# Comparing the direct normal form and multiple scales methods through frequency detuning

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## Abstract

Approximate analytical methods, such as the multiple scales (MS) and direct normal form (DNF) techniques, have been used extensively for investigating nonlinear mechanical structures, due to their ability to offer insight into the system dynamics. A comparison of their accuracy has not previously been undertaken, so is addressed in this paper. This is achieved by computing the backbone curves of two systems: the single-degree-of-freedom Duffing oscillator and a non-symmetric, two-degree-of-freedom oscillator. The DNF method includes an inherent detuning, which can be physically interpreted as a series expansion about the natural frequencies of the underlying linear system and has previously been shown to increase its accuracy. In contrast, there is no such inbuilt detuning for MS, although one may be, and usually is, included. This paper investigates the use of the DNF detuning as the chosen detuning in the MS method as a way of equating the two techniques, demonstrating that the two can be made to give identical results up to \(\varepsilon ^2\) order. For the examples considered here, the resulting predictions are more accurate than those provided by the standard MS technique. Wolfram Mathematica scripts implementing these methods have been provided to be used in conjunction with this paper to illustrate their practicality.

## Keywords

Nonlinear Vibration Normal form Multiple scales## 1 Introduction

In recent years, there has been substantial interest in the study of backbone curves, due to their utility in studying lightly damped nonlinear vibrations in multi-degree-of-freedom (MDOF) mechanical structures. The motivation for this paper comes from observations made by the authors when comparing backbone curves found using the multiple scales (MS) method (see, for instance, [1]) and those found using the normal form method, defined in [2].

The normal form method in [2] was developed as a technique that can be applied *directly* to systems of weakly coupled second-order nonlinear differential equations. This concept is not entirely uncommon, having previously been proposed in [3], but it is the matrix formulation proposed in [2] that is considered particularly beneficial to the current work. We will call this the “direct” normal form (DNF) method^{1} in order to differentiate it from the “classical” method described, for example, by Jezequel and Lamarque [4], Arnold [5], Murdock [6], Kahn and Zarmi [7] and Nayfeh [8]; the latter is not investigated here, as similar comparisons have previously been made, for example, in [2].

In recent years, the DNF method (and other normal form methods similar to this) has been used extensively to capture the responses of nonlinear systems. This includes, but is not limited to, describing modal interactions and bifurcations in backbone curves [9, 10, 11, 12], recognising out-of-unison resonance in a taut cable [13], reduced-order modelling [14], nonlinear system identification [15, 16], investigating aeroelastic systems under fluid flow [17, 18], exploring applicability conditions for nonlinear superposition [19], and quantifying the significance of nonlinear normal modes [20]. In contrast with the recent development of the DNF method, the MS method is well established in the literature, with thorough discussions regarding its development readily available, for example, in [21, 22, 23, 24, 25, 26].

Perturbation methods require the repeated application of a number of steps, building up an increasingly accurate solution by addressing smaller terms in each repetition. In the practical application of these methods, the steps can require significant computational effort and produce increasingly complex expressions, which can, arguably, hide the mathematical insight gained from employing such a technique. In this paper, we consider the “accuracy” of these methods by assessing the result after one or two repetitions of their respective steps. It is generally recognised that these techniques converge to the correct solution with many repetitions, so can ultimately be considered as precise as each other.

A contributing factor in the accuracy of the DNF method, as shown in [27], is the frequency detuning which arises in its formulation. In physical terms, this can be interpreted as a series expansion around the natural frequency of the underlying linear system. This is not naturally present in the MS technique; however, several examples of alternative detunings, applied to MS technique, can be found in the literature [28, 29, 30, 31, 32, 33]. The attempt that most closely resembles the detuning of the DNF method is found in [32], although this proposed detuning is only employed in a small number of papers, such as [34, 35]. In [32], an \(\varepsilon \)-expansion is applied, not only to time, as is standard, but also to frequency. The paper presents the updated frequency–amplitude relationships and suggests that they appear more accurate, although it was not possible for this to be verified with numerical data. The motivation for expanding the frequency is solely to remove the secular terms in the response, and so the technique lacks the physical motivation that is present in the DNF method, as described in detail in the current paper.

