Wave Propagation in Exponentially Varying Cross‐Section Rods and Vibration Analysis

Nikkhah‐Bahrami, Mansour; Loghmani, Masih; Pooyanfar, Mostafa

doi:10.1063/1.2991051

Cited by 9 publications

(8 citation statements)

References 9 publications

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“…Many previous papers have concerned the effects of a linear boundary condition on wave reflection in uniform and non-uniform continuous structures such as rods, plates, Euler-Bernoulli beams and Timoshenko beams (e.g. [1][2][3][4][5][6][7]). The wave propagation approach has also been widely used to study vibrations of nanorods [8] and nanobeams [9] with cracks.…”

Section: Introductionmentioning

confidence: 99%

Reflection of waves in a waveguide from a boundary with nonlinear stiffness: application to axial and flexural vibrations

2022

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The reflection of time-harmonic waves in a waveguide with a nonlinear boundary stiffness is considered with applications to rods and beams. Incident waves at frequencies that are multiples of a fundamental frequency give rise to reflected propagating and nearfield waves at the same frequencies. An infinite set of equations is developed for the reflection coefficients, which depend on the amplitudes and phases of the incident waves. Nonlinear boundary conditions are applied, and equations is truncated by using the harmonic balance method and solved numerically. The case of zero linear boundary stiffness, i.e. essential nonlinearity, is studied. First, the case where there is only one incident wave is considered. An approximate solution is found when retaining only two reflected waves. Numerical examples are presented, energy being seen to leak into the higher harmonics. The minimum magnitudes of the reflection coefficients of axial and flexural vibrational waves at the fundamental frequency and the maximum energy that can leak into the higher harmonics are determined. Accuracy and convergence when retaining different numbers of reflected harmonics are illustrated. The case of two incident waves is then considered. Multiple incident waves affect the leakage of energy to higher harmonics and can have a significant effect on the reflection coefficient for the fundamental harmonic. With some parameters, a much lower reflection coefficient is obtained for the wave at the fundamental frequency as compared to the case of one incident wave. It is seen that with two incident flexural waves, the reflection coefficients can be multi-valued for certain values of the system parameters. A numerical study is performed to show the region of multiple solutions.

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Section: Introductionmentioning

confidence: 99%

Reflection of waves in a waveguide from a boundary with nonlinear stiffness: application to axial and flexural vibrations

2022

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“…Harland et al [25] and Mei [26] presented a systematic technique for describing wave transmission and reflection in uniform 1D waveguides. Lee et al [27], Nikkhah-Bahrami et al [28], and Loghmani [29] used a wave approach to analyse the non-uniform 1D waveguides for which analytical solution is available, such as polynomial or exponential cross-section. In this article, the above-mentioned wave propagation method is applied by the authors to present a modified wave propagation method for the calculation of frequencies and mode shapes of 1D waveguides with an arbitrary variable cross-section for which no analytical solution is available.…”

Section: Introductionmentioning

confidence: 99%

A modified wave approach for calculation of natural frequencies and mode shapes in arbitrary non-uniform one-dimensional waveguides

Khoshbayani-Arani

Rasekh-Saleh

Nikkhah‐Bahrami

2011

Proceedings of the Institution of Mechanical Engineers, Part C:

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In this article, using the wave propagation method, the natural frequencies and mode shapes of an arbitrary non-uniform one-dimensional waveguide are calculated. The non-uniform rods and beams are partitioned into several continuous segments with constant cross-sections, for which there exists an exact analytical solution. At the end of each segment, waves in positive and negative directions are obtained in terms of waves at initial segment and subsequently, the calculations of the mode shapes become simple. By satisfying the boundary conditions, the characteristic equation is obtained and natural frequencies are calculated for both the arbitrary non-uniform rod and beam. Also, by adding waves in positive and negative directions at the end point of each segment, the mode shapes are obtained. To verify the modified wave method presented here, the frequencies and mode shapes of the rod and the beam with a polynomial cross-section having an exact analytical solution are compared and have been proven to be of high accuracy. Besides, comparisons of finite element method are also included. Therefore, this method can also be used to calculate the natural frequencies and mode shapes of rods and beams with any arbitrary variable cross-section for which no analytical solution is available. For the ‘Modified Wave Approach’ developed here, dimensions of transmission matrix remain constant if the number of segments is increased, while in general wave propagation method, dimensions of transmission matrix increase upon increasing the number of segments. Besides this novelty, this method has the advantage that it gives all the natural frequencies and mode shapes, unlike other approximate methods such as weighted residual, Rayleigh–Ritz, and finite difference methods which have their own shortcomings such as limited number of natural frequencies. Also, since each segment has an exact analytical solution, in contrast to other approximate methods, much higher accuracy is obtained even with only a few number of partitions.

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“…They also expressed the solutions in matrix form and introduced the transfer matrices which have broader application and could be coded by computer. By using the Myklstad method and writing transfer matrices for continuous elements, one can calculate natural frequencies of composite shafts or beams [16]. Q.S.Li and colleagues [9] analyzed the longitudinal vibrations of rods having exponential cross-sections.…”

Section: Introductionmentioning

confidence: 99%

“…Researchers have studied wave propagations in elastic solids [1 to 4]. Lee [6], Mei [11], Nikkhah-Bahrami [15] and their colleagues, studied the longitudinal vibration of non-uniform rods by wave methods and they extended the work of Harland [17]. Because the analytical solutions to longitudinal vibrations in rods, having variable crosssections are complex and in many cases impossible, other methods such as approximate solutions are also used.…”

Section: Introductionmentioning

confidence: 99%

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A Modified Wave Approach for the Calculation of Natural Frequencies and Mode Shapes of the Rods with Variable Cross Section

Arani

Saleh

Bahrami

2011

AMM

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Analytical solutions for vibration analysis of the rods with variable cross section are in general complex and in many cases impossible. On the other hand, approximate methods such as the weighted residual, Rayleigh-Ritz and finite difference methods also have their own shortcomings such as a limited number of natural frequencies and accuracy. Using the wave propagation method, the structure is partitioned into several continuous segments with constant cross-section, for which there exists an exact analytical solution. Waves in positive and negative directions at the entrance of each segment are obtained in terms of waves at the initial segment. Then, by satisfying the boundary conditions, the characteristic equation is obtained and all natural frequencies are calculated. By adding waves in positive and negative directions at each point, the shape modes are obtained. To verify this modified method, the frequencies and mode shapes of a rod with polynomial cross section, which has an exact analytical solution, are compared and have proven to be of highly accuracy. Therefore, this method can also be used to calculate the natural frequencies and its mode shapes of the rods with variable cross section for which no analytical solution is available.

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Wave Propagation in Exponentially Varying Cross‐Section Rods and Vibration Analysis

Cited by 9 publications

References 9 publications

Reflection of waves in a waveguide from a boundary with nonlinear stiffness: application to axial and flexural vibrations

Reflection of waves in a waveguide from a boundary with nonlinear stiffness: application to axial and flexural vibrations

A modified wave approach for calculation of natural frequencies and mode shapes in arbitrary non-uniform one-dimensional waveguides

A Modified Wave Approach for the Calculation of Natural Frequencies and Mode Shapes of the Rods with Variable Cross Section

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