Abstract
The present study deals with the response of a damped Mathieu equation with hard constant external loading. A second-order perturbation analysis using the method of multiple scales (MMS) unfolds resonances and stability. Non-resonant and low-frequency quasi-static responses are examined. Under constant loading, primary resonances are captured with a first-order analysis, but are accurately described with the second-order analysis. The response magnitude is of order , where is the small bookkeeping parameter, but can become arbitrarily large due to a small denominator as the Mathieu system approaches the primary instability wedge. A superharmonic resonance of order two is unfolded with the second-order MMS. The magnitude of this response is of order and grows with the strength of parametric excitation squared. An nth-order multiple scales analysis under hard constant loading will indicate conditions of superharmonic resonances of order n. Subharmonic resonances do not produce a non-zero steady-state harmonic, but have the instability property known to the regular Mathieu equation. Analytical expressions for predicting the magnitude of responses are presented and compared with numerical results for a specific set of system parameters. In all cases, the second-order analysis accommodates slow time-scale effects, which enable responses of order or . The behavior of this system could be relevant to applications such as large wind-turbine blades and parametric amplifiers.
1 Introduction
This paper represents a second part of the analysis of the Mathieu equation with hard loading. The first part of the study on hard excitation dealt with cyclic loading at the same frequency as the parametric excitation [1]. It also includes a more thorough introduction to the system, most of which is not repeated here. This second part of the study focuses on a constant load.
It has been noted that Eq. (1) is linear and the solution can be expressed as a sum of homogeneous and particular solutions, as q = qh + qp. The particular solution, qp, by linearity, is simply scaled by F0. By linearity, superposition holds for cases when the excitation has additional terms. Thus, effects of this constant loading and previous studies with cyclic loading [1,14] can be directly combined. The qh term exhibits the behavior of the standard Mathieu equation, including instability tongues based at frequency ratios Ω/ω = 2/N, N being a positive integer [15–17], and stable regions with quantifiable decay rates and frequencies [18].
In this work, we analyze Eq. (1) by using second-order expansions of the method of multiple scales (MMS) [15,19], with the aim of capturing resonances that occur due to interactions between the constant load and the cyclic stiffness. A superharmonic resonance of order two had been observed through numerical simulations [2], and also in the system with added non-linearity [6]. The current work characterizes this resonance in the loaded regular Mathieu equation, as well as a primary resonance, and describes the roles of parameters. Initial elements of this work were introduced in Ref. [20], and are more thoroughly developed here, with numerical validations.
2 Initial Analysis of Responses to Constant Loading
In this section, we give a preview of behavior through an example frequency sweep. We then initialize the perturbation analysis, look at non-resonant responses, and then setup the analysis for studying the resonant behavior.
Equation (1) can be non-dimensionalized by letting τ = ωt and q = (F0/ω2)y to obtain , which reveals three independent parameters as δ = γ/ω2, and ρ = Ω/ω. We will analyze Eq. (1), the more familiar form relative to the standard oscillator, and treat it as non-dimensional with ω = 1 and parameters μ, γ, and Ω (and with F0 producing only scaling effects by linearity).
2.1 Preview of Behavior.
Figure 1 gives a preview of potential resonances to be analyzed. The figure shows response “amplitude,” taken as the maximum value of the steady-state response, under simulated frequency sweeps, with F0 = 1, ω = 1, μ = 0.25, and Part (a) shows γ = 0.9, 1.0, and 1.1. At each frequency, Eq. (1) is solved numerically, and allowed to settle close to the steady-state. At a fixed frequency, the response consists of an oscillation about a non-zero mean value, which can be interpreted as a reference equilibrium displacement due to the constant load without parametric excitation. There is significant response behavior at Ω = 1/2, 1, and 2. The superharmonic resonance is considerably smaller than the primary resonance. However, it does demonstrate a significant increase in the oscillatory part of the response, relative to the nearby non-resonant conditions, and could be important if, e.g., reliability and fatigue were a concern. The subharmonic response at Ω = 2 is actually not a large steady-state resonant response, but the result of an instability due to the parametric excitation. Part (b) shows the superharmonic and primary resonant amplitudes (the vertical axis removes the static displacement), for γ = 1, 2, and 3. The plot indicates that increasing γ increases the response at all frequencies, particularly at the resonances. We will analyze these resonances and describe more details about their characteristics. Part (c) shows superharmonic resonances for γ = 3 with decreasing damping. The damping affects resonances, but not off-resonance amplitudes. Superharmonics of orders two, three, and four are observed, the latter of which is only observed if damping is very small. This is consistent with the observations of these and additional superharmonics in numerical simulations of the undamped case [2]. In the following, the resonances of orders one and two will be uncovered and characterized with a perturbation analysis.
