Nonlinear forced vibrations of rotating anisotropic beams

This work deals with forced vibration of nonlinear rotating anisotropic beams with uniform cross sections. Coupling the Galerkin method with the balance harmonic method, the nonlinear intrinsic and geometrically exact equations of motion for anisotropic beams subjected to large displacements, are converted into a static formulation. This latter is treated with two continuation methods. The first one is the asymptotic-numerical method, where power series expansions and Padé approximants are used to represent the generalized vector of displacement and the frequency. The second one is the pseudo-arclength continuation method. Numerical tests dealing with isotropic and anisotropic beams are considered. The natural frequencies obtained for prismatic beams are compared with the literature. Response curves are obtained and the nonlinearity is investigated for various geometrical conditions, excitation amplitudes and kinematical conditions. The nonlinearity related to the angular speed for prismatic isotropic beam is thus identified. The stability of the solution branch is examined, in the frequency domain using the Floquet theory.     

Nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate

The nonlinear vibration of an isotropic cantilever plate with viscoelastic laminate is investigated in this article. Based on the Von Karman’s nonlinear geometry and using the methods of multiple scales and finite difference, the dimensionless nonlinear equations of motion are analyzed and solved. The solvability condition of nonlinear equations is obtained by eliminating secular terms and, finally, nonlinear natural frequencies and mode-shapes are obtained. Knowing that the linear vibration of this type of plate does not have exact solution, Ritz method is employed to obtain semi-analytical nonlinear mode-shapes of transverse vibration of this plate. Airy stress function and Galerkin method are employed to reduce nonlinear PDEs into an ODE of duffing type. Stability of plate and chaotic behavior are investigated by Runge–Kutta method. Poincare section diagrams are in good agreement with results of Lyapunov criteria.     

Vibration of two beams connected by nonlinear isolators: analytical and experimental study

To study the coupling vibration of nonlinear isolators and flexible bodies, test rigs of two flexible beams connected by wire mesh isolators are constructed and investigated both experimentally and analytically. A five-parameter polynomial model of wire mesh isolators is derived by identifying parameters in the frequency domain with the sine-sweep test. For obtaining the parameters that are valid in a wide range of frequency, a numerically assisted identification method is developed. With this model, the vibration of two flexible beams connected by wire mesh isolators is studied. The frequency response is obtained analytically by employing the Green’s function method and harmonic balance method. Sine-sweep test results with three test rigs show good coherence with the corresponding numerical results. With obtained experimental results and numerical results, effect of connection parameters is studied in detail. It is found that traditional design rules for isolators are no longer effective and the coupling vibration must be investigated in the design phase. Another phenomenon is that the damping has a function of weakening the effect of nonlinear stiffness. Nonlinear stiffness and nonlinear damping can decrease the transmissibility along with the increase of the excitation level.     

Effect of transverse shears on complex nonlinear vibrations of elastic beams

Models of geometrically nonlinear Euler-Bernoulli, Timoshenko, and Sheremet’ev-Pelekh beams under alternating transverse loading were constructed using the variational principle and the hypothesis method. The obtained differential equation systems were analyzed based on nonlinear dynamics and the qualitative theory of differential equations with using the finite difference method with the approximation O(h²) and the Bubnov-Galerkin finite element method. It is shown that for a relative thickness λ ⩽ 50, accounting for the rotation and bending of the beam normal leads to a significant change in the beam vibration modes.     

Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system

Thermo-mechanical vibrations of a simply supported spring-mass-beam system are investigated analytically in this paper. Taking into account the thermal effects, the nonlinear equations of motion and internal/external boundary conditions are derived through Hamilton’s principle and constitutive relations. Under quasi-static assumptions, the equations governing the longitudinal motion are transformed into functions of transverse displacements, which results in three integro-partial differential equations with coupling terms. These are solved using the direct multiple-scale method, leading to closed-form solutions for the mode functions, nonlinear natural frequencies and frequency–response curves of the system. The influence of system parameters on the linear and nonlinear natural frequencies, mode functions, and frequency–response curves is studied through numerical parametric analysis. It is shown that the vibration characteristics depend on the mid-plane stretching, intra-span spring, point mass, and temperature change.     

