Optimum design criteria for elastic structures subject to random dynamic loads

A new stochastic optimization method, which makes use of a constraint on structural reliability, is proposed for structures subject to dynamic random loads. A minimum weight problem is posed, in which a constraint condition imposes that the failure probability must be smaller than a given admissible level. The failure is determined by the first crossing outside the safe domain of a suitable structural response vector. The method is used to find the optimal shape of an elastic vertical column supporting a fixed mass positioned on the top, subject to a Gaussian filtered stationary stochastic horizontal acceleration process. The column, with variable annular cross-section, is described by a deterministic elastic multi-degree-of-freedom system. It is assumed that failure is reached when its lateral displacement exceeds an acceptable threshold value. Under this constraint, the structural weight is minimized and the optimal shape is determined for different structural conditions.     

Optimum design of truss structures undergoing large deflections subject to a system stability constraint

A structural optimization algorithm is developed for shallow trusses undergoing large deflections subject to a system stability constraint. The method combines the non‐linear buckling analysis, through displacement control technique, with the optimality criteria approach. Four examples illustrate the procedure and allow the results obtained to be compared with those in the literature. It is shown that a design based on the generalized eigenvalue problem (linear buckling) highly underestimates the optimum mass for these types of structures so a design based on the linear buckling analysis can result in catastrophic failure. In one of the design examples the stresses in the elements, in the optimum design, exceed the allowable stresses, pointing out the need for a design that accounts for both non‐linear buckling and stress constraints. Copyright © 2000 John Wiley & Sons, Ltd.     

A model to evaluate dynamic stability of imperfection sensitive shells

The paper deals with the perturbation energy concept and its application to stability of imperfection sensitive structures under time-dependent loads. The evaluation of the stability is based on energy norms, which may be used for investigation of safety against buckling. Starting from a stable state of equilibrium the presented procedure allows to decide whether the structure stays for a certain load history within critical bounds, which separate the motion round the prebuckling state from a motion in the postbuckling region. The stability is proved by comparing the critical energy calculated by a static analysis and the load induced energy. Applications to a truss and a spherical shell illustrate the variety of the phenomena in dynamic buckling behaviour of elastic structures in case of impulsive loading and the accuracy of the proposed method.     

Experimental investigations on the dynamic plastic buckling of a slender beam subject to axial impact

Experimental investigations on the dynamic plastic impact buckling of slender beams were conducted. A 10 kg free-fall impact hammer applied an axial impulsive load to aluminium (6061-T6) and cold rolled steel beams (CR1018) with a fixed-slide bearing boundary condition. Two rectangular cross-sections (50.8 mm×3.175 mm and 50.8 mm×4.7625 mm) with effective slenderness ratios ranging from 75 to 300 were considered. A total of 94 tests were conducted and the deformed buckled geometry for each impact event was profiled. A normalized response parameter was defined to characterize the observed dynamic buckling behaviour, which related the axial position and transverse amplitude of the peak buckle with respect to local stiffness and natural frequency characteristics. Simplified theoretical expressions defining the critical buckle wavelength provided a reasonable bound limit with respect to the experimental data.     

Simple models for the stability of a crack in a cantilever beam subject to impact

The use of simple models to study the structural dynamics and crack stability of a cracked cantilever beam subject to impact is demonstrated. A rigid-perfectly plastic material model is used, and the J-integral and tearing modulus concepts are employed in simple forms. Concise algebraic expressions provide means for validating some of the assumptions made, and they are shown to be capable of capturing the salient phenomena involved in the problem.     

The stability of a penny-shaped crack in a solid subject to an applied tensile stress

The stability of a penny-shaped crack in a solid subject to an applied tensile stress is investigated using a model in which plastic deformation is assumed to be confined to the plane of the crack, the growth criterion being that a constant crack tip opening angle is maintained at the crack tip during growth. The results are compared with those for the analogous two-dimensional crack problem. It is shown that a penny-shaped crack is appreciably more stable than a two-dimensional crack, when the applied stress levels are high relative to the yield stress.

