Continuous Mechanical Systems Research Articles

A theoretical approach to the interaction between buckling and resonance instabilities

This article deals with the interaction between buckling and resonance instabilities of mechanical systems. Taking into account the effect of geometric nonlinearity in the equations of motion through the geometric stiffness matrix, the problem is reduced to a generalized eigenproblem where both the loading multiplier and the natural frequency of the system are unknown. According to this approach, all of the forms of instabilities intermediate between those of pure buckling and pure forced resonance can be investigated. Numerous examples are analyzed, including discrete mechanical systems with one to n degrees of freedom, continuous mechanical systems, such as oscillating deflected beams subjected to a compressive axial load, as well as oscillating beams subjected to lateral–torsional buckling. A general finite element procedure is also outlined, with the possibility to apply the proposed approach to any general bi- or tri-dimensional framed structure. The proposed results provide a new insight in the interpretation of coupled phenomena such as flutter instability of long-span or high-rise structures.

Geometry and dynamics in the Fresnel transforms of discrete systems

Free propagation in continuous optical and mechanical systems is generated by the momentum-squared operator and results in a shear of the phase space plane along the position coordinate. We examine three discrete versions of the Fresnel transform in periodic systems through their Wigner function on a toroidal phase space. But since it is topologically impossible to continuously and globally shear a torus, we examine a fourth version of the Fresnel transform on a spherical phase space, in a model based on the Lie algebra of angular momentum, where the corresponding Fresnel transform wrings the sphere.

POMs analysis of randomly vibrating systems obtained from Karhunen–Loève expansion

The Karhunen–Loève (KL) decomposition establishes that a 2D random field can be expanded as a series involving a sequence of deterministic orthogonal functions with orthogonal random coefficients. The proper orthogonal decomposition (POD) method consists in detecting spatially coherent modes in the dynamics of a spatio-temporally varying system by diagonalizing the spatial correlation function given by an averaging operator. The KL expansion is applied here to the responses of randomly excited vibrating systems with a view to performing a POD in separated-variables (time and space) form. Discrete and continuous mechanical systems are considered in this study as well as stationary and transient (non-stationary) responses. An averaging operator involving time and ensemble averages is proposed to draw up the POD in separated-variables form from the associated KL expansion. The result obtained using this approach agrees with the classical POD in the case of deterministic or ergodic random signals. The associated proper orthogonal modes are interpreted in case of linear and nonlinear vibrating systems subjected to white noise excitation in terms of normal modes.

SLOW HIGH-FREQUENCY EFFECTS IN MECHANICS: PROBLEMS, SOLUTIONS, POTENTIALS

Strong high-frequency excitation (HFE) may change the "slow" (i.e. effective or average) properties of mechanical systems, e.g. their stiffness, natural frequencies, equilibriums, equilibrium stability, and bifurcation paths. This tutorial describes three general HFE effects: Stiffening — an apparent change in the stiffness associated with an equilibrium; Biasing — a tendency for a system to move towards a particular state which does not exist or is unstable without HFE; and Smoothening — a tendency for discontinuities to be apparently smeared out by HFE. The effects and a method for analyzing them are introduced, first in terms of simple physical examples, and then in generalized form for mathematical models covering broad classes of discrete and continuous mechanical systems. Several application examples are summarized. Three mathematical tools for analyzing HFE effects are described and compared: The Method of Direct Separation of Motions, the Method of Averaging, and the Method of Multiple Scales. The tutorial concludes with a suggestion that more vibration experts, researchers and students should be aware of HFE effects, for the benefit of general vibration troubleshooting, and also for furthering the creation of innovative technical devices and processes utilizing HFE effects.

Hybrid graphs in modelling and analysis of discrete–continuous mechanical systems

The scope of the paper is a method of modelling vibrating mechanical systems with discrete–continuous distribution of parameters and couplings by means of hybrid graphs. The method is focused on broadening hybrid graphs formalism by mechanical systems and improving their description and design methods so that the mathematical formalism can reflect the essence of the problem involved in the designation of dynamic characteristics of such systems.

Computer-aided synthesis of discrete–continuous subsystems of machines with the assumed frequency spectrum represented by graphs

The problem of synthesising discrete physical, mechanical, electrical and electronic systems is widely developed in scientific research. However, it is the problem to find examples of methods of synthesising discrete–continuous mechanical systems. In the paper a foundation of the synthesis such systems has been presented, and its exact formulation and solution is still to deal with.

