Steady-state periodic solutions of the nonlinear wave propagation problem of a one-dimensional lattice using a new methodology with an incremental harmonic balance method that handles time delays

Wang, Xuefeng; Zhu, Weidong; Liu, Mao

doi:10.1007/s11071-020-05535-4

Cited by 10 publications

(8 citation statements)

References 28 publications

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“…A set of the model parameters are given in Table 4, which will be used in the calculation and analysis of the diatomic chain system. The frequency ωn is a reference frequency that is given in [36], and the ratio of the system frequency to the reference frequency show that the convergence rate of given frequencies is faster (M ≥3) than that of given wave vectors (M ≥5), which is consistent with that of the wave problem in Section 3.1.…”

supporting

confidence: 71%

“…and according to the conclusion of improved IHB method in 1D monatomic lattices [36], the steady-state solution of the j-th unit can be written as…”

Section: Transformation Of Multi-dof Wave Problems To Ddesmentioning

confidence: 99%

“…Wang [36] developed a new IHB method to solve wave problems of 1D nonlinear single-degreeof-freedom (single-DoF) systems. Using the new IHB method, Song [37] studied elastic wave propagations and control in 2-DoF systems.…”

Section: Introductionmentioning

confidence: 99%

“…Thus, this method significantly reduces the number of basis functions, as well as the dimension of general coordinates, in a IHB method, and is capable to solve wave responses at a given excitation frequency. At present, the method was used to calculate wave responses of 1D single-DoF and two-DoF systems [36,37]. Multi-degree-of-freedom (multi-DoF) systems are more general and complicated due to coupling of DoFs.…”

Section: Introductionmentioning

confidence: 99%

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Steady-State Wave Responses of One-Dimensional Multi-Degree-of-Freedom Lattices with Strong Nonlinearities by an Improved Incremental Harmonic Balance Method

Wang

Zhao

et al. 2022

Preprint

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The nonlinearity plays an important role in modulation of wave propagation characteristics, such as the adjustment of band structures by media nonlinearities. There was a modified incremental harmonic balance (IHB) method developed to hand single- and two-degree-of-freedom (2-DoF) wave propagation problems with nonlinearities. When the nonlinearity is strong, high-order basis functions are used to achieve accurate solutions, which requires complex formula derivation and high computational effort, especially for a general multi-DoF wave problem. To handle the issue, an improved IHB method is developed for solving multi-DoF wave problems with strong nonlinearities. In the method, construction of analytical Jacobian matrices of delay differential equations that are transformed from original wave problems is formulated for any DoFs and expansion orders, by which no manual derivation and numerical integration are needed. Thus, computational effort can be significantly reduced, and convergence analysis of solutions obtained from the IHB method with different expansion orders can be conducted to determine the minimum order of stable solutions, which has not been done for wave problems with strong nonlinearities. Moreover, the algorithm to solve nonlinear wave problems with given frequencies and unknown wave vectors is newly proposed in this work based on the improved IHB method, which is can be a more efficient approach if external excitations are given. In this work, one-dimensional diatomic lattices with Hertz contact law and cubic spring models are studied as examples, where dispersion curves and bandgap properties are generated. Results show that it is necessary to use high-order Fourier functions to obtain accurate steady-state wave responses when strong nonlinearities exist, which was not explicitly recognized in past works without convergence analysis for wave problems. Results also show that the convergence rate of the method with use of a fixed frequency is faster than that of a fixed wave vector.

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supporting

confidence: 71%

“…and according to the conclusion of improved IHB method in 1D monatomic lattices [36], the steady-state solution of the j-th unit can be written as…”

Section: Transformation Of Multi-dof Wave Problems To Ddesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Steady-State Wave Responses of One-Dimensional Multi-Degree-of-Freedom Lattices with Strong Nonlinearities by an Improved Incremental Harmonic Balance Method

Wang

Zhao

et al. 2022

Preprint

Self Cite

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show abstract

“…In recent years, the nonlinear phononic crystals with further phenomenal properties have been introduced and the nonlinearity-induced properties of such structures were investigated from several points of view [17][18][19][20][21]. The studies revealed that nonlinear phononic crystals can show exquisite phenomena such as amplitude-induced stop-bands [22,23], internal resonance [24], and non-reciprocity [25][26][27][28] which were not observed in linear phononic crystals.…”

Section: Introductionmentioning

confidence: 99%

Manipulation of wave motion in smart nonlinear phononic crystals made of shape memory alloys

2021

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Thanks to the functional role of shape memory alloys (SMAs) in controlling the mechanical behavior of structures, researchers have started investigating the possibility of manipulating wave motion in phononic crystals using SMAs. While SMAs were used before to tune the wave propagation in linear phononic crystals, in this work, we aim to extend their utilization to nonlinear lattices. For this purpose, SMA helical springs are used to manipulate the dispersion curves and the location of stop-bands in weakly nonlinear monoatomic and diatomic lattice chains. Using Brinson’s formulation to describe the thermo-mechanical behavior of SMA wires and Lindstedt-Poincaré method to solve the derived governing equations, closed-form nonlinear dispersion relations in monoatomic and diatomic lattice chains are obtained and the effects of temperature-induced phase transformation and stiffness nonlinearity on the wave propagation are investigated. The results reveal that the dispersion curves of a weakly nonlinear monoatomic chain are formed at lower frequencies through the austenite-to-martensite phase transformation. Similarly, both the acoustic and optical branches of a diatomic lattice are moved to lower frequencies during the phase transformation in the cooling process. Therefore, the generated stop-bands in nonlinear diatomic lattices are also moved to lower frequencies. In addition, using auxiliary SMA ground springs, new classes of nonlinear monoatomic and diatomic chains exhibiting additional low-frequency attenuation zones are introduced. These low-frequency stop-bands are tunable and their frequency range can be modulated by exploiting the temperature-induced phase transformation in the SMA springs. The results obtained from analytic formulations are verified by numerical calculations and an excellent agreement is observed. Such tunability and the potential for adding stop-bands in low frequencies reveal that SMAs can be very helpful in designing nonlinear phononic and acoustic devices, such as vibration mitigators and wave filters with pre-defined attenuation zones.

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Nonlinear wave dispersion in monoatomic chains with lumped and distributed masses: discrete and continuum models

Ghavanloo,

El-Borgi

2024

Appl. Math. Mech.-Engl. Ed.

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Steady-state periodic solutions of the nonlinear wave propagation problem of a one-dimensional lattice using a new methodology with an incremental harmonic balance method that handles time delays

Cited by 10 publications

References 28 publications

Steady-State Wave Responses of One-Dimensional Multi-Degree-of-Freedom Lattices with Strong Nonlinearities by an Improved Incremental Harmonic Balance Method

Steady-State Wave Responses of One-Dimensional Multi-Degree-of-Freedom Lattices with Strong Nonlinearities by an Improved Incremental Harmonic Balance Method

Manipulation of wave motion in smart nonlinear phononic crystals made of shape memory alloys

Nonlinear wave dispersion in monoatomic chains with lumped and distributed masses: discrete and continuum models

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