scholarly journals Nonlinear Models for Transverse Forced Vibration of Axially Moving Viscoelastic Beams

2011 ◽  
Vol 18 (1-2) ◽  
pp. 281-287 ◽  
Author(s):  
Hu Ding ◽  
Li-Qun Chen
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Steady State ◽  
Transverse Vibration ◽  
Nonlinear Models ◽  
Transverse Motion ◽  
Steady State Response ◽  
Partial Differential ◽  
State Response ◽  
Axially Moving

Nonlinear models of transverse vibration of axially moving viscoelastic beams subjected external transverse loads via steady-state periodical response are numerically investigated. An integro-partial-differential equation and a partial-differential equation of transverse motion can be derived respectively from a model of the coupled planar vibration for an axially moving beam. The finite difference scheme is developed to calculate steady-state response for the model of coupled planar and the two models of transverse motion under the simple support boundary. Numerical results indicate that the amplitude of the steady-state response for the model of coupled vibration and two models of transverse vibration predict qualitatively the same tendencies with the changing parameters and the integro-partial-differential equation gives results more closely to the coupled planar vibration.

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.

Steady-State Transverse Response in Coupled Planar Vibration of Axially Moving Viscoelastic Beams

2010 ◽  
Vol 132 (1) ◽  
Author(s):  
Li-Qun Chen ◽  
Hu Ding
Keyword(s):  
Steady State ◽  
Transverse Vibration ◽  
Nonlinear Models ◽  
Numerical Results ◽  
State Response ◽  
Planar Vibration ◽  
Transverse Loads ◽  
Steady State Responses ◽  
Axially Moving ◽  
Transverse Response

Steady-state periodical response is investigated for planar vibration of axially moving viscoelastic beams subjected external transverse loads. A model of the coupled planar vibration is established by introducing a coordinate transform. The model can reduce to two nonlinear models of transverse vibration. The finite difference scheme is developed to calculate steady-state response numerically. Numerical results demonstrate there are steady-state periodic responses in transverse vibration, and resonance occurs if the external load frequency approaches the linear natural frequencies. The effect of material parameters and excitation parameters on the amplitude of the steady-state responses are examined. Numerical results also indicate that the model of coupled vibration and two models of transverse vibration predict qualitatively the same tendencies with the changing parameters, and the two models of transverse vibration yield satisfactory results.

Periodic Steady State Response and Fatigue Analysis of a Nonlinear City Bus Model

2009 ◽  
Author(s):  
George Valsamos ◽  
Christos Theodosiou ◽  
Sotirios Natsiavas
Keyword(s):  
Steady State ◽  
Degrees Of Freedom ◽  
Nonlinear Models ◽  
Equations Of Motion ◽  
High Stress ◽  
Direct Determination ◽  
Accurate Information ◽  
Steady State Response ◽  
State Response ◽  
Periodic Steady State

Dynamic response related to fatigue prediction of an urban bus is investigated. First, a quite complete model subjected to road excitation is employed in order to extract sufficiently reliable and accurate information in a fast way. The bus model is set up by applying the finite element method, resulting to an excessive number of degrees of freedom. In addition, the bus suspension units involve nonlinear characterstics. A step towards alleviating this difficulty is the application of an appropriate coordinate transformation, causing a drastic reduction in the dimension of the final set of the equations of motion. This allows the application of a systematic numerical methodology leading to direct determination of periodic steady state response of nonlinear models subjected to periodic excitation. Next, typical results were obtained for excitation resulting from selected urban road profiles. These profiles have either a known form or known statistical properties, expressed by an appropriate spatial power spectral density function. In all cases examined, the emphasis was put on investigating ride response. The main attention was focused on identifying areas of the bus suspension and frame subsystems where high stress levels are developed. This information is based on the idea of a nonlinear transfer function and provides the basis for applying suitable criteria in order to perform analyses leading to prediction of fatigue failure.

scholarly journals Geometric shape features extraction using a steady state partial differential equation system

2019 ◽  
Vol 6 (4) ◽  
pp. 647-656 ◽  
Author(s):  
Takayuki Yamada
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Steady State ◽  
Analytical Solution ◽  
Geometric Shape ◽  
Geometrical Shape ◽  
Shape Features ◽  
Partial Differential ◽  
Topological Constraint ◽  
Numerical Examples

Abstract A unified method for extracting geometric shape features from binary image data using a steady-state partial differential equation (PDE) system as a boundary value problem is presented in this paper. The PDE and functions are formulated to extract the thickness, orientation, and skeleton simultaneously. The main advantage of the proposed method is that the orientation is defined without derivatives and thickness computation is not imposed a topological constraint on the target shape. A one-dimensional analytical solution is provided to validate the proposed method. In addition, two-dimensional numerical examples are presented to confirm the usefulness of the proposed method. Highlights A steady state partial differential equation for extraction of geometrical shape features is formulated. The functions for geometrical shape features are formulated by the solution of the proposed PDE. Analytical solution is provided in one-dimension. Numerical examples are provided in two-dimension.

scholarly journals On closed-form solutions of some nonlinear partial differential equations

2000 ◽  
Vol 23 (2) ◽  
pp. 81-88 ◽  
Author(s):  
S. S. Okoya
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Wave Equation ◽  
Steady State ◽  
Closed Form ◽  
Nonlinear Wave Equation ◽  
Nonlinear Partial Differential Equations ◽  
Partial Differential ◽  
Closed Form Solutions ◽  
Thermal Explosion Theory

This paper is devoted to closed-form solutions of the partial differential equation:θxx+θyy+δexp(θ)=0, which arises in the steady state thermal explosion theory. We find simple exact solutions of the formθ(x,y)=Φ(F(x)+G(y)), andθ(x,y)=Φ(f(x+y)+g(x-y)). Also, we study the corresponding nonlinear wave equation.

Inversion-Based and Optimal Feedforward Control for Population Dynamics with Input Constraints and Self-Competition in Chemostat Reactor Applications

2020 ◽  
Author(s):  
Anna-Carina Kurth ◽  
Kevin Schmidt ◽  
Oliver Sawodny
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Steady State ◽  
Feedforward Control ◽  
Reference Trajectory ◽  
Input Constraints ◽  
Partial Differential ◽  
First Order ◽  
Physical Limitations ◽  
Chemostat Reactor

Abstract Through chemostat reactors, organisms can be observed under laboratory conditions. Hereby, the reactor contains the biomass, whose growth can be controlled via the dilution rate respectively the speed of a pump. Due to physical limitations, input constraints need to be considered. The population density in the reactor can be described by a hyperbolic nonlinear integro partial differential equation of first order. The steady-states and generalized eigenvalues and -modes of these integro partial differential equation are determined. In order to track a desired reference trajectory an optimal and an inversion-based feedforward control are designed. For the optimal feedforward control, the singular arc of the control is calculated and a switching strategy is stated, which explicitly considers the input constraints. For the inversion-based feedforward control, the integro partial differential equation is first linearized around the steady-state. To comply with the input constraints a control system simulator is designed. For the simulation model, the integro partial differential equation is approximated using Galerkin's method. Simulations show the functionality of the designed controls and provide the basis for comparison. The inversion-based feedforward control operates well near the steady-state, whereas the performance of the optimal feedforward control is not bounded to the proximity to the steady-state.

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