On Large Amplitude Vibrations of the Softening Duffing Oscillator at Low Excitation Frequencies—Some Fundamental Considerations

Abstract

The Duffing equation containing a cubic nonlinearity is probably the most popular example of a nonlinear oscillator. For its harmonically excited, slightly damped, and softening version, stationary large amplitude solutions at subcritical excitation frequencies are obtained when standard semi-analytical methods like Harmonic Balance or Perturbation Analysis are applied. These solutions have the shape of a nose in the amplitude-frequency diagram. In prior work, it has been observed that these solutions may contain large errors and that high ansatz orders may be necessary when applying the Harmonic Balance or other semi-analytical methods to make them converge. Some of these solutions are observed to be asymptotically stable, while in most cases, they are unstable. The current paper aims to give a descriptive explanation for this behavior of the nose solutions, which is mainly related to the exact solution of the free undamped vibrations. Based on this, approximations of the nose solutions are calculated with a procedure combining properties of Perturbation Analysis and Harmonic Balance. Therein, the exact solution of the free undamped vibrations is taken as the zeroth approximation, while higher-order solution parts are calculated by balancing the harmonics, and the phase shift of the zeroth approximation is calculated by a residuum minimization. This method just requires the solution of a system of linear algebraic equations, while systems of nonlinear algebraic equations have to be solved in the case of directly applying Harmonic Balance.

1. Introduction

The Duffing equation, adding a cubic term to a linear oscillator, is probably the most considered classical nonlinear oscillator in mechanics [1]. In the same book [1], several technical applications are given, which can be described by the Duffing equation, e.g., multistable energy harvesting systems, vibration dampers (“bubble mount”), or electric circuits with saturating elements. Current publications on applications of the Duffing oscillator are, e.g., [2] for the control of solar cells and [3] for energy harvesting.

There are several classical semi-analytic solutions methods applied for the Duffing equation’s approximate solution like Harmonic Balance, Lindstedt–Poincaré Perturbation Analysis, or Multiple Time Scales. Corresponding results can be found in many textbooks on Nonlinear Dynamics like references [1,4,5]. A detailed investigation of the solution behavior of the differential equation, based on analytical and numerical methods, was conducted by Szemplińska-Stupnicka and also by Nayfeh. Their work addresses, among other issues, the question of the stability of the solutions, the bifurcation behavior, and the occurrence of chaos [6,7,8]. Even nowadays, there are still numerous publications on these topics, like references [9,10,11], or adding additional influences like multiple excitation frequencies [12] or noise [13].

Regarding the development of semi-analytical methods used for the calculation of the solutions, Urabe’s work should be particularly mentioned in the context of the scope of the present article. In reference [14] and subsequent publications of the same author, the theoretical foundations and application examples for the method of Harmonic Balance are presented.

The present paper aims to investigate a special type of stationary solutions occurring in the Duffing oscillator in case of slight damping and harmonic excitation. These solutions possess large amplitudes at subcritical excitation frequencies and are often hard to calculate with classical approximation techniques. The paper considers a descriptive explanation for the existence of these solutions and calculates them by applying a method combining properties of Perturbation Analysis and Harmonic Balance.

2. Fundamentals of the Softening Duffing Oscillator

The Duffing equation considered in this paper reads as follows:

$$x″\left(t\right)+2D{\omega }_0 x\prime \left(t\right)+{{\omega }_0}^2 x\left(t\right)+\gamma x^3\left(t\right)=f \cos \left(\Omega t\right).$$

(1)

Herein, x denotes the displacement, ${\omega }_0$ the circular eigenfrequency of the linear system, $D$ the damping ratio, γ the coefficient of the cubic nonlinearity, f the excitation amplitude, $\Omega$ the circular excitation frequency, and $t$ the time. All these quantities are considered dimensionless, and $()\prime$ denotes the derivative with respect to dimensionless time $t$. As we consider the softening case, γ is negative. In the following, the parameters ${\omega }_0$, $D$, γ, and f are considered as constant in the respective parameter set, while we vary the excitation circular frequency $\Omega$.