Flapwise bending vibration analysis of a rotating double-tapered Timoshenko beam

In this study, free vibration analysis of a rotating, double-tapered Timoshenko beam that undergoes flapwise bending vibration is performed. At the beginning of the study, the kinetic- and potential energy expressions of this beam model are derived using several explanatory tables and figures. In the following section, Hamilton’s principle is applied to the derived energy expressions to obtain the governing differential equations of motion and the boundary conditions. The parameters for the hub radius, rotational speed, shear deformation, slenderness ratio, and taper ratios are incorporated into the equations of motion. In the solution, an efficient mathematical technique, called the differential transform method (DTM), is used to solve the governing differential equations of motion. Using the computer package Mathematica the effects of the incorporated parameters on the natural frequencies are investigated and the results are tabulated in several tables and graphics.     

Numerical analysis of the isolation of the vibration due to Rayleigh waves by using pile rows in the poroelastic medium

The isolation of the vibration due to harmonic Rayleigh waves using pile rows embedded in a saturated poroelastic half-space is investigated in this study. Based on Biot’s theory and the potential function method, the free field solution for Rayleigh waves along the surface of the poroelastic half-space is derived first. The fundamental solution for a harmonic circular patch load applied in the poroelastic half-space are obtained in terms of Biot’s theory and the integral transform method. Using Muki’s method and the fundamental solution for the circular patch load as well as the Rayleigh waves solution for the poroelastic half-space, the second kind of Fredholm integral equations in the frequency domain for pile rows are derived. Numerical solution of the integral equations yields the dynamic response of the pile–soil system to incident Rayleigh waves. Influences of various parameters on the vibration isolation effect of piles rows are investigated numerically. Numerical results suggest that for the same vibration source, the same pile rows will produce a better vibration isolation effect for the poroelastic medium than for a single phase elastic medium. Also, stiffer piles tend to have better vibration isolation effect than flexible piles. Moreover, the pile length and the spacing between neighboring piles in each pile row have significant influence on the vibration isolation effect of pile rows.     

Chaotic vibrations of beams on nonlinear elastic foundations subjected to reciprocating loads

Chaotic vibration of beams resting on a foundation with nonlinear stiffness is investigated in this paper. Cosine–cosine function is employed in modeling of the reciprocating load. The equation of motion is derived and solved to obtain corresponding Poincaré section in phase–space. Lyapunov exponent as a criterion for chaos indication is obtained. Dynamic behavior of the beam is examined in resonance condition. Homoclinic orbits are captured and their corresponding Melnikov's functions are established. A parametric study is then carried out and effects of linear and nonlinear parameters on the chaotic behavior of the system are studied.     

Stability and vibration of a helical rod constrained by a cylinder

The stability and vibration of an elastic rod with a circular cross section under the constraint of a cylinder is discussed. The differential equations of dynamics of the constrained rod are established with Euler’s angles as variables describing the attitude of the cross section. The existence conditions of helical equilibrium under constraint are discussed as a special configuration of the rod. The stability of the helical equilibrium is discussed in the realms of statics and dynamics, respectively. Necessary conditions for the stability of helical rod are derived in space domain and time domain, and the difference and relationship between Lyapunov’s and Euler’s stability concepts are discussed. The free frequency of flexural vibration of the helical rod with cylinder constraint is obtained in analytical form. The project supported by the National Natural Science Foundation of China (10472067). The English text was polished by Yunming Chen.     

On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method

This paper deals with the capabilities of linear and nonlinear beam theories in predicting the dynamic response of an elastically supported thin beam traversed by a moving mass. To this end, the discrete equations of motion are developed based on Lagrange’s equations via reproducing kernel particle method (RKPM). For a particular case of a simply supported beam, Galerkin method is also employed to verify the results obtained by RKPM, and a reasonably good agreement is achieved. Variations of the maximum dynamic deflection and bending moment associated with the linear and nonlinear beam theories are investigated in terms of moving mass weight and velocity for various beam boundary conditions. It is demonstrated that for majority of the moving mass velocities, the differences between the results of linear and nonlinear analyses become remarkable as the moving mass weight increases, particularly for high levels of moving mass velocity. Except for the cantilever beam, the nonlinear beam theory predicts higher possibility of moving mass separation from the base beam compared to the linear one. Furthermore, the accuracy levels of the linear beam theory are determined for thin beams under large deflections and small rotations as a function of moving mass weight and velocity in various boundary conditions.     