Further attempts to detune the MS method have been proposed, though a number of these focus on the forced case in which it is common practice to perturb the forcing frequency [28, 29]. A more thorough investigation is given in [30], and a comparison of the MS method and the generalised method of averaging can be found in [31]. Additionally, the detuning applied in [32] has also been applied to the Lindstedt–Poincaré method of strained parameters and the generalised method of averaging, with these detuned methods producing identical truncated results [33].

In this paper, a comparison on the DNF and MS techniques is provided, with emphasis placed on the detuning used. Specifically, in Sect. 2, the two techniques are briefly outlined and compared using the Duffing oscillator as an example system, a system which is adopted in [27, 31, 32, 33]. The two techniques are equated by introducing a detuning step, which is physically interpreted as a perturbation about the response frequency rather than the linear frequency, into the MS technique in Sect. 3. The detuning approach employed in the DNF method will be applied in the MS method, and it will be shown that doing so allows the two methods to be equated. By considering a more general detuning, it is shown that using MS both the fundamental and the harmonic response predictions are affected by the detuning. This is in contrast to the DNF technique, in which only the harmonic response changes. In Sect. 4, the techniques are compared for a two-mode system, where it is shown that the techniques give the same results if the MS method is modified to include the detuning. Conclusions are drawn in Sect. 5.

## 2 Approximate methods

*x*denotes the displacement, \(\omega _n\) represents the linear natural frequency, and \(n_x(x)\) is a nonlinear term. For both techniques, the nonlinear term is assumed to be small. Here, this is indicated by \(\varepsilon \), which may be thought of as a bookkeeping parameter that allows the relative size of terms to be tracked [8]. As such, \(\varepsilon \) is taken to have a value of unity, such that it does not alter the equations. The application of the techniques is described as a series of steps, with the Duffing oscillator (\(n_x(x)=\alpha x^3\)) being used as an example.

### 2.1 Direct normal form

*u*, that can be solved exactly by using the following form for the solution, which assumes that system will respond as a single harmonic

*t*and \(\phi _0\) denotes the phase of the response. Once

*u*has been found, the harmonics of the response can be recovered using the transform equation.

*q*, where \(q=x\) for SDOF systems. This means that

*x*could be used instead of

*q*in the following equations. However,

*q*has been kept to allow easier comparison with the MDOF case discussed in Sect. 4. The transform may be summarised as

*q*from the original differential equation in Eq. (3) using the transform and then simplifying using the transformed equation of motion, the \(\varepsilon ^i\) balance equation is given by the homological equation:

*excitation*of these equations, which defines the vectors \(\mathbf {u}_i^*\), is given as

*Step *1\(_{NF}\) The substitution \(q = u = u_p + u_m\) is made in the nonlinear term to give \(n_q(q) = n_q(u_p + u_m) = \mathbf {n}_{e1}\mathbf {u}^*_1\). Here, \(\mathbf {n}_{e1}\) contains coefficient values and \(\mathbf {u}^*_1\) is defined above.

*Step*2\(_{NF}\) Using Eq. (2), the variables \(u_p\) and \(u_m\) in \(\mathbf {u}_1^*\) are written as a series of complex exponentials in time. The resulting vector is double differentiated with respect to time. The second derivative with respect to time can be expressed as a Hadamard product (\(\circ \)); \(\mathrm {d}^2 \mathbf {u}_1^*/\mathrm {d} t^2 = -\mathbf {dd} \circ \mathbf {u}_1^*\). Further details on this are given in Appendix A.

*Step*3\(_{NF}\) Now, \(\mathbf {h}_1\) and \(\mathbf {n}_{u1}\) may be found using

### 2.2 Multiple scales

The method of multiple scales is an established technique that is discussed at length in the literature (for example, see [22, 23, 26, 31, 32] and references therein), and here we provide a brief summary of this technique to form a basis on which modifications can be discussed later.

Following this review of the method, in Sect. 2.3, solutions found using the frequency detuning proposed in [32] will be presented; a more thorough investigation is given in Sect. 3, in which a comparison will be made between this detuning and that used in the DNF method.

