## 4 Approximate Solution for Forced Vibrations Based on Exact Solution of Free Vibrations
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 repisolation solution. The same assumptions are also applied in reference [22], i.e., a nonlinear equation for the 0 th 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 \(\psi\) 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 \(x_{0}(t)\) with arbitrary phase shift \(\psi\) is inserted into Equation (22), and for \(x_{1}(t)\), 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 [PERSON]'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 \(x_{0}(t)\). 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 \(\psi\) compared to the solution plotted in Figure 6, this Fourier expansion of \(x_{0}(t)\) is inserted into Equation (17), the solution \(x_{1}(t)\) 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.