Influences of the stiffness and mass of coupler on mode localization of simply supported continuous multispan beams with couplers on supports are studied theoretically and the results are confirmed by examples. Measures for degree of mode localization, for sensitivity to mode localization, and for coupling strength are defined and their usefulnesses are verified. Through this study, it is shown that the mass of coupler makes the structure sensitive to mode localization especially in higher modes while the stiffness does in all modes. A new type of delocalization phenomenon is observed for the first time in some modes for which mode localization does not occur or is very weak although structural disturbances are severe.
Many engineering structures are made of identical substructures connected by couplers, shaping into periodic structures. However, in real structures, no substructures will be perfectly identical since no one can be free from manufacturing errors and damages. For some periodic structures, the presence of small irregularities breaking the assumption of structural identity may significantly affect mode shapes of the structures and may be a cause of mode localization phenomenon. Mode localization phenomenon is that under conditions of weak internal coupling, the mode shapes undergo dramatic changes to become strongly localized when small disorder is introduced into periodic structures. The specific part of a localized mode has significantly larger deflection relative to the rest of it. This means that the vibration energy of the mode is confined to that region, and so the structure may suffer unpredicted local damages and the performance of a motion controller of the structure may decrease abruptly. For that reason, it is important to know the characteristics of mode localization of the structure.
As a simple model of structures suffering mode localization, a spring-mass system consisting of two substructures and a coupler connecting them is considered in the start of analysis. It is qualitatively shown that the stiffness and mass of coupler make the system sensitive to mode localization. The influence of mass is more pronounce when the natural frequency of the system is high while the influence of stiffness does not depend on the natural frequency. Under the condition that the natural frequency of the system is close to that of the coupler, corresponding mode does not sensitive to mode localization although the stiffness and/or mass of coupler are large.
To study characteristics of mode localization of multispan beams and influences of the stiffness and mass of coupler on it, transfer matrix equation and boundary contitions are driven and approximated. Using the general form of characteristic equation driven from the approximated transfer matrix and boundary conditions, the influences of the stiffness and mass of coupler and span length variation on eigenvalues of the system are studied. The span length variation shifts the eigenvalues periodically and the periods are functions of the amount of variation and the initial span length. The sensitivity of eigenvalue to span length variation is proportional to the eigenvalue.
Using maximum amplitudes of vibration of substructures, a measure for degree of mode localization is defined. The proposed measure for degree of mode localization is very useful to describe degrees of mode localization in practical structures such as general multispan beams although it is hard to use in analytical study.
The sensitivity to mode localization of the multispan beam is defined as derivatives of the modified span response ratio with respect to the span length variation. The span response ratio is a classical measure for degree of mode localization and given by relationship between the vibration amplitudes of adjacent two spans. The sensitivity to mode localization is proportional to the eigenvalue of the system and to the stiffness of coupler. The mass of coupler plays important role in the sensitivity to mode localization. The sensitivity is also proportional to the mass multiplied by the fourth power of eigenvalue. However the sign of the term of mass is negative while that of the stiffness positive, so a certain combination of the stiffness, the mass, and the eigenvalue makes the sensitivity zero resulting in delocalization. For delocalized mode, mode localization does not occur or is very weak although the stiffness and/or the mass of coupler are large and structural disturbances are severe.
Mode localization phenomenon can be explained using the wave propagation constant since the normal modes are also waves forming standing waves. According to the attenuation and phase change of wave propagation constant, frequency domain is divided into stop-bands and pass-bands. The influences of the stiffness and the mass of coupler and the span length variation on the wave propagation constant of the multispan beam are studied. Shortening the span length caused to increase of widths of the bands and lengthening caused to decrease them. By the span length variation, some eigenfrequencies on pass-bands initially of the system are moved to stop-bands and corresponding modes are become localized. If the attenuation of the stop-band to where the eigenfrequency is moved is large, corresponding mode is significantly localized one. And narrow pass-band implies that the eigenfrequency on it easily moves to stop-bands. The coupling strength defined in this study gives the insight into the influences of the stiffness and mass of coupler on wave propagation constant for given frequency. It is shown, in this study, that the large stiffness of coupler results in the large attenuation of stop-bands and narrow pass-bands in lower frequency band. However the mass of coupler caused to increase the attenuation in higher frequency band and decrease the widths of pass-bands, and the influence is increased with square of frequency. From the results, one can say that the weak coupling, precondition for drastic occurrence of mode localization, is caused by the large stiffness of coupler in lower frequency band and by the mass of it in higher frequency band. On a certain frequency, the coupling strength has maximum and the modes near the frequency are delocalized ones.
As an example structure of general multispan beams, a simply supported continuous three-span beams with couplers having a rotational stiffness and a rotational mass are considered in the last chapter of analysis although in each section the two-span beam examples are included already. The analysis of mode localization of three-span beam gives the same results with those of two-span beam.