Free vibration of axially loaded Reddy-Bickford beam on elastic soil using the differential transform method

Yesilce, Yusuf,Catal, Seval Techno-Press 2009 Structural Engineering and Mechanics, An Int'l Jou Vol.31 No.4

The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko beams on elastic soil is plenty, but the free vibration analysis of Reddy-Bickford beams on elastic soil with/without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded Reddy-Bickford beam on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton's principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of one end fixed and the other end simply supported Reddy-Bickford beam on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed and the mode shapes are presented in graphs.

Differential transform method for free vibration analysis of a moving beam

Yesilce, Yusuf Techno-Press 2010 Structural Engineering and Mechanics, An Int'l Jou Vol.35 No.5

In this study, the Differential Transform Method (DTM) is employed in order to solve the governing differential equation of a moving Bernoulli-Euler beam with axial force effect and investigate its free flexural vibration characteristics. The free vibration analysis of a moving Bernoulli-Euler beam using DTM has not been investigated by any of the studies in open literature so far. At first, the terms are found directly from the analytical solution of the differential equation that describes the deformations of the cross-section according to Bernoulli-Euler beam theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the differential equation of the motion. The calculated natural frequencies of the moving beams with various combinations of boundary conditions using DTM are tabulated in several tables and are compared with the results of the analytical solution where a very good agreement is observed.

Differential transform method and numerical assembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and rotary inertias

Yesilce, Yusuf Techno-Press 2015 Structural Engineering and Mechanics, An Int'l Jou Vol.53 No.3

Multiple-step beams carrying intermediate lumped masses with/without rotary inertias are widely used in engineering applications, but in the literature for free vibration analysis of such structural systems; Bernoulli-Euler Beam Theory (BEBT) without axial force effect is used. The literature regarding the free vibration analysis of Bernoulli-Euler single-span beams carrying a number of spring-mass systems, Bernoulli-Euler multiple-step and multi-span beams carrying multiple spring-mass systems and multiple point masses are plenty, but that of Timoshenko multiple-step beams carrying intermediate lumped masses and/or rotary inertias with axial force effect is fewer. The purpose of this paper is to utilize Numerical Assembly Technique (NAT) and Differential Transform Method (DTM) to determine the exact natural frequencies and mode shapes of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and/or rotary inertias. The model allows analyzing the influence of the shear and axial force effects, intermediate lumped masses and rotary inertias on the free vibration analysis of the multiple-step beams by using Timoshenko Beam Theory (TBT). At first, the coefficient matrices for the intermediate lumped mass with rotary inertia, the step change in cross-section, left-end support and right-end support of the multiple-step Timoshenko beam are derived from the analytical solution. After the derivation of the coefficient matrices, NAT is used to establish the overall coefficient matrix for the whole vibrating system. Finally, equating the overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the differential equations of the motion. The calculated natural frequencies of Timoshenko multiple-step beam carrying intermediate lumped masses and/or rotary inertias for the different values of axial force are given in tables. The first five mode shapes are presented in graphs. The effects of axial force, intermediate lumped masses and rotary inertias on the free vibration analysis of Timoshenko multiple-step beam are investigated.

Differential transform method for free vibration analysis of a moving beam

Yusuf Yesilce 국제구조공학회 2010 Structural Engineering and Mechanics, An Int'l Jou Vol.35 No.5

In this study, the Differential Transform Method (DTM) is employed in order to solve the governing differential equation of a moving Bernoulli-Euler beam with axial force effect and investigate its free flexural vibration characteristics. The free vibration analysis of a moving Bernoulli-Euler beam using DTM has not been investigated by any of the studies in open literature so far. At first, the terms are found directly from the analytical solution of the differential equation that describes the deformations of the cross-section according to Bernoulli-Euler beam theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the differential equation of the motion. The calculated natural frequencies of the moving beams with various combinations of boundary conditions using DTM are tabulated in several tables and are compared with the results of the analytical solution where a very good agreement is observed.

