Approximate Decoupling of the Equations of Motion of Large Flexible Structures

1989 ◽  
Author(s):  
S. M. Shahruz ◽  
F. Ma
Keyword(s):  
Equations Of Motion ◽  
Flexible Structures

scholarly journals Integration of Nonlinear Models of Flexible Body Deformation in Multibody System Dynamics

2013 ◽  
Vol 9 (1) ◽  
Author(s):  
Martin Schulze ◽  
Stefan Dietz ◽  
Bernhard Burgermeister ◽  
Andrey Tuganov ◽  
Holger Lang ◽  
...  
Keyword(s):  
Degrees Of Freedom ◽  
Multibody System ◽  
Nonlinear Models ◽  
Equations Of Motion ◽  
Flexible Structures ◽  
General Purpose ◽  
Cosserat Rod ◽  
Rod Model ◽  
Flexible Deformation ◽  
Run Up

Current challenges in industrial multibody system simulation are often beyond the classical range of application of existing industrial simulation tools. The present paper describes an extension of a recursive order-n multibody system (MBS) formulation to nonlinear models of flexible deformation that are of particular interest in the dynamical simulation of wind turbines. The floating frame of reference representation of flexible bodies is generalized to nonlinear structural models by a straightforward transformation of the equations of motion (EoM). The approach is discussed in detail for the integration of a recently developed discrete Cosserat rod model representing beamlike flexible structures into a general purpose MBS software package. For an efficient static and dynamic simulation, the solvers of the MBS software are adapted to the resulting class of MBS models that are characterized by a large number of degrees of freedom, stiffness, and high frequency components. As a practical example, the run-up of a simplified three-bladed wind turbine is studied where the dynamic deformations of the three blades are calculated by the Cosserat rod model.

Order-Tuned Vibration Absorbers for Cyclic Rotating Flexible Structures

2005 ◽  
Author(s):  
Brian J. Olson ◽  
Steve W. Shaw ◽  
Christophe Pierre
Keyword(s):  
Equations Of Motion ◽  
Flexible Structures ◽  
Closed Form Solution ◽  
Nonlinear Effects ◽  
Form Solution ◽  
Vibration Absorbers ◽  
Bladed Disk ◽  
Tuned Vibration Absorbers ◽  
Engine Order ◽  
Bladed Disk Assembly

This paper investigates the use of order-tuned absorbers to attenuate vibrations of flexible blades in a bladed disk assembly subjected to engine order excitation. The blades are modeled by a cyclic chain of N oscillators, and a single vibration absorber is fitted to each blade. These absorbers exploit the centrifugal field arising from rotation so that they are tuned to a given order of rotation, rather than to a fixed frequency. A standard change of coordinates based on the cyclic symmetry of the system essentially decouples the governing equations of motion, yielding a closed form solution for the steady-state response of the overall system. These results show that optimal reduction of blade vibrations is achieved by tuning the absorbers to the excitation order n, but that the resulting system is highly sensitive to small perturbations. Intentional detuning (meaning that the absorbers are slightly over- or under-tuned relative to n) can be implemented to improve the robustness of the design. It is shown that by slightly undertuning the absorbers there are no system resonances near the excitation order of interest and that the resulting system is robust to mistuning (i.e., small random uncertainties in the system parameters) of the absorbers and/or blades. These results offer a basic understanding of the dynamics of a bladed disk assembly fitted with order-tuned vibration absorbers, and serve as a first step to the investigation of more realistic models, where, for example, imperfections and nonlinear effects are considered, and multi-DOF and general-path absorbers are employed.

Pendulum as Vibration Absorber for Flexible Structures: Experiments and Theory

1996 ◽  
Vol 118 (4) ◽  
pp. 558-566 ◽  
Author(s):  
O. Cuvalci ◽  
A. Ertas
Keyword(s):  
Energy Transfer ◽  
Coupling Effect ◽  
Equations Of Motion ◽  
Flexible Structures ◽  
Experimental Results ◽  
Vibration Absorber ◽  
Pendulum System ◽  
Nonlinear Equations Of Motion ◽  
Tip Mass ◽  
Sinusoidal Excitation

The dynamic response of a beam-tip mass-pendulum system subjected to a sinusoidal excitation is investigated. A simple pendulum mounted to a tip mass of a beam is used as a vibration absorber. The nonlinear equations of motion are developed to investigate the autoparametric interaction between the first two modes of the system. The nonlinear terms appear due to the curvature of the beam and the coupling effect between the beam and pendulum. Complete energy transfer between modes is shown to occur when the beam frequency is twice the pendulum frequency. Experimental results are compared with a theoretical solution obtained using numerical integration. The experimental results are in qualitative agreement with the theory.

