Second-order Differential Equation Of Motion Research Articles

Harmonic-wavelet approach for response spectrum estimation of vehicle and bridge systems with uncertain parameters subjected to stochastic excitation

Vehicle and bridge systems (VBSs) are subjected to stochastic excitations. Significant research effort has been devoted to the stochastic response analysis of VBSs. In most previous studies, VBSs have been treated as deterministic systems with certain parameters. However, the real VBSs always have uncertain parameters, such as the mass and spring stiffness of the vehicles and elastic modulus of the bridge material, which significantly affect the response power spectrum (RPS) estimation. Moreover, the equations of motion of the VBSs are essentially second-order differential equations with time-dependent and random coefficient matrices, and the excitation-response spectral relation cannot be directly described using the frequency response functions. These properties render the available methods unsuitable for the RPS estimation of the VBSs when both structural uncertainty and stochastic excitations are considered. In the present study, a harmonic-wavelet approach is proposed to estimate the time-varying RPSs of the VBSs with random parameters. The second-order differential equations of motion of VBSs with random parameters are first transformed into linear algebraic equations with random coefficient matrices using a harmonic-wavelet analysis technique; then, the excitation-response spectral relation of VBSs is derived based on these linear algebraic equations. Finally, the RPSs of VBSs with uncertain parameters are solved using the perturbation technique. The accuracy of the proposed method was verified by comparing it with the Monte Carlo method using a numerical example of railway VBS. The numerical simulation results show that the parameter uncertainties significantly affect the RPSs of the VBSs and increase the amplitude of the time-varying RPSs. The uncertainty of vehicle stiffness was the most sensitive factor for vehicle RPSs, and the bridge elastic modulus was the most sensitive factor for the bridge RPSs.

Comparative assessment of the contending force and placement methods for weightless sagging cables

Sagging elastic cables are inherently nonlinear structures due to the presence of self-weight apart from the presence of conventional physical and geometrical nonlinearities. Recently, authors have proposed a force method of weightless sagging elastic cables with linear tension–extension relations and undergoing small elastic displacements to focus explicitly on their characteristic ‘configurational’ nonlinearity. This force method is based upon the rate-type constitutive equations and third-order differential equations of motion. Existing displacement method is reconstructed here for the physically linear weightless sagging elastic cables undergoing finite elastic displacements. In place of nodal forces in force method, deformed state nodal placements are the primary variables in this placement method. Rate-type constitutive equations for two-node weightless planar sagging elastic cables are derived. The objective of the present paper is to compare and contrast these two contending methods in reference to their underlying hypotheses, analytical reach, and quantitative predictions of structural response. Possible equivalence of the proposed third-order and the popular incremental second-order differential equation of motion are explored. Theoretical significance of the present paper is critically evaluated.

Multi-component scalar fields and the complete factorization of its equations of motion

In the paper by Bazeia D. et al., EPL, 119 (2017) 61002, the authors demonstrate the equivalence between the second-order differential equation of motion and a family of first-order differential equations of Bogomolnyi type for the cases of single real and complex scalar field theories with non-canonical dynamics. The goal of this paper is to demonstrate that this equivalence is also valid for a more general classes of real scalar field models. We start the paper by demonstrating the equivalence in a single real scalar model. The first goal is to generalize the equivalence presented in papers by Bazeia et al. to a single real scalar field model without a specific form for its Lagrangian. The second goal is to use the setup presented in the first demonstration to show that this equivalence can be achieved also in a real multi-component scalar field model again without a specific form for its Lagrangian. The main goal of this paper is to show that this equivalence can be achieved in real scalar field scenarios that can be standard, or non-standard, with single, or multi-component, scalar fields.

Wavelet-Galerkin method for reconstruction of structural dynamic responses

In this article, the wavelet-Galerkin method is adopted for calculating structural dynamic responses under the framework of multiresolution analysis. At first, the second-order differential equation of motion is transformed into wavelet domain through function expansion with scaling function at the chosen resolution level. The number of basis function is uniquely determined, and the corresponding coefficients are approximated by numerical quadrature method. And then the two-term connection coefficients are derived and computed at various scales. After this, the dynamic responses of the system with single degree of freedom subjected to sinusoidal excitation are obtained and validated. Subsequently, the proposed method is extended to multidegree-of-freedom systems based on mode-superposition method. Finally, the parametric study is performed, and the simulation results show that the identified dynamic responses match the true responses very well with appropriate choice of the resolution level and the wavelet order.