THE ROLE OF EIGENPARAMETER GRADIENTS IN THE DETECTION OF PERTURBATIONS IN DISCRETE LINEAR SYSTEMS

Continuous mechanical systems can be characterized through analysis of discrete linear models. Such models can provide approximations for the eigenvalues and eigenvectors (collectively, “eigenparameters”). Although the eigenparameters do not qualify as measures for the state of the system in response to specific loading and boundary conditions, they do reflect the identity of the system, and this in itself has important applications. In particular, the eigenparameters can be used to study the sensitivity of a system to perturbations, due perhaps to damage incurred by one or more discrete elements. These studies can rationalize the choice and weighting of eigenparameters for system identification strategies, damage detection algorithms, and damage assessment methods. To this end, this paper develops a set of sensitivity coefficients based on gradients of the eigenparameters. Sensitivities are normalized with respect to that of the harmonic oscillator, and generalized to include the mode vectors through the definition of a figure of merit. Analytical and numerical examples based on appropriate elements are used to illustrate the utility of the approach.

Reliability analysis of continuous mechanical systems using multistate fault trees

A method is presented which employs multistate fault trees to determine the reliability of a system wholly or partially comprised of continuous random variables. The advantage of this method over Monte Carlo simulation is that an algebraic expression for system reliability can be generated which, in turn, can be used to evaluate the reliability of the system corresponding to various probability distributions. A four-bar function generating mechanism and a single-shoe drum brake are considered as illustrative examples which compare the results of the present approach to those obtained by the Monte Carlo method.

Eigensolutions of non‐conservative continuous mechanical systems computed by use of a matrix iteration method

AbstractTwo‐sided Ostrowski matrix iteration is applied to solve eigenproblems of non‐conservative continuous systems in structural dynamics starting from known eigensolutions of a related conservative system. Eigensolution paths are followed as the parameters of the system governing its non‐conservative behaviour are increased from zero. No discretization is performed and no truncation errors are introduced. The real eigenvalues of the related conservative problem are calculated by use of the Wittrick–Williams algorithm. The corresponding eigenvectors are obtained with an inverse iteration method using eigenfrequency‐dependent trial eigenvectors. Residues of system harmonic transfer functions are computed using the eigenvalues and eigenvectors together with eigenvector normalization constants (modal Foss dampings). These residues are needed in a transient analysis. The method is applied to the generalized Beck column stability problem. Critical loads are calculated. Harmonic flexibilities of subcritically loaded columns in transverse vibration are studied. Applications to control problems are foreseen.

Open‐loop vibration control of elastic structural elements

AbstractThis paper discusses a general technique for vibration control of simple, continuous mechanical systems. The discussion involves giving the expressions for the response of an illustrative element (fixed‐free bars) to a sinusoidal disturbance and then formulating the problem of open‐loop vibration control of this system as a mathematical programming problem. The final step uses numerical optimization techniques to find the control forces. When these techniques are used, the amplitude of the system's steady state response is minimized. The discussion also shows that while this method effectively controls the vibration, it has a great deal of flexibility in accomplishing its stated purpose.

Integral covariance analysis for random vibrations of linear continuous mechanical systems

In this paper the method of integral covariance analysis is generalized. It yields new closed-form solutions in random vibrations of a large class of continuous systems under uncorrected loadings. These solutions may be of great importance since in addition to their own merits they can also be used to check numerical and approximate methods, among other things.

Changes in fat concentration of human milk during delivery by intermittent bolus and continuous mechanical pump infusion

The changes in fat concentration and cumulative fat losses that occur during the delivery of human milk using two different continuous infusion systems were compared with the changes in fat concentration during simulated intermittent gavage or bolus feedings. With both mechanical pumps the largest cumulative fat losses and the greatest decreases in fat concentrations occurred at the slowest infusion rates. State of homogenization of the milk generally made little difference in the changes in fat concentration using the syringe pump, whereas homogenizing the milk increased the fat concentration significantly with the roller pump. With the syringe pump the positioning of the syringe tip (horizontal or vertical) made no difference in fat concentration at an infusion rate of 1 ml/hr, whereas at 4 and 7 ml/hr the fat concentration was increased significantly by keeping the syringe tip vertical. With either mechanical pump a large fat bolus was delivered during the eighth and final hour of infusion if the milk remaining in the tubing was recovered by using air infusion at the same infusion rate. Intermittent bolus delivery of human milk resulted in no significant loss of human milk fat, no changes in fat concentration, and no terminal delivery of a large fat load. Thus intermittent bolus feedings are preferred over continuous mechanical pump infusion systems for the delivery of human milk to low-birth-weight infants.

The helix as a mechanical filter — an example of continuous mechanical systems analyzed by use of the fourier time transform and complex frequency response

The helix as a mechanical filter — an example of continuous mechanical systems analyzed by use of the fourier time transform and complex frequency response

Lagrangian formalism for vibrations in some damped continuous systems

While dissipative effects in discrete and continuous mechanical systems have usually been treated, in the Lagrangian formalism, via the introduction of either the Rayleigh dissipative function or fictitious co-ordinates and fields, subsequent work has shown that certain discrete-mass, damped systems can be treated entirely within Lagrangian dynamics. Here this method is extended to vibrations in some linearly-damped continuous systems, including transverse waves on a string, longitudinal vibrations in a rod, and to damped acoustic waves in a viscous fluid in narrow channels.