2.2 Second-Order Multiple Scales Approach.
We can immediately identify two resonances in which Ω contributes to secular terms (Ω ≈ ω and Ω ≈ 2ω), in addition to the non-resonant case. A superharmonic resonance with Ω ≈ ω/2 will be detected when extending the non-resonant case to the second-order of . Other superharmonic resonance conditions would be revealed at higher orders of expansion. For example, the third-order expansion reveals resonance cases at Ω ≈ ω/3 and Ω ≈ 2ω/3. This can be seen by considering a higher-order expansion with a non-resonant condition until the nth-order. Equation (4) shows that q0 has a constant term and a frequency ω. Therefore the q0 cos ΩT0 term in the equation of Eq. (3) produces exponentials with frequencies of ±Ω and ±Ω ± ω, which then appear in the q1 solution. The q1cos ΩT0 term in the equation of Eq. (3) produces frequencies of ±Ω ± Ω and ±Ω ± Ω ± ω, where we see the possible resonance case of Ω ≈ ω/2 in the first term. These frequencies persist in the non-resonant q2 solution. Continuing without resonances to the nth-order, the equation produces terms of (n occurrences of ± Ω) and (n occurrences of ± Ω) ± ω. The former term is from the constant load and indicates a possible resonance at Ω ≈ ω/n and the latter indicates a possible resonance at Ω ≈ 2ω/n, which is consistent with the unforced Mathieu instability pattern. The case of Fig. 1 suggests that these resonances diminish markedly in the presence of damping. A full nth-order analysis would lead to information about the steady-state responses.
The non-resonant and resonant cases are analyzed with two orders of multiple scale expansions below.
2.3 Non-Resonant Case.
3 Resonant Behavior
Cases of primary resonance, for which Ω ≈ ω, and subharmonic resonance, for which Ω ≈ 2ω, are apparent. A superharmonic resonance will be uncovered later. We will look at these cases in the following.
3.1 Primary Resonance.
3.1.1 First-Order Analysis of Primary Resonance.
3.1.2 Second-Order Analysis of Primary Resonance.
3.1.3 Discussion and Numerical Examples.
Expression (24) makes use of the zeroth-order approximation plus the first-order correction. The correction term, being a constant contribution plus a harmonic of frequency 2Ω, is reminiscent of the effect of a quadratic non-linearity. Based on Eq. (22), and indeed the result (16) of the coarser first-order analysis, we see that the resonance is “nominally” of order one, relative to the bookkeeping parameter . However, as noted, as the instability wedge is approached, the resonant response can be arbitrarily large via a small denominator.
Figure 2(a) shows the amplitude of the frequency response near primary resonance for values of γ ranging from γ = 1 to γ = 4, with and F0 = 1. The solid lines are the numerically simulated sweeps, and the dashed lines are the amplitudes obtained from the approximation of Eq. (24). The response amplitudes were taken as the maximum value of |q(t)| during a single period at steady-state. In the figure, as γ increases, the associated curves increase in amplitude. The error between the simulations and the asymptotic approximation from the second-order perturbation analysis is visually imperceptible, even for γ = 4, which is very close to the instability (near γ = 4.48) for this set of parameters [14], at which the forced solution becomes unbounded. In contrast, while the first-order perturbation uncovers the resonance condition, the approximated amplitude has notable error, both in magnitude and peak location, unless γ is sufficiently small.
Figure 2(b) shows the results of the first-order multiple scales (dotted curve), as well as the q0 (dash-dot) and (dashed) approximations from the second-order multiple scales, the latter of which has a nearly perfect match in comparison to the numerical solutions (solid) in a sweep with γ = 3.
Figure 2(c) indicates the amplitude a of the primary harmonic of the peak response versus γ. The solid curve shows the amplitude as predicted from Eqs. (22) and (23) of the second-order perturbation analysis, while the dashed line graphs the prediction from Eq. (16) of the first-order analysis. They are agreeable for γ up to one or two, and then diverge. The circles denote the primary harmonic amplitudes from the fast Fourier transforms (FFTs) of the steady-state numerical solutions at the associated values of γ. The periodic solutions were sampled at 100 samples per period, which being an integer, led to FFT spectra without leakage distortions. The accuracy of the second-order perturbation is once again striking, for all stable values of γ, especially its values push the interpretation of order-one in the analysis.
