Analysis of axial waves in viscoelastic complex structural-acoustic systems: Theory and experiment

It is well known that when a solid elastic bar is excited with an axial harmonic force, a set of resonances whose frequencies are equidistant is produced. The intensities and amplitudes of these resonances depend on the energy dissipation present in the bar. However, a lesser-known phenomenon occurs when the bar has a notch near one of its ends so that the resulting system is a sequence of two coupled bars, one small and one long. The perturbation produced by this modification drastically changes the intensity distribution of the resonances, giving rise to a profile of such distribution that resembles a Lorentzian function that is very wide and centered at a frequency close to the resonant frequency of the small bar. What makes this phenomenon particularly interesting is that it occurs in a large number of physical systems: mechanical, classical and quantum electromagnetic, nuclear, etc., where a structure with a very dense eigenspectrum (called a sea of states) is coupled to a system with a low density of eigenvalues, which causes the so-called strength function phenomenon. The intensity distribution of the resonances follows a profile that some authors have called giant resonance, but we will not use this name. Instead, we will use the enveloping function of quasi-Lorentzian shape (EFQS) or doorway state phenomenon (DSP). In a previous experimental and numerical study,¹ the dependence of the EFQS on the different parameters of a cylindrical aluminum bar with a notch was determined. In the present work, we derive analytical expressions that confirm these results. Closed formulas are derived, which give the parameters of the EFQS in terms of energy loss due to internal friction and coupling with the atmosphere.

Specifically, the first objective of this paper is to report the results of experimental measurements and derive analytical expressions for the EFQS of the spectral response of coupled bar systems subject to axial oscillations. This kind of EFQS has been discussed little in the literature. Indeed, while the shape of the curve of a single resonance is a widely discussed topic, the shape of the envelope of a family of resonances is not a sufficiently discussed topic. It is important to study these envelopes because that is what an observer detects when using instruments with an insufficient resolution to see the details of the phenomenon studied. He will not be able to see individual resonances but only one very broad resonance. These envelopes are also important when the only thing that matters is knowing the coarse response of the studied system.

On the one hand, in the case of a single resonance, it is usually found that on sufficiently simple systems and as the energy dissipation tends to zero, the form of the square of the resonance curve has the standard form of a Lorentzian. On the other hand, in the case of the envelopes, the few works that have studied them have provided rather incomplete information. Among the few works that study these envelopes is the one in Ref. 2 and those that are cited there. These works study how the states of systems governed by quantum mechanics having a special state (the doorway state) are distributed. They prove that when the density of the energy spectrum is equal to a constant the distribution is a Breit–Wigner function, which essentially is the same as a Lorentzian function. However, in general, the energy spectra of the systems do not meet this hypothesis, and therefore, that prediction may be wrong. Moreover, these discussions do not provide information about the values that the parameters that characterize the Lorentzian should have.

Additionally, as mentioned above, those studies were only in the context of the quantum mechanics. So, the subject, for elastic systems within the frame of classical mechanics has not been studied. This is done in the present paper. We will study systems where the laws of the elasticity are the ones that govern their movement. The systems are similar to those studied in Refs. 1 and 3–5. They consist of one, two, or three coupled bars subjected to axial vibrations. The structures were designed so that one of the bars forming the composite system has a very high spectral density when considered isolated from the other bars. The other bars have a very low density. We have experimentally observed that in the coupled bars, the spectral density is not equal to a constant. Therefore, the envelope of the single resonances should not be expected to be a perfect Lorentzian.

The second objective of this article is to discuss the connection between two quite different points of view with which the systems considered here can be studied. One of them is based on approximated approaches. Among them are, for example, statistical energy analysis (SEA),⁶ spectral element method (SEM),⁴ fuzzy structure theory (FST),^7–9 Belyaev smooth function approach (BSFA),^10–12 and hybrid method vibration analysis (HMVA).⁵ Several methods are based on the FST, in which it is considered that the system consists of a master structure that has a set of fuzzy couplings connected to it. Usually, the master structure has a small spectral density compared to the spectral density of the fuzzy couplings. In this context, the underlying hypothesis is that the details of the fuzzy couplings are not essential for the overall or macroscopic description of the system.

