Non-uniform Timoshenko Beam Research Articles

Stress-driven nonlocal integral model with discontinuities for transverse vibration of multi-cracked non-uniform Timoshenko beams with general boundary conditions

Stress-driven nonlocal integral model with discontinuities for transverse vibration of multi-cracked non-uniform Timoshenko beams with general boundary conditions

Saturated Boundary Feedback Stabilization for a Nonuniform Timoshenko Beam With Disturbances and Uncertainties

Saturated Boundary Feedback Stabilization for a Nonuniform Timoshenko Beam With Disturbances and Uncertainties

Closed-form solutions for axially non-uniform Timoshenko beams and frames under static loading

This paper presents the Green’s Functions Stiffness Method (GFSM) for solving linear elastic static problems in arbitrary axially non-uniform Timoshenko beams and frames subjected to general external loads and bending moments. The GFSM is a mesh reduction method that seamlessly integrates elements from the Stiffness Method (SM), Finite Element Method (FEM), and Green’s Functions (GFs), resulting in a highly versatile methodology for structural analysis. It incorporates fundamental concepts such as stiffness matrices, shape functions, and fixed-end forces, in line with SM and FEM frameworks. Leveraging the capabilities of GFs, the method facilitates the derivation of closed-form solutions, addressing a gap in existing methods for analyzing non-uniform reticular structures which are typically limited to simple cases like single-span beams with specific axial variations and loading scenarios. The effectiveness of the GFSM is demonstrated through three practical examples, showcasing its applicability in analyzing non-uniform beams and plane frames, thereby broadening the scope of closed-form solutions for axially non-uniform Timoshenko structures.

Exponential Stability of Non-uniform Timoshenko Beam With Indefinite Damping Under Boundary Control

Exponential Stability of Non-uniform Timoshenko Beam With Indefinite Damping Under Boundary Control

Exponential Stability of Non-uniform Timoshenko Beam With Indefinite Damping Under Boundary Control

In this paper we study the stabilization of a non-uniform Timoshenko beam with indefinite damping terms under boundary control and prove how the damping terms can affect the decay rate asymptotically. Using the theory of perturbed problems, we obtain the stability and establish the spectrum determined growth condition for the problem. Moreover, when the two damping terms are indefinite, we provide a condition to obtain  the exponential stability.

A fully nonlinear BEM-beam coupled solver for fluid–structure interactions of flexible ships in waves

A fully nonlinear hydroelastic numerical method is proposed for the fluid–structure interactions of elastic ships. The boundary value problem is solved by the three-dimensional boundary element method. To simulate the instantaneous evolution on the free surface, an effective mesh optimization solution consisting of the local coordinate system and an elastic mesh technique has been developed. The mixed Eulerian–Lagrangian method is implemented to update the fully nonlinear free surface conditions in the local coordinate system. The hull structure is simplified as a non-uniform Timoshenko beam, and the modal superposition approach is adopted to analyse the symmetric responses in the head sea. Convergence of the model is evaluated for mesh resolutions and the time step size, and then comprehensive investigations of vertical motions as well as vertical bending moment are carried out for an advancing ship in a range of wave frequencies. The numerical solutions are compared with the experimental data and a satisfactory agreement is demonstrated. The distribution of high-order harmonics and whipping responses are predicted by the amplitude spectrum, in which the water entry and water exit are considered. The high-frequency vertical bending moments are investigated, in which the effects of higher harmonics and whipping responses are addressed.

Flexural wave control via the profile modulation of non-uniform Timoshenko beams

A methodology to control the flexural wave propagation via modulating the beam profile is presented in this paper. Firstly, the power series method is utilized to theoretically solve the flexural waves in a non-uniform Timoshenko beam with arbitrarily contoured profiles. It is demonstrated that this method is effective for most of inhomogeneous beams as long as the convergence criterion about the beam thickness is satisfied in advance. After validation by comparing the theoretical results with those from the finite element analysis for different thickness variations, lenses are designed by using the quadratic thickness variation pattern with the aim of focusing the wave emitted from a point source on one or two particular positions. Additionally, according to the generalized Snell's law, the thickness-induced phase modulation of the flexural wave is proved from the views of theoretical analysis and numerical simulations, based on which lenses for focusing a plane flexural wave are also realized. It is illustrated through systematic analysis in the frequency domain that all of the lenses designed can exhibit good performances with the evidently increased energy at actual focal positions and the focusing sizes smaller than one working wavelength, which has great potentials in versatile engineering applications.

