Resonant Dynamics in a Rotordynamic System With Nonlinear Inertial Coupling and Shaft Anisotropy

2005 ◽  
Author(s):  
D. Dane Quinn
Keyword(s):  
Initial Conditions ◽  
Equilibrium Points ◽  
Rotational Velocity ◽  
Equations Of Motion ◽  
Translational Motion ◽  
Stationary Solutions ◽  
Reduced Order Model ◽  
Order Model ◽  
Reduced Order ◽  
Applied Torque

The response of a nonlinear, damped Jeffcott rotor with anisotropic stiffness is considered in the presence of an unbalance. For sufficiently small external torque or large imbalance, resonance capture or rotordynamic stall can occur, whereby the rotational velocity of the shaft is unable to increase beyond the fundamental resonance between the rotational and translational motion. This phenomena provides a mechanism for energy transfer from the rotational to the translational mode. Using the method of averaging a reduced-order model is developed, valid near the resonance, that describes this resonant behavior. The equilibrium points of these averaged equations, which correspond to stationary solutions of the original equations and rotor-dynamic stall, are described as the applied torque, damping, and anisotropy vary. As the anisotropy increases, assumed to arise from increasing shaft cracks, the torque required to eliminate the possibility of stall increases. However, when the system is started with zero initial conditions, the minimum torque required to pass through the resonance is approximately constant as the anisotropy increases. The predictions from the reduced-order model are verifled against numerical simulations of the original equations of motion.

Dynamics of a waveboard simplified

2020 ◽  
Vol 476 (2244) ◽  
pp. 20200486
Author(s):  
Anirvan DasGupta
Keyword(s):  
Kinetic Energy ◽  
Equations Of Motion ◽  
Translational Motion ◽  
Rolling Resistance ◽  
Reduced Order Model ◽  
Dynamic Parameters ◽  
Order Model ◽  
Reduced Order ◽  
Propulsion Mechanism ◽  
Subsequent Conversion

A study of dynamics of a waveboard is presented. The equations of motion are derived and analysed to understand the intriguing propulsion mechanism. A reduced order model is obtained, and the contributions of different terms are clearly brought out. The geometry of the castor wheels is found to play a key role in the conversion of the twisting oscillatory motion of the rider to the forward translational motion. The process of periodic gain in potential energy and its subsequent conversion to kinetic energy aids the propulsion. Interestingly, the dynamic analysis reveals that the efficacy of this propulsion mechanism tapers off as the speed increases. Rolling resistance in the wheels ultimately limits the speed of the device. The effect of various geometric and dynamic parameters on the motion and forces are studied, and optimality of design is indicated.

Reduced Order Model of Two Coupled MEMS Parallel Cantilever Resonators Under DC and AC Voltage Near Natural Frequency

2012 ◽  
Author(s):  
Dumitru I. Caruntu ◽  
Kyle N. Taylor
Keyword(s):  
Frequency Response ◽  
Natural Frequency ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Reduced Order Model ◽  
Lagrange Equations ◽  
Order Model ◽  
Reduced Order ◽  
Dc Voltage ◽  
Ground Plate

This paper deals with a system of two coupled parallel identical MEMS cantilever resonators and a ground plate. Alternating Current (AC) and Direct Current (DC) voltages are applied between the first resonator and ground plate, and a DC voltage applied between the resonators. The AC voltage frequency is near natural frequency of the resonators. The electrostatic forces produced by voltages are nonlinear. System equations of motion are obtained using Lagrange equations, then nondimensionalized. The Method of Multiple Scales (MMS) is used to find the steady state frequency response. The Reduced Order Model (ROM) is used to validate MMS results. Matlab is used to find cantilever frequency response of the resonator tip. The DC voltage between resonators is showed to significantly influence the response of the first resonator.

scholarly journals New approximations, and policy implications, from a delayed dynamic model of a fast pandemic

2020 ◽  
Author(s):  
C. P. Vyasarayani ◽  
Anindya Chatterjee
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Reduced Order Model ◽  
Policy Implications ◽  
Galerkin Projection ◽  
Order Model ◽  
Reduced Order ◽  
Wave Approximation ◽  
Long Wave ◽  
Long Wave Approximation