The other point of view uses an exact approach in which phenomena appear analogous to those discovered in the 1940s and, later, were found to be present in other systems, both quantum^2,13–26 and classical.^1,3,27–29 These are the strength function phenomenon, doorway states, and the EFQS. This approach constitutes an exact and alternative description for coupled bars to the approximate studies based on the fuzzy structure theory mentioned above. Unfortunately, very little has been discussed in the literature about the connection between these two points of view. As in the case of the fuzzy structure theory, the doorway states appear when two or more systems, which have very different spectral density, are brought into interaction, one with a low and usually simple state density, while the other with a high and usually more complicated density of states forms a “sea” of states. As these systems interact, the states of the system with a low-density spectrum act as doorway states to the coupled system. When a doorway is excited, then, as time goes by, the injected energy is distributed very efficiently among the eigenstates of the composite system whose eigenvalues are close to that of the doorway. In addition, the amplitude with which each of them is excited has a quasi-Lorentzian envelope, giving rise to an EFQS.

This article discusses the connection between these two points of view. Experimental results are presented, and two analytical formulations are discussed: one of them an exact formulation and the other an approximated formulation. These mathematical expressions explicitly show that the details of the attached fuzzy couplings have little influence on the gross response of the coupled system, which confirms the hypothesis assumed in the fuzzy structure theory.

In Sec. II, we discuss the structure of the frequency spectrum of the system shown in Fig. 1. Section III deals with the derivation of analytical expressions for the different resonances studied here, including closed formulas for the width (FWHM) of the resonance curves. We also present numerical and experimental results. In particular, Subsections III A and III B focus on the EFQS and on the common resonances, respectively, that exist in bars with a groove. In Subsection III C, it is shown that in bars without a groove, there are no EFQS, so only common resonances are studied. Finally, in Sec. IV, we draw the conclusions.

Here, we discuss the behavior of the system shown in Fig. 1 when excited by compressional waves. The system consists of a thin metallic complex bar formed by three circular cylinders of lengths L, ɛ, and ℓ as shown. The radii of the cylinders are, respectively, r_L, r_ɛ = ηr_L, and r_ℓ = r_L, with η a real number between 0 and 1. The two cylinders of radius r_L have lengths that fulfill L ≫ ℓ. Therefore, the cylinder of length L has a higher spectral density than the others. The small cylinder of length ɛ is called groove, and it is shorter and narrower than the others.

The bar was excited at the right extreme z₃ = L + ɛ + ℓ by an axial force F_ext(t) of the form

for different values of frequency Ω. The response of the bar was studied by analyzing the amplitude of its oscillations at its left end z₀ = 0.

For convenience, in what follows, we will use the name of common resonance (or single resonance) interchangeably to refer to either the oscillation of the bar itself when an oscillatory force is applied with a frequency equal to one of its natural frequencies or the mathematical expression that gives the amplitude of the oscillation as a function of frequency in a range of frequencies around the natural frequency or the plot of this function. This plot will also be called the resonance curve. Furthermore, when we have a function for which it is not certain that it is a Lorentzian but whose graphical representation is similar, we will say that it is a quasi-Lorentzian function.

In Fig. 2, we plot five different curves associated with the response of the bar. We will first discuss the dashed blue curve. This dashed blue line and the blue lines of Figs. 3 and 4 were calculated using Eq. (7), which was taken from a previous paper,¹ omitting the transient [Eq. (21) of that reference]. Its derivation is briefly reproduced in the  Appendix. In Sec. III, three of the other curves will be discussed. The fifth curve (small blue circles) shows the experimental results. The dashed blue curve is, indeed, a sequence of the dashed blue vertical curves whose shape is approximately the same as that of a very narrow Lorentzian. So, each dashed blue vertical line in Fig. 2 is actually a couple of lines, one going up and the other going down. This sequence of quasi-Lorentzians apparently separated from each other actually forms a single continuous line made up of them joined at the bottom. As an example, in Figs. 3 and 4, several of these curves were calculated and are plotted as blue lines with a very elongated horizontal axis scale to clearly show their shape. Each of these curves corresponds to an axial vibrational resonance of the bar, and it is what we call common or single resonance. The horizontal axis shows the values of the frequency f of the exciting force of Eq. (1). The corresponding values of the square amplitude of the acceleration at z₀ = 0 are shown on the vertical axis. At this point, it is convenient to make the following warning: for simplicity, in our analytical discussion, the frequency will be expressed in radians per second and will be denoted as Ω. On the other hand, in the discussion of our figures, the frequency will be expressed in hertz and will be denoted as f, with f = Ω/2π. This is justified because usually, our figures will display readings taken directly from the devices of the laboratory.