Investigation of the nonlinear slamming-induced whipping response of ships using a fully-coupled hydroelastoplastic method

Slamming-induced whipping is traditionally computed as the response of a linear-elastic structural model. However, in order to investigate the consequence of whipping on the hull girder’s collapse, the hydro-structure interaction must be performed in a fully-coupled approach where the nonlinearities of both domains are considered. Therefore, this paper presents a new approach developed for solving the fully-coupled hydroelastoplastic problem. The structural part is modeled as two non-uniform Timoshenko beams, connected with a nonlinear hinge, described by the nonlinear relation between the internal bending moment and the relative rotation angle. The hydrodynamic problem is solved using the 3D boundary element method, and the exact coupling between the structural and the 3D hydrodynamic models is achieved by constructing the hydrodynamic boundary value problem for each shape function of the finite element beam model. The hydroelastoplastic response is calculated using a hybrid nonlinear time-domain approach, allowing for very fast computation of the nonlinear whipping response. Finally, the nonlinear whipping response is calculated on a broad range of ships and it is compared to the linear whipping response in order to derive the whipping effectiveness coefficients. It is shown that the nonlinear structural behavior has a very small influence on the whipping response, and thus, the effectiveness of whipping should not be reduced.

Symplectic Transfer-Matrix Method for Bending of Nonuniform Timoshenko Beams on Elastic Foundations

AbstractThis paper proposes a new symplectic transfer-matrix method (STMM) to solve the bending problem of nonuniform beams. The STMM is a hybrid method that combines the symplectic dual solution s...

Nonuniform Isospectrals of Uniform Timoshenko Beams

Spectrally equivalent systems are those that have the same free vibration natural frequencies for a given boundary condition. In this paper, we establish isospectrality between certain classes of nonuniform Timoshenko beams with a given uniform Timoshenko beam. We apply a transformation to convert the nondimensional coupled nonuniform Timoshenko beam equations from the frame of reference to a hypothetical frame of reference. The transformed equations are then combined by eliminating one of the variables. Specific material and geometric properties are chosen, and a few auxiliary variables are introduced to convert the transformed equation into the required form. If the coefficients of the transformed equation match with the required uniform equation, then the nonuniform beam is said to be isospectral to the uniform beam. The boundary configurations also change during this transformation. We present the constraints under which they are preserved. Frequency equivalence of the beams is confirmed by the finite element method. For the considered cases, examples of beams having a rectangular cross-section are also presented.

Impact of Vlasov Foundation Parameters on the Deflection of a Non-uniform Timoshenko Beam Subject to a Moving Load

Subject of the study is a Timoshenko beam with transverse-variable Young’s modulus. Problem of vibrations of the beam resting on an inertial Vlasov foundation and subjected to a moving force is solved analytically. The Timoshenko beam’s eigenproblem is discussed and the physical sense of the additional band of natural vibrations and the corresponding critical frequency is analytically explained. The impact of the foundation and its parameters on the Timoshenko beam’s vibrations forced by moving force is investigated. Dynamic factors relating to beam deflections are analyzed. Two vibration cases are discussed: forced vibrations (when a moving force is applied to the beam) and free vibrations (after the moving force has been left the beam). The damping effect on vibration is taken into account in the problem solution. The results indicate that appropriate selection of the foundation’s parameters allows for the beam deflection’s significant reduction, while the impact of the shear coefficient in the foundation on the reduction is more pronounced than the impact of other factors.

A new approach for free vibration analysis of nonuniform tall building structures with axial force effects

SummaryDynamic analysis of beam‐like structures is significantly important in modeling actual cases such as tall buildings and several other related applications as well. This article studies free vibration analysis of tall buildings with nonuniform cross‐section structures. A novel and simple approach is presented to solve natural frequencies of free vibration of cantilevered tall structures with variable flexural rigidity and mass densities. These systems could be replaced by a cantilever Timoshenko beam with varying cross‐sections. The governing partial differential equation for vibration of a nonuniform Timoshenko beam under variable axial loads is transformed with varying coefficients to its weak form of integral equations. Natural frequencies can be determined by requiring the resulting integral equation, which has a nontrivial solution. The presented method in this study has fast convergence. Including high accuracy for the obtained numerical results as well. Numerical examples including framed tube as well as tube‐in‐tube structures are carried out in the study and compared with available results in the literature, and also with the results obtained from finite element analysis in order to show the accuracy of the proposed method in the study. Obtained results indicate that the presented method in this study is powerful enough for the free vibration analysis of tall buildings.

