scholarly journals The Influence of Higher-Order Mode Shapes for Reduced-Order Models of Electrostatically Actuated Microbeams

2010 ◽  
Vol 77 (4) ◽  
Author(s):  
Stefanie Gutschmidt
Keyword(s):  
System Modeling ◽  
Mode Shapes ◽  
Approximation Procedure ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Advantages And Disadvantages ◽  
Electromechanical Actuation ◽  
Nano Electromechanical System ◽  
Nonlinear Continuum

Reduced-order models for micro-electromechanical structures possess several attractive features when compared with computational approaches using, e.g., finite-element packages. However, also within the business of reduced-order modeling, there are different approaches that yield different results. The efficiency of such approaches has to be judged according to, first, the purposes and aims of the model and, second, according to computational expenses and modeling efforts. This paper deals specifically with the frequently asked question of how many modes have to be considered in the discretization procedure to ensure an efficient reduced-order model. A consistent nonlinear continuum model is employed to describe a doubly clamped microbeam subject to two cases of electromechanical actuation. The analysis, confined to the static behavior, concentrates on two discretization techniques and addresses the differences between the final reduced-order models, accordingly. The results show significant differences with respect to the number of implemented linear-undamped mode shape functions, which are used as basis functions in the approximation procedure. This is demonstrated for the two mentioned distinct excitation schemes of the doubly clamped microbeam. The purposes of this paper are twofold. First, it draws attention to the differences between reduced-order models, which have been discretized one way or the other according to investigation goals and purposes. Second, it serves as a guideline for future micro- and nano-electromechanical system modeling by elaborating the advantages and disadvantages of both techniques.

Voltage Response of Superharmonic Resonance of Electrostatically Actuated MEMS Resonators: Comparison Between Boundary Value Problem Model and Reduced Order Model

2018 ◽  
Author(s):  
Julio Beatriz ◽  
Martin Botello ◽  
Dumitru I. Caruntu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Value ◽  
Mode Shapes ◽  
Voltage Response ◽  
Order Model ◽  
Partial Differential ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Modes Of Vibration

This paper deals with the voltage response of electrostatically actuated NEMS resonators at superharmonic resonance. In this work a comparison between Boundary Value Problem (BVP) model, and Reduced Order Model (ROM) is conducted for this type of resonance. BVP model is developed from the partial differential equation by replacing the time derivatives with finite differences. So, the partial differential equation is replaced by a sequence of boundary value problems, one for each step in time. Matlab’s function bvp4c is used to numerically integrate the BVPs. ROMs are based on Galerkin procedure and use the mode shapes of the resonator as a basis of functions. Therefore, the partial differential equation is replaced by a system of differential equations in time. The number of the equations in the system is equal to the number of mode shapes (or modes of vibration) used in the ROM. One mode of vibration ROM is solved using the method of multiple scales. Two modes of vibration ROM is numerically integrated using Matlab’s function ode15s in order to obtain time responses, and a continuation and bifurcation analysis is conducted using AUTO 07P. The effects of different nonlinearities in the system on the voltage response are reported. This work shows that BVP model is a valid method to predict the voltage response of a micro/nano cantilevers.

Accuracies of Reduced Order Models of a Bladed Rotor With Geometric Mistuning

2013 ◽  
Vol 136 (7) ◽  
Author(s):  
Yasharth Bhartiya ◽  
Alok Sinha
Keyword(s):  
Natural Frequencies ◽  
Domain Analysis ◽  
Forced Response ◽  
Reduced Order Model ◽  
Mode Shapes ◽  
Reduced Order Models ◽  
Order Model ◽  
Reduced Order ◽  
Model Based ◽  
Bladed Rotor

The results from a reduced order model based on frequency mistuning are compared with those from recently developed modified modal domain analysis (MMDA). For the academic bladed rotor considered in this paper, the frequency mistuning analysis is unable to capture the effects of geometric mistuning, whereas MMDA provides accurate estimates of natural frequencies, mode shapes, and forced response.