The blue dashed line and the small blue circles in Fig. 2 are the theoretical and experimental results, respectively. Both results were previously reported.¹ The dashed blue line reproduces several of the observed effects. The curve is a refinement of the results reported in that reference since a higher precision was used here. This allowed the widths of the 18 common resonances shown in Figs. 3 and 4 to be determined with a great precision. Their values are plotted in Fig. 8 by means of green stars. It is observed that the theoretical widths are much smaller than the experimental widths. However, as we will see later, these differences do not affect the envelope curve of the theoretical and experimental common resonances (black curve).

In Fig. 2, it is seen that the intensity of the common resonances is not uniform, being greater for those around the frequency f_MAX = 53.582 59 kHz, which is a value near to the first resonance frequency f_ℓ = v_c/2ℓ = 50.0 kHz of an isolated cylinder equal to the short cylinder of length ℓ. The value of f_MAX depends on the parameter η in such a way that when η → 0 (that is, when the interaction between the cylinders of length L and ℓ tends to zero), the value of f_MAX gets closer and closer to f_ℓ. Therefore, the complex bar shown in Fig. 1 seems to be intrinsically more efficient in absorbing energy for frequencies near the eigenfrequency f_ℓ. To construct this plot, an effective value of η was used (η_eff = 0.160 ≠ η). The use of an effective η is needed to take into account the nesting (or stretching) effect of cylinder 2 on cylinders 1 and 3 when the bar is being compressed (or stretched) during the axial vibrations.³⁰ The exact position of the dashed blue quasi-Lorentzian curves is very sensitive to small variations of η_eff.

The very particular bell-shaped appearance in which the intensities of the resonances are distributed is the so-called strength function phenomenon, and it occurs when a resonance of the master structure is embedded inside the set of resonances of the attached structures as is the case of the complex bar shown in Fig. 1. In Fig. 2, the envelope of the common resonances, i.e., the EFQS, is represented by the black curve. As mentioned in the section titled Introduction, this envelope is detected as a single very wide resonance without an internal structure when an observer uses instruments with a low resolution;¹⁷ see Figs. 6 and 7 of that reference.

We see then that the formulation derived in Ref. 1 predicts, through a numerical calculation, the existence of the strength function phenomenon. However, in that reference, no analytical expressions were derived for it, which would allow a formal prediction of them and establish analytical relationships between the parameters that characterize them.

Since in the previous paper,^1,3 there was no analytic expression for the envelope curve, it was constructed after numerically determining the set of common resonances. Once all these resonances were determined, their envelope was built by fitting a function, whose shape was suggested by the intensity and distribution of these resonances, using the least square method. The suggested shape was that of a Lorentzian. Something similar was done to determine the envelope of the set of the experimental resonant curves. It was found that both the family of the calculated resonant curves and the family of the experimental resonant curves admit the same envelope. In contrast, in the present paper, the envelope function shown in Fig. 2 was not obtained by any curve fitting. It was calculated with the formula derived below.

The intensities of the common resonances for a homogeneous bar without a groove follow a very different pattern as can be seen by comparing Figs. 2 and 5. In Fig. 5, the vertical dashed blue lines are again very narrow quasi-Lorentzian curves and correspond to the common resonances of the homogeneous bar. We see in Fig. 5 that the intensities for a bar without a groove decreases monotonically. As a matter a fact, the intensity goes as 1/Ω² for all values of Ω. So, the strength function phenomenon is not present. The procedure to obtain this plot will be discussed later in connection with the exact expression (7) with η = 1. In addition, in this case, an expression for the envelope curve (represented by a black line in Fig. 5) is obtained.

Before presenting the derivation of the formalism, we should make the following comments. The narrow quasi-Lorentzians associated with common resonances and the wider quasi-Lorentzian associated with the enveloping function have very different characteristics. The first ones correspond to real oscillations of the elastic body when it is excited by an oscillatory force of frequency equal to one of its normal frequencies. These resonances appear in bars both with and without grooves. On the other hand, the second ones appear only in bars with a groove. Another difference is associated with energy. It is well known in the different fields of physics that as the dissipation of energy increases, the width of the curve of a common resonance also increases. Instead, according to the numerical results,¹ the width of an EFQS is not affected by the loss of energy. As a consequence, the energy loss cannot be obtained by analyzing the EFQS only. Nevertheless, for a given bar with a groove, the presence of an EFQS indicates that in the zone around the EFQS, the energy is absorbed more efficiently by the states whose eigenfrequency is near or equal to the frequency of a normal mode of the master structure.

