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Technical Briefs

# Quadratic Mode Shape Components From Linear Finite Element Analysis

[+] Author and Article Information
L. H. van Zyl

E. H. Mathews

Centre for Research and Continued Engineering Development,  North West University, Suite 90, Private Bag X30, 0040 Pretoria, South Africa

J. Vib. Acoust 134(1), 014501 (Dec 22, 2011) (8 pages) doi:10.1115/1.4004681 History: Received July 15, 2010; Revised March 16, 2011; Published December 22, 2011; Online December 22, 2011

## Abstract

Points on a vibrating structure move along curved paths rather than straight lines; however, this is largely ignored in modal analysis. Applications where the curved path of motion cannot be ignored include beamlike structures in rotating systems, e.g., helicopter rotor blades, compressor and turbine blades, and even robot arms. In most aeroelastic applications the curvature of the motion is of no consequence. The flutter analysis of T-tails is one notable exception due to the steady-state trim load on the horizontal stabilizer. Modal basis buckling analyses can also be performed when taking the curved path of motion into account. The effective application of quadratic mode shape components to capture the essential kinematics has been shown by several researchers. The usual method of computing the quadratic mode shape components for general structures employs multiple nonlinear static analyses for each component. It is shown here how the quadratic mode shape components for general structures can be obtained using linear static analysis. The derivation is based on energy principles. Only one linear static load case is required for each quadratic component. The method is illustrated for truss structures and applied to nonlinear static analyses of a linear and a geometrically nonlinear structure. The modal method results are compared to finite element nonlinear static analysis results. The proposed method for calculating quadratic mode shape components produces credible results and offers several advantages over the earlier method, viz., the use of linear analysis instead of nonlinear analysis, fewer load cases per quadratic mode shape component, and user-independence.

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## Figures

Figure 5

Static deflection of the nonsymmetric beam: (a) linear finite element analysis (matlab code), (b) linear modal basis analysis, (c) quadratic modal basis analysis, (d) node 8 vertical displacement comparison with msc /nastran nonlinear analysis

Figure 1

First three modes of the cantilever beam

Figure 2

Mode 1 of the two-dimensional tower: (a) linear mode shape, s = 5, (b) quadratic mode shape component, (c) parabolic mode shape, (d) elastic energy versus generalized coordinate

Figure 3

Linear static analysis: (a) linear finite element analysis (matlab code), (b) linear modal basis analysis, (c) quadratic modal basis analysis, (d) node 15 horizontal displacement comparison with msc /nastran nonlinear analysis

Figure 4

Mode 1 of the nonsymmetric beam: (a) linear mode shape component, s = 5 (matlab finite element code), (b) quadratic mode shape component, (c) parabolic mode shape, (d) elastic energy versus generalized coordinate

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