Switched Stiffness Vibration Controllers for Fluidic Flexible Matrix Composites

2009 ◽  
Author(s):  
Amir Lotfi-Gaskarimahalle ◽  
Christopher D. Rahn
Keyword(s):  
Equations Of Motion ◽  
Active Vibration Control ◽  
Flexible Structures ◽  
Variable Stiffness ◽  
Flow Coefficient ◽  
Step Response ◽  
Semiactive Control ◽  
Matrix Composites ◽  
Mass System ◽  
3D Elasticity

This paper investigates semi-active vibration control using Fluidic Flexible Matrix Composites (F2MC) as variable stiffness components of flexible structures. The stiffness of F2MC tubes can be dynamically switched from soft to stiff by opening and closing an on/off valve. Fiber reinforcement of the F2MC tube changes the internal volume when externally loaded. With an open valve, the fluid in the tube is free to move in or out of the tube, so the stiffness is low. When the valve is closed, the high bulk modulus fluid resists volume change and produces high stiffness. The equations of motion of an F2MC-mass system is derived using a 3D elasticity model and the energy method. The stability of the unforced dynamic system is proven using a Lyapunov approach. To capture the important system parameters, nondimensional full order and reduced order models are developed. A Zero Vibration (ZV) state switch technique is introduced that suppresses vibration in finite time, and is compared to conventional Skyhook semiactive control. The ITAE performance of the controllers is optimized by adjusting the open valve flow coefficient. Simulation results show that the optimal ZV controller outperforms the optimal Skyhook controller by 13% and 60% for impulse and step response, respectively.

Multibody Dynamics Incorporating Deployment of Flexible Structures

1996 ◽  
Vol 118 (2) ◽  
pp. 237-241 ◽  
Author(s):  
S. S. K. Tadikonda ◽  
R. P. Singh ◽  
S. Stornelli
Keyword(s):  
Multibody Dynamics ◽  
Solution Method ◽  
Equations Of Motion ◽  
Flexible Structures ◽  
Open Loop ◽  
Space Structures ◽  
Shape Functions ◽  
Recursive Solution ◽  
Multi Body ◽  
Free Beam

The equations of motion for a flexible structure during deployment from and retraction into a base that is part of an open-loop multi-body chain are derived. The eigenfunctions of a fixed-free beam are used as the shape functions and their properties are exploited to express various domain integral terms as explicit functions of the instantaneous deployed length. The essential contributions of the present paper are the modeling of flexible body deployment with mass transfer and a recursive solution method for the dynamics. The deployment or retraction of space structures such as the SAFE Extension Mast can be simulated using this model. The model is presented in a format that is suitable for implementation in multibody dynamics codes.

Active Vibration Control of Beams By Combining Precompressed Layer Damping and ACLD Treatment: Theory and Experimental Implementation

2011 ◽  
Vol 133 (6) ◽  
Author(s):  
Sanjiv Kumar ◽  
Rakesh Sehgal ◽  
Rajiv Kumar
Keyword(s):  
Equations Of Motion ◽  
Active Vibration Control ◽  
Flexible Structures ◽  
Constrained Layer Damping ◽  
Linear Quadratic ◽  
Active Constrained Layer Damping ◽  
Active Vibration ◽  
Passive Technique ◽  
Failure Conditions ◽  
Passive Constrained Layer Damping

By attaching initially stressed poly vinyl chloride (PVC) layers on the flexible structures, necessary passive damping can be provided. Using passive constrained layers on these PVC layers, the efficiency can be made even better than ordinary passive constrained layer damping (PCLD) treatment. By using stressed PVC layers, a rich performance in case of circuit failure conditions is always available. An active constraining layer further enhances the damping performance of this passive technique. Precompressed layer damping treatment augmented with active constrained layer damping (ACLD) treatment has been suggested, which has many desirable features as compared to existing pretensed layer damping treatment. Such enhancement in damping performance is not possible by conventional ACLD as well as PCLD techniques. The effect of initial strain (compressive or tensile) and other parameters of the PVC layers on the vibration characteristics of flexible structure have been investigated. The Hamilton principle in conjunction with finite element method is used to derive the differential equations of motion. Using proportional feedback controllers, the complex closed loop eigenvalue problem is developed and solved numerically. The effectiveness of the proposed technique has been validated experimentally using a digital linear quadratic Gaussian controller.