Natural frequencies of the elastically end restrained non-uniform Timoshenko beam using the power series method

In recent years, the transverse vibration of the uniform Timoshenko beam has become a highly significant research topic. The Timoshenko beam (TB) model is suitable for describing the behavior of the beams under high-frequency excitation load when the wavelength approaches the thickness of the beam because it considers the shear deformation and the rotational inertia effects. In this paper, the natural frequencies of the elastically end constrained nonuniform TB is discussed. For this reason, the nonuniform TB is modeled by an exact and direct modeling technique. The semi-analytical solution based on the power series method is adopted in technique. The effect of the spring supports, different boundary conditions, the non-uniformity in the cross-section of the TB, and other parameters are assessed. The results are shown that the nonuniformity in the cross-section of the beam is impressed on the natural frequencies. On the other hand, the presented formulation not only effectively is easy to use, but also provides the reasonable solution of non-uniform TB, carrying various boundary conditions.

Development of a numerical approach for the prediction of thrust generated by a bio-mimetic propulsion system

ABSTRACTThis paper presents a numerical approach for the prediction of thrust generated by a fish-type biomimetic propulsion system. We model the fish as a non-uniform Timoshenko Beam (TB) with varying Young’s modulus and moment of inertia along its length from head to tail. The non-uniform Timoshenko beam is assumed to be a cantilever with the fixed end at the head and free end at the tail and the corresponding mode shapes are determined. Based on the mode shapes, the thrust is calculated with respect to the input parameters such as wave velocity, propulsive velocity and wavelength. The method developed here is shown to be useful for the design of bio-inspired propulsive systems for marine vehicles and also for gaining a better understanding propulsive mechanics of a robotic fish. In results, the thrust delivered by the simple flapping beam is identified and the relationship between the non-dimensional frequencies is determined.

Dynamic stiffness matrix for free vibration analysis of flexure hinges based on non-uniform Timoshenko beam

As a typical non-uniform beam with complex varying cross-section and large curvature, free vibration analysis of flexure hinges in the state-of-the-art compliant mechanisms is challenging due to their complicated and even unsolvable governing differential equation. Investigation on the free vibration analysis of non-uniform beams was attractive in the past but still with a few types of specific beams or approximate solutions. The contribution of this paper is to formulate the dynamic stiffness matrix for generic non-uniform beams and applied for the first time to the free vibration analysis of flexure hinges. Firstly, the vibration solution of general non-uniform Timoshenko beam is derived by using a recursive integral in power series of circular frequency but not the traditional way in the well-known method of Frobenious. Then, the dynamic stiffness matrix of non-uniform Timoshenko beam is derived based on the variational principle. At last, the new method is applied for the free vibration analysis of flexure hinges and flexure-based compliant mechanisms. Several case studies demonstrate the advantage of the proposed approach in terms of high accuracy, generality and high efficiency (only one element is needed). The obtained results are useful for dynamic design of flexure hinges and can serve as a benchmark to examine the accuracy of other numerical solutions. Due to the closed form and universality of the proposed method, it is also of benefit for free vibration analysis of other non-uniform beams with all kinds of varying cross-section or functionally graded beams.

Buckling of nonuniform and axially functionally graded nonlocal Timoshenko nanobeams on Winkler-Pasternak foundation

Buckling of axially functionally graded and nonuniform Timoshenko beams is examined based on the nonlocal Timoshenko beam theory. Small scale effect on the buckling is taken into account via the length scale parameter often referred as the nonlocal parameter. The material properties vary in the axial direction and the nanobeam is modelled as a nonuniform Timoshenko beam which rests on a Winkler-Pasternak foundation. Rayleigh quotients for the buckling load are derived for Euler-Bernoulli and Timoshenko beams. Chebyshev polynomials based on the Rayleigh-Ritz method is used to obtain the numerical solution of the problem for a combination of clamped, simply supported and free boundary conditions. Accuracy of the method is verified by comparing the buckling loads obtained by the present approach with those available in the literature. Numerical results for the problem are given in the form of contour plots to study the effect of nonlocal parameter, cross-sectional non-uniformity, axial grading and the boundary conditions on the buckling loads.

