Refined models of elastic beams undergoing large in-plane motions: Theory and experiment

Accurate mechanical models of elastic beams undergoing large in-plane motions are discussed theoretically and experimentally. Employing the geometrically exact theory of rods with appropriate kinematic assumptions and asymptotic arguments, two approximate models are obtained—a relaxed model and its constrained version—that describe extensional and bending motions and neglect shear deformations. These models are shown to be suitable to predict, via an asymptotic approach, closed-form nonlinear motions of beams with general boundary conditions and, in particular, with boundary conditions that longitudinally constrain the motions. On the other hand, for axially unrestrained or weakly restrained beams, an inextensible and unshearable model is presented that describes bending motions only. The perturbations about the reference configuration up to third order are consistently derived for all beam models. Closed-form solutions of the responses to primary-resonance excitations are obtained via an asymptotic treatment of the governing equations of motion for two different beam configurations; namely, hinged–hinged (axially restrained) and simply supported (axially unrestrained) beams. In particular, considering the present theory and the existing theories, variations of the frequency–response curves with the beam slenderness or the relative boundary mass are investigated for the lowest modes. The fidelity of the proposed nonlinear models is ascertained comparing the theoretically obtained frequency–response curves of the first mode with those experimentally obtained.     

Simply supported elastic beams under parametric excitation

In this paper, the nonlinear characteristics of the parametric resonance of simply supported elastic beams are investigated. Considering a geometrically exact formulation for unsharable and inextensible elastic beams subject to support motions, the integral-partial-differential equation of motion is obtained. The third-order perturbation of the equation of motion is then determined in a form amenable to an asymptotic treatment. The method of multiple scales is used to obtain the equations that describe the modulation of the amplitude and phase of parametric-resonance motions. The stability and bifurcations of the system are investigated considering, in particular, the frequency-response function. Furthermore, experimental results are shown to confirm the theoretically predicted stability and bifurcations.     

Non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams

An axially moving visco-elastic Rayleigh beam with cubic non-linearity is considered, and the governing partial-differential equation of motion for large amplitude vibration is derived through geometrical, constitutive, and dynamical relations. By directly applying the method of multiple scales to the governing equations of motion, and considering the solvability condition, the linear and non-linear frequencies and mode shapes of the system are analytically formulated. In the presence of damping terms, it can be seen that the amplitude is exponentially time-dependent, and as a result, the non-linear natural frequencies of the system will be time-dependent. For the resonance case, through considering the solvability condition and Routh–Hurwitz criterion, the stability conditions are developed analytically. Eventually, the effects of system parameters on the vibrational behavior, stability and bifurcation points of the system are investigated through parametric studies.     

An experiment and analyses on elastic waves in beams from lateral impact

A long beam is laterally loaded with a spatially uniform impulse over a length equal to three beam thicknesses. Loading is produced by short-duration magnetic pressure pulses and the response is measured with strain gages. Strain measurements are compared with predictions from beam theories and a two-dimensional numerical analysis.     

数值仿真轴向运动黏弹性梁非线性参激振动

分别通过两种直接数值方法研究速度变化的经典边界条件下轴向运动黏弹性梁参数振动的稳定性。在控制方程的推导中,采用物质导数黏弹性本构关系和只对时间取偏导数的黏弹性本构关系;分别运用有限差分法和微分求积法对两种经典边界下轴向变速运动黏弹性梁的非线性控制方程求数值解,计算得到梁中点非线性参数振动的稳定稳态响应。数值结果表明,两种黏弹性本构关系对应的稳态响应存在明显差别,同时发现两种直接数值方法的仿真结果基本吻合,证明数值仿真具有较高精度。     

Non-linear interactions in imperfect beams at veering

Non-linear interactions in a hinged-hinged uniform moderately curved beam with a torsional spring at one end are investigated. The two-mode interaction is a one-to-one autoparametric resonance activated in the vicinity of veering of the frequencies of the lowest two modes and results from the non-linear stretching of the beam centerline. The excitation is a base acceleration that is involved in a primary resonance with either the first mode only or with both modes. The ensuing non-linear responses and their stability are studied by computing force- and frequency-response curves via bifurcation analysis tools. Both the sensitivity of the internal resonance detuning—the gap between the veering frequencies—and the linear modal structure are investigated by varying the rise of the beam half-sinusoidal rest configuration and the torsional spring constant. The internal and external resonance detunings are varied accordingly to construct the non-linear system response curves. The beam mixed-mode response is shown to undergo several bifurcations, including Hopf and homoclinic bifurcations, along with the phenomenon of frequency island generation and mode localization.     

Non-linear oscillators under parametric and external poisson pulses

The extended Itô calculus for non-normal excitations is applied in order to study the response behaviour of some non-linear oscillators subjected to Poisson pulses. The results obtained show that the non-normality of the input can strongly affect the response, so that, in general, it can not be neglected.     

