The generalized harmonic balance method for determining the combination resonance in the parametric dynamic systems

Szemplińska-Stupnicka, W.

doi:10.1016/s0022-460x(78)80043-1

Cited by 55 publications

(13 citation statements)

References 17 publications

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“…Bolotin's method [1] is very common for obtaining the primary instability regions, but it cannot be used for solving combination resonance instability problems. Different researchers [28][29][30][31] have proposed different methods for solving combination resonance instability problems. Iwatsubo et al [32] studied the simple and combination resonances of columns under periodic axial loads.…”

Section: Introductionmentioning

confidence: 99%

Parametric combination resonance instability characteristics of laminated composite curved panels with circular cutout subjected to non-uniform loading with damping

Udar

Datta

2007

International Journal of Mechanical Sciences

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Section: Introductionmentioning

confidence: 99%

Parametric combination resonance instability characteristics of laminated composite curved panels with circular cutout subjected to non-uniform loading with damping

Udar

Datta

2007

International Journal of Mechanical Sciences

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“…They could be detected either by using perturbation methods, involving the small parameter concept, 6,29 or by assuming that the solution on the boundary of the combination resonance is a two-harmonics component function of time for all the coordinates and applying the HBM. 1,11 To determine simple-type instabilities, it will be assumed that the solution of Equation 12 has the form 1,2,11…”

Section: Equation 12mentioning

confidence: 99%

“…References 2, 6, 8 and 9), the harmonic balance method (HBM; e.g. References 1,11,[13][14][15][16][17][18][21][22][23] and numerical integration methods (e.g. References 5 and 7 for brief descriptions of numerical methods to solve the non-linear differential equations of motion and to determine the monodromy matrix, and References 34 and 35 for more detailed descriptions of numerical solution of non-linear differential equations).…”

Section: Introductionmentioning

confidence: 99%

Stability of multi-degree-of-freedom Duffing oscillators

Ribeiro¹

2009

Proceedings of the Institution of Civil Engineers - Engineering

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Different methods of examining the stability of periodic solutions of non-linear multi-degree-of-freedom systems are compared in this paper. The methods are particularly suited for solutions computed with the harmonic balance method and the non-linear systems considered are represented by systems of coupled Duffing-type equations; that is, with cubic non-linear expressions. The latter systems appear frequently in models of thin-walled structures, particularly in straight beams and flat plates. In the first method reviewed in this paper the harmonic balance procedure is applied to define an eigenvalue problem, from which the characteristic exponents are determined. If the real part of any of these characteristic exponents is greater than zero, then the solution is unstable. The second method relies on the sign of the determinant of a Jacobian matrix; a matrix that is also often used to solve the equations of motion. The last method is based on a perturbation procedure and with it a closed-form expression for stable regions is achieved. The advantages and shortcomings of the different methods are discussed. INTRODUCTIONA non-linear dynamic system excited by a harmonic force can have periodic and non-periodic solutions. With regard to the periodic solutions, it is possible that more than one solution exists for a single excitation frequency and the initial conditions determine to which solution the system converges. An unstable steady-state solution does not occur in practice: if a solution is unstable, the system will evolve to another attractor. Hence, it is important to determine whether or not a solution is stable. The local stability can be determined by investigating the evolution of a perturbed solution that results from the addition of a small disturbance. A perturbation near an unstable equilibrium condition leads to a departure from this condition and the inverse occurs near a stable equilibrium condition. Several types and orders of instability can occur and this is a subject widely analysed 1-33 but still of considerable interest.

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“…To obtain the periodic response of the nonlinear PDE, harmonic balance method [3] and incremental harmonic balance (IHB) method [4] H T s Q evaluated at x = 1 is always equal to zero. To introduce the motion at x = 1 for the representation of w(x, t), an additional trial function e(x) = x with its generalized coordinate q(t) is used.…”

Section: Introductionmentioning

confidence: 99%

The Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Responses of a Nonlinear Partial Differential Equation With a Linear Complex Boundary Condition

Wang

Zhu

2017

Volume 8: 29th Conference on Mechanical Vibration and Noise

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The spatial and temporal harmonic balance (STHB) method is used to solve the periodic solution for a nonlinear partial dierential equation (PDE) demonstrated by a nonlinear string equation with a linear complex boundary condition, and stablity analysis is conducted for the periodic solutions using Hill's method. In order to avoid the integration procedure for discretizing the PDE to obtain the ordinary dierential equations (ODEs), spatial and temporal harmonic balance procedures are conducted simultaneously, which can be eciently achieved by the discrete sine transform and the fast Fourier transform. An additional coordinate associated with the generalized coordinates of the trial functions for the spatial discretization is introduced to make the solution satisfy all boundary conditions, and a relationship of the additional coordinate and the generalized coordinates is developed and used in the STHB method so that the test functions can be the same to the trial functions. Jacobian matrix of the harmonic balanced residual is obtained analyti- * the author's current institute is Georgia Institute of Technology, Atlanta, Georgia, 30332 cally, which can be used in Newton method for solving the periodic response. The STHB method and Jacobian matrix make the calculation of the periodic solution for the nonlinear string with a linear spring boundary condition ecient and easy to be implemented by computer programs. The relationship between the Jacobian matrix and the system matrix of the linearized ODEs are developed, so that one can directly obtain the Toeplitz form of the system matrix, and Hill's method can be used to analyze the stability with the eigenvalues of the Toeplitz-form system matrix without the derivation of the ODEs. The frequency curve of the periodic solutions is obtained and their stability is indicated by the method in this work.

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The generalized harmonic balance method for determining the combination resonance in the parametric dynamic systems

Cited by 55 publications

References 17 publications

Parametric combination resonance instability characteristics of laminated composite curved panels with circular cutout subjected to non-uniform loading with damping

Parametric combination resonance instability characteristics of laminated composite curved panels with circular cutout subjected to non-uniform loading with damping

Stability of multi-degree-of-freedom Duffing oscillators

The Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Responses of a Nonlinear Partial Differential Equation With a Linear Complex Boundary Condition

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