The model used in this paper to describe the energy dissipation is the Voigt model for axial waves in elastic bodies.^31,32 The model introduces a parameter λ called coefficient of internal friction or coefficient of viscosity. The following equation is the expression for the acceleration at the left end of the bar when it is excited on the right end by the external force F_ext(t) given by Eq. (1). The expression was taken from Ref. 1, where the transitory part has been eliminated. The procedure to obtain this expression is briefly summarized in the  Appendix,

where

where i² = −1. Taking the real part of the right-hand side member Eq. (2), one obtains

where

represents the acceleration amplitude and Re G and Im G represent the real and imaginary parts of G, respectively.

We shall now derive from Eq. (7) the analytical expression for the envelope that we have denoted as the EFQS. This task, as we will see, is neither straightforward nor trivial since the strength function and the EFQS are not explicitly exhibited in the original formulation.

The analytical form of Re G and Im G functions has a very large number of terms. However, taking into account that for the values used in the experiment, E ≫ Ωλ ⇒ Ω²λ² + E² ≈ E², b = λ Ω/E ≪ 1 ⇒ bⁿ ≈ 0 for n ≥ 2, etc., it is possible to make a selection of the significant terms and considerably simplify the expressions. Then, the following approximations are valid:

and Eq. (7) reduces to

where

in which

Here,

Figure 2 show the plots of $A2$ and $Aapprox2$ as a function of frequency f. These plots were built using Eqs. (7) and (11), respectively. The plot of $A2$ is the dashed blue line, and the plot of $Aapprox2$ is the red line. It is clear that the approximate function is very close to the exact one. The figure shows that the same common resonances appear in both graphs. In addition, they are at the same place and with the same intensity. Note that the coefficient λ appears as a linear factor in Eqs. (18)–(21) through the parameter b.

In these equations, two quotients appear and one (ΩL/v_c) is much larger than the other (Ωℓ/v_c). In addition, it is clear that the number of oscillations of the trigonometric functions whose argument contains ΩL/v_c is much larger than when its argument contains Ωℓ/v_c. The trigonometric functions with argument ΩL/v_c repeat their value with opposite sign every time the frequency increases, πv_c/L = 4341 rad/s = 690.9 Hz, which is the average separation between the vertical lines in Fig. 2. This behavior can be explicitly seen in the function Im G′, in which all the arguments of the trigonometric functions contain ΩL/v_c. In Fig. 6, it is seen that, indeed, the variations of Im G′ occur with the same “frequency” as the variations of $Aapprox$⁠. However, one can also see that the zeros of Im G′ are very close to the position of the maxima of $Aapprox$⁠, which fix the points where the envelope forming the EFQS must pass. This property is crucial to obtain the envelope we are looking for, that is, the envelope of the squared common resonance curves. We denote it as $Cgia(Ω)$⁠.

On the other hand, Re G′ is the product of one function that oscillates slowly in space with another that has a large spatial frequency. The first is the modulating function, and the second is the modulated function. Therefore, to obtain $Cgia(Ω)$⁠, the factor $sin2arctanpcps+ΩLvc$ of Eq. (12) is set equal to its maximum value and the resulting function is introduced in the square of Eq. (11) with Im G′ = 0. One obtains

which is one of the expressions we wanted to obtain and is, in fact, one of the most important results of this paper. In Fig. 2, an excellent agreement between $A2$⁠, $Aapprox2$⁠, and their envelope $CgiaΩ$ is seen.

In order to show that a quasi-Lorentzian function is a good approximation to the function given by Eq. (26), we use the Taylor expansion for $ps2$ and $pc2$ of Eqs. (14) and (15) around their minima to obtain

where

Here, n = 1, 2, 3, …, is the index that counts the different doorway states of the system. Using Eqs. (27)–(29) in Eq. (26), one obtains the following approximate expression for $Cgia(Ω)$⁠:

which can be rewritten as

where

Expression (31) is the other expression for the EFQS we were looking for. It is similar to that of a Lorentzian function except that here, α^gia and β^gia are not constants but complicated functions of frequency Ω. Therefore, $Capproxgia(Ω)$ is not an exact Lorentzian function. It is plotted in Fig. 2 by means of magenta points. As we can see, its plot is very similar to a Lorentzian. As it is well known, in a perfect Lorentzian, its full width at half maximum (FWHM) is equal to 2β. However, in our case, we cannot use this equality because β is not constant. Nevertheless, plotting the β^gia function given by Eq. (34) in a frequency interval where the EFQS is appreciably different from zero yields a curve varying between 1.01 and 1.07 kHz, which implies that β^gia varies very slowly compared to $(Ω−αgia)2$⁠. Consequently, β^gia behaves almost as a constant in that interval. Thus, 2β^gia evaluated at the maximum of the EFQS is approximately equal to FWHM of the EFQS with a value 2.1 kHz. Thus,

The envelope shown in Fig. 2 obtained from the analytical expression (26) is indistinguishable from the envelope obtained numerically in Ref. 1. In addition, it was shown in that reference that both the numerical and experimental calculations admit the same envelope curve with the same FWHM^gia; therefore, the above expression reproduces the experimental observation.

From Eqs. (18)–(21), one concludes that the quotients appearing in Eq. (34) do not depend on b. Therefore, β^gia is independent of λ, and consequently, the strength function phenomenon is not a dissipative effect.

Here, we derive a simple and compact function that describes each common resonance. This function will be denoted by $Ccom(Ω)$⁠. Note that in this case, $Ccom(Ω)$ is not an envelope as was in the case of $Cgia(Ω)$ of the EFQS, but rather, the curve itself is associated with a common resonance of the system. It should also be noted that we already have the exact function [Eq. (7)] [or alternatively its approximate form, Eq. (11)] that describes all the common resonances. Then, in principle, one can write that $Ccom(Ω)$ is equal to the square of $A$ or of $Aapprox$⁠. Nevertheless, since neither [Eq. (7)] nor Eq. (11) explicitly shows the value of the width of the resonances and neither show the relationship between this width and the coefficient λ, they are not the expressions we want. Furthermore, these expressions are not as simple or compact as desired. What will be done then is to transform Eq. (11) to obtain a much simpler but approximately equivalent function.

First of all, it should be noted that there are several causes why the bar could dissipate energy when it is oscillating in compression. One of them is the threads used to hang the bar. To analyze this effect, the threads were placed in different places, including in the nodes of vibration where, of course, the influence of the supports is minimal. Essentially, the same resonance curves were obtained in all cases, so this effect can be neglected.

On the other hand, the effect of the air surrounding the bar is not negligible, but its effect can be incorporated into our theoretical description by means of an effective internal friction coefficient as the following two experiments show. In the first one, the bar was placed inside a vacuum chamber, and as the air was removed, the width of the resonances was measured. In the rest of our work, this experiment had to be done with a shorter bar than the one considered due to our limitation of not having a vacuum chamber that could hold a bar of length L + ℓ + ɛ = 3.66 m. Therefore, a bar of only 0.5 m was used. The results are shown in Fig. 7. The vertical axis shows the width of the resonance associated with the compressional mode that has two nodes. The horizontal axis shows the pressure in the chamber. It can be seen that as the air pressure within the chamber decreases, the points tend asymptotically to a horizontal line, which is considerably above the abscissa axis. This means that the bar continues to dissipate energy even though it finds itself in a vacuum. This dissipation must be due to the internal friction in the bar, and it is clear that this effect will be present regardless of the size of the bar. So, it is concluded that the width of the resonances has at least two origins. Therefore, it is to be expected that if the width of a resonance is calculated from expression (7) with a λ value associated only with the internal friction, the experimental width will not be reproduced.

Figure 8 shows the results of the second experiment in which we return to analyze the original bar outside of the vacuum chamber. The figure shows the width values of the 18 resonances considered in Figs. 3 and 4. These widths were calculated for three different values of λ as indicated in the inset. The calculations were done with formula (7). The corresponding widths measured in the laboratory are also shown (blue circles). The value λ = 0.01 Pa s is the one suggested by Auld.³² It is seen that with this value (green stars), the predictions of Eq. (7) are appreciably smaller than the measured widths. This difference, as already mentioned, is no surprising and is not necessarily because the value of λ set by Auld is incorrect, but rather because Eq. (7) was derived considering only the internal friction and not the effects of the air. However, Fig. 8 also shows that it is possible to use an effective λ that takes into account the two effects simultaneously. The yellow asterisks show the prediction of Eq. (7) using λ = 0.12 Pa s, which reproduces very well the linear fitting (yellow line) of the values measured in the laboratory. In what follows, the width of the resonances due to a given λ is what we are interested in describing.