In the simplest case, the approximate solution of Equation (1) reads as follows:

$$x\left(t\right)=C·\cos \left(\Omega t-\varphi \right)$$

(2)

with amplitude $C$ and phase shift $\varphi$. Plotting $C\left(\Omega \right)$ results in the well-known behavior of the Duffing oscillator with bending of the resonance peak near $\Omega ={\omega }_0$ to the right in the stiffening $(\gamma >0)$ and to the left $(\gamma <0)$ in the softening case. If damping is respectively small and the excitation is large enough, a region of multiple solutions (small and large amplitude asymptotically stable, medium amplitude unstable) in the resonance peak can also be observed. This solution branch is called the regular solution in the following. A corresponding example is sketched in Figure 1 for the following parameter set:

$${\omega }_0=1; \gamma =-0.1; f=0.08; D=0.022$$

(3)

i.e., a softening case with $\gamma <0$, and the regular solution branch is sketched in orange while the results were obtained using Harmonic Balance.

In addition to the regular solution with bending of the resonance peak to the left, large amplitude solutions for lower excitation circular frequencies (in the case of parameter set (3) up to $\Omega =0.65$) with the shape of a nose can be observed in the softening case only and are also found sketched in many textbooks. This solution branch will be denoted as the nose solution in the following, and the larger amplitude part is marked in blue while the lower amplitude part is marked in green in Figure 1. Validity and stability of the different solutions are discussed afterwards.

In the following, we will consider some properties of these nose solutions. Therefore, we introduce higher-order solutions by Harmonic Balance with the following ansatz:

$$x\left(t\right)=a_0+{\displaystyle \sum }_{i=1}^n{a}_i\cos \left(i\Omega t\right)+b_i\sin \left(i\Omega t\right)$$

(4)

Ansatz (4) is introduced into Equation (1), and according to the ansatz order $n$, the first $n$ harmonics and the constant term are balanced while the higher-order harmonics $n+1, \dots , 3n$ (according to the cubic nonlinearity) are neglected, considering them to be small. E.g., in references [15,16], an error criterion was introduced based on the neglected terms:

$${\displaystyle \sum }_{i=n+1}^{3n}{\tilde{a}}_i\cos \left(i\Omega t\right)+{\tilde{b}}_i\sin \left(i\Omega t\right)$$

(5)

which can be calculated as a function of time if the coefficients $a_i$, $b_i$ in Equation (4) have been calculated. Based on this, in reference [15], a discussion on the errors of different types of solutions of the Duffing Equation (1) has been performed. In general, therefore the maximum in time of the absolute value of the neglected term Equation (5) is taken. This error criterion was also applied, e.g., in reference [16] and other publications of the first two authors and it may be normalized, e.g., by the maximum of the corresponding solution $x\left(t\right)$. Other error criteria are studied, e.g., in reference [17].

The general outcome of the analysis of the solutions for low ansatz orders, e.g., in reference [15] is that, beside the solutions sketched in Figure 1 with zero mean value, there exist solutions with nonzero mean value, i.e., $a_0\ne 0$. These unstable solutions represent forced vibrations around the nonzero equilibrium positions of the softening Duffing oscillator. These solutions are not considered in the present paper.

Considering the solutions with zero mean value, it is found that the regular solution contains, in general, small errors, while the nose solutions show large errors. This resulted in the assumption that these solutions can be considered as artifacts, e.g., in reference [15], as the fundamental assumption of the Harmonic Balance method—neglected terms are small—is not fulfilled. Therefore, it can be expected that the solution calculated by low-order Harmonic Balance differs largely from the sought-after solution of the differential equation and is therefore only a solution of the method but not of the considered system. In fact, later investigations of the problem, e.g., in reference [18] show that higher orders in the Harmonic Balance are necessary to make the nose solutions converge and that the shape of the nose, its amplitudes, and its frequency range of existence vary in parts largely compared to the low-order solution in Figure 1. Based on the methods in reference [18], Figure 2 shows corresponding results (in order to improve readability, the number of plotted points has been drastically reduced by the Ramer–Douglas–Peucker algorithm with a max. distance threshold of 10⁻⁵). In reference [18], several a posteriori measures are discussed in order to detect and characterize the artifact behavior. For this, a mathematical (Supplementary Materials) definition of artifact solutions is introduced and several residual, geometric, algebraic, and solver-related error measures are studied. The utilized solvers are the so-called algebraic HBM in conjunction with a quasi-Gauss–Newton method for the numerical path following the frequency response curves.