AbstractWe study an SEIQR (Susceptible-Exposed-Infectious-Quarantined-Recovered) model for an infectious disease, with time delays for latency and an asymptomatic phase. For fast pandemics where nobody has prior immunity and everyone has immunity after recovery, the SEIQR model decouples into two nonlinear delay differential equations (DDEs) with five parameters. One parameter is set to unity by scaling time. The subcase of perfect quarantining and zero self-recovery before quarantine, with two free parameters, is examined first. The method of multiple scales yields a hyperbolic tangent solution; and a long-wave approximation yields a first order ordinary differential equation (ODE). With imperfect quarantining and nonzero self-recovery, the long-wave approximation is a second order ODE. These three approximations each capture the full outbreak, from infinitesimal initiation to final saturation. Low-dimensional dynamics in the DDEs is demonstrated using a six state non-delayed reduced order model obtained by Galerkin projection. Numerical solutions from the reduced order model match the DDE over a range of parameter choices and initial conditions. Finally, stability analysis and numerics show how correctly executed time-varying social distancing, within the present model, can cut the number of affected people by almost half. Alternatively, faster detection followed by near-certain quarantining can potentially be even more effective.

scholarly journals Model Reduction for Kinetic Models of Biological Systems

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 863
Author(s):  
Neveen Ali Eshtewy ◽  
Lena Scholz
Keyword(s):  
Nonlinear Systems ◽  
Model Reduction ◽  
Kinetic Models ◽  
Initial Conditions ◽  
Reduced Order Model ◽  
Order System ◽  
High Dimensional ◽  
Order Model ◽  
Model Order ◽  
Reduced Order

High dimensionality continues to be a challenge in computational systems biology. The kinetic models of many phenomena of interest are high-dimensional and complex, resulting in large computational effort in the simulation. Model order reduction (MOR) is a mathematical technique that is used to reduce the computational complexity of high-dimensional systems by approximation with lower dimensional systems, while retaining the important information and properties of the full order system. Proper orthogonal decomposition (POD) is a method based on Galerkin projection that can be used for reducing the model order. POD is considered an optimal linear approach since it obtains the minimum squared distance between the original model and its reduced representation. However, POD may represent a restriction for nonlinear systems. By applying the POD method for nonlinear systems, the complexity to solve the nonlinear term still remains that of the full order model. To overcome the complexity for nonlinear terms in the dynamical system, an approach called the discrete empirical interpolation method (DEIM) can be used. In this paper, we discuss model reduction by POD and DEIM to reduce the order of kinetic models of biological systems and illustrate the approaches on some examples. Additional computational costs for setting up the reduced order system pay off for large-scale systems. In general, a reduced model should not be expected to yield good approximations if different initial conditions are used from that used to produce the reduced order model. We used the POD method of a kinetic model with different initial conditions to compute the reduced model. This reduced order model is able to predict the full order model for a variety of different initial conditions.

scholarly journals A Reduced-Order Model for the Vibration Analysis of Mistuned Blade–Disc–Shaft Assembly

2019 ◽  
Vol 9 (22) ◽  
pp. 4762
Author(s):  
Wang ◽  
Bi ◽  
Zheng
Keyword(s):  
Vibration Analysis ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Normal Modes ◽  
Order Reduction ◽  
Symmetry Analysis ◽  
Reduced Order Model ◽  
Cyclic Symmetry ◽  
Order Model ◽  
Reduced Order

An effective reduced-order model is presented in this paper for the vibration analysis of a mistuned blade–disc–shaft assembly considering the flexibility of the shaft and the rotordynamic effects. For the sake of accurate modeling and quantitative analysis, three-dimensional (3D) finite element models were employed in obtaining the governing equations of motion with the Coriolis force, centrifugal stiffening, and spin softening effects taken into account. Then, an efficient model order reduction technique based on the coordinate projection by normal modes of tuned assembly and cyclic symmetry analysis was developed for mistuned blade–disc–shaft assembly. The criterion of whether one matrix could be incorporated in cyclic symmetry analysis is presented. During the modeling, the mistuning in blade and disc was taken into account and dealt with independently. In mistuning projection, the blade and disc parts were both projected onto their tuned counterparts of the sector model, where the boundary conditions were set to be fixed and free, respectively. Finally, an example of a blade–disc–shaft assembly was employed to validate the effectiveness of the presented method in free and forced vibration analysis.

A Nonlinear Reduced-Order Model for Electrostatic MEMS

2003 ◽  
Author(s):  
Eihab M. Abdel-Rahman ◽  
Mohammad I. Younis ◽  
Ali H. Nayfeh
Keyword(s):  
Equations Of Motion ◽  
Computational Cost ◽  
Design Tool ◽  
Reduced Order Model ◽  
Mode Shapes ◽  
Design Parameters ◽  
Mems Devices ◽  
Order Model ◽  
Reduced Order ◽  
Galerkin Procedure

