The dynamic analysis of nonuniformly pretwisted Timoshenko beams with elastic boundary conditions

The coupled governing differential equations and the general elastic boundary conditions for the coupled bending–bending forced vibration of a nonuniform pretwisted Timoshenko beam are derived by Hamilton's principle. The closed-form static solution for the general system is obtained. The relation between the static solution and the field transfer matrix is derived. Further, a simple and accurate modified transfer matrix method for studying the dynamic behavior of a Timoshenko beam with arbitrary pretwist is presented. The relation between the steady solution and the frequency equation is revealed. The systems of Rayleigh and Bernoulli–Euler beams can be easily examined by taking the corresponding limiting procedures. The results are compared with those in the literature. Finally, the effects of the shear deformation, the rotary inertia, the ratio of bending rigidities, and the pretwist angle on the natural frequencies are investigated.     

Closed-form solutions for crack detection problem of Timoshenko beams with various boundary conditions

An analytical approach for crack identification procedure in uniform beams with an open edge crack, based on bending vibration measurements, is developed in this research. The cracked beam is modeled as two segments connected by a rotational mass-less linear elastic spring with sectional flexibility, and each segment of the continuous beam is assumed to obey Timoshenko beam theory. The method is based on the assumption that the equivalent spring stiffness does not depend on the frequency of vibration, and may be obtained from fracture mechanics. Six various boundary conditions (i.e., simply supported, simple–clamped, clamped–clamped, simple–free shear, clamped–free shear, and cantilever beam) are considered in this research. Considering appropriate compatibility requirements at the cracked section and the corresponding boundary conditions, closed-form expressions for the characteristic equation of each of the six cracked beams are reached. The results provide simple expressions for the characteristic equations, which are functions of circular natural frequencies, crack location, and crack depth. Methods for solving forward solutions (i.e., determination of natural frequencies of beams knowing the crack parameters) are discussed and verified through a large number of finite-element analyses. By knowing the natural frequencies in bending vibrations, it is possible to study the inverse problem in which the crack location and the sectional flexibility may be determined using the characteristic equation. The crack depth is then computed using the relationship between the sectional flexibility and the crack depth. The proposed analytical method is also validated using numerical studies on cracked beam examples with different boundary conditions. There is quite encouraging agreement between the results of the present study and those numerically obtained by the finite-element method.     

Coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams

A dynamic transfer matrix method of determining the natural frequencies and mode shapes of axially loaded thin-walled Timoshenko beams has been presented. In the analysis the effects of axial force, warping stiffness, shear deformation and rotary inertia are taken into account and a continuous model is used. The bending vibration is restricted to one direction. The dynamic transfer matrix is derived by directly solving the governing differential equations of motion for coupled bending and torsional vibration of axially loaded thin-walled Timoshenko beams. Two illustrative examples are worked out to show the effects of axial force, warping stiffness, shear deformation and rotary inertia on the natural frequencies and mode shapes of the thin-walled beams. Numerical results demonstrate the satisfactory accuracy and effectiveness of the presented method.     

Output feedback boundary control of an axially moving system with input saturation constraint

This paper is concerned with boundary control for an axially moving belt system with high acceleration/deceleration subject to the input saturation constraint. The dynamics of belt system is expressed by a nonhomogeneous hyperbolic partial differential equation coupled with an ordinary differential equation. First, state feedback boundary control is designed for the case that the boundary states of the belt system can be measured. Subsequently, output feedback boundary control is developed when some of the system states can not be accurately obtained. The well-posedness and the uniformly bounded stability of the closed-loop system are achieved through rigorous mathematical analysis. In addition, high-gain observers are utilized to estimate those unmeasurable states, the auxiliary system is introduced to eliminate the constraint effects of the input saturation, and the disturbance observer is adopted to cope with unknown boundary disturbance. Finally, the control performance of the belt system is illustrated by carrying out numerical simulations.     

