The Exact Solutions for the Natural Frequencies and Mode Shapes of Non-Uniform Beams With Multiple Spring–mass Systems

Chen, D.-W.; Wu, Jong‐Shyong

doi:10.1006/jsvi.2001.4156

Cited by 45 publications

(15 citation statements)

References 17 publications

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“…The literature concerning free vibration analyses of uniform or non-uniform beams carrying single or multiple spring-mass systems is plenty [1][2][3][4][5][6][7][8][9][10][11][12][13]. However, among the existing reports [1][2][3][4][5][6][7][8][9][10][11][12][13], most of them neglect the mass of helical spring of each spring-mass system except Refs.…”

Section: Introductionmentioning

confidence: 95%

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Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered

Hsu

2007

International Journal of Mechanical Sciences

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Section: Introductionmentioning

confidence: 95%

“…However, among the existing reports [1][2][3][4][5][6][7][8][9][10][11][12][13], most of them neglect the mass of helical spring of each spring-mass system except Refs. [10][11][12][13].…”

Section: Introductionmentioning

confidence: 96%

Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered

Hsu

2007

International Journal of Mechanical Sciences

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“…Broad surveys and literature on these subjects can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14].…”

Section: Introductionmentioning

confidence: 99%

Steady State Response and Stability of an Elastically Restrained Tapered Beam

Abdel-Jaber¹,

Al-Qaisa²,

Beale³

2014

Volume 10 Number 1

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An analytical method for the study of the nonlinear forced vibrations and their stabilities of an elastically restrained tapered cantilever beam due to a direct periodic excitation is developed. The method of harmonic balance is used to study the steady state frequency response of the beam system for different values of physical parameters such as the root translational and rotational stiffness and the beam taper ratio. Results are presented for the first three modes of vibration. The stability of the frequency response for some selected values of the physical parameters is investigated, i.e. the regions on the frequency response curves at which the solution may bifurcate and then culminate into chaos. The qualitative features of the solutions are studied and identified using phase plane, Poincare maps and Fast Fourier Transform. The results are presented, discussed and conclusions on the elastically restrained tapered beam nonlinear dynamics are drawn.Keywords: Forced vibration, tapered beam, elastically restrained, stability, period doubling, chaos INTRODUCTIONBeams with continuous changes in cross sectional area along with the beam axis, i.e. tapered beams, have been extensively used in many civil and mechanical engineering structures such as; such as offshore structure piles, oil platform supports, oil-loading terminals, tower structures, bridges, lighting standards, rotating tapered blades and moving/rotating arms. Tapered beams are also increasingly being used in the construction industry, due to their particular specifications and unique ability to combine efficiency, economy and aesthetic architectural needs. They have the capability to optimise weight and strength despite the cost of fabrication of tapered members.Since tapered beams can model many engineering structures that require variable stiffness along the beam axis "length", designs and analyses of such structures have attracted considerable interest and many researchers have directed their efforts and much research have been devoted towards the mathematical modelling and dynamic behaviour.Studying the dynamic analysis and vibrational behaviour of tapered beams enables the predetermination of undesirable behaviour such as resonance, large vibration levels and unstable vibration behaviour. The results of which will help designers and engineers predict the dynamic behaviour and enable them to impose suitable vibration control strategies to achieve optimum and safe behaviour.Having realized the vital role of the accurate mathematical modelling of tapered beams and having recognized their importance in studying dynamic behaviour researchers published many studies and reported analysis to deal with calculating natural frequencies and modes shapes either for small or large amplitude vibrations. Broad surveys and literature on these subjects can be found in [1][2][3][4][5][6][7][8][9][10][11][12][13][14].

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“…Most of the pertinent literature is directed towards the calculation of linear natural frequencies and mode shapes [1][2][3][4][5][6] with different end conditions and with attached inertia elements at the free end of the beam. In [7], a simple formulation for the large amplitude free vibrations of tapered beams was presented.…”

Section: Introductionmentioning

confidence: 99%

“…But in many cases, the deflection in structures may reach large values and consequently, using the linear vibration assumption is not valid. In order to take into consideration the nonlinearities arised due to large deformations, the nonlinear vibration theory must be used to predict with high accuracy the dynamic behavior like; natural frequencies and dynamic responses.In this paper, the non-linear planar large amplitude free vibration of a tapered cantilever beam is studied for two cases; double tapered beam and single tapered 'wedge shaped beam".Most of the pertinent literature is directed towards the calculation of linear natural frequencies and mode shapes [1][2][3][4][5][6] with different end conditions and with attached inertia elements at the free end of the beam. In [7], a simple formulation for the large amplitude free vibrations of tapered beams was presented.…”

mentioning

confidence: 99%

Nonlinear Natural Frequencies of a Tapered Cantilever Beam

Abdel-Jaber¹,

Al-Qaisia

2009

Volume 5 Number 3

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ABSTRACT:The non-linear natural frequencies of the first three modes of a clamped tapered beam are investigated. The mathematical model is derived using the Euler-Lagrange method and the continuous system is discretized using the assumed mode method. The resulted uni-modal nonlinear equation of motion was solved using the harmonic balance (HB) to obtain approximate analytical expressions for the nonlinear natural frequencies. Results were obtained for two types of taper; double taper, i.e. the beam width and thickness are varied linearly along the beam axis and single taper "wedge shaped beams", i.e. the variation is in thickness only. The effects of vibration amplitude and taper ratio on the nonlinear natural frequencies for the first three modes are obtained and presented in non-dimensional form.Keywords: Nonlinear, free vibration, harmonic balance, tapered beam, cantilever beam INTRODUCTIONIt is known that a lot of engineering structures can be modeled as beam. Some can be modeled as tapered beams, such as piles, fixed-type platforms, tower structures, high buildings and robot arms.In general, due to various excitation loads "wind and waves", high aspect ratio and flexibility such structures might have large deformations and deflections. The prediction of the dynamic behavior is extremely important during the design process.The linear vibration theory predicts the natural frequencies to be independent of the amplitude. But in many cases, the deflection in structures may reach large values and consequently, using the linear vibration assumption is not valid. In order to take into consideration the nonlinearities arised due to large deformations, the nonlinear vibration theory must be used to predict with high accuracy the dynamic behavior like; natural frequencies and dynamic responses.In this paper, the non-linear planar large amplitude free vibration of a tapered cantilever beam is studied for two cases; double tapered beam and single tapered 'wedge shaped beam".Most of the pertinent literature is directed towards the calculation of linear natural frequencies and mode shapes [1][2][3][4][5][6] with different end conditions and with attached inertia elements at the free end of the beam. In [7], a simple formulation for the large amplitude free vibrations of tapered beams was presented. The method is based on an iterative numerical scheme to obtain results for tapered beams with rectangular and circular cross sections.The objective of the present work is to extend the results and analysis obtained in [8] to study the non-linear planar large amplitude free vibrations of a cantilever tapered beam for the cases of a double taper beam and a single taper "wedge shaped beam". The mathematical model is derived

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The Exact Solutions for the Natural Frequencies and Mode Shapes of Non-Uniform Beams With Multiple Spring–mass Systems

Cited by 45 publications

References 17 publications

Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered

Free vibration analyses of simply supported beams carrying multiple point masses and spring-mass systems with mass of each helical spring considered

Steady State Response and Stability of an Elastically Restrained Tapered Beam

Nonlinear Natural Frequencies of a Tapered Cantilever Beam

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