Quasi-periodic motion is an oscillation of a dynamic system characterized by m frequencies that are incommensurable with one another. In this work, new incremental harmonic balance (IHB) method only two time scales, where the frequencies, referred to as fundamental frequency, and other interval distance adjacent proposed for quasi-periodic motions axially moving beam three-to-one internal resonance under single-tone external excitation. It found frequency every located around or its integer...

Abstract Quasi-periodic motions can be numerically found in piecewise-linear systems, however, their characteristics have not been well understood. To illustrate this, an incremental harmonic balance (IHB) method with two timescales is extended this work to analyze quasi-periodic of a non-smooth dynamic system, i.e., gear transmission system piecewise linearities stiffness. The simplified four degree-of-freedom nonlinear model by using lumped mass method. Nonlinear governing equations the...

Considering the elasticity of gear solid bodies, load applied to teeth will force theoretically separated get into engaging state in advance. This phenomenon is named as extended tooth contact (ETC). Effects ETC directly influence time-varying mesh stiffness pairs and subsequently alter nonlinear dynamic characteristics transmission systems. Time-vary stiffness, considering effects ETC, thus introduced model system. Periodic motions a system are discussed detail this work. The analytical...

Nonlinear dynamic responses of an Euler–Bernoulli beam attached to a rotating rigid hub with constant angular velocity under the gravity load are investigated. The slope angle centroid line is used describe its motion, and nonlinear integro-partial differential equation that governs motion hub-beam system derived using Hamilton's principle. Spatially discretized governing equations Lagrange's based on expressions kinetic potential energies system, yielding set second-order ordinary combined...

A modified two-timescale incremental harmonic balance (IHB) method is introduced to obtain quasi-periodic responses of nonlinear dynamic systems with combinations two incommensurate base frequencies. Truncated Fourier coefficients residual vectors algebraic equations are obtained by a frequency mapping-fast transform procedure, and complex two-dimensional (2D) integration avoided. Jacobian matrices approximated Broyden's resulting solved. These modifications lead significant reduction...

Various bifurcation phenomena in a nonlinear curved beam subjected to base harmonic excitation, which is governed by coupled equation with both quadratic and cubic nonlinearities, are investigated using the incremental balance (IHB) method. The partial differential that governs motion of given Hamilton’s principle. A spatially discretized governing derived Galerkin’s method, yielding set second-order ordinary different equations. high-dimensional model can take coupling into account derived....

Abstract Periodic and period-doubling vibrations in a gear transmission system subjected to forced excitations with multipiecewise linear functions are investigated this work by using the incremental harmonic balance (IHB) method. The nonlinear ordinary differential equations that govern vibration of formulated employing Newton's second law. Analytical results reveal abundant interesting phenomena, including jumps, bifurcations, softening-spring behaviors, primary, super-harmonic,...

This article analyzes nonlinear vibration of an axially moving beam subject to periodic lateral forces. The governing equation the is developed using Hamilton's Principle. stable and unstable solutions are obtained by employing multivariable Floquet theory incremental harmonic balance (IHB) method. In solution procedure, Hsu's method applied for computing transition matrix at end one period. effects internal resonance on responses discussed. from IHB in good agreement with results numerical...