Time Derivative Of Displacement Research Articles

Location of eigenmodes of Euler-Bernoulli beam model under fully non-dissipative boundary conditions.

The Euler-Bernoulli beam model with non-conservative feedback-type boundary conditions is investigated. Components of the two-dimensional input vector are shear and moment at the right end, and components of the observation vector are time derivative of displacement and slope at the right end. The boundary matrix containing four control parameters relates input and observation. The following results are presented: (i) if one and only one of the control parameters is positive and the rest of them are equal to zero, then the set of the eigenmodes is located in the open left half-plane of the complex plane, which means that all eigenmodes are stable; (ii) if the diagonal elements of the boundary matrix are positive and off-diagonal elements are zeros, then the set of the eigenmodes is located in the open left half-plane, which implies stability of all eigenmodes; (iii) specific combinations of the diagonal and off-diagonal elements have been found to ensure the stability results. To prove the results, two special relations between the eigenmodes and mode shapes of the non-self-adjoint problem and clamped-free self-adjoint problem have been established.

To believe the past or to trust the future

A small ball rolls down from a quarter-circle to a frictionless plane. What will be themagnitude of the normal force when the ball arrives at the tangent point of the circle andthe plane? According to the centripetal force formula, the normal force will be3mg when the curvature radius of the circle is considered, but will bemg instead when the curvature radius of the plane is considered. Which one is the correctanswer? The difficulty is that Newton’s second law requires the second time derivative ofdisplacement to be continuous; however, this condition is not fulfilled at the tangent point.In this paper we will discuss several possible solutions to this problem in detail.

A rational approach to mass matrix diagonalization in two‐dimensional elastodynamics

AbstractA variationally consistent methodology is presented, which yields diagonal mass matrices in two‐dimensional elastodynamic problems. The proposed approach avoids ad hoc procedures and applies to arbitrary quadrilateral and triangular finite elements. As a starting point, a modified variational principle in elastodynamics is used. The time derivatives of displacements, the velocities, and the momentum type variables are assumed as independent variables and are approximated using piecewise linear or constant functions and combinations of piecewise constant polynomials and Dirac distributions. It is proved that the proposed methodology ensures consistency and stability. Copyright © 2004 John Wiley & Sons, Ltd.

Simple nonlinear jerk functions with periodic solutions

The appearances of jerk, the third time derivative of displacement, in several papers of mathematical and physical content are reviewed. Two simple nonlinear jerk functions, all of whose solutions are periodic, are introduced and solved. The first has period dependent on initial acceleration but independent of amplitude, and is related to the second-order cubic oscillator. The second, with intrinsically third-order dynamics, has period dependent also on initial displacement. A possible bearing on chaotic jerk functions is mooted.

Source mechanisms and near-source wave propagation from broadband seismograms

Recording seismic events at teleseismic distances with broadband and high dynamic range instruments provides new high-quality data that allow us to interpret in more detail the complexity of seismic rupture as well as the heterogeneous structure of the medium surrounding the source where waves are initially propagating. Wave propagation analysis is performed by ray tracing in a local cartesian coordinate system near the source and in a global spherical coordinate system when waves enter the mantle. Seismograms are constructed at each station for a propagation in a 2.5-D medium. Many phases can be included and separately analyzed; this is one of the major advantages of ray tracing compared to other wave propagation techniques. We have studied four earthquakes, the 1988 Spitak Armenia Earthquake (Ms = 6.9), the 1990 Iran earthquake (Ms = 7.7), the 1990 romanian earthquake (Ms = 5.8) and the 1992 Erzincan, Turkey earthquake (Ms = 6.8). These earthquakes exhibit in different ways the complexity of the rupture and the signature of the medium surrounding the source. The use of velocity seismograms, the time derivative of displacement, increases the difficulty of the fit between synthetic seismograms and real seismograms but provides clear evidence for a need of careful time delay estimations of the different converted phases. We find that understanding of the seismic rupture as well as the influence of the medium surrounding the source for teleseismically recorded earthquakes requires a multi-stop procedure: starting with ground displacement seismograms, one is able to give a first description of the rupture as well as of the first-order influence of the medium. Then, considering the ground velocity seismograms makes the fit more difficult to obtain but increases our sensitivity to the rupture process and early converted phases. With increasing number of worldwide broadband stations, a complex rupture description is possible independently of field observations, which can be used to check the adequacy of such complicated models.