We present an analytical approach and a reduced-order model (macromodel) to investigate the behavior of electrically actuated microbeam-based MEMS. The macromodel provides an effective and accurate design tool for this class of MEMS devices. The macromodel is obtained by discretizing the distributed-parameter system using a Galerkin procedure into a finite-degree-of-freedom system consisting of ordinary-differential equations in time. The macromodel accounts for moderately large deflections, dynamic loads, and the coupling between the mechanical and electrical forces. It accounts for linear and nonlinear elastic restoring forces and the nonlinear electric forces generated by the capacitors. A new technique is developed to represent the electric force in the equations of motion. The new approach allows the use of few linear-undamped mode shapes of a microbeam in its straight position as basis functions in a Galerkin procedure. The macromodel is validated by comparing its results with experimental results and finiteelement solutions available in the literature. Our approach shows attractive features compared to finite-element softwares used in the literature. It is robust over the whole device operation range up to the instability limit of the device (i.e., pull-in). Moreover, it has low computational cost and allows for an easier understanding of the influence of the various design parameters. As a result, it can be of significant benefit to the development of MEMS design software.

A Reduced-Order Model for an Oscillating Hydrofoil Near the Free Surface

2015 ◽  
Author(s):  
Rory C. Kennedy ◽  
Dillon Helfers ◽  
Yin Lu Young
Keyword(s):  
Free Surface ◽  
High Speed ◽  
Equations Of Motion ◽  
Operating Conditions ◽  
Reduced Order Model ◽  
Order Model ◽  
Disturbing Force ◽  
Reduced Order ◽  
Resonance Frequencies ◽  
Force Components

The first objective of this work is to numerically investigate how the proximity to the free surface influences the hydrodynamic response and susceptibility to cavitation of a hydrofoil undergoing controlled pitching oscillations, for high-speed full-scale operating conditions. A second objective is to develop a time-domain Reduced Order Model (ROM) to predict the unsteady hydrodynamic loads (for rapid exploration of the design space and for real-time active/passive actuation/control). The ROM delineates the fluid-structure interaction (FSI) forces into fluid inertial, damping, and disturbing force components, and only predicts the primary oscillation frequency. In addition to predicting the unsteady loads, when coupled with the solid equations of motion, the ROM can also be used to calculate the natural resonance frequencies and damping characteristics with consideration for viscous and free surface effects. This will allow designers to better predict and control the dynamic response of lifting surfaces operating near the free surface.

scholarly journals A Hybrid Reduced-Order Model for the Aeroelastic Analysis of Flexible Subsonic Wings—A Parametric Assessment

Aerospace ◽  
2018 ◽  
Vol 5 (3) ◽  
pp. 76 ◽  
Author(s):  
Marco Berci ◽  
Rauno Cavallaro
Keyword(s):  
Structural Parameters ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Unsteady Aerodynamics ◽  
Reduced Order Model ◽  
Order Model ◽  
Flexible Wings ◽  
Reduced Order ◽  
Aeroelastic Analysis ◽  
Strip Theory

A hybrid reduced-order model for the aeroelastic analysis of flexible subsonic wings with arbitrary planform is presented within a generalised quasi-analytical formulation, where a slender beam is considered as the linear structural dynamics model. A modified strip theory is proposed for modelling the unsteady aerodynamics of the wing in incompressible flow, where thin aerofoil theory is corrected by a higher-fidelity model in order to account for three-dimensional effects on both distribution and deficiency of the sectional air load. Given a unit angle of attack, approximate expressions for the lift decay and build-up are then adopted within a linear framework, where the two effects are separately calculated and later combined. Finally, a modal approach is employed to write the generalised equations of motion in state-space form. Numerical results were obtained and critically discussed for the aeroelastic stability analysis of a uniform rectangular wing, with respect to the relevant aerodynamic and structural parameters. The proposed hybrid model provides sound theoretical insights and is well suited as an efficient parametric reduced-order aeroelastic tool for the preliminary multidisciplinary design and optimisation of flexible wings in the subsonic regime.

Reduced Order Model and Decoupled Control Architecture for Two Manipulators Holding a Rigid Object

1991 ◽  
Vol 113 (4) ◽  
pp. 646-654 ◽  
Author(s):  
A. J. Koivo ◽  
M. A. Unseren
Keyword(s):  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Functional Relation ◽  
Contact Forces ◽  
Reduced Order Model ◽  
Dynamical Model ◽  
Control Architecture ◽  
Order Model ◽  
Rigid Object ◽  
Reduced Order

A dynamical model and a control architecture are developed for the closed chain motion of two N-joint manipulators holding a rigid object in a three-dimensional workspace. Dynamic and kinematic constraints are determined and combined with the equations of motion of the manipulators to obtain a dynamical model of the entire system in the joint space. Reduced order equations of motion and a functional relation for the generalized contact forces are derived. The problem of solving the reduced order model for the forward and inverse dynamics is discussed. Control laws are determined for the reduced order model so as to decouple the force and position (velocity) controlled degrees of freedom (DOF). A simulation example is presented to illustrate the approach.

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