---

Résumé On étudie la stabilité d'une fissure circulaire noyée dans un solide soumis à une contrainte de traction en utilisant un modèle dans lequel on suppose que la déformation plastique est limitée au plan de la fissure, le critère de croissance étant que se maintient à l'extrémité de la fissure un angle d'ouverture de fissure constant au cours de cette croissance. On compare les résultats avec ceux obtenus dans un problème analogue d'une fissure à deux dimensions. On montre qu'une fissure circulaire noyée présente une stabilité plus grande qu'une fissure à deux dimensions, dès lors que les niveaux de contrainte appliquée sont élevés en regard de la limite d'élasticité.

     

矩形薄板在面内随机参数激励下的随机稳定性与分岔研究

本文根据小挠度薄板的弹性理论建立了矩形薄板的受面内随机激励的振动模型,并用Galerkin变分法将其化简为常微分非线性动力学方程。又利用拟不可积Hamilton平均理论将方程等价为一个一维的Ito随机微分方程,并通过计算系统的最大Lyapunov指数来研究系统的局部随机稳定性,同时利用奇异边界理论研究了模型的全局稳定性,最后通过稳态概率密度函数的形状研究了系统参数对发生的随机Hopf分岔现象的影响,发现随机Hopf分岔在两个关键值附近发生,数值模拟结果验证了理论分析的正确性。     

Finite element investigations on the dynamic plastic buckling of a slender beam subject to axial impact

The dynamic plastic buckling response of slender beams, with geometric imperfections, subject to an axial impact is modelled by the finite element method. One of the primary objectives for the current study was the development of an accurate numerical model, which was validated by comparison with experimental investigations. A sensitivity analysis was conducted that considered the element aspect ratio, element formulation, constitutive relationship, boundary condition and type of geometric imperfection. The buckling response was defined by a normalized modal parameter as a function of the effective slenderness ratio. The normalized modal parameter accounted for the axial position and amplitude of the transverse peak buckle with respect to the local stiffness characteristics and natural frequency parameters. For the parameters considered, the influence of modelled boundary conditions and element formulation on the computed buckling response was established. In relative terms, the aspect ratio, imperfection model and contact mechanics was of secondary importance on the modal behaviour.     

Resonator/Oscillator response to liquid loading

The resonant frequency of a thickness-shear mode resonator operated in contact with a fluid was measured with a network analyzer and with an oscillator circuit. The network analyzer measures changes in the device's intrinsic resonant frequency, which varies linearly with (ρη)(1/2), where ρ and η are liquid density and viscosity, respectively. The resonator/oscillator combination, however, responds differently to liquid loading than the resonator alone. By applying the operating constraints of the oscillator to an equivalent-circuit model for the liquid-loaded resonator, the response of the resonator/oscillator pair can be determined. By properly tuning the resonator/oscillator pair, the dynamic range of the response can be extended and made more linear, closely tracking the response of the resonator alone. This allows the system to measure higher viscosity and higher density liquids with greater accuracy.     

轴向载荷下超空泡航行体动力稳定性的数值研究

考虑超空泡航行体头部受到的轴向扰动载荷作用,推导了水下航行体动力失稳控制方程,进而利用Bolotin方法对水下航行体的动力稳定性问题进行了数值计算,得到了相应的临界频率曲线,并分析了载荷参数和航行体参数变化对主动力不稳定区域的影响规律,计算结果可为水下航行体的安全评估提供一定的理论依据。     

Numerical solution of the dynamic stability problems

A novel integral equation technique is employed for the analysis of dynamic stability problems. The governing equation of the linearized parametric resonance problem is transformed into an integral equation. The kernel of the integral equation is computed as the influence function for the deflection and/or bending moment of a corresponding beam. The highest derivative of the governing function (in our case fourth derivative of the displacement function) is chosen as the basic unknown. Using the formal analogy with the differential equation of the beam flexure this highest derivative is comprehended as some unknown transverse ‘load’. The distribution of this ‘load’ is a priori assumed to be polygonal. Using elementary methods of structural analysis, the displacements due to the assumed ‘load’ are determined. These displacements, arrayed into a square matrix, approximate the kernel of the governing integral equation. The subsequent procedure via Hill's determinant is a conventional one. The results prove to be accurate enough even for a very modest number of points of integration. This reflects the fact that the method is based on numerical integration rather than on numerical differentiation.     

Line crack subject to shear

Field equations of nonlocal elasticity are solved to determine the state of stress in the neighborhood of a line crack in an elastic plate subject to a uniform shear at the surface of the crack tip. A fracture criterion based on the maximum shear stress gives the critical value of the applied shear for which the crack becomes unstable. Cohesive stress necessary to break the atomic bonds is calculated for brittle materials.