Free and transient responses of linear complex stiffness system by Hilbert transform and convolution integral

This paper addresses the free and transient responses of a SDOF linear complex stiffness system by making use of the Hilbert transform and the convolution integral. Because the second-order differential equation of motion having the complex stiffness give rise to the conjugate complex eigen values, its time-domain analysis using the standard time integration scheme suffers from the numerical instability and divergence. In order to overcome this problem, the transient response of the linear complex stiffness system is obtained by the convolution integral of a green function which corresponds to the unit-impulse free vibration response of the complex system. The damped free vibration of the complex system is theoretically derived by making use of the state-space formulation and the Hilbert transform. The convolution integral is implemented by piecewise-linearly interpolating the external force and by superimposing the transient responses of discretized piecewise impulse forces. The numerical experiments are carried out to verify the proposed time-domain analysis method, and the correlation between the real and imaginary parts in the free and transient responses is also investigated.

Exact vibration solutions for cross-ply laminated plates with two opposite edges simply supported using refined theories of variable order

Abstract This paper presents exact solutions for free vibration of rectangular cross-ply laminated plates with at least one pair of opposite edges simply supported using refined kinematic theories of variable order. Exact natural frequencies are obtained using an efficient and unified formulation where the solving set of second-order differential equations of motion and related boundary conditions are expressed at layer level in terms of so-called fundamental nuclei having invariant properties with respect to the order of the plate theory. The nuclei are then appropriately expanded according to the number of layers and the order of the theory and the resulting equations are transformed into a first-order model whose solution is obtained by using the state space concept. In this way, the mathematical effort needed to derive analytical solutions is highly reduced. Both higher-order equivalent single-layer and layer-wise theories are considered in this study. Comparisons with other exact solutions are presented and useful benchmark frequency results for symmetric and un-symmetric cross-ply laminates are provided.

Nonlinear Periodic Response of Composite Curved Beam Subjected to Symmetric and Antisymmetric Mode Excitation

The periodic response of cross-ply composite curved beams subjected to harmonic excitation with frequency in the neighborhood of symmetric and antisymmetric linear free vibration modes is investigated. The analysis is carried out using higher-order shear deformation theory based finite element method (FEM). The governing equations are integrated using Newmark’s time marching coupled with shooting technique and arc-length continuation. Shooting method is used to solve the second-order differential equations of motion directly without converting them to the first-order differential equations. This approach is computationally efficient as the banded nature of equations is retained. A detailed study revealed that the response of antisymmetrically excited beams has contribution of higher antisymmetric as well as symmetric modes whereas the response of symmetrically excited beams has the significant participation of the higher symmetric modes except for the excitation in the neighborhood of first symmetric mode. The beam excited in the neighborhood of first symmetric mode has an additional branch corresponding to significant participation of first antisymmetric mode due to two-to-one internal resonance. Furthermore, for the beams excited in the neighborhood of higher modes, the peak response amplitude becomes less than that of the beam excited in the neighborhood of first mode but vibration behavior is drastically different due to the presence of subharmonics and higher harmonics. Two-to-one internal resonance between second antisymmetric mode and first symmetric mode is predicted for the first time.

Semi-empirical magnetostrictive delay line modelling

In this paper, analogous approach as commonly used at the electric lines was adopted to model the magneostrictive delay line (MDL) operating on amorphous ribbons and wires like that of composition Fe 78Si 7B 15. Particularly, the damping and deterioration of the propagating magnetostrictive wave along the sample and their relation to the intrinsic material properties were of primary interest. Two damping factors taken into the consideration lead to a second-order differential equation of motion that could be solved analytically for instance of a rectangular-pulse excitation. The Laplace transform and convolution, in most of cases as a discrete procedure, has to be utilized in all other cases. Theoretical assessment confronted with some experimental results is showing a fairly good agreement.

USE OF OPTIMAL QUASI– VELOCITY TRAJECTORIES

The paper presents optimization of the generalized velocity components (GVC) in order to obtain the energy efficient profile. The GVC arise from decomposition of the system mass matrix and the corresponding equations of motion are first-order. The obtained optimal quantities show some properties of the system that are not observable if the system is described using the second-order differential equations of motion. As a result, an insight into the system dynamics can be given and nonlinearity evaluation and reduction is possible. The presented approach was tested analytically and in simulation on a 2 DOF planar mechanical system.