Effects of shear deformation and rotary inertia on the natural frequencies of axially loaded beams

The purpose of this study is to investigate how the axial load in beams influences the relationships between the natural frequencies and the effects of shear deformation and rotary inertia. Four beam theories are considered in this study. Finite element equations of motion for the beams under a tensile load are formulated to allow the application of various axial loads as well as to impose any type of boundary conditions. The results demonstrate that the stiffening effect by a tensile load may not reduce the frequency error of the Euler beam theory, unlike the results reported in other studies.     

Identification of pseudo-natural frequencies of an axially moving cantilever beam using a subspace-based algorithm

This study focuses on extraction of frequency information of a linear time-varying system using free response data. Frequency information is obtained from the pseudo-modal parameters that were defined in a previous study. A subspace-based identification algorithm is introduced. An improved version is proposed to make the algorithm less sensitive to measurement noise. An axially moving cantilever beam is used as the experimental system. A dynamic model is presented to show that lateral vibration of the axially moving cantilever beam is governed by a linear time-varying model. A computer simulation is conducted to compare the true pseudo-modal parameters and approximate ones that can be identified using the improved algorithm. The experimental study focuses on the capabilities of the algorithm and the factors that affect the identification results. A method of grouping identified structural pseudo-natural frequencies is proposed. Limitations of the algorithm are discussed.     

Free vibration analysis of beams with non-ideal clamped boundary conditions

A non-ideal boundary condition is modeled as a linear combination of the ideal simply supported and the ideal clamped boundary conditions with the weighting factors k and 1-k, respectively. The proposed non-ideal boundary model is applied to the free vibration analyses of Euler-Bernoulli beam and Timoshenko beam. The free vibration analysis of the Euler-Bernoulli beam is carried out analytically, and the pseudospectral method is employed to accommodate the non-ideal boundary conditions in the analysis of the free vibration of Timoshenko beam. For the free vibration with the non-ideal boundary condition at one end and the free boundary condition at the other end, the natural frequencies of the beam decrease as k increases. The free vibration where both the ends of a beam are restrained by the non-ideal boundary conditions is also considered. It is found that when the non-ideal boundary conditions are close to the ideal clamped boundary conditions the natural frequencies are reduced noticeably as k increases. When the non-ideal boundary conditions are close to the ideal simply supported boundary conditions, however, the natural frequencies hardly change as k varies, which indicate that the proposed boundary condition model is more suitable to the non-ideal boundary condition close to the ideal clamped boundary condition.     

Transient response of Timoshenko beams with discontinuities of cross-section

Based on the Timoshenko beam theory, flexural elastic wave propagation in beams is analysed. The method of characteristics is used for the numerical solution of the problem. The effect of reflections in finite beams with discontinuities of cross-section is considered. Stability and convergence of the numerical solution are discussed.     

An investigation on effects of traveling mass with variable velocity on nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions

In this paper, the nonlinear dynamic response of an inclined Timoshenko beam with different boundary conditions subjected to a traveling mass with variable velocity is investigated. The nonlinear coupled partial differential equations of motion for the bending rotation of cross-section, longitudinal and transverse displacements are derived using Hamilton’s principle. These nonlinear coupled PDEs are solved by applying Galerkin’s method to obtain dynamic response of the beam under the act of a moving mass. The appropriate parametric studies by taking into account the effects of the magnitude of the traveling mass, the velocity of the traveling mass with a constant acceleration/deceleration and effect of different beam’s boundary conditions are carried out. The beams’ large deflection has been captured by including the stretching effect of its mid-surface. It was seen that the existence of quadratic-cubic nonlinear terms in the governing coupled PDEs of motion renders hardening/stiffening behavior on the dynamic responses of the beam when traversed by a moving mass. In addition, the obtained nonlinear results are compared with those from the linear analysis.     

Coupled, thermoelastic, large amplitude vibrations of Timoshenko beams

The main goal of this work is to develop accurate and time efficient numerical approaches to study geometrically nonlinear vibrations of moderately thick beams under the combined action of mechanical and thermal loads. The influence of the thermal loading duration and thermal loading amplitude on the response of the structures is studied. Beams subjected to short heat flux and mechanical harmonic loading, with the excitation frequencies close to the natural frequencies of the beams, are investigated.     