While, in many textbooks on nonlinear vibrations, the low-order nose solutions (e.g., $n=1$), as in Figure 1, are sketched without further considerations, as well as the lower part of the nose being denoted as unstable and the upper part as asymptotically stable, a general stability analysis using Floquet theory shows that most parts of the nose are unstable. This can also be observed in Figure 2 in part A, where only small parts of the upper part of the nose solution close to the nose tip are asymptotically stable, while all other parts are unstable. Whether there exist asymptotically stable nose parts depends on the respective parameter set. While the considered parameter set in reference [18] does not show any asymptotically stable parts, there are asymptotically stable parts in the parameter set considered in reference [19] and as mentioned in the parameter set (3), considered here as a first example.

Based on the analysis so far, we want to address the following questions in this paper:

• Is there a descriptive, respectively ostensive explanation for the existence of the nose solution and is there also a descriptive explanation for why these solutions are partly hard to calculate in Harmonic Balance?
• How can we explain the fact that only (small) parts of the nose solutions are asymptotically stable, depending on the parameter set, while sometimes there are no stable parts in these isolated solutions?
• Can the corresponding insights be used for a semi-analytic method of calculating the nose solutions?

The authors are aware that the addressed problem (i.e., the Duffing equation) is more than 100 years old and has been subject of numerous examinations. Most single details of the following reasoning are known, and the basic single steps of the calculus have already been applied. On the other hand, the authors are, to the best of their knowledge, not aware of a similar description giving insight in the characteristics of the nose solutions.

The following parts of the paper are structured as follows: First, the exact solution of the free undamped vibrations of the softening Duffing oscillator, well known from the literature, is considered, as this is the starting point of the illustrative reasoning on the behavior of the nose solutions. Based on the restoring characteristic, a very simple criterion of (non-) existence of stable parts of the nose solution is also found. Finally, based on the prior results, a combination of Perturbation Analysis and harmonic solutions is introduced for calculating the nose solution, which requires just the solution of linear algebraic equations systems and not of nonlinear ones.

3. Exact Solution for Free Undamped Vibrations and Conclusions for Forced Vibrations

As part of the descriptive explanation of the observations announced in the previous section, we consider the exact solution in the case of free undamped vibrations:

$$x″\left(t\right)+{{\omega }_0}^2 x\left(t\right)+\gamma x^3\left(t\right)=0$$

(6)

.

This solution can be found in numerous textbooks and papers, often generalized for arbitrary conservative nonlinear restoring or the pendulum equation, e.g., [5]. It is, in fact, based thereon that Equation (6) describes a conservative system, i.e., conservation of energy gives the following:

$${\displaystyle \frac{1}{2}}{x\prime }^2+{\displaystyle \frac{1}{2}}{\omega }_0{x}^2+{\displaystyle \frac{1}{4}}\gamma x^4=E_0=\mathrm{const}.$$

(7)

with total energy $E_0$ depending on the chosen amplitude due to initial conditions. The total energy for a free undamped oscillation with amplitude A is given according to (7) for a state with zero velocity and therefore only potential energy by

$$E_0={{\omega }_0}^2 A^2+\gamma A^4$$

(8)

From Equation (7) follows

$${\displaystyle \frac{\mathrm{d}x}{\mathrm{d}t}}=x^{\prime }=\pm \sqrt{2\left(E_0-{\displaystyle \frac{1}{2}}{{\omega }_0}^2{x}^2-{\displaystyle \frac{1}{4}}\gamma x^4\right)}$$