Dynamic Response of a Nonuniform Timoshenko Beam with Elastic Supports, Subjected to a Moving Spring-Mass System

This paper is concerned with the dynamic response of a nonuniform Timoshenko beam with elastic supports subjected to a moving spring-mass system. The modal orthogonality of nonuniform Timoshenko beams and the corresponding overall matrix of undetermined coefficients are derived. Then the natural frequencies and mode shapes of nonuniform Timoshenko beams are obtained by the Runge–Kutta method and cubic spline interpolation method. By using the Newmark-[Formula: see text] method and the mode summation method, the vibration equation of Timoshenko beams subjected to a moving spring-mass system was established. A comparison of results between the proposed method and finite element method reveals that this method possesses favorable accuracy for dynamic response analysis. In numerical examples, the effects of the support spring and moving spring-mass system on Timoshenko beams have been examined in detail.

Theoretical and Experimental Study on Nonlinear Hydroelastic Responses and Slamming Loads of Ship Advancing in Regular Waves

Wave loads estimation and structural strength evaluation are the fundamental work at the ship design stage. The hydroelastic responses and slamming strength issues are also concerned especially for large‐scale high‐speed ships sailing in harsh waves. To accurately predict the wave‐induced motions and loads acting on the ship sailing in regular waves, a fully coupled 3D time‐domain nonlinear hydroelasticity theory is developed in this paper. The vibration modal characteristics of the flexible hull structure derived by the 3D finite element method (FEM) and simplified 1D nonuniform Timoshenko beam theory are firstly described. The hydrostatic restoring force and hydrodynamic wave force are calculated on the real‐time wetted surface of hull to address geometric nonlinearity due to the steep wave and large amplitude motions. The bow slamming and green water loads acting on the ship in severe regular waves are estimated by the momentum impact method and dam‐breaking method, respectively. Moreover, a small‐scaled segmented ship model is designed, constructed, and tested in a laboratory wave basin to validate the hydroelasticity algorithm. The results predicted by theoretical and experimental approaches are systemically compared and analyzed. Finally, future work for predictions of ship hydroelasticity and slamming loads in irregular waves is prospected.

Vibration analysis of a Timoshenko non-uniform nanobeam based on nonlocal theory: An analytical solution

In this article free vibration of a Timoshenko nanobeam with variable cross-section is investigated using nonlocal elasticity theory within the scope of continuum mechanics. Small scale effects are modelled after Eringen’s nonlocal elasticity theory while the non-uniformity is presented by exponentially varying width through the beam length with constant thickness. Analytical solution is achieved for both Timoshenko beams and nanobeams with different boundary conditions including both ends being simply-supported (S-S), both ends being clamped (C-C) and one end clamped other free (C-F). It is shown that section variation accompanying small scale effects has a noticeable effect on natural frequencies of non-uniform Timoshenko beams at nano-scale. In order to illustrate these effects, Natural frequencies of single-layered graphene nano-ribbons (GNRs) with various boundary conditions are obtained for different nonlocal and non-uniform parameter which shows a great sensitivity to non-uniformity in different shape modes.

Free vibration analysis of cracked Timoshenko beams carrying spring-mass systems

In this paper, an analytical approach is proposed for determining vibration characteristics of cracked non-uniform continuous Timoshenko beam carrying an arbitrary number of spring-mass systems. This method is based on the Timoshenko beam theory, transfer matrix method and numerical assembly method to obtain natural frequencies and mode shapes. Firstly, the beam is considered to be divided into several segments by spring-mass systems and support points, and four undetermined coefficients of vibration modal function are contained in each sub-segment. The undetermined coefficient matrices at springmass systems and pinned supports are obtained by using equilibrium and continuity conditions. Then, the overall matrix of undetermined coefficients for the whole vibration system is obtained by the numerical assembly technique. The natural frequencies and mode shapes of a cracked non-uniform continuous Timoshenko beam carrying an arbitrary number of springmass systems are obtained from the overall matrix combined with half-interval method and Runge-Kutta method. Finally, two numerical examples are used to verify the validity and reliability of this method, and the effects of cracks on the transverse vibration mode shapes and the rotational mode shapes are compared. The influences of the crack location, depth, position of spring-mass system and other parameters on natural frequencies of non-uniform continuous Timoshenko beam are discussed.