Modal truncation damping in reduced modal analysis of piecewise linear continuum structures

2021 ◽  
pp. 1-24
Author(s):  
Róbert K. Németh ◽  
Borbála B. Geleji
Keyword(s):  
Modal Analysis ◽  
Piecewise Linear ◽  
Continuum Structures ◽  
Modal Truncation

scholarly journals The impacting cantilever: modal non-convergence and the importance of stiffness matching

2013 ◽  
Vol 371 (1993) ◽  
pp. 20120434 ◽  
Author(s):  
John Melcher ◽  
Alan R. Champneys ◽  
David J. Wagg
Keyword(s):  
Modal Analysis ◽  
Degrees Of Freedom ◽  
Applied Mathematics ◽  
Point Of View ◽  
Higher Order Modes ◽  
Intermittent Contact ◽  
Bilinear Formulation ◽  
Modal Truncation ◽  
The Impact ◽  
Impact Models

The problem of an Euler–Bernoulli cantilever beam whose free end impacts with a point constraint is revisited from the point of view of modal analysis. It is shown that there is non-uniqueness of consistent impact laws for a given modal truncation. Moreover, taking an N -mode compliant, bilinear formulation and passing to the rigid limit leads to a sequence of impact models that does not converge as . The dynamics of such truncated models are studied numerically and found to give rise to quite different dynamics depending on the number of degrees of freedom taken. The simulations are compared with results from simple experiments that show a propensity for multiple-tap dynamics, in which higher-order modes lead to rapidly cycling intermittent contact. The conclusion reached is that, to derive an accurate model, one needs to avoid the impact limit altogether, and take sufficiently many modes in the formulation to match the actual stiffness of the constraining stop. mechanical engineering, applied mathematics

A Hybrid Approach for the Modal Analysis of Continuous Systems With Localized Nonlinear Constraints

2010 ◽  
Author(s):  
M. R. Brake
Keyword(s):  
Boundary Conditions ◽  
Modal Analysis ◽  
Piecewise Linear ◽  
Hybrid Approach ◽  
Constitutive Relationship ◽  
Leaf Spring ◽  
Continuous Systems ◽  
Superposition Method ◽  
Arbitrary Boundary ◽  
Linear Force

The analysis of continuous systems with nonlinearities in their domain have previously been limited to either numerical approaches, or analytical methods that are constrained in the parameter space, boundary conditions, or order of the system. The present analysis develops a robust method for studying continuous systems with arbitrary boundary conditions and nonlinearities using the assumption that the nonlinear constraint can be modeled with a piecewise-linear force-deflection constitutive relationship. Under this assumption, a superposition method is used to generate homogeneous boundary conditions, and modal analysis is used to find the displacement of the system in each state of the piecewise-linear nonlinearity. In order to map across each nonlinearity in the piecewise-linear force-deflection profile, a variational calculus approach is taken that minimizes the L2 energy norm between the previous and current states. To illustrate this method, a leaf spring coupled with a connector pin immersed in a viscous fluid is modeled as a beam with a piecewise-linear constraint. From the results of the convergence and parameter studies, a high correlation between the finite-time Lyapunov exponents and the contact time per period of the excitation is observed. The parameter studies also indicate that when the system’s parameters are changed in order to reduce the magnitude of the velocity impact between the leaf spring and connector pin, the extent of the regions over which a chaotic response is observed increases.

Modal Analysis of Rotor on Piecewise Linear Journal Bearings Under Seismic Excitation

1999 ◽  
Vol 121 (2) ◽  
pp. 190-196 ◽  
Author(s):  
B. J. Gaganis ◽  
A. K. Zisimopoulos ◽  
P. G. Nikolakopoulos ◽  
C. A. Papadopoulos
Keyword(s):  
Modal Analysis ◽  
Dynamic Properties ◽  
Piecewise Linear ◽  
Computer Time ◽  
Seismic Excitation ◽  
Damping Coefficients ◽  
Highly Nonlinear ◽  
Rotor Bearing ◽  
Stiffness And Damping ◽  
Bearing System