(9)

By separation of variables, the time $t$ can be computed as a function of the displacement $x$. Integrating from $x\left(t=0\right)=0$ to $x\left(t\right)$, we find

$$t={{\displaystyle \int }}_0^t\mathrm{d}\widehat{t}={{\displaystyle \int }}_0^x{\displaystyle \frac{\mathrm{d}\widehat{x}}{\pm \sqrt{2(E_0-\frac{1}{2}{{\omega }_0}^2{\widehat{x}}^2-\frac{1}{4}\gamma {\widehat{x}}^4}}},$$

(10)

which is the incomplete elliptic integral of the first kind. Evaluation of Equation (10) with limits $0$ and $A$ gives a quarter of the period $T$, from which the fundamental circular frequency $\omega$ of the free vibrations can be calculated by

$$\omega ={\displaystyle \frac{\pi }{2}}{\displaystyle \frac{1}{{{\displaystyle \int }}_0^A\frac{\mathrm{d}\widehat{x}}{\pm \sqrt{2(E_0-\frac{1}{2}{{\omega }_0}^2{\widehat{x}}^2-\frac{1}{4}\gamma {\widehat{x}}^4}} }}$$

(11)

In the following, the inverse function $x\left(t\right)$ to (10) shall be calculated, which requires the usage of Jacobi elliptic functions. Using the prefactor:

$$c=\sqrt{{\displaystyle \frac{2}{A^2\gamma +2{\omega }_0^2}}}$$

(12)

as well as the parameters:

$$\phi =\arcsin \left({\displaystyle \frac{x}{A}}\right)$$

(13)

$$m=-{\displaystyle \frac{A^2\gamma }{A^2\gamma +2{\omega }_0^2}}$$

(14)

Equation (10) can be written in compact form as

$$t\left(x;A,{\omega }_0\right)=cF\left(\phi ,m\right)$$

(15)

with the incomplete elliptic integral of the first kind F. The displacement $x$ is then given as the inverse from Equation (15), which is denoted as Jacobi Amplitude G and can be written as

$$x\left(t;A,{\omega }_0\right)=A\sin \mathrm{G}\left({\displaystyle \frac{t}{c}},\mathrm{m}\right).$$

(16)

For ${\omega }_0=1; \gamma =-0.1$, according to parameter set (3), the restoring characteristic $R\left(x\right)={{\omega }_0}^{2}x+\gamma x^3$ is plotted in Figure 3 (left) and $\omega \left(A\right)$ in Figure 3 (right).

For two fundamental circular frequencies $\omega$ of the free vibrations and corresponding amplitudes A, x(t) is plotted in Figure 4. Corresponding Fourier expansions up to order 15 are given in

Table 1. Fourier coefficients of $x\left(t\right)$ for $\omega =0.5; A=3.01$ (left) and $\omega =0.2; A=3.16$ (right); $x\left(t=0\right)=0$.

	$\mathit{\omega }=0.5; \mathit{A}=3.01$	$\mathit{\omega }=0.2; \mathit{A}=3.16$
i	sin	sin
1	$3.29618$	$3.89686$
3	$0.31806$	$1.01404$
5	$0.034323$	$0.39263$
7	$0.003708$	$0.15988$
9	$0.0004006$	$0.06564$
11	$0.00004329$	$0.026987$
13	$4.67785\times {10}^{-6}$	$0.011097$
15	$5.05431\times {10}^{-7}$	$0.004563$

. Due to the cubic nonlinearity, only odd terms do not vanish and due to the chosen initial conditions; only the sin terms occur.

