Steady-state response of beams supported on non-linear springs.

AIAA Journal ◽  
1966 ◽  
Vol 4 (10) ◽  
pp. 1863-1864 ◽  
Author(s):  
A. V. SRINIVASAN
Keyword(s):  
Steady State ◽  
Steady State Response ◽  
State Response ◽  
Non Linear

Transient Response of Inductrack Systems for Maglev Transport: Part II—Solution and Dynamic Analysis

2020 ◽  
Vol 142 (3) ◽  
Author(s):  
Ruiyang Wang ◽  
Bingen Yang
Keyword(s):  
Steady State ◽  
Dynamic Analysis ◽  
Transient Response ◽  
Numerical Procedure ◽  
Steady State Response ◽  
Governing Equations ◽  
Transient Model ◽  
State Response ◽  
Proposed Model ◽  
Non Linear

Abstract In Part I of this two-part paper, a new benchmark transient model of Inductrack systems is developed. In this Part II, the proposed model, which is governed by a set of non-linear integro-differential governing equations, is used to predict the dynamic response of Inductrack systems. In the development, a state-space representation of the non-linear governing equations is established and a numerical procedure with a specific moving circuit window for transient solutions is designed. The dynamic analysis of Inductrack systems with the proposed model has two major tasks. First, the proposed model is validated through comparison with the noted steady-state results in the literature. Second, the transient response of an Inductrack system is simulated and analyzed in several typical dynamic scenarios. The steady-state response results predicted by the new model agree with those obtained in the previous studies. On the other hand, the transient response simulation results reveal that an ideal steady-state response can hardly exist in those investigated dynamic scenarios. It is believed that the newly developed transient model provides a useful tool for dynamic analysis of Inductrack systems and for in-depth understanding of the complicated electro-magneto-mechanical interactions in this type of dynamic systems.

scholarly journals Principal Parametric Resonance of Axially Accelerating Viscoelastic Beams: Multi-Scale Analysis and Differential Quadrature Verification

2012 ◽  
Vol 19 (4) ◽  
pp. 527-543 ◽  
Author(s):  
Li-Qun Chen ◽  
Hu Ding ◽  
C.W. Lim
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Steady State ◽  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Steady State Response ◽  
Governing Equations ◽  
State Response ◽  
Non Linear ◽  
Axial Speed

Transverse non-linear vibration is investigated in principal parametric resonance of an axially accelerating viscoelastic beam. The axial speed is characterized as a simple harmonic variation about a constant mean speed. The material time derivative is used in the viscoelastic constitutive relation. The transverse motion can be governed by a non-linear partial-differential equation or a non-linear integro-partial-differential equation. The method of multiple scales is applied to the governing equations to determine steady-state responses. It is confirmed that the mode uninvolved in the resonance has no effect on the steady-state response. The differential quadrature schemes are developed to verify results via the method of multiple scales. It is demonstrated that the straight equilibrium configuration becomes unstable and a stable steady-state emerges when the axial speed variation frequency is close to twice any linear natural frequency. The results derived for two governing equations are qualitatively the same, but quantitatively different. Numerical simulations are presented to examine the effects of the mean speed and the variation of the amplitude of the axial speed, the dynamic viscosity, the non-linear coefficients, and the boundary constraint stiffness on the instability interval and the steady-state response amplitude.

scholarly journals On the Steady State Response of a Cantilever Beam Partially Immersed in a Fluid and Carrying an Intermediate Mass

2000 ◽  
Vol 7 (4) ◽  
pp. 179-194 ◽  
Author(s):  
A.A. Al-Qaisia ◽  
M.N. Hamdan ◽  
B.O. Al-Bedoor
Keyword(s):  
Steady State ◽  
Harmonic Balance ◽  
Equation Of Motion ◽  
Steady State Response ◽  
Intermediate Mass ◽  
Assumed Mode Method ◽  
State Response ◽  
Non Linear ◽  
Mode Method ◽  
Assumed Mode

This paper presents a study on the nonlinear steady state response of a slender beam partially immersed in a fluid and carrying an intermediate mass. The model is developed based on the large deformation theory with the constraint of inextensible beam, which is valid for most engineering structures. The Lagrangian dynamics in conjunction with the assumed mode method is utilized in deriving the non-linear unimodal temporal equation of motion. The distributed and concentrated sinusoidal loads are accounted for in a consistent manner using the assumed mode method. The non-linear equation of motion is, analytically, solved using the single term harmonic balance (SHB) and the two terms harmonic balance (2HB) methods. The stability of the system, under various loading conditions, is investigated. The results are presented, discussed and some conclusions on the partially immersed beam nonlinear dynamics are extracted.

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