A New Mistuning Identification Method Based on the Subset of Nominal System Modes Method

2020 ◽  
Vol 142 (2) ◽  
Author(s):  
Christian U. Waldherr ◽  
Patrick Buchwald ◽  
Damian M. Vogt
Keyword(s):  
Finite Element ◽  
Input Data ◽  
Measurement Data ◽  
Computational Effort ◽  
Mode Shapes ◽  
Reduced Order Models ◽  
Identification Procedure ◽  
Reduced Order ◽  
Nominal System ◽  
Identification Method

Abstract The mistuning problem of quasi-periodic structures has been the subject of numerous scientific investigations for more than 50 years. Researchers developed reduced-order models to reduce the computational costs of mistuning investigations including finite element models. One question which has also high practical relevance is the identification of mistuning based on modal properties. In this work, a new identification method based on the subset of nominal system modes method (SNM) is presented. Different to existing identification methods where usually the blade stiffness of each sector is scaled by a scalar value, N identification parameters are used to adapt the modal blade stiffness of each sector. The input data for the identification procedure consist solely of the mistuned natural frequencies of the investigated mode family as well as of the corresponding mistuned mode shapes in the form of one degree-of-freedom per sector. The reduction basis consists of the tuned mode shapes of the investigated mode family. Furthermore, the proposed identification method allows for the inclusion of centrifugal effects like stress stiffening and spin softening without additional computational effort. From this point of view, the presented method is also appropriate to handle centrifugal effects in reduced-order models using a minimum set of input data compared to existing methods. The power of the new identification method is demonstrated on the example of an axial compressor blisk. Finite element calculations including geometrical mistuning provide the database for the identification procedure. The correct functioning of the identification method including measurement noise is also validated to show the applicability to a case of application where real measurement data are available.

scholarly journals Component-Centric Reduced Order Modeling of the Dynamic Response of Linear Multibay Structures

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Yuting Wang ◽  
Marc P. Mignolet
Keyword(s):  
Natural Frequencies ◽  
Element Model ◽  
Fixed Number ◽  
Mode Shapes ◽  
Reduced Order Models ◽  
Reduced Order ◽  
Entire Structure ◽  
Similar Mode ◽  
Fixed Interface ◽  
Proper Orthogonal

Component-centric reduced order models (ROMs) are introduced here as small-size ROMs providing an accurate prediction of the linear response of part of a structure (the β component) without focusing on the rest of it (the α component). Craig–Bampton (CB) substructuring methods are first considered. In one method, the β component response is modeled with its fixed interface modes while the other adopts singular value eigenvectors of the β component deflections of the linear modes of the entire structure. The deflections in the α component induced by harmonic motions of these β component modes are processed by a proper orthogonal decomposition (POD) to model the α component response. A third approach starts from the linear modes of the entire structure which are dominant in the β component response. Then, the contributions of other modes in this part of the structure are approximated in terms of those of the dominant modes with close natural frequencies and similar mode shapes in the β component, i.e., these nondominant modal contributions are “lumped” onto dominant ones. This lumping permits to increase the accuracy in the β component at a fixed number of modes. The three approaches are assessed on a structural finite element model of a nine-bay panel with the modal lumping-based method yielding the most “compact” ROMs. Finally, good robustness of the ROM to changes in the β component properties (e.g., for design optimization) is demonstrated and a similar sensitivity analysis is carried out with respect to the loading under which the ROM is constructed.

The Study of Aeroelastic Reduced-Order Model Based on CFD/CSD Method and its Application

2014 ◽  
Vol 978 ◽  
pp. 131-134
Author(s):  
Rui Li ◽  
Chang Hong Tang
Keyword(s):  
Harmonic Balance ◽  
Unsteady Aerodynamics ◽  
Harmonic Balance Method ◽  
Reduced Order Model ◽  
Reduced Order Models ◽  
Order Model ◽  
Reduced Order ◽  
Advantages And Disadvantages ◽  
Aeroelastic Analysis ◽  
Proper Orthogonal

Unsteady aerodynamics research is the foundation of aeroelastic analysis. How to effectively improve the aeroelastic computational efficiency,it is the key of current research on aeroelasticity now.Reduced order models are proposed as a powerful tool to solve this problem. Analyzed the three reduced-order models for Volterra ,Proper Orthogonal Decomposition and Harmonic Balance method ,their advantages and disadvantages were pointed out. The direction of the reduced order model in the future was Proposed and some suggest was given out for its application.