---

Résumé Les équations de champ d'élasticité non locale sont résolues afin de déterminer l'état de tension au voisinage d'une fissure linéaire dans une tôle élastique soumise à cisaillement uniforme à la surface de l'extrémité d'une fissure. Un critère de rupture basé sur la tension de cisaillement maximum fournit la valeur critique du cisaillement appliqué, pour lequel la fissure devient instable. La contrainte de cohésion nécessaire pour rompre les liaisons atomiques est calculée dans le cas des matériaux fragiles.

---

---

The present work was supported by the Office of Naval Research.     

The axial motion effect on the dynamic response of a laterally vibrating frame subject to a moving load

Summary In this paper the combined effect of the longitudinal inertia and other parameters on the free and forced motion of a laterally vibrating two-bar frame is thoroughly discussed. These parameters are: joint mass and its moment of inertia, slenderness ratio as well as moment of inertia and length ratios. The field equations are those of the classical dynamic analysis with the unique exception that the effect of the axial contraction (or extension) of the bar axes is accounted for when establishing the translational kinematic continuity boundary conditions. Such an effect may lead — under certain conditions of geometrical configurations of the frame — to serious discrepancies between standard dynamic analysis and the analysis presented herein. This is confirmed by means of a thorough parametric discussion of the dynamic response of the frame acted upon by a moving load.With 7 Figures     

Noise assessment of a Fourier transform infrared spectroradiometer subject to the stability of a conventional laboratory blackbody source

A method is described for the measurement of the noise-equivalent spectral radiance (NESR) of Fourier transform infrared (FTIR) spectroradiometers at all wave numbers of a selected range. The method requires minimal detailed knowledge of the sensor and no support equipment beyond a blackbody source. The NESRs of the FTIR spectroradiometer are determined at every wave-number increment in the 700-1300 cm(-1) range, for six resolutions, with a conventional blackbody source and ensembles of differential spectra. The NESRs are well behaved and consistent with the expected dependence on resolution; however, they depend on source temperature at the highest (1 cm(-1)) and lowest (32 cm(-1)) resolutions, with little or no statistical dependence at intermediate resolutions. Residual source drift is shown to be the likely cause of the dependence at 1 cm(-1); the dependence on the source at 32 cm(-1) resolution is shown to be most probably due to photon noise. At intermediate resolutions the sensor noise is dominant.     

Asymptotic Lyapunov stability with probability one of Duffing oscillator subject to time-delayed feedback control and bounded noise excitation

The asymptotic Lyapunov stability with probability one of a Duffing system with time-delayed feedback control under bounded noise parametric excitation is studied. First, the time-delayed feedback control force is expressed approximately in terms of the system state variables without time delay. Then, the averaged Itô stochastic differential equations for the system are derived by using the stochastic averaging method and the expression for the Lyapunov exponent of the linearized averaged Itô equations is derived. It is inferred that the Lyapunov exponent so obtained is the first approximation of the largest Lyapunov exponent of the original system, and the asymptotic Lyapunov stability with probability one of the original system can be determined approximately by using the Lyapunov exponent. Finally, the effects of time delay in feedback control on the Lyapunov exponent and the stability of the system are analyzed. The theoretical results are well verified through digital simulation.     

Nonlinear transient dynamic response of a blade subject to a pulsating load in a decaying centrifugal force field

This study is concerned with an analytical solution for the nonlinear transient dynamic response of a rotating blade in a gas turbine engine experiencing dynamic unbalance due to a blade-off scenario leading to rotor unbalance. As a consequence of rotor unbalance, the blade experiences a pulsating load at its tip due to contact with the casing and a decaying centrifugal force field due to the rotor deceleration. The governing equations of motion consider the blade as an elastic Timoshenko beam of varying thickness subject to changing rotational speeds. The analysis takes into account the variation in the thickness of the beam, the coupled axial and lateral displacements of the beam as a result of Coriolis component of acceleration, shear deformation, rotary inertia, and friction resulting from the contact between the blade and the casing. Our findings indicate that the thickness profile of the beam plays a significant role in the transient response. It further reveals that the decaying centrifugal force field leads to a dramatic change in the dynamic response and the resulting forces on the blade.