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Improved reduced order solution techniques for nonlinear systems with localized nonlinearities

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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.

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Abbreviations

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

x k (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

F k (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

Y s s −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

Ω i :

ith Natural frequency

Δ:

Increment between iterations

c:

Hypersphere center

FM:

Full Model

MS:

Mode Superposition

PC:

Physical Condensation

CM:

Component Mode

ss:

Slave partition

sm:

Slave master partition

ms:

Master slave partition

mm:

Master master partition

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Authors and Affiliations

  1. Machine Design Section, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai, 600036, India

    I. R. Praveen Krishna & C. Padmanabhan

Authors
  1. I. R. Praveen Krishna
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  2. C. Padmanabhan
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Correspondence to C. Padmanabhan.

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Praveen Krishna, I.R., Padmanabhan, C. Improved reduced order solution techniques for nonlinear systems with localized nonlinearities. Nonlinear Dyn 63, 561–586 (2011). https://doi.org/10.1007/s11071-010-9820-5

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