Reduced Order Models for Blisks With Small and Large Mistuning and Friction Dampers

2016 ◽  
Author(s):  
Weihan Tang ◽  
Seunghun Baek ◽  
Bogdan I. Epureanu
Keyword(s):  
Equations Of Motion ◽  
Dynamic Responses ◽  
Mode Shapes ◽  
Reduced Order Models ◽  
Cyclic Symmetry ◽  
Friction Forces ◽  
Frictional Contacts ◽  
Reduced Order ◽  
Random Mistuning ◽  
Forced Responses

In operation, rotating bladed disks (blisks) are often subject to high levels of dynamic loading, resulting in large amplitudes of forced vibrations especially at resonance. Moreover, variations in structural properties of individual sectors, referred to as mistuning, can lead to strain energy localization and can amplify forced responses. To prevent damages caused by high cycle fatigue, various frictional damping sources are introduced to dissipate vibration energy. Due to the nonlinear behavior of frictional contacts, conventional methods to study the dynamics of the blisk-damper systems are based often on numerical time integration, which is time-consuming and can be computationally prohibitive due to the large sizes of commercial blisk models. Existing techniques for model reduction either rely heavily on cyclic symmetry of the blisk-damper system, or are based on component mode synthesis (CMS). However, in the presence of mistuning, cyclic symmetry no longer exists. Also, mistuning is random and best studied statistically. Repetitive CMS condensation for a large amount of random mistuning patterns can lead to a computationally formidable task. This paper presents a reduced-order modeling technique to efficiently capture the nonlinear dynamic responses of blisk-damper systems with both small perturbations in blade material properties (small mistuning), and significant changes in the blisk geometries (large mistuning). The reduced-order models (ROMs) are formed by projecting the blisk-damper systems onto a novel mode basis that mimics the contact behavior. This mode basis contains normal mode shapes of the mistuned blisk-damper systems with either sliding or sticking conditions enforced on the contact surfaces. These mode shapes are computed through the N-PRIME method, a technique recently developed by the authors to efficiently obtain mode shapes for blisks with simultaneous large and small mistuning. The resulting modal nonlinear equations of motion are solved by a hybrid frequency/time (HFT) domain method with continuation. In the HFT method, the contact status and friction forces are determined in the time domain by a quasi-two-dimensional contact model at each contact point, whereas the modal equations of motion are solved in the frequency domain according to a harmonic balance formulation. The forced responses computed by the proposed ROMs are validated for two systems with distinct mistuning patterns. A statistical analysis is performed to study the effectiveness of the frictional dampers under random mistuning patterns.

scholarly journals Comparison Between Calculated and Measured Free-Free Modes for a Flexible Rotor

1998 ◽  
Author(s):  
José A. Vázquez ◽  
Lloyd E. Barrett
Keyword(s):  
Natural Frequencies ◽  
Forced Response ◽  
Modal Testing ◽  
Analytical Models ◽  
Mode Shapes ◽  
Reduced Order Models ◽  
Response Functions ◽  
Flexible Rotor ◽  
Reduced Order ◽  
Response Characteristics

Many industrial machines nowadays are sold based on analysis performed on mathematical models of the rotors, bearings, substructures, and other components. The validity of the analysts therefore depends on the accuracy of the models themselves. When the rotor is available, modal testing may be used to validate the model of the rotor by comparing the calculated and measured free-free natural frequencies and mode shapes. This work presents additional tools for the verification of analytical models against experimental data. These tools use models of the rotor constructed from the measured data and the analytical model. A comparison of the first six calculated and measured free-free natural frequencies and mode shapes for a multi-mass flexible rotor is presented. The natural frequencies compare within 1.8%. The calculated and measured mode shapes were used to construct independent reduced order models of the rotor. These models were used to perform forced response and stability analyses. Forced response functions are presented comparing the forced response characteristics obtained from the two models. This provides a comparison between the measured and calculated forced response functions for the same number of modes. For the stability analysis, identical bearing models were added to both reduced order models. The eigenvalues were calculated using both models for a range of bearing stiffness and damping coefficients and were plotted for comparison.

scholarly journals Comprehensive Reduced-Order Models of Electrostatically Actuated MEMS Switches and Their Dynamics Including Impact and Bounce

2010 ◽  
Author(s):  
Michael G. Snow ◽  
Anil K. Bajaj
Keyword(s):  
Galerkin Method ◽  
Rarefied Gas ◽  
Dielectric Substrate ◽  
Contact Dynamics ◽  
Mode Shapes ◽  
Squeeze Film ◽  
Equilibrium Solutions ◽  
Reduced Order ◽  
Electrostatically Actuated ◽  
Lift Off