Formulation of a Basic Building Block Model for Interaction of High Speed Vehicles on Flexible Structures

1989 ◽  
Vol 56 (2) ◽  
pp. 451-458 ◽  
Author(s):  
L. Vu-Quoc ◽  
M. Olsson
Keyword(s):  
Nonlinear Equations ◽  
Building Block ◽  
High Speed ◽  
Traditional Approach ◽  
Equations Of Motion ◽  
A Priori ◽  
Flexible Structures ◽  
Structural Deformation ◽  
Block Model ◽  
Nonlinear Equations Of Motion

In traditional analyses of vehicle/structure interaction, especially when there are constraints between vehicle masses and the structure, vehicle nominal motion is prescribed a priori, and therefore unaffected by the structure flexibility. In this paper, a concept of nominal motion is defined, and a methodology is proposed in which the above restriction is removed, allowing the vehicle nominal motion to become unknown, and encompassing the traditional approach as a particular case. General nonlinear equations of motion of a building block model, applicable to both wheel-on-rail and magnetically levitated vehicles, are derived. These equations are simplified to a set of mildly nonlinear equations upon introducing additional assumptions — essentially on small structural deformation. An example is given to illustrate the present formulation.

Robust Time-Optimal Control of Flexible Structures With Parametric Uncertainty

1997 ◽  
Vol 119 (4) ◽  
pp. 743-748 ◽  
Author(s):  
Shin-Whar Liu ◽  
Tarunraj Singh
Keyword(s):  
Optimal Control ◽  
Equations Of Motion ◽  
Flexible Structures ◽  
Parametric Uncertainty ◽  
Time Optimal Control ◽  
Flexible Beam ◽  
Nonlinear Simulation ◽  
Switch Time ◽  
Optimal Controllers ◽  
Time Optimal

The design of robust time-optimal controllers using the sensitivity concept is presented in this paper. A parameter optimization problem is solved using the Switch Time Optimization algorithm to determine a bang-bang control profile that minimizes the maneuver time subject to the constraint that the sensitivity of the final states with respect to system parameters are zero. The proposed approach is illustrated on the benchmark floating oscillator problem and a slewing flexible beam whose equations of motion are nonlinear. Simulation results illustrate the reduction of residual vibrations of the system subject to the robust control profile, compared to the time-optimal control profile.

Vibration Suppression of Flexible Structures Utilizing Internal Resonance of Liquid Sloshing in a Nearly Square Tank

2010 ◽  
Author(s):  
Takashi Ikeda
Keyword(s):  
Frequency Response ◽  
Internal Resonance ◽  
Vibration Suppression ◽  
Equations Of Motion ◽  
Flexible Structures ◽  
Harmonic Excitation ◽  
Liquid Sloshing ◽  
Maximum Efficiency ◽  
Tuned Liquid Damper ◽  
Response Curves

This paper proposes a new idea to utilize the internal resonance of two different sloshing modes in a nearly square tank when used as a tuned liquid damper (TLD). This idea results in achieving higher efficiency of vibration suppression for flexible structures subjected to horizontal harmonic excitation. Namely, the two sloshing modes (1, 0) and (0, 1) in a nearly square tank are degenerated and hence their natural frequencies are nearly equal with each other. Because the two predominant sloshing modes are nonlinearly coupled, internal resonance is expected to occur. Galerkin’s method is used to determine the modal equations of motion for liquid sloshing. Then, van der Pol’s method is used to determine the expressions of the frequency response curves. Frequency response curves and bifurcation sets are numerically calculated. From these results, the optimal values of the size and instillation angle of the tank can be determined in order to achieve maximum efficiency of vibration suppression in a flexible structure. Experiments confirmed the validity of the theoretical analysis.

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