Calculation And Identification Of Structures With Localized Nonlinearities

Nonlinearity exists in a wide range of engineering structures and plays a key role in complicated structural behaviors such as jumps,multi-value responses,amplitude and/or frequency dependent responses,etc.Different from the well-established modal testing schemes or updating procedures for linear structures,the analysis and identification process of nonlinear structures do not have a universal solution and thus has drawn considerable attention during the past decade.First of all,we consider the reduction of nonlinear dynamic equations for structures with localized nonlinearities and a new method based on the relative coordinates is proposed.To introduce the relative coordinates into the reduction algorithm,the dynamic equations are first converted to a set of nonlinear complex equations by the Harmonic Balance Method and then reduced to the relative coordinates by the way of Linear Modal Superposition.The reduced equations are solved iteratively to obtain the responses at each frequency.Three numerical examples,i.e.,cantilever beam with a local nonlinear element,lumped mass model with a nonlinear isolator and beam-spring structure with six nonlinear elements,are presented to validate the proposed method.Results show that the proposed method can significantly improve the computation efficiency for structures with localized nonlinearities.In addition,it was also applied to study the effects of flexible mounting structures on a piecewise linear isolator.Secondly,a simple and general Equivalent Dynamic Stiffness Mapping Technique is proposed for identifying the mathematical model and the parameters of a nonlinear structural element with steady-state primary harmonic frequency response functions(FRFs).The Equivalent Dynamic Stiffness is defined as the complex ratio between the internal force and the displacement response of the unknown element.Obtained with the test data of responses' frequencies and amplitudes,the real and imaginary part of Equivalent Dynamic Stiffness are plotted as discrete points in a three dimensional space over the displacement amplitude and the frequency,which are called the real and the imaginary Equivalent Dynamic Stiffness map,respectively.These points will form a repeatable surface as the Equivalent Dynamic stiffness is only a function of the corresponding data as derived in the paper.The mathematical model of the unknown element can thus be obtained by surface-fitting these points with special functions selected by priori knowledge of the nonlinear type or with ordinary polynomials if the type of nonlinearity is not pre-known.An important merit of this technique is its capability of dealing with strong nonlinearities owning complicated frequency domain behaviors,such as jumps and breaks in resonance curves.Besides,there is no need to pre-identify the underlying linear parameters,this method uses the measured data of excitation forces and responses without requiring a strict control of the excitation force during the test,which can greatly simplify the test procedure.The proposed technique is demonstrated and validated with four classical single-degree-of-freedom(SDOF)numerical examples and one experimental example.Finally,we study the location process of nonlinear elements and intend to remove the restriction that requires full measurements of the dofs associated with nonlinearities(or its equivalent form,restrict the possible location of nonlinearities in the measured region)in location methods based on FRF data.Specifically,we introduce the Craig-Bampton reduction to project the unmeasured nonlinear forces to the measured region,and employ the accumulated reduced nonlinear forces(with clear physical meanings)as an index to locate nonlinearities.In order to improve the robustness of the whole procedure,a threshold is used to filter the modelling error and measurement noise.Apart from that,the inverse of transfer function matrix is also avoided here by a two-step process.The location procedure is demonstrated by a numerical example with twenty degree of freedom system including two nonlinearities(one cubic stiffness nonlinearity and one piecewise linear nonlinearity)and an experimental study of a clamped beam with a nonlinear spring mechanism.