7 = −9ω22 + + p6 = 9ω22 − 2 2 2 2 2 d2 d8 1 − − , p8 = 9k1 , p9 = c5 , p10 = (d4 + d6 − d2 2 2 2 1 −d8 ), p11 = (d2 + d8 − d10 − d0 ) + 25ω22 . 2 (23) AC Eqs. (20), (21), (22) are solved for Fourier coefficients c1 , c3 , c5 , c7 , d0 , d2 , d4 , d6 , d8 , d10 and the periodic solution x3 (t) and its corresponding angular frequency ω3 are found to be x3 (t) = (A + k1 + k2 ) cos ω32 t + (k3 − k1 − k2 ) cos 3(ω32 t) − k3 cos 5(ω32 t), ω32 = ω22 + ∆ω22 . 12 (24) ACCEPTED MANUSCRIPT 5. Direct Integration for Accurate Solution CR IP T Angular frequency ω3 derived from the third order approximation is more accurate than the angular frequencies ω1 , ω2 obtained from first and second approximation methods. This accuracy is noticed graphically in the Amplitude - Frequency plot shown in Figure 2. The governing Cubic Quintic Septic Duffing equation (5) is integrated directly under the initial condition x(0) = A and ẋ(0) = 0 to obtain the exact frequency ωex of the dynamical three-spring system. AN US Eq. (5) is integrated once and applying the above initial condition gives rise to 1 2 1 4 1 6 1 8 αA − βA + γA − δA = C. 2 4 6 8 (25) 0 A dx q . (26) α(A2 − x2 ) − β2 (A4 − x4 ) + γ3 (A6 − x6 ) − 4δ (A8 − x8 ) ED T (A) = Z M where C is the integration constant. One can find the solution for the period of oscillation of the governing equation by equating Eq. (25) and Eq. (5) after integration. The result is PT By substituting x = −A cos t in Eq. (26), we get T (A) = Z π 2 0 dt √ R (27) CE where R = α − βA2 − 21 δA6 − 12 δA6 cos4 t + 12 βA2 sin2 t + 14 δA6 sin2 t + 1 δA6 sin2 cos2 t. 4 AC The exact frequency which is a function of initial amplitude A is obtained from the relation ωex (A) = T2π and the exact displacement is expressed as (A) xex (t) = A cos t where Z dx q t= . (28) α(A2 − x2 ) − β2 (A4 − x4 ) + γ3 (A6 − x6 ) − 4δ (A8 − x8 ) 13 ACCEPTED MANUSCRIPT 6. Frequency Comparison CR IP T The three angular frequencies derived from the approximation method in this paper are denoted as ω1 , ω2 , ω3 respectively. The exact frequency is denoted by ωex . The frequencies calculated by the approximation method is compared with the exact frequency which is obtained by the integration of governing Duffing equation and is shown in Figure 2. CE PT ED M AN US Figure 2 depicts the frequency response of free vibration of the threespring system under QZS condition. The change in angular frequency determined using the LHBM method for various initial displacements shows that the curve for ω3 almost coalesce with the curve for ωex whereas curves ω1 and ω2 start to deviate from ωex at x = 0.36. This indicates that the third order approximate frequency ω3 provides excellent approximation with the exact frequency ωex as compared to ω1 and ω2 . The natural frequency of the system is observed from the figure and it is found as ωn = 8.44N m−1 Kg −1 . Another noticeable observation extracted from the figure is the value of the equilibrium point, which falls at xeq = 0.41. This value is exactly close to the expected value xeq = 0.4 determined from Eq (6). Also, the angular frequency is very low at the equilibrium point due to zero dynamic stiffness and increases when the system is away from the equilibrium point. The displacement upto which the angular frequency is found to be low from either sides of the equilibrium point is observed as x = 0.12 which corresponds to the displacement range 0.29 < x < 0.53 where the natural frequency and stiffness are minimum near the equilibrium point. This measurement virtually matches with the calculated range 0.28 < x < 0.52. This means that outside this typical range the dynamic stiffness increases leading to higher angular frequency. AC More generally, the natural frequency of the spring mass system depends only on the material properties called stiffness and mass. Normally, a stiffer spring or a lower mass increases the natural frequency and a softer spring or a higher mass lowers the natural frequency. For increasing spring stiffness keeping mass as constant, we find that the natural frequency (ωn ) of the system increases and is shown in Figure 3 (a). Likewise the decrease in natural frequency with increasing mass with constant stiffness is presented in Figure 3(b). 14 PT ED M AN US CR IP T ACCEPTED MANUSCRIPT AC CE Figure 2: Comparison of calculated frequencies with exact frequency of Cubic Quintic Septic Duffing Equation for α = 71.43, β = −793.65, γ = 6613.75, δ = −30619.24 15 AN US CR IP T ACCEPTED MANUSCRIPT Figure 3: (a) Increase in Natural frequency with increasing Stiffness, (b) Decrease in Natural frequency with increasing Mass CE PT ED M The periodic solution for the Cubic Quintic Septic Duffing equation for different amplitudes within the range x = 0.28 to x = 0.52 are plotted in Figure 4. These figures represent that the period of oscillation is not constant all over the range and it depends on the initial displacement of the mass in the system about the equilibrium point. As the distortion amplitude become shorter which means the amplitude near the neighbourhood of the equilibrium point, between x = 0.4 and x = 0.43 the period of oscillation increases, therefore the angular frequency decreases. Beyond x = 0.43 there is a change in period of oscillation due to the high dynamic stiffness response in the structure. AC The corresponding phase diagrams are also given in Figure 5 which shows an elongated perfect ellipse at the low frequency region indicating smooth dynamics with respect to initial amplitudes near the equilibrium point. The perfectness of the ellipse gets distorted along the vertical axis for the displacement of mass far from the equilibrium point xeq = 0.41. This is because of the oscillation characteristic of Cubic, Quintic, Septic nonlinear force springs. 