In the results in Table 1, it can be seen that for the exact solutions, the lower the fundamental circular frequency $\omega$ (and the larger the amplitude $A$) becomes, the greater the influence of the higher harmonics, and the representation of the solution by just the basic harmonics becomes more and more erroneous. Looking at Figure 3 (left), this means that the closer the amplitude comes to the unstable equilibrium position, the harder it is to approximate the restoring characteristics by a linear one. It should be mentioned again that the plotted solutions show results for $x\left(t=0\right)=0$ and corresponding velocity $x’\left(t=0\right)$ in such a way that the given ω and A are realized. Of course, the same solution with an arbitrary phase shift in time is also an exact solution of Equation (6), as there is no trigger by an external excitation.

Comparing these observations with the properties of the nose solutions in case of forced vibrations described in the section before, there are direct relations.

Question 1 from the previous section for a descriptive explanation of the nose solution can be answered in that way, that the nose solutions are a resonance phenomenon (i.e., large response compared to the excitation) in that sense, that we get, for a certain excitation frequency an amplitude which is close to the amplitude of free vibrations with same fundamental frequency. Therefore, the nose solution seems (at least for parameter sets comparable to (3)) to contain a “large portion” of free vibration, i.e., the homogeneous solution (however, with a well-defined phase shift determined by the phase of the external excitation) with a small addition of excitation and damping, so that energy is balanced. In fact, in many textbooks, “backbone” curves with the relation $\omega \left(A\right)$ (mostly based on the method used for the determination of the forced vibration, e.g., Harmonic Balance) are also sketched in the curves of forced vibrations like Figure 1. As a restriction, it should be mentioned that this seems to be valid for large parts of the nose but not for excitation circular frequencies close to zero. As can be seen in Figure 2 for $\Omega$ smaller than 0.1 in the higher-order Harmonic Balance solutions, the amplitude increases for decreasing $\Omega$ and significantly exceeds the maximum possible amplitude value (for parameter set (3)) of the free undamped solutions $\sqrt{10} \approx 3.1623$.

Also, the second part of question 1—why is it so hard to calculate this solution by Harmonic Balance—can therefore be answered. At least the “large portion” of free vibration contains large portions of higher harmonics (the lower the frequency the larger the portions) which ultimately makes it necessary to use high ansatz orders in Harmonic Balance.

Looking at the restoring characteristics in Figure 3 left and the Harmonic Balance solutions in Figure 2, there is also a very simple descriptive explanation for why only (small) parts of the nose solutions are asymptotically stable, depending on the parameter set, while there are often no stable parts in these isolated solutions, i.e., the response to question 2. Asymptotically stable solutions can be expected in regions of positive restoring, i.e., any solution exceeding the amplitude of $\sqrt{10}$ for parameter set (3) is suspicious for becoming unstable. As the amplitudes of nose solutions have almost the amplitude of the corresponding free vibrations for $\omega =\Omega$, plus an amplitude correction depending on excitation amplitude and damping, asymptotically stable solutions can be expected only at the right end of the nose, where, according to larger excitation frequency and following the previous argumentation corresponding lower amplitudes, the range of positive restoring is not exceeded. This behavior can be observed exactly when testing several parameter sets. Parameter set (3) is in fact chosen with small damping and excitation frequency so that the nose exists (i.e., it is not merged with the bent resonance peak) but turns out to reach enough to the right, that sufficiently small amplitudes result that there is a positive restoring characteristic. Figure 5 shows the basin of attraction of the asymptotically stable solutions. To determine the basin of attraction, Equation (1) with the parameter set (3) and $\Omega =0.5$ is numerically integrated for various initial conditions until a stationary solution is reached. The initial conditions used are uniformly distributed over the range shown below. In order to accelerate the large number of numerical integrations, these were executed in parallel on the GPU (Nvidia RTX 4090). Since this is significantly faster with single precision, a custom-optimized integration kernel was written. This uses the Runge Kutta 4th algorithm with a fixed step size. In convergence studies and comparison calculations with the integrator ODE45 implemented in Matlab R2024a, the time step size is adapted to $h=2 \pi /100\Omega =T/100$ so that the results are in sufficient alignment. In Figure 5, orange points denote initial conditions leading to the regular low-amplitude solution (as in Figure 1), while for initial conditions marked in white, $x\left(t\right)$ diverges to +/− infinity. The asymptotically stable nose solution has a very small basin of attraction in the upper middle part marked by blue. The three points A, B, and C denote the three stationary solutions calculated by numerical integration or Harmonic Balance (especially in case of unstable solution C), giving the corresponding $x\left(0\right)$ and $x’\left(0\right)$ as initial conditions, i.e., the stationary solution is reached by starting from these points without transient behavior. For the two asymptotically stable solutions, the point A represents the low-amplitude regular solution located in the orange area, and from this starting point, the corresponding stationary solution is reached without transient behavior. The same holds for B, representing the larger-amplitude asymptotically stable nose solution in the small blue area. Point C lies on the separatrix of the basins of attraction of the two asymptotically stable solutions and represents the lower-amplitude unstable nose solution, and it is calculated by the Harmonic Balance method. So, the remaining question is the third one: can the corresponding insights be used for a semi-analytic method of calculating the nose solutions, which will be addressed in the next section.