Explicit formulation for natural frequencies of double-beam system with arbitrary boundary conditions

In this paper, free transverse vibration of two parallel beams connected through Winkler type elastic layer is investigated. Euler-Bernoulli beam hypothesis has been applied and it is assumed that boundary conditions of upper and lower beams are similar while arbitrary without any limitation even for non-ideal boundary conditions. Material properties and cross-section geometry of beams could be different from each other. The motion of the system is described by a homogeneous set of two partial differential equations, which is solved by using the classical Bernoulli-Fourier method. Explicit expressions are derived for the natural frequencies. In order to verify accuracy of results, the problem once again solved using modified Adomian decomposition method. Comparison between results indicates excellent accuracy of proposed formulation for any arbitrary boundary conditions. Derived explicit formulation is simplest method to determine natural frequencies of double-beam systems with high level of accuracy in comparison with other methods in literature.     

Stabilization of an axially moving accelerated/decelerated system via an adaptive boundary control

In this study, an adaptive boundary control is developed for vibration suppression of an axially moving accelerated/decelerated belt system. The dynamic model of the belt system is represented by partial-ordinary differential equations with consideration of the high acceleration/deceleration and unknown distributed disturbance. By utilizing adaptive technique and Lyapunov-based back stepping method, an adaptive boundary control is proposed for vibration suppression of the belt system, a disturbance observer is introduced to attenuate the effects of unknown boundary disturbance, the adaptive law is developed to handle parametric uncertainties and the S-curve acceleration/deceleration method is adopted to plan the belt׳s speed. With the proposed control scheme, the well-posedness and stability of the closed-loop system are mathematically demonstrated. Simulations are displayed to illustrate the effectiveness of the proposed control.     

Corrugated plate formed by side extrusion with two coaxial rams moving at different speeds

Plane strain side extrusion with two rams moving along the same line but with different speeds is considered using slip-line field theory and the effect on exit velocity of the material, of variation in the ram speeds, is deduced. It is demonstrated that by suitably programming the rams, sheet materials with corrugated shape can be formed.     

Influence of boundary conditions and turntable speeds on the stability of hydrostatic oil cavity

The flow, bearing, and carrying capacity of the cycloidal hydrostatic oil cavity in hydrostatic turntable systems are numerically simulated, considering the rotation speeds of a turntable from 0 to 5 m/s and different boundary conditions. The vortex effect is weakened, and the stability of the oil cavity is enhanced with the increase in lubricant viscosity. However, the increase in inlet speed, depth, and inlet radius of the oil cavity causes the vortex effect to increase and the stability of oil cavity to reduce. With the increase in the oil film thickness, the carrying capacity of the oil cavity diminishes. The oil cavity pressure increases along the direction of the motion of the turntable; it is distributed unevenly because of the rotation of the turntable. With the increase in turntable speed, the location and size of the vortex scope in the oil cavity flow field and the strength of the vortex near the entrance gradually weaken and move away from the entry. The distribution of pressure is determined by the locations of the vortex. When the vortex is close to the wall, the wall pressure increases at its location. Otherwise, the wall pressure decreases first and then increases after the center of the vortex.     

Vibration analysis of cross-ply laminated beams with general boundary conditions by Ritz method

The present study is concerned with the vibration analysis of cross-ply laminated beams subjected to different sets of boundary conditions. The analysis is based on a three-degree-of-freedom shear deformable beam theory. The continuity conditions between layers of symmetric cross-ply laminated beams are satisfied by the use of the shape function incorporated into the theory which also unifies the 1D shear deformable beam theories developed previously. The governing equations are obtained by means of Hamilton's principle. Six different combinations of free, clamped and simply supported edge boundary conditions are considered. The free vibration frequencies are obtained by applying the Ritz method where the three displacement components are expressed in a series of simple algebraic polynomials. The numerical results obtained for different length-to-thickness ratios and lay-ups are presented and compared with results available in the literature.     

