Stochastic Analysis of a 1-D System With Fractional Damping of Order 1/2

2002 ◽  
Vol 124 (3) ◽  
pp. 454-460 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
White Noise ◽  
Stochastic Analysis ◽  
Damping Ratio ◽  
Expansion Method ◽  
Stochastic Dynamic ◽  
Stochastic Response ◽  
Analytical Technique ◽  
Damped System ◽  
Duhamel Integral ◽  
General Response

This paper presents an analytical technique for the analysis of a stochastic dynamic system whose damping behavior is described by a fractional derivative of order 1/2. In this approach, an eigenvector expansion method is used to obtain the response of the system. The properties of Laplace transforms of convolution integrals are used to write a set of general Duhamel integral type expressions for the response of the system. The general response contains two parts, namely zero state and zero input. For a stochastic analysis, the input force is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic characteristics, namely the variance and covariance responses of the system. Closed form stochastic response expressions are obtained for white noise. Numerical results are presented to show the stochastic response of a fractionally damped system subjected to white noise. Results show that stochastic response of the fractionally damped system oscillates even when the damping ratio is greater than its critical value.

An Analytical Scheme for Stochastic Dynamic Systems Containing Fractional Derivatives

1999 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
White Noise ◽  
Expansion Method ◽  
Stochastic Dynamic ◽  
Stochastic Response ◽  
Analytical Technique ◽  
Analytical Scheme ◽  
Damped System ◽  
Duhamel Integral ◽  
General Response ◽  
Stochastic Dynamic Systems

Abstract This paper presents an analytical technique for the analysis of a stochastic dynamic system whose damping behavior is described by a fractional derivative of order 1/2. In this approach, an eigenvector expansion method proposed by Suarez and Shokooh is used to obtain the response of the system. The properties of Laplace transforms of convolution integrals are used to write a set of general Duhamel integral type expressions. The general response contains two parts, namely zero state and zero input. For a stochastic analysis the input force is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic characteristics of the system. Closed form stochastic response expressions are obtained for white noise. Numerical results are presented to show the stochastic response of a fractionally damped system subjected to white noise.

Analytical Solution for Stochastic Response of a Fractionally Damped Beam

2004 ◽  
Vol 126 (4) ◽  
pp. 561-566 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Stochastic Analysis ◽  
Normal Modes ◽  
Continuous Beam ◽  
Stochastic Response ◽  
Analytical Technique ◽  
System A ◽  
Laplace Transform Technique ◽  
Duhamel Integral ◽  
Fractional Green’S Function ◽  
Response Expression

This paper presents a general analytical technique for stochastic analysis of a continuous beam whose damping characteristic is described using a fractional derivative model. In this formulation, the normal-mode approach is used to reduce the differential equation of a fractionally damped continuous beam into a set of infinite equations, each of which describes the dynamics of a fractionally damped spring-mass-damper system. A Laplace transform technique is used to obtain the fractional Green’s function and a Duhamel integral-type expression for the system’s response. The response expression contains two parts, namely, zero state and zero input. For a stochastic analysis, the input force is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic characteristics of the system. Closed-form stochastic response expressions are obtained for white noise for two cases, and numerical results are presented for one of the cases. The approach can be extended to all those systems for which the existence of normal modes is guaranteed.

Stochastic Analysis of a Fractionally Damped Beam

2001 ◽  
Author(s):  
Om P. Agrawal
Keyword(s):  
Stochastic Analysis ◽  
Normal Modes ◽  
Continuous Beam ◽  
Analytical Technique ◽  
System A ◽  
Laplace Transform Technique ◽  
Mass Damper ◽  
Duhamel Integral ◽  
Fractional Green’S Function ◽  
Response Expression

Abstract This paper presents a general analytical technique for stochastic analysis of a continuous beam whose damping characteristic is described using a fractional derivative model. In this formulation, the normal-mode approach is used to reduce the differential equation of a fractionally damped continuous beam into a set of infinite equations each of which describes the dynamics of a fractionally damped spring-mass-damper system. A Laplace transform technique is used to obtain the fractional Green’s function and a Duhamel integral type expression for the system’s response. The response expression contains two parts, namely zero state and zero input. For a stochastic analysis, the input force is treated as a random process with specified mean and correlation functions. An expectation operator is applied on a set of expressions to obtain the stochastic characteristics of the system. Closed form stochastic response expressions are obtained for White noise. The approach can be extended to all those systems for which the existence of normal modes is guaranteed.