4. Approximate Solution for Forced Vibrations Based on Exact Solution of Free Vibrations

In the previous section, it was argued that the forced vibrations in the nose solutions seem to be free undamped vibrations with a small correction in amplitude and triggering of the phase, which makes it an obvious object of applying Perturbation Analysis. When determining curves like those in Figure 1 as a solution of Equation (1), Lindstedt–Poincaré Perturbation Analysis is often applied, claiming that damping and excitation as well as the cubic term are small and excitation frequency Ω and linear natural frequency Ω₀ are almost equal. This reflects the expectation, that the response amplitude is large compared to the excitation amplitude, which holds if damping is small and we are close to resonance.

In the present case, it is still true that we have small damping and (compared to the response amplitude) small excitation amplitude. Regarding, considering the magnitude of the cubic term for the nose solutions and looking at the restoring characteristic, they have the same magnitude as the linear restoring term. So, it is not reasonable to assume this term to be small anymore. As we will see, application of the Lindstedt–Poincaré technique, i.e., expansion of a characteristic frequency parameter, is no longer necessary here.

As a result, the starting point of the analysis is the following equation:

$$x^{″}+\epsilon \delta x^{\prime }+{{\omega }_0}^{2}x+\gamma x^3=\epsilon \widehat{f}\cos \left(\Omega t\right)$$

(17)

with

$$\delta ={\displaystyle \frac{2D{\omega }_0}{\epsilon }};\widehat{f}={\displaystyle \frac{f}{\epsilon }}$$

(18)

and $\epsilon$ as a small parameter. We expand $x\left(t\right)$ up to the first order by

$$x=x_0+\epsilon x_1+\dots$$

(19)

Inserting Equation (19) into Equation (17) results in

$$\begin{gathered} x_0^{″}+\epsilon x_1^{″}+\dots +\epsilon \delta x_0^{\prime }+\dots +{{\omega }_0}^2\left(x_0+\epsilon x_1+\dots \right) \\ +\gamma \left({x_0}^3+3\gamma {x_0}^2{x}_1+\dots \right)=\epsilon \widehat{f}\cos \left(\Omega t\right) \end{gathered}$$

(20)

Sorting with respect to order of $\epsilon$ gives

$${\epsilon }^0: x_0^{″}+{{\omega }_0}^2{x}_0+\gamma {x_0}^3=0$$

(21)

and

$${\epsilon }^1: x_1^{″}+{{\omega }_0}^2{x}_1+3\gamma {x_0}^2{x}_1=-\delta x_0^{\prime }+\widehat{f}\cos \left(\Omega t\right)$$

(22)

Obviously, the solution of Equation (21) is the exact solution of the free undamped vibrations discussed in the previous section.

