Abstract
The cubic-quintic Duffing oscillator of a system with strong quadratic damping and forcing is considered. We give 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. The approximant allows us to estimate the points at which the solution crosses the horizontal axis.
1. Introduction
In this paper, some novel analytical and numerical techniques are introduced for analyzing and solving nonlinear ordinary differential equations (NODEs) that are associated to some strongly nonlinear oscillators such as a quadratically damped cubic-quintic Duffing equation. There are many numerical and analytical approaches that were applied for solving the second-order nonlinear oscillator equations. For instance, both the homotopy perturbation method (HBM) and MTS technique were applied for analyzing a forced Van der Pol (VdP) generalized oscillator to obtain the amplitudes of the forced harmonic and super and subharmonic oscillatory states[1]. Also, Melnikov’s method was employed for analyzing a mVdPD equation and deriving analytical criteria for the appearance of horseshoe chaos in chemical oscillations [2]. He et al. [3] used the Poincare–Lindstedt technique (PLT) for solving and analyzing the hybrid Rayleigh–Van der Pol–Duffing equation. Moreover, the homotopy analysis method (HAM) was employed for analyzing the DVdP oscillator [4]. Both methods of differentiable dynamics and Lie symmetry reduction method were devoted for analyzing the DVdP-type oscillator [5]. The 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 and forced cubic-quintic Duffing oscillator:
The quadratically damped oscillator (1) is never critically damped or overdamped. In the absence of damping and forcing, we obtain the cubic-quintic Duffing equation
Equation (2) admits exact analytical solution that is expressed in terms of the Jacobian elliptic functions. Other solution methods may be found in [1–8].
2. Solution Procedure
2.1. First Case: Undamped and Unforced Cubic-Quintic Duffing Equation
Let us consider the i.v.p.
Assume the ansatzwhere the function is the solution to some Duffing equation
The numbers and are determined from the initial conditions. Observe that
From (5), it follows that
Observe that
We havewhere
Eliminating , , , , and from the system gives
The values for , , and are
The value of is found from the condition :
On the other hand, the solution to i.v.p. (5) is given bywhere
The numberis called the discriminant to Duffing equation (5). In the case when , this solution may also be written in the formwhere
Suppose that . Define
Since , necessarily . From the equalityit is evident that .Then, the solution to i.v.p. (5) readswhere
Observe that
Example 1. Let us consider the i.v.p.The exact solution is given bywhereSee Figure 1.
2.2. Solution by Means of He’s Frequency Method
Let
He’s method assumes the solution in the ansatz form
The frequency is evaluated by means of the formula
Then,
The number is found from the initial condition so that
2.3. Solution by Means of a Simple Trigonometric Ansatz
As in He’s approach, we assume the solution in ansatz form (28) so that
We choose the frequency so that
Then,
This last formula looks like He’s formula (30). The difference is . This suggests to consider the following -parameter solution:where the number is a solution to the sextic
The number is chosen in order to get as small residual error as possible.
2.4. Solution by Means of an Improved Trigonometric Ansatz
2.4.1. First Improved Ansatz
Let us consider the i.v.p.
Assume the ansatz
Let
The numbers , and are found from the conditions
The frequency is found from the quadratic equation
2.4.2. Second Improved Ansatz
Let us consider the i.v.p.
Assume the ansatz
The numbers , and are found from the conditions
The frequency is found from the cubic equation
2.5. Homotopy Method
Consider the homotopyand assume the solution in then ansatz formwhere .
Plugging the expression for into and equating the coefficients of will give an ode system. Solving this system so that no secular terms appear will give the following expressions: y.
2.6. Second Case: Quadratically Damped and Unforced Oscillator
Our aim is to give approximate analytical solution to i.v.p. (1). Define the residual function as follows:
2.6.1. First Approach
Assume the ansatz
Let . We have
We will choose the function so that
Then,where
The value of is found from the initial condition , and it is a solution to the sextic
The number is a free parameter that is chosen in order to minimize the residual error. In particular, when , we obtain approximate trigonometric solution to the undamped cubic-quintic Duffing equation
Example 2. Let us consider the i.v.p.The approximate analytical solution for iswhere the function is given by (52) with , , , , and (see Figure 2).
The obtained results may be applied to solve the pendulum equation with quadratic dampingIndeed, we may use the approximationand then we replace i.v.p. (58) with the i.v.p.
Example 3. Let us consider the i.v.p.The optimal value for is . The value of is (see Figure3).
2.6.2. Second Approach
Let us consider the i.v.p.
Assume the ansatzwhere the function is the exact solution to the i.v.p.
The numbers and are free parameters that are chosen in order to minimize the residual error
Observe that when , and then we obtain the exact solution to the undamped and unforced cubic-quintic oscillator (3). So, we expect accurate approximate analytical solution for small . This approach is more accurate, but here the solution involves elliptic functions and the solution is not elementary.
2.6.3. Third Approach
Let us consider the i.v.p.
Assume the ansatz
Proceeding in a similar way as in the first approach, we may choose
Here we have two free parameters and that are chosen in order to get as less residual error as possible. The numbers and are determined from the initial conditions as follows:
The number is a solution to the octic
2.7. Third Case: Quadratically Damped and Forced Oscillator
Let us consider the i.v.p.
Assume the ansatzwhere the function is a solution to some i.v.p.
The suitable constants , , , , and are to be determined. Define the residual function
The expression contains many terms. Equating some coefficients of , , , and (), we obtain the following algebraic system:
From equations (77)–(79), we obtain
Decupling equations (75) and (76) by means of their eliminants gives
We choose the least in magnitude real root to quintic (81) and the least in magnitude real root to quintic (82). Assuming that the forces and are small in magnitude, we have the following approximations for these roots:
More precise approximations are
3. Further Applications
Suppose we are given a quadratically damped oscillatorwhere is an odd continuous function. Suppose that . Then, we may approximate the function by means of the following quintic polynomial:where
Then, the i.v.p. (85) is reduced to the i.v.p. (1).
Example 4. Consider the motion of a satellite along a path that is equidistant from two identical massive stars with mutually interacting gravitational fields. If the distance between the two stars is 2d and the coordinate of the satellite motion is , then the equation of motion of the satellite is given aswhere is the mass of a star and the restoring force isThe nonlinear restoring force is an irrational force because of the bottom square root. The restoring force spikes near the origin. The spikes indicate the point when the satellite is most influenced by the mutual gravitational field of the stars. Away from the origin, the restoring force decreases gradually and approaches the horizontal axis asymptotically. This means that the satellite is far away from the stars and experiences a much smaller gravitational force.
4. Conclusions
We have obtained approximate analytical solutions to the quadratically damped Duffing oscillator equation by means of an elementary approach. We introduced a parameter technique that allowed us to optimize the obtained solution. The results are also valid for the linear quadratically damped oscillator . Also, a more general quadratically damped oscillator may be solved for any odd parity function . We also show the way to solve quadratically damped forced oscillators having the form for any continuous functions and with .
The quadratically damped cubic-quintic oscillator having both forcing term and quadratic damping term has been analyzed analytically using some highly accurate approaches. The proposed analytical techniques may be applied to solve other strongly nonlinear oscillators.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This study was supported by Universidad Nacional de Colombia.