Figure 3 shows the amplitude a and phase β of the resonant term in Eq. (24). Although the peak amplitude is offset from Ω = 1, particularly for larger values of γ, the π/2 phase crossover remains close to Ω = 1, and is hardly affected by γ. The response is oppositely phased from the parametric excitation at lower excitation frequencies, meaning that as the stiffness tightens, the mass deflection reduces, similar to the quasi-static trend. Resonance goes through a phase transition, and at high frequencies, the mass moves oppositely from the quasi-static trend (in phase with the cyclic stiffness).
While peak amplitude ap0 of the first-order analysis suggests that the peak scales inversely with the damping coefficient μ, the behavior is less obvious with the amplitude a from Eq. (23) of the second-order analysis. Indeed, the denominator can be zero at the stability transition. For the case of γ = 2, Fig. 4 shows the amplitude and phase versus the excitation frequency for values of μ ranging from 0.25 in the lowest curve, to 0.05 in the curve which peaks off the scale. The phase β demonstrates a regular transition from π to 0 with a crossover at the resonant frequency. The transition becomes more step-like as μ decreases.
Figure 5 shows the peak amplitude of the resonant harmonic versus μ for γ = 2. The solid curve is the amplitude predicted from Eqs. (22) and (23) of the second-order perturbation analysis, and the dashed curve is from Eq. (16) of the first-order analysis. The circles were obtained from the primary harmonic in the FFTs of the simulated responses. The simulations were conducted at the peak frequency as predicted from a numerical evaluation of Eqs. (22) and (23). When μ = 0.04, the tip of the instability wedge (not shown) dips to a value of γ = 2. Thus in the figure, the theoretical solid curve approaches a vertical asymptote at about μ = 0.04. The first-order analysis does not uncover the instability wedge, and thus the dashed line in the figure has a vertical asymptote at μ = 0, the undamped condition.
The peak amplitude amax as a function of μ and γ can be investigated. In Fig. 6, is plotted versus γ and μ. The value of is obtained by numerically evaluating Eqs. (22) and (23) of the second-order perturbation at each value of γ and μ. The parameter range for which the response can be unstable near primary resonance is the black region on the plot. The shaded regions vary from darker shades for lower amplitudes, to lighter shades for higher amplitudes. The peak amplitudes grow as the unstable region is approached in this parameter space.
3.2 Subharmonic Resonance.
We follow the same procedure to show the existence of instability at the subharmonic resonance for Eq. (1). Letting , the solvability condition is obtained by eliminating secular terms from Eq. (5) at , solving for q1 from Eq. (5) after secular terms are removed, inserting q1 into the last of Eq. (3), and then eliminating secular terms from the part of Eq. (3). The subsequent the solvability conditions are
These are the same solvability conditions as were achieved in the analysis of subharmonic resonance of order 1/2 in the case of harmonic forcing [1]. Indeed, in both cases the excitation is not involved in the slow flow. The analysis reveals a stability wedge consistent with previous analyses of the unforced Mathieu equation.
3.3 Superharmonic Resonance of Order Two.
This resonance is not apparent after the first-order of perturbation expansions, namely Eq. (5). So at of the analysis, we use the non-resonant case, and then continue with a second-order analysis.
3.3.1 Second-Order Perturbation Analysis.
From the Jacobian of Eq. (30), it can be shown that the response is stable if μ > 0. It is known that there is a very slender instability wedge based on this resonance condition, but it is not captured by this second-order perturbation analysis.
3.3.2 Numerical Examples.
Figure 7 compares a numerical solution of Eq. (1) in the solid line with q0(t) in the dash-dotted line, and from Eq. (33) in the dashed line. In this case, γ = 2, with F0 = 1 and the parameter values used in the other figures. While q0 provides a rough estimate of the response, it only accommodates the resonating harmonic and the constant bias term, while accommodates harmonics at 3Ω and Ω, and provides an excellent approximation of the true solution, leading to excellent peak-to-peak amplitude information.
Figure 8(a) shows the amplitude of the frequency response near this superharmonic resonance for values of γ ranging from γ = 1 to γ = 4, with F0 = 1 and the parameter values used in the other figures. The solid lines are the numerically simulated sweeps, and the dashed lines are the amplitudes garnered from the first-order approximation in Eq. (33). The response amplitudes were taken as the maximum value of |q(t)| over a period at steady-state. In the figure, as γ increases, the associated curves increase in amplitude value.
