Construction Of Analytical Approximate Solutions To Strongly Nonlinear Oscillation Systems

The vibration phenomena in natural sciences and engineering techniques can be described by differential equations in theory of vibration. Theory of vibration can be classified into the theory of linear vibration and the nonlinear one. For linear oscillation systems the principle of superposition holds, and the corresponding mathematical techniques are well-developed. In contrast, the general techniques of solutions for nonlinear oscillation systems are less well known, since the principle of superposition does not apply for them. Therefore, in order to deal with nonlinear oscillation systems, new techniques should be developed. Analytical (approximate) solutions can supply explicit expressions of the solution and allow the direct discussion of the influence of parameters and initial conditions on the solution. The analytical approximate methods are important methods. The perturbation method is one of the most commonly used analytical techniques for solving nonlinear oscillations with a small parameter. However, the use of perturbation theory in many important practical problems is invalid, or it simply breaks down for parameters beyond a certain specified range. In many cases, one can apply the harmonic balance method and the Krylov method to determine analytical approximate periods and periodic solutions to the conservative single-degree-of-freedom nonlinear oscillation systems with odd nonlinearity, even these analytical approximations are valid for larger amplitude. However, applying the method of harmonic balance and the Krylov method to construct higher-order analytical approximate solutions are very difficult, since they require analytical solution of algebraic equation(s) with very complex nonlinearity.In this dissertation, some methods for constructing analytical approximate periodic solutions to strongly nonlinear oscillation systems are presented.Consider the following nonlinear oscillation system governed by where a dot represents derivative with respect to t ; the restoring force function - f ( u)satisfies uf (u )>0 for u≠0. Let V (u ) =∫f(u )du be the potential energy of the system and suppose it reach its minimum at u = 0. For the case of f (u )is odd nonlinear function of u , the system will oscillate between symmetric limits [ - A, A]. For the case of f (u )being general nonlinear function of u , the system will oscillate between asymmetric limits [-B,A] where both -B(B>0) and A have the same energy level, i.e., V (- B) =V( A). (2)By introducing a new independent variableÏ„=ωt, and equation (1) can be rewritten asΩu″+ f(u) = 0, u ( 0)= A, u′(0 )= 0 (3) where a prime represents derivative with respect toÏ„, andΩ=ω². The new independent variable is chosen in such a way that the solution to Eq. (3), is a periodic function ofÏ„of period 2Ï€. Here, both the frequencyω=Ω^1/2 , period T = 2Ï€/ωand periodic solution u (Ï„) depend on A . 1. Case of odd nonlinear oscillation systems [ f ( ?u ) = ? f (u )]1) Improved L-P perturbation methodFor Equation (3), by addingφu to the left side, balancing this term by addingεφu to the right side, and replacing f(u) on the left side by -εf(u) on the right side, we obtain Hereε=1 is the bookkeeping quantity used to formally order the sequence of linear equations that replace the original nonlinear differential equation, andφis the unknown linear spring constant of the neighboring linear system to be determined.Following the usual L-P perturbation method, u andω2can be expanded as where U_k(Ï„) andω_k² are the k thlevel approximations to the solutions, and u_i(Ï„) andφ_i are the i thlevel terms in the corresponding expansions. The u_i must be periodic inÏ„with period 2Ï€, and the initial conditions for the ui are Substituting Equation (5) into Equation (4), expanding the resulting expression into the power series ofε, and equating coefficients of the like power ofεlead to a set of linear equations where subscript u denotes derivative of f with respect to u . Here, the equations for higher orders terms ui ( i≥4)are omitted. All equations above can be solved sequentially.The basic rule of determiningφis to make the neighboring linear system being the nearest to the original nonlinear system. Two new methods are presented as followsI SHB MethodφH = a1/A (8) whereII Combined Method whereω_a is an analytical approximate frequency to Equation (1) computed by others analytical approximate methods.1) Newton-harmonic balance MethodLet the period solution u (Ï„) and squareΩof frequency of Equation (3) be expressed as whereΩ_k, u_k(Ï„) are the main part of the solution;Δu_k,ΔΩ_k are the correction part of the solution, andΔu_k are periodic functions ofÏ„of period 2Ï€. Substituting Equation (10) into Equation (3) and linearizing with respect to the correction termsΔu_k andΔΩ_k lead to where subscript u denotes derivative of f with respect to u , and the ?u kandΔΩ_k will be solved by the HB method.When the restoring force -f (u )is an odd nonlinear function of u , we take u₁(Ï„) andΔu_k in Equation (10) as follows u₁ (Ï„)= AcosÏ„, (12)2. Case of general nonlinear oscillation systems [ f (-u )≠-f (u)]1) Improved L-P perturbation method based on the adding terms techniqueFor Equation (3), by addingφ_u+ (?) to the left side, balancing this term by addingε(φu+ (?)) to the right side, and replacing f(u) on the left side by -εf(u) on the right side, we obtain Hereε=1 is the bookkeeping quantity used to formally order the sequence of linear equations that replace the original nonlinear differential equation, andφ, (?) are the unknown linear constants of the neighboring linear system to be determined.For determining the relation betweenφand ? , the left and right amplitude of oscillation of the corresponding linear system are set as ? B and A , respectively. Then, we have Substituting Equations (5) and (15) into Equation (14), expanding the resulting expression into the power series ofε, and equating coefficients of the like power ofεlead to a set of linear equations where subscript u denotes derivative of f with respect to u . Here, the equations for higher orders terms ui ( i≥3)are omitted. All equations above can be solved sequentially.The rule of determining the linear term coefficientφis similar to that of the odd nonlinear oscillation system, and the corresponding methods can be applied with only minor modification.2) Newton-harmonic balance method based on selecting reasonable initial approximationWhen the restoring force -f(u)is general nonlinear function of u , we take u₁(Ï„) andΔu_k in Equation (10) as follows 3) Improved L-P perturbation method and Newton-harmonic balance method based on the splitting techniqueFor Equation (1), the two odd nonlinear oscillating systems which oscillate between symmetric limits [- H ,H] are introduced where withλ=±1. Here we set H = A forλ= 1, and H = B forλ= -1 , respectively.Applying the improved L-P perturbation method and Newton-harmonic balance method to the odd nonlinear oscillation systems (19), respectively, we may achieve the corresponding analytical approximate periods and the periodic solutions T_n⁺¹ A , u_n⁺¹(t) and T_n^-1(B), u_n^-1(t)(n = 1,2,3). Utilizing these analytical approximate solutions, we can construct the corresponding the n th (n = 1,2,3) analytical approximate period and periodic solution as follows and Applying the methods mentioned above, we have studied a series of strongly nonlinear oscillatory problems, including Duffing-Harmornic oscillator, the motion equation of a particle attached to the center of a stretched elastic wire, mixed-parity nonlinear Duffing equation, motion equation of a current-carrying conductor, nonlinear oscillation equation with the piecewise restoring force, free vibration equation of the system with linear and nonlinear springs in series, nonlinear jerk equation, free vibration equation of a two-degree-of-freedom system, free vibration equation of laminated thin plates and the double-well Duffing equation etc. Their high accurate analytical approximate periodic solutions have been constructed.These methods presented in this dissertation are very simple in principle and very easy to be applied. They do not require non-zero linear part of restoring force and small parameters. The most interesting features of these methods are its simplicity and its excellent accuracy. These analytical approximate periods and corresponding periodic solutions are valid for small as well as large amplitudes of oscillation, including the case of amplitude of oscillation tending to infinity.