Naser Mostaghel

Representations of Coulomb friction for dynamic analysis

In many engineering problems involving friction, the friction is treated as Coulomb friction, where the magnitude of the friction force is constant but its direction is always opposite to that of the sliding velocity. In dynamic problems, the direction of the sliding velocity can change quite often. The many changes of the velocity direction cause many discontinuities in the friction force, complicating the process of evaluating the response of systems involving friction. In this paper, it is shown that the discontinuous Coulomb friction force can be represented by at least four different continuous functions. Each of these functions involves one constant that controls the level of accuracy of that functions representation of the friction force. The accuracy of the various representations is verified by comparing the response of a single degree freedom system, obtained through numerical solutions utilizing these representations, with an exact analytical solution.

Buckling of rotating rods and plates

Abstract The elastic instability of two basic structural elements, rods and plates, when subjected to compressive stresses arising from steady rotation is considered. The resulting eigenvalue problems are defined and in each case numerical procedures are employed which determine the critical rotational speed and its dependence on the relevant parameters.

Inversion of Ramberg–Osgood equation and description of hysteresis loops

Abstract The Ramberg–Osgood equation has been approximately inverted. Four orders of approximations providing progressively more accurate inversions are considered. The second order inversion is used to develop closed-form relations for stress in terms of strain and for cyclic stress in terms of cyclic strain. Explicit relations between the cyclic-strength coefficient and the cyclic strain-hardening exponent and between the cyclic strain-hardening exponent and the elastic modulus are developed. Application to nine different engineering metals shows that the proposed approximate inversion provides a powerful tool for describing stress–strain relations. Examples of hysteresis loops for a number of strain time-histories are presented. Also, it is shown that augmenting the strain time-history with fictitious strain segments to close all the hysteresis loops yields the same stress–strain relation as that obtained through the rain-flow counting method.

Performance analysis of aseismic base isolation systems for a multi-story building

Abstract A comparative study of the performances of several leading base isolation devices including the Rubber Bearing with and without lead core, the Sliding-Joint, the French System, and the Resilient-Friction isolator with and without upper sliding plate for a multi-story building under seismic ground accelerations is carried out. Several earthquake ground accelerations including El Centro 1940, Pacoima Dam 1971, and Mexico City 1985 earthquakes are considered. The acceleration, the base displacement and the structural deflection response spectra for the structure with different base isolation systems under various conditions are obtained and discussed. Several sensitivity analyses for variations in the parameters of the isolators and/or structure are also carried out. The effectiveness of various base isolators under a variety of conditions are studied and advantages and disadvantages of different systems are discussed. The results show that the acceleration transmitted and the column stresses of the structure can be effectively reduced by using properly designed base isolation systems. The results of the sensitivity analyses show the friction-type isolation systems are less sensitive to the variations in the amplitude and frequency content of the ground excitation in comparison to the Rubber Bearing base isolators. The study also shows that the peak accelerations transmitted by the friction-type isolators are not significantly affected by small variations in friction coefficient. Furthermore, variations in mass ratio and damping of the structure or the isolator do not significantly affect the peak responses.

On the stability of a class of non-stationary non-linear random systems

The stability of the trivial equilibrium solution of a class of non-linear differential equations with a non-stationary absolutely integrable random coefficient is studied. The mean square stability of random systems is discussed and stability theorems of Bertram and Sarachik (1959) are stated. It is shown that the equilibrium null solution is mean square stable under some mild conditions. Some examples are also considered.

On the stability of Hill's equation☆

By means of a modification of Malkins method, the stability of the solutions of Hills equation is studied. The procedure utilized is such that secular terms never appear in the development. From a first approximation, a simple expression is obtained for the boundaries of the stable regions in parameter space. A second approximation is also presented and applications are made to Mathieus equation.

Stability and Upper Bound to the Response of Tall Structures to Earthquake Support Motion

Abstract A vertical cantilever beam model of tall structures is considered. The mean square stability of the structures subjected to vertical ground acceleration is studied via the Liapunov method of stability analysis. Due to the presence of self-weight of the structure, the equation of motion cannot be decoupled by the use of usual normal modes, and the partial differential equation of vibration must be employed directly in the stability analysis. The stability of the tall structure is analyzed, and some bounds on the deflection are obtained and discussed.

Smooth site dependent spectra

Abstract Based on the characteristic site period and the presumed peak ground acceleration, a method is proposed for the construction of smooth site dependent spectra. Smooth spectra are plotted and compared with spectra of many actual records. To show the generality of the proposed method the records used for comparison are selected to have small, intermediate and relatively large predominant periods with various intensities from relatively small, intermediate, to large events. The agreement in all cases is very good. The proposed site dependent spectra are also compared with Newmarks and the Regulatory Guide 1.60 spectra. The comparison is favorable for a limited range of characteristic site periods.

The Role of Time in Cosmic Expansion

By treating time as an independent variable free of space-time the role of time in cosmic expansion is clarified. We show that this treatment of time is consistent with General Relativity, and addresses the quandaries of dark or vacuum energy. We consider the current of time to be composed of many time waves. As the current flows, the number of its waves keeps increasing. It is shown that the cumulative sum of the periods of these waves represents the stretching-time, the redshift, Z represents the stretching velocity, and the quantity Z 2 / t represents the stretching acceleration of the stretching-time. By isolating time from space-time we find a simple equation which is developed based only on time and its kinematics. The validity of this equation is confirmed first through the conformity of its predictions with Einstein’s three predictions, namely the precession of Mercury’s orbit, the bending of light by the sun’s gravity, and the gravitational time dilation. Second, its validity is further confirmed through its consistency with three different sets of observational data as well as with the recent LIGO/Virgo gravitational waves measurement. It is shown that the flow of stretching-time is propelled by the energy released at the big bang. Further, the Hubble constant is estimated analytically. Also a possible source and the quantity of what is called dark energy are identified. It is concluded that the time model may clear the way to a quantum mechanical description of the cosmos.