Differential transform method and numerical assembly technique for free vibration analysis of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and rotary inertias

Yusuf Yesilce 국제구조공학회 2015 Structural Engineering and Mechanics, An Int'l Jou Vol.53 No.3

Multiple-step beams carrying intermediate lumped masses with/without rotary inertias are widely used in engineering applications, but in the literature for free vibration analysis of such structural systems; Bernoulli-Euler Beam Theory (BEBT) without axial force effect is used. The literature regarding the free vibration analysis of Bernoulli-Euler single-span beams carrying a number of spring-mass systems, Bernoulli-Euler multiple-step and multi-span beams carrying multiple spring-mass systems and multiple point masses are plenty, but that of Timoshenko multiple-step beams carrying intermediate lumped massesand/or rotary inertias with axial force effect is fewer. The purpose of this paper is to utilize Numerical Assembly Technique (NAT) and Differential Transform Method (DTM) to determine the exact natural frequencies and mode shapes of the axial-loaded Timoshenko multiple-step beam carrying a number of intermediate lumped masses and/or rotary inertias. The model allows analyzing the influence of the shear and axial force effects, intermediate lumped masses and rotary inertias on the free vibration analysis of the multiple-step beams by using Timoshenko Beam Theory (TBT). At first, the coefficient matrices for the intermediate lumped mass with rotary inertia, the step change in cross-section, left-end support and right-end support of the multiple-step Timoshenko beam are derived from the analytical solution. After the derivation of the coefficient matrices, NAT is used to establish the overall coefficient matrix for the whole vibrating system. Finally, equating the overall coefficient matrix to zero one determines the natural frequencies of the vibrating system and substituting the corresponding values of integration constants into the related eigenfunctions one determines the associated mode shapes. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the differential equations of the motion. Thecalculated natural frequencies of Timoshenko multiple-step beam carrying intermediate lumped masses and/or rotary inertias for the different values of axial force are given in tables. The first five mode shapes are presented in graphs. The effects of axial force, intermediate lumped masses and rotary inertias on the free vibration analysis of Timoshenko multiple-step beam are investigated.

Free vibration of axially loaded Reddy-Bickford beam on elastic soil using the differential transform method

Yusuf Yesilce,Seval Catal 국제구조공학회 2009 Structural Engineering and Mechanics, An Int'l Jou Vol.31 No.4

The literature regarding the free vibration analysis of Bernoulli-Euler and Timoshenko beams on elastic soil is plenty, but the free vibration analysis of Reddy-Bickford beams on elastic soil with/ without axial force effect using the Differential Transform Method (DTM) has not been investigated by any of the studies in open literature so far. In this study, the free vibration analysis of axially loaded Reddy-Bickford beam on elastic soil is carried out by using DTM. The model has six degrees of freedom at the two ends, one transverse displacement and two rotations, and the end forces are a shear force and two end moments in this study. The governing differential equations of motion of the rectangular beam in free vibration are derived using Hamilton’s principle and considering rotatory inertia. Parameters for the relative stiffness, stiffness ratio and nondimensionalized multiplication factor for the axial compressive force are incorporated into the equations of motion in order to investigate their effects on the natural frequencies. At first, the terms are found directly from the analytical solutions of the differential equations that describe the deformations of the cross-section according to the high-order theory. After the analytical solution, an efficient and easy mathematical technique called DTM is used to solve the governing differential equations of the motion. The calculated natural frequencies of one end fixed and the other end simply supported Reddy-Bickford beam on elastic soil using DTM are tabulated in several tables and figures and are compared with the results of the analytical solution where a very good agreement is observed and the mode shapes are presented in graphs.

Investigation of natural frequencies of multi-bay and multi-storey frames using a single variable shear deformation theory

Bozyigit, Baran,Yesilce, Yusuf Techno-Press 2018 Structural Engineering and Mechanics, An Int'l Jou Vol.65 No.1

This study concerns about calculating exact natural frequencies of frames using a single variable shear deformation theory (SVSDT) which considers the parabolic shear stress distribution across the cross section. Free vibration analyses are performed for multi-bay, multi-storey and multi-bay multi-storey type frame structures. Dynamic stiffness formulations are derived and used to obtain first five natural frequencies of frames. Different beam and column cross sections are considered to reveal their effects on free vibration analysis. The calculated natural frequencies are tabulated with the results obtained using Euler-Bernoulli Beam Theory (EBT) and Timoshenko Beam Theory (TBT). Moreover, the effects of inner and outer columns on natural frequencies are compared for multi-bay frames. Several mode shapes are plotted.