Using Eqs. (11) and (13), we get

where $ReG′2$ is given by Eq. (12). Now, what matters is not the modulator, but the modulated function. Approximating the function $sin2arctanqcqs+ΩLvc$ by the first term of its Taylor series around $Ω=vcLmπ−arctanqcqs$⁠, with m = 1, 2, 3 …, expression (36) reduces to

where

Equation (37) is the expression for the common resonance we were looking for. As was the case of the EFQS, it is similar to that of a Lorentzian function except that α^com and β^com are not constants but complicated functions of frequency Ω. The red points in Figs. 3 and 4 are plots of $Ccom(Ω)$ corresponding to the 18 common resonances of Fig. 2. As we can see, although $Ccom(Ω)$ is not an exact Lorentzian function, its plots (red points) are very similar to it. Furthermore, when they are compared with $A2$ given by Eq. (7) (blue continuous lines), an excellent agreement is evident. Using the same arguments that led to Eqs. (34) and (35), we can establish a similar result for the common resonances, that is,

From Eqs. (12), (14), (15), and (18)–(21), it can be seen that Re G′ depends linearly on b. Therefore, from Eqs. (3) and (39), it follows that β^com is a linear function of λ. Consequently, the width of the common resonances predicted by this model is due to a dissipative effect. Thus, our formulation explicitly reproduces the following two experimental observations:

1. The width of the common resonances depends on the value of λ.

2. The width of the EFQS is independent of λ.

Thus, the EFQS is unaffected by changes in the width of the common resonances. It is also unaffected due to the changes in the atmospheric pressure surrounding the bar. The latter can be concluded by looking at Fig. 8, which shows that by changing the pressure, there is a change in the width of the common resonances, but this, as already said, does not affect the EFQS.

Furthermore, the numerical analysis of the behavior of the function defined in expression (7) shows that the length of the longest bar has a negligible effect on the EFQS (except when the value of η is very small as compared to the value we used). Figure 9 shows that varying the length L changes the separation between the common resonances, but the envelope in all four figures is practically the same. The small differences are only noticeable when doing a more detailed analysis of these envelopes. In the language of the fuzzy structure theory, we can say that the details of the fuzzy couplings (in this case, the length of the longest bar, the internal friction coefficient, and the atmospheric pressure) do not have an important influence on the global or macroscopic behavior of the system. In contrast, the changes in the length of the shortest bar (i.e., the changes in the master structure) do affect the EFQS. It has been observed, in fact, that by changing the length ℓ, the EFQS change their position.

In the previous discussion, we have considered a bar composed of three coupled cylinders because the objective was to analyze the experimental observations and the theoretical study that were made on the system discussed in Refs. 1 and 3. However, the EFQS, doorway states, and the strength function phenomenon also exist in simpler bars consisting of only two coupled cylinders. In fact, if one bar shown in Fig. 1 is removed, for example, the central bar that joins the two end bars of lengths 3.607 and 0.0498 m, it makes the first narrower and the bar shown in Fig. 10 is obtained. Then, when calculating the response of this new system, one obtains Fig. 11(a). The vertical axis, as before, shows the values of the square amplitude of the acceleration at the left end of the system as a function of frequency when the system is excited by the force given in Eq. (1). The presence of three EFQS is observed. For the calculation, Eq. (7) was used, taking one of the lengths equal to zero to adapt it to the case of only two cylinders.

We have also used our formulation to compare the predictions of the fuzzy structure theory with the exact calculation. For this, we have considered again the bar shown in Fig. 10 but now with the values used in Ref. 4. The results are shown in Figs. 11(b) and 11(c). The presence of two EFQS is observed. Figure 11(c) should be compared with fig. 4 of Ref. 4, where the results of the fuzzy structure theory are presented. Both studies show that the response of the system is appreciably higher around the frequencies 39 and 75 kHz. This comparison convinces us of the usefulness of the fuzzy structure theory and confirms its validity. Nevertheless, as expected, the details are very different. However, that should not cause any concern because, from the beginning, the fuzzy structure theory establishes that its objective is to obtain global results leaving aside the details. Therefore, the comparison is very satisfactory.

