The Application of the Ritz Averaging Method to Determining the Response of Systems with Time Varying Stiffness to Harmonic Excitation

1980 ◽  
Vol 102 (2) ◽  
pp. 384-390 ◽  
Author(s):  
M. Benton ◽  
A. Seireg
Keyword(s):  
Linear System ◽  
Averaging Method ◽  
Approximate Solutions ◽  
Harmonic Excitation ◽  
Time Varying ◽  
Parametric Vibrations ◽  
Nonlinear Characteristics ◽  
Closed Form Solutions ◽  
Mathieu’S Equation ◽  
Stiffness Variation

Parametric vibrations occur in many mechanical systems such as gears where the stiffness variation and external excitations generally occur at integer multiples of the rotational speed. This paper describes a procedure based on the Ritz Averaging Method for developing closed form solutions for the response of such systems to harmonic excitations. Although the method is illustrated in the paper by the case of a linear system with harmonic stiffness fluctuation (defined by Mathieu’s equation) it can be readily applied to determine approximate solutions for systems with nonlinear characteristics and any periodic variations of parameters.

A Resonance Chart for Damped Vibration

1955 ◽  
Vol 59 (540) ◽  
pp. 850-852 ◽  
Author(s):  
R. E. D. Bishop
Keyword(s):  
Linear System ◽  
Convenient Method ◽  
Harmonic Excitation ◽  
Degree Of Freedom ◽  
Viscous Damping ◽  
Similar Form ◽  
Hysteretic Damping ◽  
Resonance Curves ◽  
Phase Angles

A convenient method is pointed out for calculating the response of a damped linear system with one degree of freedom to harmonic excitation. Results of such calculations are usually represented by the familiar “ resonance curves ”—one curve being plotted for each intensity of damping. These curves are not particularly convenient to use and Yates has overcome several of their defects by throwing them into a nomographic form. Yates' nomogram is based upon the concept of viscous damping and it does not give the information of a conventional set of resonance curves in that it relates to the velocity of vibration. By changing over to hysteretic damping, a nomogram of somewhat similar form may be constructed such that it gives amplitudes and phase angles of displacements while retaining the advantages, over resonance curves, of this form of representation.

In-Plane Parametric Vibrations of Curved Bellows Subjected to Oscillating Internal Fluid Pressure Excitation

2004 ◽  
Author(s):  
Masahiro Watanabe ◽  
Eiji Tachibana ◽  
Nobuyuki Kobayashi
Keyword(s):  
Stability Analysis ◽  
Natural Frequencies ◽  
Parametric Instability ◽  
Fluid Pressure ◽  
Parametric Vibrations ◽  
Internal Fluid ◽  
Mathieu’S Equation ◽  
Stability Maps ◽  
Plane Direction ◽  
Pressure Excitation

This paper deals with the theoretical stability analysis of in-plane parametric vibrations of a curved bellows subjected to periodic internal fluid pressure excitation. The curved bellows studied in this paper are fixed at both ends rigidly, and are excited by the periodic internal fluid pressure. In the theoretical stability analysis, the governing equation of the curved bellows subjected to periodic internal fluid pressure excitation is derived as a Mathieu’s equation by using finite element method (FEM). Natural frequencies of the curved bellows are examined and stability maps are presented for in-plane parametric instability. It is found that the natural frequencies of the curved bellows decrease with increasing the static internal fluid pressure and buckling occurs due to high internal fluid pressure. It is also found that two types of parametric vibrations, longitudinal and transverse vibrations, occur to the curved bellows in-plane direction due to the periodic internal fluid pressure excitation. Moreover, effects of axis curvature on the parametric instability regions are examined theoretically.

scholarly journals A Modified Techniques of Fractional-Order Cauchy-Reaction Diffusion Equation via Shehu Transform

2021 ◽  
Vol 2021 ◽  
pp. 1-15
Author(s):  
Mounirah Areshi ◽  
A. M. Zidan ◽  
Rasool Shah ◽  
Kamsing Nonlaopon
Keyword(s):  
Fractional Order ◽  
Reaction Diffusion ◽  
Approximate Solutions ◽  
Transformation Method ◽  
Reaction Diffusion Equations ◽  
Reaction Diffusion Equation ◽  
Closed Form Solutions ◽  
Homotopy Perturbation ◽  
In Series ◽  
The Given

In this article, the iterative transformation method and homotopy perturbation transformation method are applied to calculate the solution of time-fractional Cauchy-reaction diffusion equations. In this technique, Shehu transformation is combined of the iteration and the homotopy perturbation techniques. Four examples are examined to show validation and the efficacy of the present methods. The approximate solutions achieved by the suggested methods indicate that the approach is easy to apply to the given problems. Moreover, the solution in series form has the desire rate of convergence and provides closed-form solutions. It is noted that the procedure can be modified in other directions of fractional order problems. These solutions show that the current technique is very straightforward and helpful to perform in applied sciences.

scholarly journals State Estimation with Reduced-Order Observer and Adaptive-LQR Control of Time Varying Linear System