A corresponding sorting of small and non-small terms resulting in the free undamped vibrations of the Duffing oscillator or another conservative nonlinear oscillator with known exact solution as the zeroth approximation is in fact used for systems with strong nonlinearities (as we have here in the considered amplitude range). A corresponding overview is given in reference [20], describing in detail how to apply the Jacobi elliptic functions for the solution of nonlinear oscillatory problems. One example described therein is the Elliptic Harmonic Balance Method, where the Jacobi elliptic functions replacing cosine and sine as ansatz functions, and there is also a section in reference [20] dealing with systems with harmonic excitation. One paper cited therein is reference [21], which in fact considers (beside other examples), with the slightly damped and harmonically excited pendulum equation, a case very close to the case considered in the present paper. The analysis in reference [21] is in fact based on the assumption of small excitation and damping, as in Equation (17), and the (right end of the) nose solution is also considered, called there penisolation solution. The same assumptions are also applied in reference [22], i.e., a nonlinear equation for the 0th order is also obtained.

In reference [21], for the Jacobian elliptic function corresponding amplitude, phase and modulus are determined by an (improved) averaging method, where it is assumed that amplitude, phase, and modulus change slowly in time and therefore can be averaged. The proposed method finally ends up, after several analysis steps, in a nonlinear set of algebraic equations for the unknown quantities.

Also, in reference [23], Jacobian elliptic functions are used as the starting point for deriving solutions for forced vibrations of the Duffing oscillator by investigating what form an excitation must take to determine exact solutions in the externally excited undamped case. In reference [24], this method was extended and applied to a system of two coupled Duffing oscillators. Subsequently the same author applied this for the determination of peak curves for nonlinear oscillators [22].

Compared to this, we chose the following method to find the solution. The exact solution of the homogeneous Equation (21), with fundamental circular frequency being the excitation circular frequency, is combined with an arbitrary phase shift ψ and expanded in a Fourier series with sufficient order. A drawback at this stage, of course, is that we do not know already how to choose this phase shift of the exact homogeneous solution. For this, we will later on need an additional attempt based on an error minimization. The expanded solution $x_0\left(t\right)$ with arbitrary phase shift ψ is inserted into Equation (22), and for $x_1\left(t\right)$, a Fourier expansion is also applied, balancing the harmonics in Equation (22) up to the order of the ansatz. The big difference compared to directly applying the Harmonic Balance to Equation (1) or applying the averaging method from [21] is that Equation (22) is linear in $x_1$, i.e., the resulting system of equations is also linear, compared to the necessity of solving large systems of nonlinear equations when directly applying the Harmonic Balance. Of course, only stationary solutions can be calculated with the present method, i.e., amplitude and phase are not even allowed to change slowly in time. Initial investigations on computational time needed shows that the effort increases approximately linear with higher expansion order in the Fourier series. Applying Harmonic Balance alone to a linear problem can also be reached by other prior approximation steps like in reference [25] with Newton’s iteration method.

Finally, as in Equation (5), the magnitude of the neglected terms in solving Equation (22) is used to determine the “correct” phase in the solution $x_0\left(t\right)$.

The described calculation is now demonstrated by parameter set (3) and $\Omega =0.5$. The larger amplitude nose solution in that case is asymptotically stable while the other one is unstable. For the expansion of the exact solution, the Fourier terms up to order 15 are taken, which are sketched in Figure 6 for the same initial conditions as in Figure 4. As described before, for an arbitrary phase shift ψ compared to the solution plotted in Figure 6, this Fourier expansion of $x_0\left(t\right)$ is inserted into Equation (17), the solution $x_1\left(t\right)$ is calculated for the respective phase shift with the same Fourier order, and the residua of the Harmonic Balance of Equation (22) are considered.

As can be seen in Figure 7, there are two phase shifts, for which the corresponding residua are almost zero. The corresponding solutions $x\left(t\right)=x_0\left(t\right)+\epsilon x_1\left(t\right)$ are plotted in Figure 8. These solutions in fact show a high agreement with the nose solutions obtained by Harmonic Balance and, in case of the asymptotically stable solution, also with numerical integration performed with NDSolve in Mathematica, Figure 9.