A survey of equations of motion in terms of inertial quasi-velocities for serial manipulators

In this work we compare equations of motion using the so-called inertial quasi-velocities. As a result of these velocities we obtain two first-order decoupled equations of motion instead of one second-order differential equation of motion. The methods presented here, solve in a way, the problem of nonlinear dynamic decoupling. The first and the second method result from diagonalized Lagrangian robot dynamics (Jain and Rodriguez, IEEE Trans Robot Autom 11:571–584, 1995) and are known as normalized and unnormalized quasi-velocities. The third method described by Junkins and Schaub (J Astronaut Sci 45:279–295, 1997) offers eigenfactor quasi-coordinate velocities formulation for multibody dynamics. As a consequence of using transformation given by Loduha and Ravani (Trans ASME J Appl Mech 62:216–222, 1995) we obtain decoupled equations of motion in terms of modified generalized velocity components. Here we limit all these methods to serial manipulators. The novelty of this paper consists in physical interpretation of the quasi-velocities and discussion concerning equations of motion, the kinetic energy shaping, relationship between each of them and properties useful for simulation and control purposes. Also forward dynamics algorithms and their computational complexity in terms of new velocities are given. Simulation results illustrate the theoretical investigations. We conclude that all methods offer interesting possibilities for dynamic simulation and future control investigations.

Inertia of amoebic cell locomotion as an emergent collective property of the cellular dynamics

Amoebic cells are ubiquitous in many species and have been used as model systems to study the eukaryotic cellular locomotion. We construct a model of amoebic cells on two-dimensional grids, which describes sensing, cell status, and locomotion in a unified way. We show that the averaged position of simulated cells is described by a second-order differential equation of motion and that the mechanical pushing at the initial moment boosts the cell movement, which continues after the cell is released from the pushing. These "inertialike" features suggest the possibility of Newtonian-type motions in chemical distributions of the signaling molecule. We show, as an example, the possibility of rotating motion in a "centripetal" distribution. The observed inertial motion is an emergent collective dynamics, which is controlled by diffusive and chemical processes in the cell.

New approaches for non-classically damped system eigenanalysis

This paper presents three new approaches for solving eigenvalue problems of non-classically damped linear dynamics systems with fewer calculations than the conventional state vector approach. In the latter, the second-order differential equation of motion is converted into a first-order system by doubling the size of the matrices. The new approaches simplify the approach and reduce the number of calculations. The mathematical formulations for the proposed approaches are presented and the numerical results compared with the existing method by solving a sample problem with different damping properties. Of the three proposed approaches, the expansion approach was found to be the simplest and fastest to compute. Copyright © 2005 John Wiley & Sons, Ltd.

On Uncoupling and Solving the Equations of Motion of Vibrating Linear Discrete Systems

The equations of motion governing the vibrations of a linear, viscously damped, discrete system are generally mutually coupled. This article examines the problem, when the viscous damping is nonclassical, of how best to uncouple and solve by approximation the governing second-order differential equations of motion. It is shown that when the equations of motion are expressed in normal coordinates, the equations can then be transformed by an orthogonal coordinate transformation to a new generalized coordinate system in which a bound on the relative error introduced in the response by discarding all the coupling terms is a minimum. This approach extends the applicability of undamped modal analysis to certain types of nonclassically damped systems. The analytical results and the effectiveness of the proposed method are illustrated with four examples taken from other previously published approaches to the stated problem.

Mathematical Model of a Railway Pneumatic Brake System With Varying Cylinder Capacity Effects

Governing equations for the analysis of pressure transient are derived from the principle of conservation of mass and momentum for a pneumatic brake system, which consists of a train pipe connected to a number of linear actuators (brake cylinders with piston displacement). The governing one-dimensional non-linear partial differential equations for the train pipe, non-linear ordinary differential equations for the brake cylinders, and second-order differential equation of motion for piston displacement are solved to determine the pressure transients in the brake system for a step change in pressure at the inlet. The governing equations are nondimensionalized and reduced to a set of ordinary nonlinear differential difference equations and integrated by standard numerical methods. The flow is considered isothermal, and the friction effects for turbulent and laminar flow are evaluated by quasi-steady state approximation. The auxiliary reservoir volume effect is also included. The results are compared with the experimental data obtained on a brake test rig.