Numerical and analytical study of seismic response of structural systems with new formulation using energy and impact methods

This paper presents two simple and robust technique for response estimating of single-degree-of-freedom (SDOF) structural systems. The impulse method, because it is formulated based on the fundamentals of dynamics; especially, the linear impulse concept, and also the energy method, because the main idea of this method is inspired by energy conservation principles. These methods can strongly cope with linear damped systems for which damping ratio ζ is greater than 0.01. Assessment of SDOF dynamic systems under any arbitrary excitations is easily possible through the proposed methods. There is no error propagation through the solving process. The numerical example reveals the simplicity and robustness of the new technique compared to Duhamel’s integral and similar techniques. Finally, a numerical example is investigated to demonstrate the efficiency of the algorithms. The most famous record of El Centro ground motion is applied to the systems. The obtained results show that the new algorithm works exactly enough to compete with a conventional method such as the Duhamel integration method and the Newmark-β method. A comparison between the results of this method with the solution methods used by other researchers is shown to be a good match.

The Response of a SDOF System Driven by a Periodic Random Force

This work deals with the response of a linear undamped SDOF system exposed to a force with random amplitude, phase shift, or their combination. The first two moments, the mean value and the variance, of the response will be determined analytically through the Duhamel's integral, and compared to the numerical Monte Carlo simulations. Integration of associated equations of motion will be performed by the Newmark method of average acceleration.

A Revised Approach for One-Dimensional Time-Dependent Heat Conduction in a Slab

Classical Green’s and Duhamel’s integral formulas are enforced for the solution of one dimensional heat conduction in a slab, under general boundary conditions of the first kind. Two alternative numerical approximations are proposed, both characterized by fast convergent behavior. We first consider caloric functions with arbitrary piecewise continuous boundary conditions, and show that standard solutions based on Fourier series do not converge uniformly on the domain. Here, uniform convergence is achieved by integrations by parts. An alternative approach based on the Laplace transform is also presented, and this is shown to have an excellent convergence rate also when discontinuities are present at the boundaries. In both cases, numerical experiments illustrate the improvement of the convergence rate with respect to standard methods.

Expansion of Indicial Function Approximations for 2-D Subsonic Compressible Aerodynamic Loads

This article deals with the new generation of proper Mach dependent exponential approximations of the indicial aerodynamic functions toward the aeroelastic formulation of 2-D lifting surface in the subsonic compressible flow. The indicial lift response is a useful starting point in the development of a general time-domain unsteady aerodynamic theory. By definition, an indicial function is the response to a disturbance that is applied instantaneously at time zero and held constant thereafter; that is a disturbance given by a step function. If the indicial response is known, then the unsteady loads to arbitrary changes in angle of attack can be obtained through the superposition of indicial responses using Duhamel’s integral. The indicial functions have been used to modify the circulatory part of the lifting force and pitching moment in unsteady compressible aerodynamic models. The coefficients of the approximation are obtained with an indirect approach by relating numerical results obtained for oscillating airfoil in the frequency domain back into the time domain. compressible and supersonic flight speed regimes. Exponential approximations of the subsonic compressible indicial functions in the existing research works are available only in limited Mach numbers (M = 0.5, 0.6, 0.7, 0.8). In the present study, a novel exponential approximation is developed which represent the coefficients of approximations as functions of Mach number (0.5 < M < 0.8).

Simulation of the Dynamic Behaviour of a Thin-Walled Meshing Gear Using Duhamel’s Integral

Aircraft transmissions have the peculiar characteristics of light structures and high operating speeds, therefore relatively low flexural natural frequencies and high excitation frequencies due to rotation and meshing. Resonance vibrations can create serious problems of malfunctions and even catastrophic failures. A reliable numerical model is surely a convenient means to perform preliminary simulations to identify the most critical resonance conditions and evaluate the effect of structural modifications on the dynamic behaviour of the component in the design development phase. Most numerical investigations found in the literature are carried out on simplified models of the rotating bodies likened to discs to reduce the computational effort. In this work a novel approach based on the application of Duhamel’s integral for the determination of the dynamic behaviour of a rotating gear subject to meshing forces has been developed to obtain more reliable results with a realistic model at an affordable computational cost. The gear response to dynamic excitation is obtained by the determination of its response to impulse using a single 3D finite element transient analysis taking afterwards into account the effect of the gear rotation. Subsequently, the Duhamel’s integral is applied using the tooth load time history in order to simulate as realistically as possible the gear load conditions. This paper presents the case of a real bi-helicoidal gear. A test bench was simulated measuring the displacement observed by some non-rotating virtual displacement sensors, located near the gear rim and disc. The signal was processed identifying the most critical rotating speeds on the basis of its RMS value. The numerical Campbell speed/frequency diagrams are in good agreement with experimental results.