The approach builds on the standard perturbation method in which the response is split into a series of terms with reducing significance \(x=X_0+\varepsilon X_1 + \varepsilon ^2 X_2 + \cdots \). In MS, each of these time-dependent components are treated as functions of multiple timescales.

*T*, and \(T_s\), are treated as independent variables, such that derivatives with respect to

*t*can be expressed

*x*, firstly the \(\varepsilon ^0\) order balance in Eq. (21) is solved to give

*Step*1\(_{MS}\) The resonant terms, i.e. those that respond at \(\tau =\omega _n t\) in Eq. (23), are removed and equated, writing

*where*\({\mathrm{Res}}\{n_{x}(X_0)\}\)

*represents the resonant terms in*\(n_{x}(X_0)\).

*This equation is then solved to find*\(A(T,T_s)\)

*and*\(\phi (T,T_s)\).

*Step*2\(_{MS}\) The remaining terms in Eq. (23),

*where*\({\mathrm{NRes}}\{n_{x}(X_0)\}\)

*represents the non-resonant terms in*\(n_{x}(X_0)\)

*are now considered. Here the right-hand side may be viewed as an “excitation” of a linear dynamic system in*\(X_1\)

*which can be solved to generate harmonic responses terms in*

*x*.

### 2.3 Duffing oscillator backbone curves

Summary of approximate solutions and expressions for backbone curves for the undamped, unforced Duffing oscillator

Technique | Amplitude of fundamental | Amplitude of third harmonic | Amplitude of fifth harmonic |
---|---|---|---|

Direct normal form | \(A_c\) | \(\displaystyle {\varepsilon \frac{ \alpha }{32\omega _r^2}A_c^3\left( 1+\varepsilon \frac{3\alpha }{32\omega _r^2}A_c^2\right) }\) | \(\displaystyle {\varepsilon ^2\frac{\alpha ^2}{512\omega _r^4}A_c^5} \) |

Multiple scales | \(A_c\) | \(\displaystyle {\varepsilon \frac{ \alpha }{32\omega _n^2}A_c^3\Big (1 - \varepsilon \frac{21\alpha }{32\omega _n^2}A_c^2 \Big )}\) | \(\displaystyle {\varepsilon ^2\frac{\alpha ^2 }{1024\omega _n^4}A_c^5}\) |

The results displayed in Fig. 1 demonstrate the differences that can occur when a detuning is applied to the square of \(\omega _r\), as opposed to directly to the linear term, and provide motivation for the application of the DNF detuning in the MS method, as described in Sect. 3.2. In particular, in contrast to the explicit form for the MS relationship, the DNF method gives an implicit equation in \(\omega _r^2\). This can be easily rearranged to give a quadratic equation in \(\omega _r^2\) which is easily solved and square rooted to give an explicit equation for \(\omega _r\). This process becomes more complicated at higher orders of \(\varepsilon \), at which point it is possible that either a Taylor expansion or numerical continuation would need to be used. That being said, the accuracy of the curves in Fig. 1 suggests that it is unlikely that these higher orders would be necessary to obtain a strong approximation of the true solution.

## 3 Equating the techniques

In this section, we compare the derivations of the DNF and MS approaches. To do this, we first consider frequency detuning. The importance of this step for the DNF method was assessed in [27], in which the Duffing oscillator was used to demonstrate that it is this detuning which increases the accuracy of the technique in comparison with the classical normal form method. In light of the fact that perturbation methods repeat a specific set of steps to find a solution, as demonstrated in Sect. 2, we consider whether the same approach may be used in the MS method to improve the agreement with the DNF method at the same number of repetitions.

It should be noted that it is possible to introduce the intrinsic time-dependent amplitudes of the MS method to the DNF technique to allow transient behaviour to be captured. This is not investigated further here, as this paper focuses on the unforced, undamped behaviour of systems.