Comparisons of experimental and theoretical frequencies for rectangular plates with various boundary conditions and added masses

This paper is concerned with a vibration analysis of rectangular plates with masses mounted on various locations. The edges of the plates may either be clamped or simply supported. The study is particularly useful in the understanding of the vibration of printed circuit boards used in the electronics industry. An energy method is developed to obtain analytical frequencies of the plates with various edge support conditions. The analytical procedure using the Rayleigh-Ritz approach is adopted in which each of single and multiple trigonometric series terms is used to represent the shape function. Two experimental methods, a spectrum analyser and a TV-holographic system, are used to study the behaviour of the plate vibrations. The holographic image produced at the corresponding mode frequencies by using the TV-holography technique has been applied to verify the frequency spectra obtained from the spectrum analyser. The experimental results have been used to illustrate the validity of the analytical model. The comparisons show that the analytical model predicts natural frequencies reasonably well, in which the EM 100-term model is suggested for vibration plates with higher modes or heavier loads.     

Active disturbance rejection boundary control of Timoshenko beam with tip mass

In this paper, the active disturbance rejection control (ADRC) is utilized to stabilize the vibration of perturbed Timoshenko beam model with tip mass. The boundary control design is based on a hybrid PDE–ODE model, and is accompanied with designing a high-gain extended state observer (ESO) that is used to estimate the boundary disturbances. By transforming the model into the appropriate state space, the semigroup theory is employed to prove the well-posedness of the closed-loop system. To this end, it is proved by a frequency domain method that the semigroup generated by the system operator is exponentially stable, which allows to conclude the boundedness of perturbed closed-loop system response. The stability of the closed-loop system is further analyzed using the Lyapunov approach. Simulation results are presented to illustrate the efficacy of the suggested method.     

The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution

This paper introduces the discrete singular convolution algorithm for vibration analysis of rectangular plates with mixed boundary conditions. A unified scheme is proposed for the treatment of simply supported, clamped and transversely supported (with nonuniform elastic rotational restraint) boundary conditions. The robustness and reliability of the present approach are tested by a number of numerical experiments. All results agree well with those in the literature.     

Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance

The aim of the study described in this paper is to investigate the forced dynamics of an axially moving viscoelastic beam. The governing equation of motion is obtained via Newton's second law of motion and constitutive relations. The viscoelastic beam material is constituted by the Kelvin–Voigt, a two-parameter rheological model, energy dissipation mechanism, in which material, not partial, time derivative is employed in the viscoelastic constitutive relation. The dimensionless partial differential equation of motion is discretized using Galerkin's scheme with hinged–hinged beam eigenfunctions as the basis functions. The resulting set of nonlinear ordinary differential equations is then solved using the pseudo-arclength continuation technique and a direct time integration. For the system with the axial speed in the sub-critical regime, the response of the system is examined when possessing an internal resonance and when not. By employing a direct time integration, it is shown how the bifurcation diagrams of the system are modified by the presence of the dissipation terms—i.e. by both the time-dependant and steady (due the simultaneous presence of the axial speed and the energy dissipation mechanism) energy dissipation terms. Moreover, the amplitude–frequency responses and bifurcation diagrams of Poincaré maps are presented for several values of the system parameters.     

A comparative study of the eigenvalue solutions for mass-loaded beams under classical boundary conditions

A comparative study of the eigenfrequency analysis for an Euler–Bernoulli beam carrying a concentrated mass at an arbitrary location is presented in this short note. The dimensionless frequency equation for different combinations of classical boundary conditions is obtained by satisfying the differential equations of motion and by imposing the corresponding boundary and compatibility conditions. Two formulation methods have been commonly used for the boundary-value problem. One is to adopt a single frame originated from the beam's left-end, while another is by dual frames associated with the concentrated mass. It is found that the forms derived by dual frames are more compact than the corresponding expressions by using the single frame. Nevertheless, the comparison for all the cases shows that the dual-frame expressions need more time to obtain the same set of eigenvalues if compared with the time by using the single-frame expressions.