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Nonlinear Dynamics

, Volume 63, Issue 4, pp 561–586 | Cite as

Improved reduced order solution techniques for nonlinear systems with localized nonlinearities

  • I. R. Praveen Krishna
  • C. Padmanabhan
Original Paper

Abstract

This paper examines the modeling and solution of large-order nonlinear systems with continuous nonlinearities which are spatially localized. This localization is exploited by a combined component mode synthesis (CMS)—dynamic substructuring approach for efficient model reduction. A new ordering method for the Fourier coefficients used in the Harmonic Balance Method (HBM) is proposed. This allows the calculation of the slave dynamic flexibility matrix, using simple analytical expressions thus saving considerable computational effort by avoiding inverse calculation. This procedure is also capable of handling proportional damping. A hypersphere-based continuation technique is used to trace the solution, and hence track bifurcations since it has the advantage that the augmented Jacobian matrix remains square. The reduced system is also solved using a time-variational method (TVM) which generates sparse Jacobian matrices when compared with HBM. Several systems including those with parametric excitation and internal resonances are solved to demonstrate the capability of the proposed schemes. A comparison of these techniques and their effectiveness in solving extremely strong nonlinear systems with continuous nonlinearities is discussed.

Keywords

Large nonlinear systems Mode superposition Dynamic condensation Component mode synthesis Harmonic balance Time variational method QR LU SVD Powell hybrid method Continuation Hypersphere 

Nomenclature

HBM

Harmonic Balance Method

TVM

Time Variational Method

DOF

Degrees-of-Freedom

M

Mass matrix

C

Damping matrix

K

Stiffness matrix

\(\ddot{\mathbf{x}}\)

Acceleration vector

\(\mathbf{\dot{x}}\)

Velocity vector

t

Time

x

Displacement vector

\(\mathbf{f}(\mathbf{x},\dot{\mathbf{x}})\)

Nonlinear force vector

F(t)

External excitation vector

xk(t)

Approximated Fourier series expansion of displacement vector for kth DOF

\(\tilde{x}_{k0}\)

DC term of the Fourier series expansion of displacement kth DOF

\(\tilde{x}_{kn}^{c}\)

Coefficient of the cosine term of Fourier series expansion of displacement kth DOF

\(\tilde{x}_{kn}^{s}\)

Coefficient of the sine term of Fourier series expansion of displacement kth DOF

Fk(t)

Approximated Fourier series expansion of external force vector applied on kth DOF

\(\tilde{F}_{k0}\)

DC term of the Fourier series of external force vector kth DOF

\(\tilde{F}_{kn}^{c}\)

Coefficient of the cosine term of Fourier series of external force vector kth DOF

\(\tilde{F}_{kn}^{s}\)

Coefficient of the sine term of Fourier series expansion of external force vector kth DOF

\(\mathbf{f}_{k}(\mathbf{x},\dot{\mathbf{x}})\)

Approximated Fourier series expansion of nonlinear force vector kth DOF

\(\tilde{f}_{k0}\)

DC term of the Fourier series of expansion of nonlinear force vector kth DOF

\(\tilde{f}_{kn}^{c}\)

Coefficient of the cosine term of Fourier series of nonlinear force vector kth DOF

\(\tilde{f}_{kn}^{s}\)

Coefficient of the sine term of Fourier series expansion of nonlinear force vector kth DOF

\(\mathbf{f}_{\mathrm{p}k}(\mathbf{x},\dot{\mathbf{x}},\mathbf {t})\)

Approximated Fourier series expansion of parametric excitation vector kth DOF

\(\tilde{f}_{\mathrm{p}k0}\)

DC term of the Fourier series of expansion of parametric excitation vector kth DOF

\(\tilde{f}_{\mathrm{p}kn}^{c}\)

Coefficient of the cosine term of Fourier series of parametric excitation vector kth DOF

\(\tilde{f}_{\mathrm{p}kn}^{s}\)

Coefficient of the sine term of Fourier series expansion of parametric excitation vector kth DOF

α

Nonlinear/Parametric excitation coefficient

R(γ)

Residue vector

Y

Fourier/ Time variational Admittance Matrix

Yss−1

Slave Flexibility Matrix

J

Jacobian matrix

ε

Convergence tolerance

\(\hat{\mathbf{f}}\)

Nonlinear force TVM coefficients

\(\hat{\mathbf{x}}\)

Displacement TVM coefficients

\(\hat{\mathbf{F}}\)

External force TVM coefficients

U

Eigen vector matrix

φ

Retained Mode matrix

ψ

Constraint mode matrix

ζ

Damping ratio

ω

Excitation frequency

Ωi

ith Natural frequency

Δ

Increment between iterations

c

Hypersphere center

FM

Full Model

MS

Mode Superposition

PC

Physical Condensation

CM

Component Mode

Subscripts

ss

Slave partition

sm

Slave master partition

ms

Master slave partition

mm

Master master partition

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Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Machine Design Section, Department of Mechanical EngineeringIndian Institute of Technology MadrasChennaiIndia

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