On linearization of the stiffness characteristics of flexible beams made of physically nonlinear materials

The geometry of flexible beams that are made of a physically nonlinear material and have a nearly linear load-deflection characteristic is identified for a wide range of monotonic and harmonic loads. The geometrically nonlinear beam equations are used. The physically nonlinear behavior of the material is described using a unified viscoplastic theory. A beam thickness criterion is formulated to provide nearly linear stiffness characteristic of the beam in the case of significant deflections and physically nonlinear deformations of the beam’s outer layers __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 85–92, February 2006.     

Generalized variational principle of dynamic analysis on naturally curved and twisted box beams for anisotropic materials

An incomplete generalized variational functional for naturally curved and twisted composite box beams with complete constrained boundaries at two ends is established by means of Lagrange multiplier method. The equations of motion governing the dynamic behavior of the beams and corresponding boundary conditions are derived from the stationary condition of the functional. The non-classical influences relevant to the beams are those due to transverse shear deformations, torsion-related warping and several elastic couplings that can arise in composite beams. In order to demonstrate the correctness of the theory developed the natural frequencies and normal mode shapes of the beams under in-plane free vibration are evaluated and compared with the results using PATRAN’s beam elements.     

Nonlinear dynamics of an inclined beam subjected to a moving load

In this paper, the nonlinear dynamic response of an inclined pinned-pinned beam with a constant cross section, finite length subjected to a concentrated vertical force traveling with a constant velocity is investigated. The study is focused on the mode summation method and also on frequency analysis of the governing PDEs equations of motion. Furthermore, the steady-state response is studied by applying the multiple scales method. The nonlinear response of the beam is obtained by solving two coupled nonlinear PDEs governing equations of planar motion for both longitudinal and transverse oscillations of the beam. The dynamic magnification factor and normalized time histories of mid-pint of the beam are obtained for various load velocity ratios and the outcome results have been illustrated and compared to the results with those obtained from traditional linear solution. The appropriate parametric study considering the effects of the linear viscous damping, the velocity of the traveling load, beam inclination angle under zero or nonzero axial load are carried out to capture the influence of the effect of large deflections caused by stretching effects due to the beam’s immovable ends. It was seen that quadratic nonlinearity renders the softening effect on the dynamic response of the beam under the act of traveling load. Also in the case where the object leaves the inclined beam, its planar motion path is derived and the targeting accuracy is investigated and compared with those from the rigid solution assumption. Moreover, the stability analysis of steady-state response for the modes equations having quadratic nonlinearity was carried out and it was observed from the frequency response curves that for the considered parameters in the case of internal-external primary resonance, both saturation phenomenon and jump phenomenon can be predicted for the longitudinal excitation.     

Nonlinear dynamic behaviors of a rotor-labyrinth seal system

The nonlinear model of rotor-labyrinth seal system is established using Muszynska’s nonlinear seal forces. We deal with dynamic behaviors of the unbalanced rotor-seal system with sliding bearing based on the adopted model and Newmark integration method. The influence of the labyrinth seal one the nonlinear characteristics of the rotor system is analyzed by the bifurcation diagrams and Poincare’ maps. Various phenomena in the rotor-seal system, such as periodic motion, double-periodic motion, quasi-periodic motion and Hopf bifurcation are investigated and the stability is judged by Floquet theory and bifurcation theorem. The influence of parameters on the critical instability speed of the rotor-seal system is also included.     

Post-critical behavior of damped beam columns with variable cross section subjected to distributed follower forces