Applying this method to other excitation circular frequencies $\Omega ,$ the following observations can be made. For increasing $\Omega ,$ i.e., going towards the nose tip, the method provides the expected behavior with converging amplitudes of the two nose solutions. The minima of the residuum increase significantly when reaching the nose tip and do not almost reach zero anymore (Figure 10 and Figure 11), i.e., the nose tip can be identified easily.

As a restriction, it was mentioned already in Section 2 that, as can be seen in Figure 2, the amplitude calculated with higher-order Harmonic Balance significantly exceeds the maximum possible amplitude for $\Omega <0.1$. As a consequence, the fundamental assumption of the Perturbation Analysis applied here, namely the “large portion” of free undamped vibrations, seems not to be valid any more for $\Omega <0.1$. In fact, it can be observed that the Perturbation Analysis applied here fails for $\Omega <0.1$ for parameter set (3).

Additionally, the described method shall be applied to the following parameter set:

$${\omega }_0=1; \gamma =-2; f=0.05; D=0.062$$

(23)

which differs significantly compared to (3) with respect to the much larger nonlinearity $\gamma$, which results in lower amplitudes, but is comparable to (3) with respect to small excitation and small damping, i.e., it fulfills the preconditions made for the Perturbation Analysis in this chapter or, e.g., in reference [21]. As is also the same as in (3), the range of Ω for the existence of the nose solution and of the nose tip is almost the same as before. The corresponding restoring characteristic and the fundamental circular frequency of the free undamped vibrations can be found in Figure 12, and a corresponding comparison of the Perturbation Analysis method and numerical integration is shown in Figure 13 for Ω = 0.5, again showing high agreement. It should also be mentioned that an asymptotically stable nose solution is again found here, as the displacement of positive restoring is not exceeded.

Finally, the results are graphically summarized in Figure 14, illustrating the results for parameter set (3). In particular, the deviation between the low-order $n=1$ Harmonic Balance nose solution compared with the solution of the analysis considered in the present paper is shown as well as the asymptotically stable region of the nose solution in comparison with the positive restoring border-line.

5. Conclusions

For harmonic excitation and slight damping, stationary large amplitude solutions at subcritical excitation frequencies are obtained for the softening Duffing oscillator. These solutions have the shape of a nose in the amplitude-frequency diagram. In prior work, it has been observed that these solutions may contain large errors and that high ansatz orders may be necessary, as well as the fact that, in most cases, these solutions are unstable.

The present paper aims to find descriptive explanations for this by addressing the questions given in Section 2 on the existence and accurate calculation of the nose solution as well as its stability.

Those questions could be answered in the following way. With respect to the explanation of existence, the nose solutions are, in a large frequency range, a resonance phenomenon (i.e., large response compared to the excitation), in the sense that we get, for a certain excitation frequency, an amplitude which is close to the amplitude of free vibrations with same fundamental frequency. The nose solution contains a “large portion” of the homogeneous solution with a small addition of excitation and damping so that energy is balanced. This behavior is systematically used by methods based on Jacobian elliptic functions as reviewed in reference [20]. Therefore, we need higher ansatz orders in (classical) Harmonic Balance as the homogeneous solution contains large portions of higher harmonics (the lower the frequency, the larger the higher portions).

The question on stability can simply be answered by looking at the restoring characteristics in Figure 3 left and the Harmonic Balance solutions in Figure 2. Asymptotically stable solutions can be expected in regions of positive restoring. As the amplitudes of nose solutions have almost the amplitude of the corresponding free vibrations for $\omega =\Omega$, asymptotically stable solutions can (only) be expected at the right end of the nose, where, according to the larger excitation frequency and corresponding lower amplitudes, the range of positive restoring is not exceeded.

Finally, as response to the question of how to alternatively calculate these solutions, a semi-analytic method considering the dominating portion of the homogeneous solution and combining Perturbation Analysis, Harmonic Balance, and residuum minimization was introduced, needing only the solution of linear algebraic equations. The method showed good results in finding the nose solutions.