Figure 8(b) depicts the peak amplitude a of the resonant term versus γ as obtained from Eq. (32) of the perturbation analysis (solid curve), and from the primary peak of the spectra of numerical solutions (circles). The numerical solutions had an integer number samples (100) per period such that there were no leakage distortions in the FFT computations. The numerical solutions were at the peak frequency calculated from Eq. (32), which is a theoretical estimate. Thus the real peak could be slightly higher than those obtained from the numerical simulations. The analytical expression of the amplitude in Eq. (31) predicts a quadratic dependence on γ, and has a denominator that is positive. Simulations at values of γ ≥ 8.5 suggest a capacity for unbounded responses. Indeed, the Mathieu equation is known to have a slender instability wedge based at Ω = ω/2, and it is likely that the steady-state solutions become unbounded when encroaching upon the instability wedge, as discussed for primary resonance. The second-order perturbation does not unfold this superharmonic instability wedge. Accordingly the amplitude expression does not have a denominator that can go to zero, and thus does not capture this unbounded behavior. As indicated in Sec. 2.2, a fourth-order MMS might capture the second-order superharmonic instability wedge, and therefore may accommodate the nearby arbitrarily large amplitudes, as was seen in the second-order MMS analysis for primary resonance in Sec. 3.1.2. Nonetheless, the asymptotic perturbation analysis is appropriate for small and order-one values of γ, and the values of γ discussed here push these limits.
4 Conclusion
Motivated by the example application of large wind-turbine blades under steady conditions, in which gravity induces cyclic stiffening and aerodynamics provides a mean load, we have looked at the linear Mathieu equation with combined parametric and constant loading. There is a homogeneous solution which has all of the characteristics shown in classical studies of the Mathieu equation, and a particular solution which was then the focus of our attention. Constant loading interacts with cyclic stiffness to produce oscillatory responses. In this study, we performed first- and second-order multiple-scales analyses to describe the resonances.
A primary resonance is detected with a first-order perturbation expansion, which provides an estimate of the response amplitude for small parametric excitation. A second-order analysis, however, generated very accurate results for larger parametric excitation, nearly up to the point of instability. The magnitude of the resonance peak is nominally , although it can reach rather large values via a small denominator. The second-order multiple scales accommodated behavior at the slowest time scale () to properly capture this resonance. The analysis shows that the resonant amplitude simultaneously becomes unbounded and unstable, as the homogeneous Mathieu destabilizes, similar to 1:1 primary and 2:1 subharmonic parametric amplification.
A superharmonic resonance of order two was also revealed with the second-order multiple-scales method. The resonance peak is nominally of order , but still produces noticeably larger responses than the surrounding non-resonant behavior, and thus can be significant if vibration or reliability is a concern. Smaller superharmonics of orders three and four were observed to exist in numerical solutions if damping is small enough, or γ is large enough, but were not revealed with the second-order perturbation analysis. Examination of an nth order multiple scales analysis indicated that hard constant loading will reveal conditions of superharmonic resonances of order n, and independent of external loading, resonant conditions will occur when Ω = 2ω/n, presumably uncovering potential superharmonic instability wedges known to exist in the unforced Mathieu.
The superharmonic resonance of order two was seen in simulations to become large and then unbounded as the parameters presumably approach the instability wedge. However, the second-order perturbation analysis does not capture this very slender and unobtrusive wedge, and likewise does not describe the unbounded aspect of the resonance amplitude, although the asymptotic analysis holds well if γ is not large. It is probable that the resonance goes unbounded at the approach to the instability boundary, as in the primary resonance.
The possible existence of superharmonic resonance may be important to large wind turbines, as they are designed to operate at frequencies well below the first natural frequency, and a small resonance can induce a response that is significant compared to the non-resonant oscillator. The constant load does not induce any subharmonic resonances, although the analysis consistently models the subharmonic instability due to parametric effects, only.
Acknowledgment
This material is based on work supported by National Science Foundation (CBET-0933292, CMMI-1335177). Any opinions, findings, and conclusions or recommendations expressed are those of the authors and do not necessarily reflect the views of the NSF.
Conflict of Interest
There are no conflicts of interest.
Data Availability Statement
The authors attest that all data for this study are included in the paper.