Differential transform method and Adomian decomposition method for free vibration analysis of fluid conveying Timoshenko pipeline

Baran Bozyigit,Yusuf Yesilce,Seval Catal 국제구조공학회 2017 Structural Engineering and Mechanics, An Int'l Jou Vol.62 No.1

The free vibration analysis of fluid conveying Timoshenko pipeline with different boundary conditions using Differential Transform Method (DTM) and Adomian Decomposition Method (ADM) has not been investigated by any of the studies in open literature so far. Natural frequencies, modes and critical fluid velocity of the pipelines on different supports are analyzed based on Timoshenko model by using DTM and ADM in this study. At first, the governing differential equations of motion of fluid conveying Timoshenko pipeline in free vibration are derived. Parameter for the nondimensionalized multiplication factor for the fluid velocity is incorporated into the equations of motion in order to investigate its effects on the natural frequencies. For solution, the terms are found directly from the analytical solution of the differential equation that describes the deformations of the cross-section according to Timoshenko beam theory. After the analytical solution, the efficient and easy mathematical techniques called DTM and ADM are used to solve the governing differential equations of the motion, respectively. The calculated natural frequencies of fluid conveying Timoshenko pipelines with various combinations of boundary conditions using DTM and ADM are tabulated in several tables and figures and are compared with the results of Analytical Method (ANM) where a very good agreement is observed. Finally, the critical fluid velocities are calculated for different boundary conditions and the first five mode shapes are presented in graphs.

Dynamic stiffness approach and differential transformation for free vibration analysis of a moving Reddy-Bickford beam

Baran Bozyigit,Yusuf Yesilce 국제구조공학회 2016 Structural Engineering and Mechanics, An Int'l Jou Vol.58 No.5

In this study, the free vibration analysis of axially moving beams is investigated according to Reddy-Bickford beam theory (RBT) by using dynamic stiffness method (DSM) and differential transform method (DTM). First of all, the governing differential equations of motion in free vibration are derived by using Hamilton’s principle. The nondimensionalised multiplication factors for axial speed and axial tensile force are used to investigate their effects on natural frequencies. The natural frequencies are calculated by solving differential equations using analytical method (ANM). After the ANM solution, the governing equations of motion of axially moving Reddy-Bickford beams are solved by using DTM which is based on Finite Taylor Series. Besides DTM, DSM is used to obtain natural frequencies of moving Reddy-Bickford beams. DSM solution is performed via Wittrick-Williams algorithm. For different boundary conditions, the first three natural frequencies that calculated by using DTM and DSM are tabulated in tables and are compared with the results of ANM where a very good proximity is observed. The first three mode shapes and normalised bending moment diagrams are presented in figures.

Investigation of natural frequencies of multi-bay and multi-storey frames using a single variable shear deformation theory

Baran Bozyigit,Yusuf Yesilce 국제구조공학회 2018 Structural Engineering and Mechanics, An Int'l Jou Vol.65 No.1

This study concerns about calculating exact natural frequencies of frames using a single variable shear deformation theory (SVSDT) which considers the parabolic shear stress distribution across the cross section. Free vibration analyses are performed for multi-bay, multi-storey and multi-bay multi-storey type frame structures. Dynamic stiffness formulations are derived and used to obtain first five natural frequencies of frames. Different beam and column cross sections are considered to reveal their effects on free vibration analysis. The calculated natural frequencies are tabulated with the results obtained using Euler-Bernoulli Beam Theory (EBT) and Timoshenko Beam Theory (TBT). Moreover, the effects of inner and outer columns on natural frequencies are compared for multi-bay frames. Several mode shapes are plotted.