A rotor bearing system is expected to exhibit large vibration amplitudes when subjected to a large seismic excitation. It is possible that these vibrations can lead to large values the eccentricity of the bearings. Then the bearing is operated in highly nonlinear region because the stiffness and the damping coefficients are nonlinear as functions of the eccentricity. To solve this problem numerical integration must be performed with high cost in computer time. The idea of this paper was to divide the nonlinear area into more areas where the stiffness and damping coefficients could be considered to be constants. In other words the bearing coefficients are considered to be piecewise constant. The excitation due to the earthquake is modelled as a movement of the base of the bearings using the El Centro data for the acceleration. Then a simplified modal analysis for each of these piecewise linear regions can be performed. The equation of motion of the rotor contains rotational speed depended terms, known as gyroscopic terms, and terms due to base excitation. The response and the variation of the dynamic properties of this complicated rotor bearing system are investigated and presented.

Modal Parameter Validation in Coupled Vibro-Acoustical System

2011 ◽  
Vol 301-303 ◽  
pp. 629-634
Author(s):  
Yi Feng Xu ◽  
Jun Wang
Keyword(s):  
Finite Element ◽  
Modal Analysis ◽  
Reciprocity Theorem ◽  
Modal Parameters ◽  
Modal Parameter ◽  
Modal Frequency ◽  
Simulation Results ◽  
Modal Truncation ◽  
Acoustic Boundary Element ◽  
Element Algorithm

The aim of this paper is to validate the modal parameters used in coupled structural finite element and acoustic boundary element algorithm to analysis the structure subjected to diffuse acoustic field. The theoretical deduction of non-symmetric coupled vibro-acoustical modal analysis was introduced firstly. In order to verify the modal truncation frequency how to affect the simulation results, based on the reciprocity theorem used in coupled FE-BE model, three different truncation frequency conditions were performed. The contrastive results show that twice the upper calculation frequency as the truncated modal frequency can make the simulation effectively and efficiently.

Exact Solution of Time History Response for Dynamic Systems With Arbitrary Viscous Damping Using Complex Modal Analysis

2007 ◽  
Author(s):  
Lonny L. Thompson ◽  
Manoj Kumar M. Chinnakonda
Keyword(s):  
Modal Analysis ◽  
State Space ◽  
Piecewise Linear ◽  
Time History ◽  
Quadrature Rules ◽  
Step Size ◽  
Time Step ◽  
Time Step Size ◽  
Complex Eigenvalues ◽  
Time History Response

A solution method for general, non-proportional damping time history response for piecewise linear loading is generalized to exact solutions which include piecewise quadratic loading. Comparisons are made to Trapezoidal and Simpson’s quadrature rules for approximating the time integral of the weighted generalized forcing function in the exact solution to the decoupled modal equations arising from state-space modal analysis of linear dynamic systems. Closed-form expressions for the weighting parameters in the quadrature formulas in terms of time-step size and complex eigenvalues are derived. The solution is obtained step-by-step from update formulas derived from the piecewise linear and quadratic interpolatory quadrature rules starting from the initial condition. An examination of error estimates for the different force interpolation methods shows convergence rates depend explicitly on the amount of damping in the system as measured by the real-part of the complex eigenvalues of the state-space modal equations and time-step size. Numerical results for a system with general, non-proportional damping, and driven by a continuous loading shows that for systems with light damping, update formulas for standard Trapezoidal and Simpson’s rule integration have comparable accuracy to the weighted piecewise linear and quadratic force interpolation update formulas, while for heavy damping, the update formulas from the weighted force interpolation quadrature rules are more accurate. Using a simple model representing a stiff system with general damping, we show that a two-step modal analysis using real-valued modal reduction followed by state-space modal analysis is shown to be an effective approach for rejecting spurious modes in the spatial discretization of a continuous system.