2020 ◽  
Vol 26 (2) ◽  
pp. 24-31
Author(s):  
Omer Aydogdu ◽  
Mehmet Latif Levent
Keyword(s):  
Optimal Control ◽  
Linear System ◽  
State Estimation ◽  
Servo System ◽  
Variable Load ◽  
Time Varying ◽  
Load Effect ◽  
Reduced Order ◽  
Lqr Control ◽  
Reduced Order Observer

In this study, a new controller design was created to increase the control performance of a variable loaded time varying linear system. For this purpose, a state estimation with reduced order observer and adaptive-LQR (Linear–Quadratic Regulator) control structure was offered. Initially, to estimate the states of the system, a reduced-order observer was designed and used with LQR control method that is one of the optimal control techniques in the servo system with initial load. Subsequently, a Lyapunov-based adaptation mechanism was added to the LQR control to provide optimal control for varying loads as a new approach in design. Thus, it was aimed to eliminate the variable load effects and to increase the stability of the system. In order to demonstrate the effectiveness of the proposed method, a variable loaded rotary servo system was modelled as a time-varying linear system and used in simulations in Matlab-Simulink environment. Based on the simulation results and performance measurements, it was observed that the proposed method increases the system performance and stability by minimizing variable load effect.

Uniform global asymptotic stability for half-linear differential systems with time-varying coefficients

2011 ◽  
Vol 141 (5) ◽  
pp. 1083-1101 ◽  
Author(s):  
Masakazu Onitsuka ◽  
Jitsuro Sugie
Keyword(s):  
Linear System ◽  
Zero Solution ◽  
Integrable Case ◽  
Time Varying ◽  
Globally Asymptotically Stable ◽  
Varying Coefficients ◽  
Linear Differential Systems ◽  
Time Varying Coefficients ◽  
Linear Differential ◽  
Image Position

The present paper deals with the following system:where p and p* are positive numbers satisfying 1/p + 1/p* = 1, and ϕq(z) = |z|q−2z for q = p or q = p*. This system is referred to as a half-linear system. We herein establish conditions on time-varying coefficients e(t), f(t), g(t) and h(t) for the zero solution to be uniformly globally asymptotically stable. If (e(t), f(t)) ≡ (h(t), g(t)), then the half-linear system is integrable. We consider two cases: the integrable case (e(t), f(t)) ≡ (h(t), g(t)) and the non-integrable case (e(t), f(t)) ≢ (h(t), g(t)). Finally, some simple examples are presented to illustrate our results.

Variable selection for joint models with time-varying coefficients

2019 ◽  
Vol 29 (1) ◽  
pp. 309-322
Author(s):  
Yujing Xie ◽  
Zangdong He ◽  
Wanzhu Tu ◽  
Zhangsheng Yu
Keyword(s):  
Variable Selection ◽  
Survival Data ◽  
Biliary Cirrhosis ◽  
Time Varying ◽  
Joint Models ◽  
Closed Form Solutions ◽  
Varying Coefficients ◽  
Shared Random Effects ◽  
Time Varying Coefficients ◽  
Time Invariant

Many clinical studies collect longitudinal and survival data concurrently. Joint models combining these two types of outcomes through shared random effects are frequently used in practical data analysis. The standard joint models assume that the coefficients for the longitudinal and survival components are time-invariant. In many applications, the assumption is overly restrictive. In this research, we extend the standard joint model to include time-varying coefficients, in both longitudinal and survival components, and we present a data-driven method for variable selection. Specifically, we use a B-spline decomposition and penalized likelihood with adaptive group LASSO to select the relevant independent variables and to distinguish the time-varying and time-invariant effects for the two model components. We use Gaussian-Legendre and Gaussian-Hermite quadratures to approximate the integrals in the absence of closed-form solutions. Simulation studies show good selection and estimation performance. Finally, we use the proposed procedure to analyze data generated by a study of primary biliary cirrhosis.

Stability of Subharmonic Oscillations in a Pipe Conveying Fluid under Harmonic Excitation

2013 ◽  
Vol 444-445 ◽  
pp. 796-800
Author(s):  
Yi Xiang Geng ◽  
Han Ze Liu
Keyword(s):  
Averaging Method ◽  
Melnikov Method ◽  
Equation Of Motion ◽  
Harmonic Excitation ◽  
Angle Variable ◽  
Galerkin Approach ◽  
Existence And Stability ◽  
The Stability ◽  
Action Angle Variable ◽  
Conveying Fluid

The existence and stability of subharmonic oscillations in a two end-fixed fluid conveying pipe whose base is subjected to a harmonic excitation are investigated. A Galerkin approach is utilized to reduce the equation of motion to a second order nonlinear differential equation. The conditions for the existence of subharmonic oscillations are given by using Melnikov method. The stability of subharmonic oscillations is discussed in detail by using action-angle variable and averaging method. It is shown that the velocity of fluid plays an important role in the stability of subharmonic oscillations.

Sign in / Sign up

Export Citation Format

Share Document