Pitch glide effect induced by a nonlinear string–barrier interaction

Nonlinear Dynamics of a Taut String with Material Nonlinearities

Journal of Vibration and Acoustics ◽

10.1115/1.1325411 ◽

2000 ◽

Vol 123 (1) ◽

pp. 53-60 ◽

Cited By ~ 5

Author(s):

M. J. Leamy ◽

O. Gottlieb

Keyword(s):

Nonlinear Dynamics ◽

Dynamic Response ◽

Natural Frequency ◽

Continuum Mechanics ◽

Finite Deformation ◽

Multiple Scales ◽

String Model ◽

Taut String ◽

Linear Material ◽

Nonlinear String

A spatial string model incorporating a nonlinear (and nonconservative) material law is proposed using finite deformation continuum mechanics. The resulting model is shown to reduce to the classical nonlinear string when a linear material law is used. The influence of material nonlinearities on the string’s dynamic response to excitation near a transverse natural frequency is shown to be small due to their appearance at high orders only. Material nonlinearities appear at low order in the equations for excitation near a longitudinal natural frequency, and a solution for this case is developed by applying a second order multiple scales method directly to the partial differential equations. The material nonlinearities are found to influence both the degree of nonlinearity in the response and its softening or hardening nature.

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

Journal of Vibration and Acoustics ◽

10.1115/1.4045775 ◽

2020 ◽

Vol 142 (3) ◽

Author(s):

Xuefeng Wang ◽

Weidong Zhu

Keyword(s):

Boundary Condition ◽

Periodic Solutions ◽

Harmonic Balance ◽

Jacobian Matrix ◽

Additional Function ◽

Complex Boundary Condition ◽

Complex Boundary ◽

Hill’S Method ◽

Sine Functions ◽

Nonlinear String

Abstract A spatial and temporal harmonic balance (STHB) method is demonstrated in this work by solving periodic solutions of a nonlinear string equation with a linear complex boundary condition, and stability analysis of the solutions is conducted by using the Hill’s method. In the STHB method, sine functions are used as basis functions in the space coordinate of the solutions, so that the spatial harmonic balance procedure can be implemented by the fast discrete sine transform. A trial function of a solution is formed by truncated sine functions and an additional function to satisfy the boundary conditions. In order to use sine functions as test functions, the method derives a relationship between the additional coordinate associated with the additional function and generalized coordinates associated with the sine functions. An analytical method to derive the Jacobian matrix of the harmonic balanced residual is also developed, and the matrix can be used in the Newton method to solve periodic solutions. The STHB procedures and analytical derivation of the Jacobian matrix make solutions of the nonlinear string equation with the linear spring boundary condition efficient and easy to be implemented by computer programs. The relationship between the Jacobian matrix and the system matrix of linearized ordinary differential equations (ODEs) that are associated with the governing partial differential equation is also developed, so that one can directly use the Hill’s method to analyze the stability of the periodic solutions without deriving the linearized ODEs. The frequency-response curve of the periodic solutions is obtained and their stability is examined.

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