16 PT ED M AN US CR IP T ACCEPTED MANUSCRIPT AC CE Figure 4: Approximate Periodic Solution of Cubic Quintic Septic Duffing Equation with α = 71.43, β = −793.65, γ = 6613.75, δ = −30619.24 for the initial displacement range X = 0.1 to 0.51 17 AC CE PT ED M AN US CR IP T ACCEPTED MANUSCRIPT Figure 5: Phase diagrams of Cubic Quintic Septic Duffing Equation with α = 71.43, β = −793.65, γ = 6613.75, δ = −30619.24 for the initial displacement range X = 0.1 to 0.51 18 M AN US CR IP T ACCEPTED MANUSCRIPT ED Figure 6: (a) Frequency response of Cubic Quintic Duffing equation, (b) Phase diagram of Cubic Quintic Duffing equation, (c) Phase diagram of Cubic Quintic Septic Duffing equation PT 7. Comparison of result with Cubic Quintic Duffing oscillator AC CE The behaviour of the solution x(t) changes for the parameter values α, β, γ and δ. If the system is modeled for the Duffing equation representing the Cubic and Quintic restoring force terms then one cannot expect the same situation discussed above as for the Cubic Quintic Septic form under same α, β, γ values. Because variation in the strength of restoring forces in the system results into variety of interesting dynamics. When δ = 0, the system turns into Cubic Quintic Duffing oscillator. Eq. (5) is written as d2 x(t) dt2 + αx(t) − βx3 (t) + γx5 (t) = 0. The Frequency - Amplitude curve of the above equation for the same system parameters is shown in the Figure 6(a). The disparity in the dynamical behaviour corresponds to the number of equilibrium points in the system. 19 ACCEPTED MANUSCRIPT AN US CR IP T Mostly, an unforced Cubic Quintic Duffing equation with appropriate system parameters can produce one, three, and five equilibrium points. Similarly a Cubic Quintic Septic Duffing equation can generate upto seven equilibrium points. In the case of the three spring system describing Cubic Quintic Duffing equation we observe only one equilibrium point at the center (0, 0) where as the Cubic Quintic Septic Duffing equation yields three equilibrium points one at the center (0, 0) and other two are at (±0.4124, 0). These are reflected in the phase portraits shown in Figures 6 (b) and 6 (c) respectively. But, the system parameters have no effect on the natural frequency of the system because the material property known as stiffness and mass of the system determine the natural frequency. In both the Duffing equations we considered same stiffness and mass. Hence, both the systems have same natural frequency as ωn = 8.44N m−1 Kg −1 . 8. Conclusion AC CE PT ED M In the present work, a nonlinear model is constructed using three springs one of which act as positive stiffness and other two springs provide a negative stiffness with the geometrical parameters d0 = 0.3, L0 = 0.5, θ0 = 530 satisfying the QZS property at the equilibrium point. The Force-Displacement characteristics of this spring configration leads to a strong nonlinear conservative Cubic Quintic Septic Duffing equation consisting of large nonlinear restoring force terms. An analytical higher order approximation technique based on the combination of linearization of governing equation with the Harmonic Balance method (LHBM) is used to solve the equation. This approach enables us to estabilish approximate expressions for angular frequency and periodic solution for different initial amplitudes about the equilibrium point. The present method provides an excellent accuracy of the approximate frequency with the exact frequency. Using this technique we observe the natural frequency of the system to be ωn = 8.44N m−1 Kg −1 and it is minimum at the equilibrium point. The change in natural frequency for varying spring stiffness and mass is also discussed here. Furthermore, a certain displacement range is noticed as 0.29 < x < 0.53 where the dynamic stiffness is found to be low corresponds to low natural frequency near the equilibrium. The periodic solution and phase plots of the three-spring system for various initial amplitudes are analysed based on the stability of the system. In addition, the results are compared with the Cubic Quintic Duffing equation in which a disparity in dynamical behaviour is found due to the variation in the number 20 ACCEPTED MANUSCRIPT of equilibrium points in the system. CR IP T In practice, numerous applications make use of Duffing oscillators are performed by powering the system with suitable excitation force. The reason is that the Duffing oscillators are more sensitive to both the friction and driving force. So, our future work is in process that referring the system to an externally excited Duffing oscillator which adds another dimension to the system. References AN US [1] Mark Schenk, Simon D. Guest, J. Mechanical Eng Sci 228 (2013) 1701. [2] A. Carrella, M.J. Brennan and T.P Waters, J. Sound Vib 301 (2007) 678. [3] Ivana Kovacic, Michael J. Brennan and Timothy P. Waters, J. Sound Vib 315 (2008) 700. M [4] S.K Lai and C.W Lim, Int. J. Comput Methods Eng Sci Mech 7 (2006) 201. ED [5] A.H Nayfeh, Perturbation Methods (1973) Wiley, New York. [6] A.H Nayfeh, Introduction to Perturbation Techniques (1993) Wiley, New York. PT [7] R.E Mickens, Oscillations in Planar Dynamic Systems (1996) World Scientific, Singapore. CE [8] A.H Nayfeh and Mook, Nonlinear Oscillations (1979) Wiley, New York. [9] P. Hagedorn, Non-linear Oscillations (1988) Clarendon, Oxford. AC [10] M.I Qaisi, J. 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