Bond graph modeling for modal dynamics of interacting lumped and distributed systems

In many practical engineering systems, it is important to understand the physics of interacting subsystems, some of which lend themselves to lumped representations while others are inherently distributed. The noise and vibration caused by the engine of an automobile is a good example. The engine, the source of the vibrational and acoustical energy, is a lumped system for typical operating speed. However, the engine sits atop the vehicle frame structure, which exhibits modal dynamics. Ultimately, the vibrational energy finds its way to surfaces that radiate acoustic energy to the surroundings. If one were to model this overall system, lumped and distributed dynamic effects would have to be included. Bond graphs are a concise pictorial representation of the energy storage, exchange, and dissipation mechanisms of dynamic engineering systems [R. C. Rosenberg and D.C. Karnopp, Introduction to Physical System Dynamics (McGraw-Hill, New York, 1983)]. Considerable work has been done in developing bond graph methods for the modal dynamics of distributed systems [D. L. Margolis, “Bond graphs, normal modes, and vehicular structures,” Veh. Syst. Dynam. 7 (1) (1978) and D. L. Margolis, “A survey of bond graph modeling for interacting lumped and distributed systems,” J. Frankin Inst. 319 (1/2) (Jan. 1985)]. The virtues of bond graph modeling are many and space is short; however, once a bond graph model has been constructed, then physical state variables are dictated, and the system can be automatically simulated using a digital computer. This applies to nonlinear as well as linear systems except that the distributed aspects of the overall system must be linear. The paper will develop bond graph modeling for lumped and distributed systems, and the procedure will be demonstrated for realistic systems.