Analysis of Stochastic Nonlinear Dynamics in the Gear Transmission System with Backlash

2015 ◽  
Vol 16 (2) ◽  
pp. 111-121 ◽  
Author(s):  
Jingyue Wang ◽  
Haotian Wang ◽  
Lixin Guo
Keyword(s):  
Time Course ◽  
Damping Ratio ◽  
Excitation Frequency ◽  
Period Doubling ◽  
Spur Gear ◽  
Transmission System ◽  
Gear Transmission ◽  
Stochastic Dynamic ◽  
Random Disturbance ◽  
Gear Transmission System

AbstractIn order to study the different backlash, gear damping ratio and random disturbance on dynamic behavior of gear transmission system, stochastic dynamic equations of the three-degree-of-freedom spur gear transmission system are established considering random disturbances of a low-frequency external excitation induced by torque fluctuation, gear damping ratio, gear backlash, excitation frequency and meshing stiffness. Using bifurcation diagram, phase diagram, time course diagram, Poincaré map and power spectrum of the system, the dynamic characteristics of the gear transmission system with different backlash under gear damping ratio changing, and the influence of the random disturbance of gear damping ratio on the bifurcation characteristic of system are analyzed. Numerical simulation shows that the gear transmission system will be from periodic motion with a noisy disturbance to chaotic-like motion by period-doubling bifurcation with decreasing gear damping ratio. In the small damping ratio range, the backlash has great effect on the motion characteristics. Random disturbance has an important effect on the bifurcation characteristics.

A novel energy dissipation outrigger system with rotational inertia damper

2018 ◽  
Vol 21 (12) ◽  
pp. 1865-1878 ◽  
Author(s):  
Liangkun Liu ◽  
Ping Tan ◽  
Haitao Ma ◽  
Weiming Yan ◽  
Fulin Zhou
Keyword(s):  
Damping Ratio ◽  
Mass Parameter ◽  
Inertial Force ◽  
Rotational Inertia ◽  
Analysis Model ◽  
Damping Force ◽  
Damped System ◽  
Modal Damping ◽  
Assembly Technique ◽  
Excellent Control

Rotational inertia damper, a novel damper, possessing the advantage of displacement amplification, has been employed in outrigger system for seismic mitigation. The equivalent analysis model composed by a uniform cantilever beam and an equivalent spring was proposed to simulate the rotational inertia damper outrigger system, by which the corresponding dynamic characteristic equation was derived based on numerical assembly technique. To gain the response of the damped system, finite element method and state space method have been utilized. Finally, the results show that the pseudo-undamped natural frequency ratios and system modal damping ratios are significantly influenced by stiffness parameter of the exterior column, while the mass parameter of the rotational inertia damper has little effect on them. The optimal damping ratio can be acquired for one mode, but it may be worse for the other mode in the same position equipping rotational inertia damper. Furthermore, numerical simulation results for the typical earthquake records have verified that the rotational inertia damper outrigger has excellent control performance in displacement as well as acceleration. A good agreement between damping force and equivalent force also suggests that the damping force of rotational inertia damper is predominant and the inertial force has no significant effect on the structure.

scholarly journals Displacement-Dependent Damping Inerter System for Seismic Response Control

2019 ◽  
Vol 10 (1) ◽  
pp. 257 ◽  
Author(s):  
Zhipeng Zhao ◽  
Ruifu Zhang ◽  
Yiyao Jiang ◽  
Dario De Domenico ◽  
Chao Pan
Keyword(s):  
Energy Dissipation ◽  
Seismic Response ◽  
Early Stage ◽  
Damping Ratio ◽  
Poor Performance ◽  
Linearization Method ◽  
Structural Deformation ◽  
Stochastic Dynamic ◽  
Control Force ◽  
The Time Domain