In this paper the post-critical behavior of beam columns with variable mass and stiffness properties subjected to follower forces arbitrarily distributed along their length in the presence of damping (both internal and external) is investigated using a complete nonlinear dynamic analysis. Although the static nonlinear analysis is more economical in computational cost, it is associated only with the loss of local stability via flutter or divergence. Thus, the nonlinear dynamic analysis is adopted in order to examine the global stability of the system. The governing equations of hyperbolic type are derived in terms of the displacements by considering (a) nonlinear response including the axial deformation, (b) nonlinear response excluding the axial deformation and (c) linear response. Moreover, as the cross-sectional properties of the beam vary along its axis, the resulting coupled nonlinear differential equations have variable coefficients. Their solution is achieved using the analog equation method (AEM) of Katsikadelis. Besides its accuracy and effectiveness, this method overcomes the shortcoming of a possible FEM solution which may experience a lack of convergence. The problems treated in this investigation include beam columns with various load distributions, such as constant, linear and parabolic. Some of the conclusions detected in studying the nonlinear dynamic stability of Beck’s column with variable cross section (Katsikadelis and Tsiatas, Nonlinear dynamic stability of damped Beck’s column with variable cross section. Int. J. Non-linear Mech. 42, 164–171, 2007), are also valid for the case of distributed loads. The important, however, finding is that the post-critical response under distributed loads depends on the law of distribution of mass and stiffness properties, which may lead also to explosive flutter (unbounded amplitude), in contrast to Beck’s column (end-tip load) where the motion is always bounded.     

Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Bernoulli–Euler beams

The dynamic transfer matrix is formulated for a straight uniform and axially loaded thin-walled Bernoulli–Euler beam element whose elastic and inertia axes are not coincident by directly solving the governing differential equations of motion of the beam element. Bernoulli–Euler beam theory is used, and the cross section of the beam does not have any symmetrical axes. The bending vibrations in two perpendicular directions are coupled with torsional vibration and the effect of warping stiffness is included. The dynamic transfer matrix method is used for calculation of exact natural frequencies and mode shapes of the nonsymmetrical thin-walled beams. Numerical results are given for a specific example of thin-walled beam under a variety of end conditions, and exact numerical solutions are tabulated for natural frequencies and solutions calculated by the other method are also tabulated for comparison. The effects of axial force and warping stiffness are also discussed.     

Dynamical analysis of axially moving plate by finite difference method

The complex natural frequencies for linear free vibrations and bifurcation and chaos for forced nonlinear vibration of axially moving viscoelastic plate are investigated in this paper. The governing partial differential equation of out-of-plane motion of the plate is derived by Newton’s second law. The finite difference method in spatial field is applied to the differential equation to study the instability due to flutter and divergence. The finite difference method in both spatial and temporal field is used in the analysis of a nonlinear partial differential equation to detect bifurcations and chaos of a nonlinear forced vibration of the system. Numerical results show that, with the increasing axially moving speed, the increasing excitation amplitude, and the decreasing viscosity coefficient, the equilibrium loses its stability and bifurcates into periodic motion, and then the periodic motion becomes chaotic motion by period-doubling bifurcation.     

Nonlinear dynamic response of an edge-cracked functionally graded Timoshenko beam under parametric excitation

This paper investigates the nonlinear flexural dynamic behavior of a clamped Timoshenko beam made of functionally graded materials (FGMs) with an open edge crack under an axial parametric excitation which is a combination of a static compressive force and a harmonic excitation force. Theoretical formulations are based on Timoshenko shear deformable beam theory, von Karman type geometric nonlinearity, and rotational spring model. Hamilton’s principle is used to derive the nonlinear partial differential equations which are transformed into nonlinear ordinary differential equation by using the Least Squares method and Galerkin technique. The nonlinear natural frequencies, steady state response, and excitation frequency-amplitude response curves are obtained by employing the Runge–Kutta method and multiple scale method, respectively. A parametric study is conducted to study the effects of material property distribution, crack depth, crack location, excitation frequency, and slenderness ratio on the nonlinear dynamic characteristics of parametrically excited, cracked FGM Timoshenko beams.     

Waves in a rod under pulsed loading

Wave processes in a semi-infinite rod located in an elastic medium under pulsed loading by an external distributed force are considered. A system of two differential equations of motion of Timoshenko’s beam theory is solved with the use of the Laplace transform in time. The resultant integrals are determined numerically. The changes in bending and bending moment over the longitudinal coordinate at different times are demonstrated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 2, pp. 178–184, March–April, 2008.     

Solution of free vibration equations of semi-rigid connected Reddy–Bickford beams resting on elastic soil using the differential transform method

The literature regarding the free vibration analysis of Bernoulli–Euler and Timoshenko beams under various supporting conditions is plenty, but the free vibration analysis of Reddy–Bickford beams with variable cross-section 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 and semi-rigid connected Reddy–Bickford beam with variable cross-section 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 semi-rigid connected Reddy–Bickford beam with variable cross-section 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.