Research Of The Orthogonalities Of Modes And Nonlinear Vibration In Cable-beam Models

Cable-beam model is a general mechanical model used to study the dynamics of complex slender structure system in the fields of aerospace,civil engineering,transportation and water conservancy.Its resonance problem is related to the safety of mast,antenna,bridge,etc.,so the cable beam coupling vibration problem has been a research hotspot in many fields.Modal analysis is an important foundation of vibration analysis,and orthogonality is an important feature of modals,especially the problem of vibration equation decoupling.In view of the lack of research on the modal orthogonality of cable-beam model and direct using it too much,this paper systematically analyzes the problem through theoretical derivation,numerical calculation and model verification.Firstly,the advantages and disadvantages and applicability of different setting methods of modes are compared.Then the expression of the orthogonality of the equivalent stiffness of the cable beam composite structure and the test formula of the orthogonality without dimension are obtained.The modes are expressed in vector form,and three modes of cable-beam separation method,drag motion method and cable-beam integral method are studied.The study found that when the simple trigonometric function modes are selected for the cable-beam separation method,each mode of the structure satisfies the orthogonal relationship of mass and stiffness.Whether the modes in the dragging motion method meet orthogonality or not are different in different situations,so it needs to be analyzed in detail.In particular,it is related to whether the ratio between the position of the cable-beam connection and the beam length is a rational number.The modals in the cable-beam integral method do not satisfy orthogonality when considering the cable sag,and do not satisfy orthogonality when the cable sag is not considered.At the same time,it is found that the absolute value of the inner product of the two modes must be less than half of the sum of their modal masses,and the dimensionless inner product occasionally has a large value,which indicates that the orthogonality between the two modes is very poor.In addition,the kinetic energy of most modal groups is concentrated on the beam,but occasionally it is concentrated on the cable or tends to be bisected.Next,the Winkler beam model and simply supported beam model are established.The subgrade coefficient k of Winkler beam is derived.Modal analysis,dual-mode discrete analysis and multi-scale method analysis are carried out.It is proved that the coupling term caused by nonorthogonal modes has a great influence on the time history curve,phase diagram,resonance frequency ratio and phase difference in the results of nonlinear dynamic analysis.If the nonorthogonality of modes is not considered,some internal resonance components will be lost.At the same time,some new phenomena such as "frequency drift" of cable-stayed bridge equivalent to Winkler foundation beam are also found when considering the influence of other cables.Finally,the conditions of multi frequency resonance are studied,and the cases of all rational numbers and Including irrational numbers are classified and discussed.The rigorous conclusions are obtained.The general methods for solving such problems are discussed and proved completely.A programmed analysis process is established: first,a group of numbers with certain properties is found,and each frequency is expressed as a rational linear combination of these numbers Then,by solving the rank of coefficient matrix and judging whether the solution of the equivalent homogeneous linear equations has constant zero component,the properties of multi frequency resonance is analyzed.Finally,according to the different results,we can accurately judge whether the selected frequency is "no multi frequency resonance","there is a kind of multi frequency resonance" or "there are infinite kinds of multi frequency resonance".In this paper,a rigorous conclusion is given for the conditions for generating multifrequency resonance,the judgment of modal orthogonality in the three methods and the influence of nonorthogonality in cable beam dynamics.It is of great scientific significance for the selection of modes,the properties of modal functions,solving of vibration equations and research on multifrequency resonance in the future research work.