Dynamic chain system with seismic damper: reduction of nonlinear model

Abstract

The efficiency of model reduction via balancing for a multimass system with a nonlinear inertial link is demonstrated. As an example, a three-mass dynamic system with a seismic damper is examined. The constraints of the seismic damper are assumed to be ideal. The order of the system of differential equations is reduced by one. It is pointed out that the difference between the numerically analyzed dynamic processes in the original and reduced systems is minor, which makes it possible to use the method for justified simplification of nonlinear systems Introduction. The destructive effects of earthquakes on various structures motivate the development of various systems such as seismic dampers to weaken such effects [4,17]. To decrease the dynamic load due to seismic accelerations, theoretical studies are carried out and methods for the analysis and synthesis of such systems with various nonlinearities are developed [12,13,20,22,24].The mechanical vibrations of buildings caused by earthquakes, mud streams, etc. can sometimes be described by systems of ordinary differential equations of high order [3,10]. Such systems may appear stiff, which would make it difficult to integrate them. This is why it was pointed out in [6,7] that the reduction procedure reducing the stiffness of the system of differential equations of motion may improve, rather than impair, the accuracy of the results. In [5], it was suggested that the major forces in the constraints of multimass systems can be identified by analyzing a simplified reduced system with a limited number of degrees of freedom.Thus, in simplifying models of elastic inertial systems of heavy machines and assemblies (rolling mills, excavators, rolling stocks, etc.), buildings and structures, it is especially important to maintain the lower natural frequencies of the spectrum, which are responsible for the major portion of the dynamic load [19,21].The modern understanding of the general principles associated with the optimization of high-dimension systems is discussed in [9], where it is pointed out that the neighborhood of the maximum in which the functional greatly exceeds the average value is rather small. Therefore, the errors of the initial data may nullify all efforts taken to find the exact solution. For example, specifying hundred parameters rounded to two decimal places may change the value of the functional severalfold.In [5], Kozhevnikov mentions then (second half of the 20th century) popular three methods to simplify multimass systems with different accuracy of approximation of the initial frequency spectrum. He believes that Baranov's method [16] is the best (most clear). This method determines the nodes of the highest vibration mode that are within each of the elastic constraints of the original system. The system is reduced to two-mass synchronous systems by dividing intermediate masses in appropriate ratios. Next, the masses of the synchronous systems are placed at the nodes to reduce the order of the differential equations of the original system by two (reduce...