Here, we analyze the bar without a groove used in Subsection III B, which was utilized to study the effect of the friction caused by the atmospheric pressure. In what follows, the envelope curve of their resonances will be obtained. Because a bar without a groove can be considered as a particular case of a bar with a groove when η = 1, we can use the expressions obtained previously and take η = 1. Then, the expression for the acceleration at the left end of the bar is again given by Eqs. (6) and (7). However, in this case, the function G is much simpler. By making η = 1 in Eq. (5), we get c₁ = 1 and c₂ = c₃ = c₄ = 0. Therefore,

where $L=L+ℓ+ε$ is the total length of the bar without a groove. Its radius will be denoted by R. Then, Eq. (42) can be written as

Using the equalities

we obtain

Since b = λ Ω/E ≪ 1, we use the following approximations:

which implies

The substitution of this into Eqs. (45) and (46) gives

With these values Eq. (11), the expression for the acceleration amplitude (denoted as $Aapprox′$⁠) becomes

This is another of the expressions that we wanted to obtain. We see that the relative maxima of $Aapprox′$ occur when $cos2ΩLvc=0$ and when $sin2ΩLvc=0$⁠. However, since the value of the factor

is negligible as compared to 1 (it is on the order of 10⁻¹⁰), it follows that the maxima of $Aapprox′$ due to the zeros of $cos2ΩLvc$ are negligible as compared to the maxima due to the zeros of $sin2ΩLvc$⁠. Therefore, significant maxima of $Aapprox′$ occur only when $sin2ΩLvc=0$⁠. These values occur with a frequency $πvc/L$ that is equal to the separation between the dashed blue vertical lines in Fig. 5. In these maxima, the value of $Aapprox′$ is

and therefore, the equation for the envelope that passes through these maxima is

This function is plotted in Fig. 5 as a black line. It is clearly shown that this line passes through the maximum values of each resonance. Since for this bar, the envelope is described by an expression that depends on the inverse square of the frequency, the strength function phenomenon is not present. Note that removing the fast oscillations from Eq. (47), setting $cos2(ΩL/vc)=1$ and $sin2(ΩL/vc)=0$⁠, for the purpose of obtaining the envelope, is similar to what was done in the general case, η ≠ 1, when the EFQS was obtained.

It is worth noting that the envelope shape for the case of bars without a groove has an important influence on the envelope shape for the case of bars with a groove. Indeed, the decreasing shape of the black curve in Fig. 5 is responsible for the asymmetry of the black curve in Fig. 2 (the left tail is higher than the right tail). This effect can be seen also in Fig. 11(b).

We will now derive an approximate and compact expression for Eq. (47) that explicitly shows its similarity to a Lorentzian function. To do this, we analyze the behavior of expression (47) in the vicinity of the natural frequencies of the bar. These are $Ωn=2πfn=πvcn/L$⁠. Therefore, only values of Ω within the interval

will be considered, being Δ a small but appropriate frequency interval. So, Ω = Ω_n ± δ, with 0 ≤ δ ≤ Δ. Then,

and

Substituting these equalities in Eq. (47), one obtains

and

where

Equation (55) is the approximate expression for the curve associated with the common resonance centered at Ω_n and it is another of the expressions that we wanted to obtain. The plot of expression (55) reproduces each of the blue lines in Fig. 5 very well. These lines were calculated with the exact formula (7) with η = 1. As was the case of the other expressions for the resonant curves, expression (55) is not an exact Lorentzian function because β′ is not a constant. However, using again the same arguments that led to Eqs. (34) and (35), we can establish the following result for the full width at half maximum, denoted as FWHM′, of the common resonances for a bar without a groove,

This result explicitly shows that the coefficient λ determines the width of the common resonances. This, of course, was to be expected based on the previous discussion of the bars with a groove.

The analytical expressions for the different resonances and their envelopes present in vibrating elastic systems, consisting of coupled rods, have been derived. The most important of these expressions is the one associated with the EFQS, that is, the analytical expression for the envelope of the common resonances for a bar with a groove, which has not been previously discussed in the literature. It was shown that in these systems, one of the rods provides the doorway state and the others the sea of states within which the external excitations are distributed, giving rise to an EFQS. This same situation, contemplated from the point of view of the fuzzy structure theory, shows that in the system of coupled rods, one of them acts as the master structure and the others as fuzzy attachments. Nevertheless, in our case, the exact expressions obtained allow us to verify that the approximate predictions of the fuzzy theory applied to the system of coupled bar are reasonable.