Modal Analysis of a Gyroscopic System With Nonlinear Constraints

2009 ◽  
Author(s):  
M. R. Brake ◽  
J. A. Wickert
Keyword(s):  
Modal Analysis ◽  
Data Storage ◽  
Chaotic Behavior ◽  
Piecewise Linear ◽  
Linear Constraint ◽  
Mapping Method ◽  
Inner Product ◽  
Mode Shapes ◽  
Design Parameters ◽  
Nonlinear Constraints

A method for the modal analysis of continuous gyroscopic systems with nonlinear constraints is developed, which assumes that the nonlinear constraints can be expressed with piecewise-linear force-deflection profiles. Using this assumption, the mode shapes and natural frequencies are found for each state, and a mapping method based on the inner product of the mode shapes is developed to map the displacement of the system between states. To illustrate this method, a model for the vibration of a traveling string in contact with a piecewise-linear constraint is developed as an analog of the interaction between magnetic tape and a guide in data storage systems. Several design parameters of the guide are considered: flange clearance, stiffness, symmetry, and the guide’s position. Critical bifurcation thresholds exist, below which the system exhibits no chaotic behavior and is dominated by period one, symmetric behavior, and above which the system contains asymmetric, higher periodic motion with windows of chaotic behavior. These bifurcation thresholds are particularly pronounced for the transport speed, flange clearance, symmetry of the force deflection profile, and guide position.

Nonlinear Modal Analysis of a Cracked Beam

1997 ◽  
Author(s):  
M. Chati ◽  
R. H. Rand ◽  
S. Mukherjee
Keyword(s):  
Linear System ◽  
Modal Analysis ◽  
Natural Frequencies ◽  
Piecewise Linear ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Degree Of Freedom ◽  
Mode Shapes ◽  
Cracked Beam ◽  
Piecewise Linear System

Abstract This paper addresses the problem of vibrations of a cracked beam. In general, the motion of such a beam can be very complex. This phenomenon can be attributed to the presence of the nonlinearity due to the opening and closing of cracks. The focus of this paper is the modal analysis of a cantilever beam with a transverse edge crack. The nonlinearity mentioned above has been modelled as a piecewise-linear system. In an attempt to define effective natural frequencies for this piecewise-linear system, the idea of a “bilinear frequency” is utilized. The bilinear frequency is obtained by computing the associated frequencies of each of the linear pieces of the piecewise-linear system. The finite element method is used to obtain the natural frequencies in each linear region. In order to better understand the essential nonlinear dynamics of the cracked beam, a piecewise-linear two degree of freedom model is studied. Perturbation methods are used to obtain the nonlinear normal modes of vibration and the associated period of the motion. Results of this piecewise-linear model problem are shown to justify the definition of the bilinear frequency as the effective natural frequency. It is therefore expected that calculating piecewise mode shapes and bilinear frequencies is useful for understanding the dynamics of the infinite degree of freedom cracked beam.

scholarly journals Breaking the Kolmogorov Barrier in Model Reduction of Fluid Flows

Fluids ◽  
2020 ◽  
Vol 5 (1) ◽  
pp. 26 ◽  
Author(s):  
Shady E. Ahmed ◽  
Omer San
Keyword(s):  
Energy Spectrum ◽  
Turbulent Flows ◽  
Turbulence Modeling ◽  
Piecewise Linear ◽  
Fluid Flows ◽  
Reduced Order ◽  
Temporal Complexity ◽  
Comparable Accuracy ◽  
The Cost ◽  
Modal Truncation

Turbulence modeling has been always a challenge, given the degree of underlying spatial and temporal complexity. In this paper, we propose the use of a partitioned reduced order modeling (ROM) approach for efficient and effective approximation of turbulent flows. A piecewise linear subspace is tailored to capture the fine flow details in addition to the larger scales. We test the partitioned ROM for a decaying two-dimensional (2D) turbulent flow, known as 2D Kraichnan turbulence. The flow is initiated using an array of random vortices, corresponding to an arbitrary energy spectrum. We show that partitioning produces more accurate and stable results than standard ROM based on a global application of modal decomposition techniques. We also demonstrate the predictive capability of partitioned ROM through an energy spectrum analysis, where the recovered energy spectrum significantly converges to the full order model’s statistics with increased partitioning. Although the proposed approach incurs increased memory requirements to store the local basis functions for each partition, we emphasize that it permits the construction of more compact ROMs (i.e., of smaller dimension) with comparable accuracy, which in turn significantly reduces the online computational burden. Therefore, we consider that partitioning acts as a converter which reduces the cost of online deployment at the expense of offline and memory costs. Finally, we investigate the application of closure modeling to account for the effects of modal truncation on ROM dynamics. We illustrate that closure techniques can help to stabilize the results in the inertial range, but over-stabilization might take place in the dissipative range.

Sign in / Sign up

Export Citation Format

Share Document