Non-linear systems under parametric white noise input: Digital simulation and response

Monte Carlo technique is constituted of three steps. Therefore, improving such technique in practice means, improving the procedure used in one of the three following steps: (i) sample paths of the stochastic input process, (ii) calculation of the outputs corresponding to the generated input samples by using methods of classical dynamics and (iii) estimating statistics of the output process from sample outputs related to the previous step. For linear and non-linear systems driven by parametric impulsive inputs such as normal or non-normal white noises, a general integration method requires a considerable reduction of the integration step when the impulse occurs, treating the impulse as a physical one, by means of a window function of finite duration. This makes Monte Carlo simulation very prohibitive from a computational time point of view. While knowing the exact jump value of the response at impulse occurring that is expressed by a numerical series, the aforementioned problem is overcome because there is no need to reduce the integration step saving computational time, reliability being equal as shown by means of a numerical example.     

Optimal elastic design of beams

According to the principle of minimum complementary energy a mathematical statement of optimal strength design problem for elastic beams is formulated in this research, which is an extremum problem of functionals with equality and inequality constraints. Further the application of the Lagrangian multiplier method yields the necessary conditions for extrema. A set of relations that must be satisfied for the optimal solution follows afterwards. This set of relations can be used to verify the optimality of a uniform strength design or any feasible elastic design. An iterative numerical method to find the optimal solution when the uniform strength design is not optimal is also presented in this paper.Project supported by the Science and Technics Fund of the Chinese National Educational Committee.     

Analysis of micropolar elastic beams

In this paper, a linear theory for the analysis of beams based on the micropolar continuum mechanics is developed. Power series expansions for the axial displacement and micro-rotation fields are assumed. The governing equations are derived by integrating the momentum and moment of momentum equations in the micropolar continuum theory. Body couples and couple stresses can be supported in this theory. After some simplifications, this theory can be reduced to the well-known Timoshenko and Euler–Bernoulli beam theories. The nature of flexural and longitudinal waves in the infinite length micropolar beam has been investigated. This theory predicts the existence of micro-rotational waves which are not present in any of the known beam theories based on the classical continuum mechanics. Also, the deformation of a cantilever beam with transverse concentrated tip loading has been studied. The pattern of deflection of the beam is similar to the classical beam theories, but couple stress and micro-rotation show an oscillatory behavior along the beam for various loadings.     

Shear stresses in elastic beams: an intrinsic approach

The theory of non-uniform flexure and torsion of Saint-Venant's beam with arbitrary multiply connected cross section is revisited in a coordinate-free form to provide a computationally convenient context. Numerical implementations, by Matlab, are performed to evaluate the maximum elastic shear stresses in beams with rectangular cross sections for different Poisson's ratios. The deviations between the maximum and mean stresses are then diagrammed to adjust the results provided by Jourawski's method.     

Non-linear vibration of variable speed rotating viscoelastic beams

Non-linear vibration of a variable speed rotating beam is analyzed in this paper. The coupled longitudinal and bending vibration of a beam is studied and the governing equations of motion, using Hamilton’s principle, are derived. The solutions of the non-linear partial differential equations of motion are discretized to the time and position functions using the Galerkin method. The multiple scales method is then utilized to obtain the first-order approximate solution. The exact first-order solution is determined for both the stationary and non-stationary rotating speeds. A very close agreement is achieved between the simulation results obtained by the numerical integration method and the first-order exact solution one. The parameter sensitivity study is carried out and the effect of different parameters including the hub radius, structural damping, acceleration, and the deceleration rates on the vibration amplitude is investigated.     

Non-linear bending of beams with uniformly distributed loads

The non-linear bending of both cantilever and simply supported beams subjected to a uniformly distributed load has been studied. The exact solutions for the slopes have been obtained and the solution for the maximum deflection and the horizontal projection of the beam length for the cantilever case are compared with a known approximate solution.     

Non-linear waves in heterogeneous elastic rods via homogenization

We consider the propagation of a planar loop on a heterogeneous elastic rod with a periodic microstructure consisting of two alternating homogeneous regions with different material properties. The analysis is carried out using a second-order homogenization theory based on a multiple scale asymptotic expansion.     

Non-linear stability analysis of imperfect thin-walled composite beams

The static non-linear behavior of thin-walled composite beams is analyzed considering the effect of initial imperfections. A simple approach is used for determining the influence of imperfection on the buckling, prebuckling and postbuckling behavior of thin-walled composite beams. The fundamental and secondary equilibrium paths of perfect and imperfect systems corresponding to a major imperfection are analyzed for the case where the perfect system has a stable symmetric bifurcation point. A geometrically non-linear theory is formulated in the context of large displacements and rotations, through the adoption of a shear deformable displacement field. An initial displacement, either in vertical or horizontal plane, is considered in presence of initial geometric imperfection. Ritz's method is applied in order to discretize the non-linear differential system and the resultant algebraic equations are solved by means of an incremental Newton-Rapshon method. The numerical results are presented for a simply supported beam subjected to axial or lateral load. It is shown in the examples that a major imperfection reduces the load-carrying capacity of thin-walled beams. The influence of this effect is analyzed for different fiber orientation angle of a symmetric balanced lamination. In addition, the postbuckling response obtained with the present beam model is compared with the results obtained with a shell finite element model (Abaqus).