As MEMS technology develops it is becoming better understood that MEMS designers must account for the large uncertainties characteristic of the relevant manufacturing processes. Uncertainty quantification tasks the designer with evaluating many different possible outcomes from the manufacturing process which creates a demand for models that are accurate and comprehensive, yet fast to evaluate. This work presents a comprehensive reduced-order model of electrostatically actuated switches incorporating a range of effects that are typically included only in FE modeling codes. Specifically, the model accounts for variable electrode geometry, stretching of centerline or large displacement effects, fringing field, squeeze film and rarefied gas damping, and allows for elastic contact with the dielectric substrate. Individual compact models for each of these effects are taken from literature and included in the model for the system. The dielectric substrate is modeled as an elastic foundation. The resulting partial differential equation for the switch modeled as a beam is discritized via a Galerkin method into ordinary differential equations for modal amplitudes. The Galerkin method uses the linear un-damped mode shapes of the beam to approximate the solution. Both cantilever and fixed-fixed type switches are analyzed. Static equilibrium solutions as a function of the applied voltage are developed along with their stability. Static pull-in voltages, first time of switch closure, and voltage for lift-off are studied with the model. To capture the contact dynamics, the contact condition is evaluated with the substrate divided into a large number of elements and the contact force is projected on to the beam basis functions. In the case of cantilever geometry and slow voltage variations, three stable regimes of contact configuration and hysteresis between them are demonstrated.

Reduced-Order Models of Blisks With Small Geometric Mistuning

2017 ◽  
Vol 139 (4) ◽  
Author(s):  
Seunghun Baek ◽  
Bogdan Epureanu
Keyword(s):  
Linear Combination ◽  
Structural Properties ◽  
Dynamic Behavior ◽  
Mode Shapes ◽  
Reduced Order Models ◽  
Bladed Disks ◽  
Reduced Order ◽  
The Past ◽  
The Difference

A technique for generating reduced-order models (ROMs) of bladed disks with small geometric mistuning is proposed. Discrepancies in structural properties (mistuning) from blade to blade can cause a significant increase in the maximum vibratory stress. The effects of mistuning have been studied over the past few decades. Many researchers have studied the dynamic behavior of mistuned bladed disks by using ROMs. Many of these techniques rely on the fact that the modes of a mistuned system can be approximated by a linear combination of modes of the corresponding tuned system. In addition, the tuned system modes have been modeled in component mode mistuning by using modal participation factors of cantilevered blade modes. Such techniques assume that mistuning can be well modeled as variations in blade-alone frequencies. However, since geometric deformations contain stiffness and mass variations, mistuning can no longer be captured by cantilevered blade modes alone. To address this, several studies have focused on large and small geometric mistuning. These studies exploited the difference between tuned (with perturbed geometry) and nominal tuned mode shapes. In this work, we extend on that approach and devote particular attention to the development of ROMs of bladed disks with small geometric mistuning. The methodology requires only sector-level calculations and therefore can be applied to highly refined, realistic models of industrial size.

A New Mistuning Identification Method Based on the Subset of Nominal System Modes Method

2019 ◽  
Author(s):  
Christian U. Waldherr ◽  
Patrick Buchwald ◽  
Damian M. Vogt
Keyword(s):  
Finite Element ◽  
Input Data ◽  
Measurement Data ◽  
Computational Effort ◽  
Mode Shapes ◽  
Reduced Order Models ◽  
Identification Procedure ◽  
Reduced Order ◽  
Nominal System ◽  
Identification Method

Abstract The mistuning problem of quasi periodic structures is subject of numerous scientific investigations for more than 50 years. Researchers developed reduced order models to reduce the computational costs of mistuning investigations including finite element models. One question which has also high practical relevance is the identification of mistuning based on modal properties. In the present work, a new identification method based on the Subset of Nominal System Modes method (SNM) is presented. The input data for the identification procedure consists solely of the mistuned natural frequencies of the investigated mode family as well as of the corresponding mistuned mode shapes in the form of one degree of freedom per sector. The reduction basis consists of the tuned mode shapes of the investigated mode family. Furthermore, the proposed identification method allows for the inclusion of centrifugal effects like stress stiffening and spin softening without additional computational effort. From this point of view the presented method is also appropriate to handle centrifugal effects in reduced order models using a minimum set of input data compared to existing methods. The powerfulness of the new identification method is demonstrated on the example of an axial compressor blisk. Finite element calculations including geometrical mistuning provide the data base for the identification procedure. The correct functioning of the identification method including measurement noise is also validated to show the applicability to a case of application where real measurement data is available.

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