### 3.1 Detuning the MS method

In the derivation of the DNF technique, a frequency detuning is applied, in which the square of the natural frequency is assumed to be detuned from the square of the response frequency such that the substitution \(\omega _n^2=\omega _r^2+\varepsilon \delta \) can be made, where \(\delta \) is introduced as a detuning parameter. This is discussed in Appendix A where, for multiple degrees of freedom, the equation is written \({\varvec{\Lambda }}={\varvec{\Gamma }}+\varepsilon {\varvec{\varDelta }}\). This allows the linear natural frequency to be replaced with the response frequency, \(\omega _r\), and a detuning term, \(\delta \), in the \(\varepsilon ^1\) relationship, Eq. (A.3), and results in coefficients in (\(\mathbf {dd}^\intercal -\omega _r^2\mathbf {1}_{1,\ell }\)) expression that are exactly zero, see \(Step 3_{NF}\).

This detuning has been discussed in [37], where it was shown that the detuning does not affect the frequency–amplitude relationship, but does improve the prediction of the third harmonic. The physical interpretation of this is associated with how the underlying linear system is defined—normally we consider the Duffing oscillator to have a linear stiffness term \(\omega _n^2 x\) (and hence a natural frequency of \(\omega _n\)), but the same result can be achieved by treating the linear stiffness term as \(\omega _r^2 x\) and modifying the nonlinear term to compensate for this, giving \(\alpha x^3+\delta x\), where \(\delta \) is a detuning parameter. Adopting the second approach can result in a smaller nonlinear term which more closely meets the key assumption that the non-linearity is of order \(\varepsilon ^1\). Note that this interpretation of the detuning does not specifically rely on the assumption that \(\delta \) is small, provided the new nonlinear term, \(\alpha x^3 + \delta x\), remains small.

### 3.2 Detuned multiple scales

*detuned multiple scales approach*(dMS). Firstly, when selecting the timescales we set the fast time as \(\omega =\omega _r\) and hence \(\tau =\omega _r t\). The result of this is that Eq. (20) is modified to

*Step*2\(_{MS}\) is applied to find the harmonics captured by \(X_1\). With the resonant terms removed, the \(\varepsilon ^1\) balance may be expressed as

As previously mentioned, a similar detuning of the MS technique is considered in [32], which introduces an \(\varepsilon \) perturbation, \(\omega ^2 = \omega _0^2+\varepsilon \omega _1+\varepsilon ^2\omega _2+\cdots \), to resolve the issues of secular terms in the response.^{2}

Once truncated to order \(\varepsilon ^1\), this expansion can be seen to be the same as that in the DNF method, though without the physical interpretation of a series expansion about the underlying natural frequency. It should be noted that, in [32], the first term is given as a square simply because it is convenient.

Note that the steps for the dMS method are illustrated in Online Resource 3.

### 3.3 Comparison of detuned multiple scales and direct normal form

It has been shown that the predicted response using the DNF method can be matched by the dMS method. Now we compare these two techniques in more detail for the case where the amplitude of response is assumed to be fixed, i.e. \(A(T)=A_c\). As with all the discussions up to this point, we will consider the \(\varepsilon ^1\) accuracy case for a SDOF system.

*u*into the transformed equation of motion and noting that, for a SDOF system, \(n_q=n_x\), we can write

*Step 3*\(_{NF}\) where \((\mathbf {dd}^\intercal -\omega _r^2\mathbf {1}_{1,\ell })\circ \mathbf {h}_1=\mathbf {n}_{e1}-\mathbf {n}_{u1}\) is considered. For the non-resonant, or harmonic, elements this equation is satisfied by setting the left-hand side of the equation equal to the values in \(\mathbf {n}_{e1}\) on an element-by-element basis. From the derivation in Appendix A, it can be seen that this solution arises from Eq. (A.3) and may be expressed as

From this, we can conclude that, at an accuracy level of \(\varepsilon ^1\), the prediction of periodic oscillations using the DNF and MS methods can be made the same. This requires the MS technique to use \(\tau =\omega _r t\), as in the DNF method, for fast time and to remove \(\omega _n\) from the equations of motion using the frequency tuning \(\omega _n^2=\omega _r^2+\varepsilon \delta \). As discussed in [37], this is justified based on the idea that the system can be linearised about a stiffness \(\omega _r^2 x\) rather than \(\omega _n^2 x\) to potentially reduce the size of the nonlinear term. This may be substituted into Eq. (41) to give the full solution to order \(\varepsilon ^1\).

