An accurate method for transcendental eigenproblems with a new criterion for eigenfrequencies

Qi, Zisong; Kennedy, David; Williams, F.W.

doi:10.1016/j.ijsolstr.2004.01.022

Cited by 14 publications

(5 citation statements)

References 19 publications

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“…(58); additionally, the individual modal contributions (51) are reported for k = 1, 2, ..., 31, while the remaining ones up to M = 131 are omitted for clarity. The two solutions (29) and (54) are in perfect agreement, substantiating the correctness of the two approaches proposed in this paper. The transmittance within the band gap is well lower than the transmittance over the remaining frequency domain, meaning that the wave attenuation properties of the infinite beam (see Figure 3) hold also for the finite beam.…”

Section: -Dof Resonatorssupporting

confidence: 80%

“…For a further insight, transmittance and FRF for the tip deflection under a unit harmonic force at the free end are reported in Figure 9 and Figure 10, respectively. Again, the exact solution (29) and the modal expansion (54) are in perfect agreement, proving the correctness of the two approaches. In this case, the modal expansion (54) represents very accurately both the transmittance and the FRF with M = 161 over the frequency domain 0-890 Hz (see Figure 10a and zoomed view in Figure 10c).…”

Section: -Dof Resonatorsmentioning

confidence: 52%

“…Several eigenvalues are close to each other, as a result of local resonance; remarkably, the algorithm proves capable of capturing also those differing by a few digits. Figure 4 shows the transmittance of the cantilever locally-resonant sandwich beam, as calculated using the exact frequency response (29) with conditions (31)- (32) and the corresponding modal representation (54) including M = 131 modes, where the coefficients χ k are given by Eq. (58); additionally, the individual modal contributions (51) are reported for k = 1, 2, ..., 31, while the remaining ones up to M = 131 are omitted for clarity.…”

Section: -Dof Resonatorsmentioning

confidence: 99%

“…Now, the interest is to calculate the FRF of the cantilever beam acted upon by a unit harmonic force applied at the free end. Figure 5a illustrates the FRF for the tip deflection over the frequency range 0-930 Hz, as computed using the exact solution (29) and the modal representation (54) for M = 131; again, the individual modal contributions (51) are reported for k = 1, 2, ..., 20 and k = 30, 31, 32. The two solutions are in perfect agreement; for the frequency range considered in Figure 5, M = 131 modes are sufficient to provide a very accurate modal representation (54) of the exact solution (29).…”

Section: -Dof Resonatorsmentioning

confidence: 99%

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Free and forced vibrations of damped locally-resonant sandwich beams

Russillo,

Failla,

Fraternali

2021

Preprint

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This paper addresses the dynamics of locally-resonant sandwich beams, where multi-degree-offreedom viscously-damped resonators are periodically distributed within the core matrix. Using an equivalent single-layer Timoshenko beam model coupled with mass-spring-dashpot subsystems representing the resonators, two solution methods are presented. The first is a direct integration method providing the exact frequency response under arbitrary loads. The second is a complex modal analysis approach obtaining exact modal impulse and frequency response functions, upon deriving appropriate orthogonality conditions for the complex modes. The challenging issue of calculating all eigenvalues, without missing anyone, is solved applying a recently-introduced contour-integral algorithm to a characteristic equation built as determinant of an exact frequency-response matrix, whose size is 4 × 4 regardless of the number of resonators. Numerical applications prove exactness and robustness of the proposed solutions.Recently, the concept of locally-resonant beam has been proposed also for sandwich beams, which are ideally suitable to host small resonators within the core matrix, featuring single or multiple degrees of freedom (DOFs). Pioneering work in this field is due to Sun and co-workers [10,[19][20][21][22][23][24]. They proposed an equivalent single-layer Timoshenko beam model coupled with mass-spring subsystems representing the resonators, investigating the dynamic behaviour under different excitations, including impact [21] and moving ones [23]. The mass-spring subsystems were considered as exerting point forces [19,20] or distributed forces over the mutual distance [10,19]. The equivalent single-layer

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Section: -Dof Resonatorssupporting

confidence: 80%

Section: -Dof Resonatorsmentioning

confidence: 52%

Section: -Dof Resonatorsmentioning

confidence: 99%

Section: -Dof Resonatorsmentioning

confidence: 99%

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Free and forced vibrations of damped locally-resonant sandwich beams

Russillo,

Failla,

Fraternali

2021

Preprint

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“…A frame used by Qi et al [3] is shown in Figure 4. The analysis has used units of kN and m, and consistent member properties are EI = 5000 kNm 2 , EA = 900 000 kN and mass per unit length = 0:035 Mgm −1 .…”

Section: Examplementioning

confidence: 99%

A note on finding roots of monotonic branching functions

Lawther

2005

Commun. Numer. Meth. Engng.

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SUMMARYSome physical problems require ÿnding roots of branching functions. Transcendental eigenvalue problems are a common example, where certain functions are known to be branching, monotonic within branches, and have root counting algorithms. Solution conventionally starts by surrounding a required root with bounds on the same branch, and then reÿning one or both of the estimates by conventional or problem-speciÿc algorithms. Here, di culties caused by branching are removed by deÿning a new function that is free of branches and has a single root. The root of this function is found by linear interpolation, with some acceleration applied if high accuracy is wanted. Construction of the new function is completely numerical, as is the root ÿnding acceleration.

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