Boundary Element Method Used in Random Response Calculation of the Plate-Beam Structure

The furnace walls of the large boilers in power plants are combined structures consisting of orthotopic plate and equally spaced beams, which are usually submitted to random vibration under the excitation of the pressure fluctuation induced by combustion in the furnace. In this paper, a numerical method based on BEM to compute the random response of the structure is offered. The agreement between the computing results and the measured data in a practical example verifies the effectiveness of the method.

Transitions Through Period Doubling Route to Chaos

The Duffing driven, damped, “softening” oscillator has been analyzed for transition through period doubling route to chaos. The forcing frequency and amplitude have been varied in time (constant sweep). The stationary 2T, 4T… chaos regions have been determined and used as the starting conditions for nonstationary regimes, consisting of the transition along the Ω(t)=Ω0±α2t,f=const., Ω-line, and along the E-line: Ω(t)=Ω0±α2t;f(t)=f0∓α2t. The results are new, revealing, puzzling and complex. The nonstationary penetration phenomena (delay, memory) has been observed for a single and two-control nonstationary parameters. The rate of penetrations tends to zero with increasing sweeps, delaying thus the nonstationary chaos relative to the stationary chaos by a constant value. A bifurcation discontinuity has been uncovered at the stationary 2T bifurcation: the 2T bifurcation discontinuity drops from the upper branches of (a, Ω) or (a, f) curves to their lower branches. The bifurcation drops occur at the different control parameter values from the response x(t) discontinuities. The stationary bifurcation discontinuities are annihilated in the nonstationary bifurcation cascade to chaos — they reside entirely on the upper or lower nonstationary branches. A puzzling drop (jump) of the chaotic bifurcation bands has been observed for reversed sweeps. Extreme sensitivity of the nonstationary bifurcations to the starting conditions manifests itself in the flip-flop (mirror image) phenomena. The knowledge of the bifurcations allows for accurate reconstruction of the spatial system itself. The results obtained may model mathematically a number of engineering and physical systems.

Enhanced Vibration Control of Ultrasonic Tooling Using Finite Element Analysis

The use of finite element analysis (FEA) in high frequency (20–40 kHz), high power ultrasonics to date has been limited. Of paramount importance to the performance of ultrasonic tooling (horns) is the accurate identification of pertinent modeshapes and frequencies. Ideally, the ultrasonic horn will vibrate in a purely axial mode with a uniform amplitude of vibration. However, spurious resonances can couple with this fundamental resonance and alter the axial vibration. This effect becomes more pronounced for ultrasonic tools with larger cross-sections. The current study examines a 4.5″ × 6″ cross-section titanium horn which is designed to resonate axially at 20 kHz. Modeshapes and frequencies from 17–23 kHz are examined experimentally and using finite element analysis. The effect of design variables — slot length, slot width, and number of slots — on modeshapes and frequency spacing is shown. An optimum configuration based on the finite element results is prescribed. The computed results are compared with actual prototype data. Excellent correlation between analytical and experimental data is found.

Vibration and Stability of an Elastic Beam Subjected to a Periodic Axial Load With Time-Dependent Displacement Excitation at Both Ends

This paper concerns the transverse vibrations and stabilities of an elastic beam simultaneously subjected to a periodic axial load, a distributed transverse load, and time-dependent displacement excitations at both ends. The equation of motion derived from Bernoulli-Euler beam theory is a fourth-order partial differential equation with periodic coefficients. To obtain approximate solutions, the method of assumed-modes is used. The unknown time-dependent function in the assumed-modes method is determined by a generalized inhomogeneous Hill’s equation. The instability regions possessed by this generalized Hill’s equation are obtained by both the perturbation technique up to the second order and the harmonic balance method. The dynamic response and the corresponding spectrum of the transversely oscillating elastic beam are calculated by the weighted-residual method.

Localized and Non-Localized Normal Modes in a Strongly Nonlinear Discrete System

The free oscillations of a strongly nonlinear, discrete oscillator are examined by computing its “nonsimilar nonlinear normal modes.” These are motions represented by curves in the configuration space of the system, and they are not encountered in classical, linear vibration theory or in existing nonlinear perturbation techniques. For an oscillator with weak coupling stiffness and “mistiming,” both localized and nonlocalized modes are detected, occurring in small neighborhoods of “degenerate” and “global” similar modes of the “tuned” system. When strong coupling is considered, only nonlocalized modes are found to exist. An interesting result of this work is the detection of mode localization in the “tuned” periodic system, a result with no counterpart in existing theories on linear mode localization.

Investigation of Parametric Resonance Instability in a Flexible Rod Slider Crank Mechanism

A direct variational approach with a floating frame is presented to derive the ordinary differential equations of motion of a flexible rod, constant crank speed slider crank mechanism. Potential energy terms contained in the derivation include beam bending energy and energy in foreshortening of the rod tip (which were selected because of the importance of these terms in a pinned-pinned rod parametric resonance). A symbolic manipulator code is used to reduce the constrained equations of motion to unconstrained nonlinear equations. A linearized version of these equations is used to explore parametric resonance stability-instability zones at low crank speeds and small deflections by a monodromy matrix technique.

On the Nonlinear Behavior of Rotating Shafts

This paper presents a nonlinear model to describe the bending behaviour of a rotating shaft, based on the general theory of a bending bar. Justification for this theoretical model has been provided by tests, the resulting curves more closely fitting observed results than those of other models.

Random Response of a Sluice Gate Support

By the use of the extended finite element method the analysis of the random response of a linear structure to a continuous excitation field, random in time and space, is presented in this paper. The extended finite element method includes the formulation for obtaining the equivalent node force power spectrum. The corresponding computer program has been produced. A random response analysis of a sluice gate support shows satisfactory agreement with the experiment results.

Solution of Dynamic Problems of Structural Elements Using Simple Polynomial Approximations

A survey of studies dealing with vibrating structural elements using simple polynomial approximations in connection with Rayleigh-Ritz or Galerkin-type methods is presented. The classical use of polynomials when solving dynamic problems of deformable bodies consists of constructing a set of coordinate functions in such a way that they satisfy at least the essential boundary conditions and that they represent “reasonably well” the deformation field of the structural element under study. An alternative and more rational procedure has been developed and used in recent years whereby orthogonal polynomials are used. A “base function” is constructed and then one generates a set of orthogonal polynomials using the Gram-Schmidt or equivalent procedure. The present paper presents comparisons of numerical results in the case of different types of vibrating structural elements Special emphasis is placed on Rayleigh’s optimization procedure which consists of taking one of the exponents of the polynomial coordinate functions as an optimization parameter “γ”. Since the calculated eigenvalues constitute upper bounds, by minimizing them with respect to “γ” one is able to optimize the eigenvalues.

Self-Excited Oscillation Analysis of a Multidegree-of-Freedom System With Cubic Non-Linearities: Dynamic Problems in Rolling Mill (IV)

In nalyzing the self-excited oscillation in drive systems of a rolling mill, almost all of the scholars didn’t consider the internal resonance and zero-frequency. We indicate that these important factors couldn’t be neglected. In this paper, dynamic model is reconstructed, and self-excited oscillation is discussed in the presence of an internal resonance. Several new conclusions are drawn. These results are useful for design of blooming mill drive systems.