### 3.4 Alternative frequency tunings

So far in this section, we have shown that the MS and DNF techniques are equivalent, to order \(\varepsilon ^1\), under the special conditions that the fast time is set to \(\tau =\omega _r t\) and the stiffness term, \(\omega _n^2 x\), in the equation of motion is rewritten as \((\omega _r^2 + \varepsilon \delta )x\), where \(\delta \) can still be viewed as a detuning parameter. However, this frequency tuning approach raises the question about the predicted response when a different detuning parameter is selected.

For the case of the DNF method, this has been addressed in [37] for both single- and multi-degree-of-freedom systems. Consider the arbitrary frequency tuning \(\omega _n^2=\omega _d^2+\delta _d\), where \(\omega _d\) is the detuned frequency with \(\omega _d=\omega _r\) for the standard technique described in 2.1. In the MDOF notation used in Appendix A, the equivalent expression is \({\varvec{\Lambda }}={\varvec{\Gamma }}_d+{\varvec{\varDelta }}_d\). In [37], it was shown that the prediction of the response at the fundamental frequency is independent of the chosen detuning at order \(\varepsilon ^1\). The reason for this is that the only change to the \(\varepsilon ^1\) balance, Eq. (A.3), is that \({\varvec{\Gamma }}_d\), a diagonal matrix of \(\omega _{ri}^2\) terms, is replaced by a diagonal matrix of \(\omega _{di}^2\) terms. The result is that, in \(Step 3_{NF}\), \(\mathbf {h}_1\) and \(\mathbf {n}_{u1}\) are now found using Eq. (10).

*u*, \(\mathbf {n}_{u1}\), also remains the same. Hence, the \(\varepsilon ^1\) order equation of motion in

*u*, and the subsequent response at the resonant frequency, is independent of the selection of \(\omega _d\). However, the harmonic response prediction is affected, as each term in this is dependent on the non-near-zero value of the bracketed term in Eq. (10). The result is that, for the Duffing oscillator, the vector for \(\mathbf {h}_1\) becomes

Figure 2 shows the DNF prediction of the response of the Duffing oscillator in terms of the first and third harmonics for a range of frequency tuning frequencies, \(\omega _d=\omega _r+(\omega _n-\omega _r)\gamma \), from \(\gamma =0\), corresponding to the standard detuning used in DNF (i.e. \(\omega _d=\omega _r\)) to \(\gamma =1\), where no detuning is used (i.e. \(\omega _d=\omega _n\)). Figure 2a shows that the prediction of the response at the resonant frequency is robust to the choice of detuning parameter; however, the third harmonic response is affected by its choice and is better captured using the standard DNF detuning (\(\gamma =0\)) than with no detuning (\(\gamma =1\)).

*A*(

*T*) and substituting the solutions into the \(X_0\) expression give

*A*(

*T*) and \(\phi (T)\) at \(t=0\). Recalling that \(\delta _d\) is defined in \(\omega _n^2=\omega _d^2+\delta _d\), this gives the response frequency \(\omega _r=\omega _n^2/(2\omega _d)+\omega _d/2+3\alpha A_c^2/(8\omega _d)\). Writing \(\omega _d=\omega _r+(\omega _n-\omega _r)\gamma \) results in the response frequency equation

Figure 3 demonstrates that varying \(\gamma \) from 0 to 1 transforms the response from the DNF/dMS to the standard MS response. For the MS technique, \(\delta _d=0\) and, hence, the frequency shift away form \(\omega _n\) is captured by \(\phi (T)\). However, for the dMS technique, \(\omega _d=\omega _r\) and so \(\phi (T)=\phi _c\) represents the fact that the \(X_0\) response is at response frequency \(\omega _r\). These represent two special cases, for a general frequency tuning with fast time \(\tau =\omega _d t\); the thin, green curves in Fig. 3 represent a continuum between these two cases. Note that the accuracy of the DNF method is only reached when the detuning from that method is used. Interestingly, the fundamental response is independent of the detuning for the DNF method, whereas this is not the case for MS.