The closed expressions for the FWHM′ of the resonance curves were also derived. The phenomenon of the strength function was also analyzed. In the case of common resonances, the internal friction coefficient λ of the Voigt model explicitly appears in the expression for FWHM^com, meaning that the width of the common resonances is due to the energy dissipation. On the other hand, in the case of the EFQS, the internal friction coefficient does not appear in the expression for FWHM^gia, meaning that the strength function phenomenon is not a dissipative effect.

The conclusions of our research can be expressed in the language of the fuzzy structure theory as follows. The effect of fuzzy couplings is to distribute the excitations applied to the master structure, among the states of the composite system, in such a way that the intensity with which they are excited has an envelope with a quasi-Lorentzian shape. The width of this quasi-Lorentzian is independent of the dissipation factor of the master structure.

If one observer attempts to measure a microscopic property with a macroscopic meter (with a low resolution), it will not be possible to distinguish the individual resonances and the observer will see the set of these resonances as a single very wide resonance.

The formulation derived here is a continuation of a previous study,¹ in which the EFQS was also discussed but without having an analytical expression to describe it. In this work, this concept is discussed from analytical, numerical, and experimental perspectives. In conclusion, for the elastic coupled bars case with a doorway state, the EFQS is fully understood.

The authors want to acknowledge Project No. DGAPA-PAPIIT IN111019.

J.A.R. thanks CONACYT-Mexico for the fellowship for doctoral studies.

E.A.C. acknowledges CONACYT-Mexico for the postdoctoral research fellowship at IFUNAM.

The authors have no conflicts to disclose.

All authors contributed equally to this work.

J. A. Rojas: Conceptualization (equal); Formal analysis (equal); Investigation (equal). A. Morales: Conceptualization (equal); Investigation (equal); Methodology (lead). L. Gutiérrez: Investigation (equal). J. A. Otero: Methodology (lead). E. A. Carrillo: Investigation (equal); Supervision (equal); Visualization (equal); Writing – original draft (supporting); Writing – review & editing (lead). G. Monsivais: Conceptualization (equal); Formal analysis (equal); Investigation (equal); Methodology (lead); Writing – original draft (lead); Writing – review & editing (supporting). J. Flores: Conceptualization (equal); Investigation (equal).

The data that support the findings of this study are available from the corresponding author upon reasonable request.

This appendix briefly reproduces the derivation of the formulas governing the compressional oscillations in a bar with a groove developed in Ref. 1. The starting point is Newton’s second law applied to the description of compressional oscillations in a circular cylinder whose cross-sectional area is equal to a,³² 

where σ is the stress, A(z, t) is the displacement in time t (suffered by the portion of material originally located at point z on the axis Z) when the bar is oscillating, S = ∂A/∂z is the strain, and F(z, t)/a is the external force applied per unit volume. The speed of the deformation is ∂A(z, t)/∂t. The speed of propagation of the waves will be denoted as v_c. Because the excitation acts continuously, the dissipation of energy plays an important role in the experiment by preventing an unlimited growth of the response. In the analytical description of the phenomenon, this effect was included by means of a model. In the literature, there are several models. In the formulation of Ref. 1, the Voigt viscoelastic model was used. It consists of assuming that the total stress applied σ is the sum of the stress associated with deformation and the stress associated with viscosity. Thus, the constitutive relation is^31–33 

where λ is the coefficient of viscosity. Substituting the definition of S in Eq. (A2), we have

Then, the equation of motion that governs the compressional vibrations, in each cylinder, is

It is easy to see that when the bar is excited at its right extreme z = L + ɛ + ℓ by the force given by Eq. (1), the function F(z, t) of the above equation must be equal to F(z, t) = h₀ sin(Ωt)δ(x − L − ɛ − ℓ)Θ(t). If we take the Laplace transform of Eq. (A4) for each cylinder, then solve the three resulting equations, and apply the boundary conditions, we obtain the Laplace transform for the function A(z, t) for the full bar,

Finally, the expression for the acceleration ∂²A(z, t)/∂t² was obtained using the Mellin inversion integral with the Bromwich contour. The expression turned out quite complicated [see Eq. (21) of Ref. 1], but if one ignores the transitory part, the following expression for the acceleration at the left end of the bar is obtained, which, although still complicated, is more manageable,

where

and

being $aL=πrL2$ the cross-sectional area of cylinders 1 and 3.