Time-Response Functions of Mechanical Networks with Inerters and Causality

This paper derives the causal time-response functions of three-parameter mechanical networks that have been reported in the literature and involve the inerter-a two-node element in which the force-output is proportional to the relative acceleration of its end-nodes. This two-terminal device is the mechanical analogue of the capacitor in a force-current/velocity-voltage analogy. The paper shows that all frequency-response functions that exhibit singularities along the real frequency axis need to be enhanced with the addition of a Dirac delta function or with its derivative depending on the strength of the singularity. In this way the real and imaginary parts of the enhanced frequency response functions are Hilbert pairs; therefore, yielding a causal time-response function in the time domain. The integral representation of the output signals offers an attractive computational alternative given that the constitutive equations of the three-parameter networks examined herein involve the third derivative of the nodal displacement which may challenge the numerical accuracy of a state-space formulation when the input signal is only available in digital form as in the case of recorded seismic accelerograms

Revisiting Schrodinger's fourth-order, real-valued wave equation and its implications to energy levels

In his seminal part IV, Ann. der Phys. Vol 81, 1926 paper, Schrodinger has developed a clear understanding about the wave equation that produces the correct quadratic dispersion relation for matter-waves and he first presents a real-valued wave equation that is 4th-order in space and 2nd-order in time. In view of the mathematical difficulties associated with the eigenvalue analysis of a 4th-order, differential equation in association with the structure of the Hamilton-Jacobi equation, Schrodinger splits the 4th-order real operator into the product of two, 2nd-order, conjugate complex operators and retains only one of the two complex operators to construct his iconic 2nd-order, complex-valued wave equation. In this paper we show that Schrodinger's original 4th-order, real-valued wave equation is a stiffer equation that produces higher energy levels than his 2nd-order, complex-valued wave equation that predicted with remarkable success the visible energy levels observed in the visible atomic line-spectra of the chemical elements. Accordingly, the 4th-order, real-valued wave equation is too stiff to predict the emitted energy levels from the electrons of the chemical elements; therefore, the paper concludes that Quantum Mechanics can only be described with the less stiff, 2nd-order complex-valued wave equation; unless in addition to the emitted visible energy there is also dark energy emitted.Comment: 22 pages, 3 figure

Modal identification of seismically isolated bridges with piers having different heights

This paper investigates the modal identification of seismically isolated bridges when the localized nonlinear behavior from the isolation bearing initiates at different times due to the uneven height of the bridge piers. More specifically, a three-span bridge supported on spherical sliding bearings is examined. Three different states of the same system with different natural periods emerge during an excitation; the linear system (LS), the partially isolated system (PIS) and the fully isolated system (FIS). Firstly, the paper identifies the time intervals that each state performs by using acceleration data. Subsequently, modal identification techniques such as the Prediction Error Method and a time-frequency wavelet analysis are applied on each interval. The LS’ results are dependable compared to the PIS which is a mildly nonlinear system. The results corresponding to the FIS suggest that it is preferable to apply the modal identification techniques on each interval independently, rather than on the entire response signal

Analyticity and causality of the three-parameter rheological models

In this paper the basic frequency response and time response functions of the three-parameter Poynting-Thomson solid and Jeffreys fluid are revisited. The two rheological models find application in several areas of rheology, structural mechanics and geophysics. The relation between the analyticity of a frequency response function and the causality of the corresponding time-response function is established by identifying all singularities at ω=0 after applying a partial fraction expansion to the frequency response functions. The strong singularity at ω=0 in the imaginary part of a frequency response function in association with the causality requirement, imposes the addition of a Dirac delta function in the real part in order to make the frequency response function well defined in the complex plane. This external intervention, which was first discovered by P.A.M. Dirac, has not received the attention it deserves in the literature of viscoelasticity and rheology. The addition of the Dirac delta function makes possible the application of time domain techniques that do not suffer from violating the premise of causality