## 4 Example: non-symmetric, two-mass oscillator

A 2DOF system is considered in this section, allowing the two methods to be compared using a more complex system, as well as examining the robustness of the frequency tuning methods.

The application of the methods is largely the same as for the Duffing oscillator considered in previous sections, so only a brief overview is given below. For brevity, solutions will only be considered to order \(\varepsilon ^1\). In addition, we provide scripts, as supplementary material, that allow the equations to be derived symbolically using Wolfram Mathematica.

### 4.1 Multiple scales

The resulting backbone curves from Eq. (64) are given in Fig. 4 and discussed in Sect. 4.3. Due to the involved process required to find the harmonics, analytical solutions for these are not given, but have been derived using Wolfram Mathematica and solved numerically to allow comparison between the techniques; this is discussed in Sect. 4.3.

### 4.2 Direct normal form

\(Steps 1_{NF}\)–\(3_{NF}\) are followed as previously described and are not shown here due to the large size of the matrices involved. Full workings are shown in [38].

### 4.3 Detuned multiple scales

As with the SDOF case, the final forms of \(q_i\) are identical, though this is not shown here for reasons of brevity.

### 4.4 Comparison of the techniques

The fundamental backbone curves for the first and second modal responses are given in Fig. 4. Four backbone curves are shown for each technique. Panels (a) and (b) correspond to the first backbone curve of the system, that is, the curve which initiates at the first natural frequency of the underlying linear system, \(\omega _{n1}\); panels (c) and (d) represent the second backbone curve. These results are comparable to those for the Duffing oscillator in Fig. 1, with the MS curve underestimating the numerical continuation results and the DNF/dMS results again remain closer to the numerical continuation results. The difference between the methods grows significantly with increasing amplitude. In particular, the MS results diverge noticeably from the numerical and DNF/dMS counterparts at higher amplitudes. As verified in [38], this is the result of the loss of influence of the higher-order terms during the linearisation of the system.

Interestingly, the third harmonic components of the backbone curves in Fig. 5 are qualitatively different from the equivalent curve for the Duffing oscillator. While the amplitudes of the third harmonics from the MS method in the SDOF case were greater than those from numerical continuation, Fig. 5 shows that the opposite is true for the 2DOF responses. This inconsistency suggests that the MS method is less robust to changes in the system compared to the DNF and dMS methods, which remains consistent across the two cases, although higher-order cases have not been considered in this study.

## 5 Conclusions and discussion

This paper presents a comparison between the multiple scales and direct normal form techniques and investigates whether the two methods can produce equivalent results. In particular, the detuning used in the DNF method was applied in the MS method to investigate whether a similar level of accuracy could be achieved. The frequency detuning, which can be physically interpreted as a way of reducing the amplitude of the nonlinear term based on adapting the effective linear stiffness, is inherent in the DNF method and has been shown to improve the prediction of the harmonic response content. In applying this detuning in the MS method, it was shown that the two methods could be equated, giving identical solutions up to \(\varepsilon ^2\) order.

The DNF is advantageous insofar as a natural detuning approach is intrinsic in its formulation, whereas this is not the case for the MS technique. It is, therefore, the decision of the user as to whether a detuning is utilised to increase the accuracy of the method. Furthermore, it has been demonstrated that the fundamental response prediction is robust to changes in detuning in the DNF method. Since this is not the case for the MS technique, we observe that there is room for further optimisation of the detuning to be applied, which could further increase the accuracy of the method.

To aid the understanding of these methods, as well as the differences in their implementation, Wolfram Mathematica files for the 2DOF case have been provided as open access data files. These closely follow the steps defined in Sect. 2 and are designed to be used in conjunction with this paper to give a practical understanding of each procedure.

## Footnotes

## Notes

### Acknowledgements

S.A. Neild gratefully acknowledges support from the EPSRC via the fellowship EP/K005375/1.

## Supplementary material

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