Analytical Approximant to a Quadratically Damped Duffing Oscillator

The Duffing oscillator of a system with strong quadratic damping is considered. We give an elementary approximate analytical solution to this oscillator in terms of exponential and trigonometric functions. We compare the analytical approximant with the Runge–Kutta numerical solution. We also solve the oscillator by menas of He's homotopy method and the famous Krylov–Bogoliubov–Mitropolsky method. The approximant allows estimating the points at which the solution crosses the horizontal axis.


Introduction
In the standard textbooks, usually the systems with linear damping are considered. Due to their simplicity and the existence of an exact analytical solution, the problem is discussed in details. Unfortunately, in reality, the systems and damping are usually not linear. In recent times, a number of articles have appeared in the literature which deal with the phenomenon of a linear oscillator subject to a quadratic damping force [1][2][3][4][5][6][7]. Most elementary textbooks deal with viscous damping for the obvious reason that it involves a linear dependance on the velocity of the oscillator and presents the simplest situation where an exact analytical treatment is possible. In general, this involves the analysis of a second-order ordinary differential equation (ODE) of the Liénard type, namely, Nonlinear equations of motion such as this are seldom addressed in intermediate instruction in classical dynamics; this one is problematic because it cannot be solved in terms of elementary functions. e principal feature associated with quadratic damping is a discontinuous jump of the damping force in the equation of motion whenever the velocity vanishes such that the frictional force always opposes the motion. In this paper, we will consider the following quadratically damped Duffing oscillator (f(x) � ε| _ x| and g(x) � αx + βx 3 ): e quadratically damped oscillator (2) is never critically damped or overdamped, and that to first order in the damping constant, the oscillation frequency is identical to the natural frequency. In the abscence of damping, we obtain the Duffing equation Equation (3) admits the exact analytical solution [8].

Solution Procedure
In what follows, we will assume that if Our aim is to give an approximate analytical solution to the i.v.p. (2). e residual function R � R(t) is defined as follows: 2.1. First Approach. e ansatz is assumed as We have Since for small t, e − 2ρt ≈ 1, we will choose the value of ω so that en, e value of c 0 is found from the initial condition x ′ (0) � _ x 0 , and it reads In the case when β ⟶ 0, we define e number ρ is a free parameter that is chosen in order to minimize the residual error.

e Krylov-Bogoliubov-Mitropolsky Method (KBM).
e Krylov-Bogoliubov-Mitropolsky Method (KBM) is a technique to give an approximate analytical solution to the weakly nonlinear second-order equation When ε � 0, the solution of (24) may be expressed as e Scientific World Journal where a and θ are constants. For the case when ε > 0 is small, Krylov and Bogoliubov (1947) assumed that the solution is still given by (25) but with time-varying a and θ and subject to the condition In the general case, the solution is assumed in the ansatz form where each u n is a periodic function of ψ with a period 2π and a and ψ are assumed to vary with time according to In order to uniquely determine A n and ψ n , we require that no u n contains cos ψ. Let N � 3. en, Here, Let us consider the i.v.p.

Fourth Approach.
We assume the ansatz We have We will choose the function f � f(t) so that 6 e Scientific World Journal en, e value of c 0 is found from the initial condition x ′ (0) � _ x 0 , and it reads In the case when β ⟶ 0, we define

Analysis and Discussion
In this section, we will compare the accuracy of the solution methods using the different approaches described in the previous section.
e approximate analytical solution using the first approach (see formula (8)) for ρ � 0.007191 is It is shown in Figure 1. e solution obtained by means of He's homotopy method (see Figure 2) equals Using the KBM, we obtained the following solution ( Figure 3): (46) Now, using the fourth approach, we get the solution (see Figure 4) Example 2. Let us consider the i.v.p.
e approximate analytical solution for ρ � 0.0084 using the fourth approach is It is shown in Figure 5. e obtained results may be applied to solve the pendulum equation with quadratic damping Indeed, we may use the approximation and then, we replace i.v.p. (43) with the i.v.p. e Scientific World Journal 9

Conclusions
We have obtained approximate analytical solutions to the quadratically damped Duffing oscillator equation by means of an elementary approach. We introduced a ρ−parameter technique that allowed us to optimize the obtained solution. e results are also valid for the linear quadratically damped oscillator € x + ε _ x| _ x| + αx � 0. A similar approach may be employed to study the quadratically damped cubic-quintic oscillator € x + ε _ x| _ x| + αx + βx 3 + cx 5 � 0. Also, a more general quadratically damped oscillator € x + ε _ x| _ x| + h(x) � 0 may be solved for any odd parity function h(x). In future work, we will study quadratically damped forced oscillators having the form € x + ε _ x| _ x| + h(x) � F(t) for any continuous functions h(x) and F(t).

Data Availability
No data were used to support this study.

Conflicts of Interest
e author declares no conflicts of interest.