TY - JOUR

T1 - Free in-plane vibration analysis of a curved beam (arch) with arbitrary various concentrated elements

AU - Wu, J. S.

AU - Lin, F. T.

AU - Shaw, H. J.

N1 - Funding Information:
This work is a part of the project under NSC 98-2221-E-006-258 supported by the National Science Council, Republic of China .

PY - 2013/8/1

Y1 - 2013/8/1

N2 - In the existing literature, the information regarding the exact solutions for free in-plane vibrations of the curved beams (or arches) carrying various concentrated elements is rare, particularly for the case with multiple attachments including eccentricities and mass moments of inertias. For this reason, this paper aims at presenting an effective approach to tackle the title problem. First of all, the un-coupled equation of motion for the circumferential displacement of an arch segment is derived. Next, based on the value of the discriminate parameter for a cubic equation, the exact solutions for the three types of roots of the un-coupled equation are determined and, corresponding to each type of roots, all displacement functions for the arch segment in terms of the real numbers (instead of the complex ones) are obtained. Finally, use of the compatible equations for the displacements and slopes together with the equilibrium equations for the forces and moments at each intermediate node and two ends of the entire curved beam, a frequency equation of the form {divides}H(ω){divides}=0 is obtained. It is found that the conventional approach by using the condition "{divides}H(ωt){divides}≤ε" to search for the approximate value of ωt is difficult even if the convergence tolerance ε is greater than 10+3 (i.e., ε>10+3) instead of less than 10-3 (i.e., ε<10-3), however, the half-interval method is one of the effective tools for solving the problem if all coefficients of the determinant {divides}H(ω){divides} are the real numbers. In addition to comparing with the existing literature, most of the numerical results obtained from the presented method are compared with those obtained from the conventional finite element method (FEM) and good agreement is achieved.

AB - In the existing literature, the information regarding the exact solutions for free in-plane vibrations of the curved beams (or arches) carrying various concentrated elements is rare, particularly for the case with multiple attachments including eccentricities and mass moments of inertias. For this reason, this paper aims at presenting an effective approach to tackle the title problem. First of all, the un-coupled equation of motion for the circumferential displacement of an arch segment is derived. Next, based on the value of the discriminate parameter for a cubic equation, the exact solutions for the three types of roots of the un-coupled equation are determined and, corresponding to each type of roots, all displacement functions for the arch segment in terms of the real numbers (instead of the complex ones) are obtained. Finally, use of the compatible equations for the displacements and slopes together with the equilibrium equations for the forces and moments at each intermediate node and two ends of the entire curved beam, a frequency equation of the form {divides}H(ω){divides}=0 is obtained. It is found that the conventional approach by using the condition "{divides}H(ωt){divides}≤ε" to search for the approximate value of ωt is difficult even if the convergence tolerance ε is greater than 10+3 (i.e., ε>10+3) instead of less than 10-3 (i.e., ε<10-3), however, the half-interval method is one of the effective tools for solving the problem if all coefficients of the determinant {divides}H(ω){divides} are the real numbers. In addition to comparing with the existing literature, most of the numerical results obtained from the presented method are compared with those obtained from the conventional finite element method (FEM) and good agreement is achieved.

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U2 - 10.1016/j.apm.2013.02.029

DO - 10.1016/j.apm.2013.02.029

M3 - Article

AN - SCOPUS:84880590599

VL - 37

SP - 7588

EP - 7610

JO - Applied Mathematical Modelling

JF - Applied Mathematical Modelling

SN - 0307-904X

IS - 14-15

ER -