Various inerter systems utilizing velocity-dependent damping for vibration control have been developed. However, a velocity-dependent damping element may exhibit relatively poor performance compared to a displacement-dependent damping element (DDE) of equivalent damping ratio, when the structural deformation is small in the early stage of the seismic response. To address this issue, the advantage of DDE in generating a larger control force in the early stage of excitation is promoted here and enhanced by a supplemental inerter-spring-system, thus realizing a proposed novel displacement-dependent damping inerter system (DDIS). First, the influence of various DDIS-parameters is carried out by resorting to the stochastic linearization method to handle non-linear terms. Then, seismic responses of the DDIS-controlled system are evaluated in the time domain taking the non-linearity into account, thus validating the accuracy of the stochastic dynamic analysis. Several design cases are considered, all of which demonstrated damping enhancement and timely control achieved by the DDIS. The results show that the energy dissipation as well as reduction of structural displacement and acceleration accomplished by the proposed system are significant. DDIS suppresses structural responses in a timely manner, as soon as the peak excitation occurs. In addition, it is demonstrated that interactions among the inerter, spring, and DDE, which constitute the damping-enhancement mechanism, lead to a higher energy-dissipation efficiency compared to the DDE alone.

Experimental Study on Water Damping Effects of Hybrid Floating Structure

2013 ◽  
Author(s):  
Youn-Ju Jeong ◽  
Young-Jun You ◽  
Du-Ho Lee ◽  
Min-Su Park
Keyword(s):  
Natural Frequency ◽  
Hybrid Model ◽  
Damping Ratio ◽  
Experimental Studies ◽  
Oscillation Period ◽  
Floating Body ◽  
Small Scale ◽  
Floating Structure ◽  
Damped System ◽  
Damping Effects

In this study, in order to evaluate water damping effects of hybrid pontoon system with cylinders, experimental studies were carried out. At first, in order to evaluate oscillatory motions, three small-scale models of hybrid, tapered, and pontoon were fabricated and tested under the still-water condition. Four acceleration gauges were attached on the top edges and acceleration of top edge were measured during the oscillation. Then, oscillatory motions of oscillation period and stabilizing time to steady-state were analyzed. Finally, based on the oscillatory motions, damping properties of the logarithmic decrement, damping ratio, and natural frequency of damped system were calculated and compared with each other. As the results of this study, it was found that hybrid model presented about 3.67 times higher decay rate of amplitude of the oscillatory motion than the pontoon model. Also, hybrid model presented about 3.67 times higher damping ratio than the pontoon model. Whereas the natural frequency of the pontoon and tapered model were nearly same with the natural frequency of undamped system, that of the hybrid model presented some difference with the that of the undamped system. In addition, periods of floating body at the wet mode presented about 1.5∼3.0 times longer periods than the dry mode, and it was expected that there was not possibility for the resonance. Therefore, it was expected that the hybrid model of this study should contribute to improve serviceability and safety of offshore floating structures as decreasing oscillatory motions.

scholarly journals Stochastic Response of Tension Leg Platform to Wave and Current Forces

1989 ◽  
Vol 111 (4) ◽  
pp. 221-230 ◽  
Author(s):  
A. Ertas ◽  
J.-H. Lee
Keyword(s):  
Constant Current ◽  
Response Analysis ◽  
Stochastic Dynamic ◽  
Superposition Method ◽  
Random Waves ◽  
Stochastic Response ◽  
Tension Leg Platform ◽  
Single Degree Of Freedom ◽  
Simulation Techniques ◽  
Wave Induced

The linear analysis in the frequency domain is presented for the surge motion of a tension leg platform (TLP) in the case of random waves only and random waves with constant current. A single-degree-of-freedom model of a TLP is employed for response. The superposition method, one of the simulation techniques, is applied to random sea wave, and the response analysis of TLP in time is developed with wave velocity and wave acceleration simulations. Wave-induced forces are calculated using the modified Morison equation, which takes into account relative motion. Computational methods for both analyses are developed, and the results of stochastic, dynamic response of the TLP, with and without the presence of current, are presented and compared.

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