250 P. Pingle and P. Avitabile 26.2 Vibrations Typically, a Vibration course can be found in most ME programs. In order to provide undergraduate students with a diverse, broad background in the major tools needed to solve typical engineering problems involving vibrations, the traditional course was modified to include an assortment of tools that might be necessary to solve a wide assortment of typical problems encountered. The foundation for this Vibrations ME course # 22.550 is laid in undergraduate Dynamic Systems ME course # 22.451. Basic concepts related to modeling of various systems, typically mechanical mass-spring-dashpot (MCK) systems are covered in the Dynamic Systems course. The basic core material related to single degree of freedom (SDOF), free and forced response, as well as sinusoidal excitations and arbitrary inputs that are presented in Vibrations 22.550, are continued in the same thought process as Dynamic Systems course. This type of teaching methodology is seen to be much more effective because students are already in sync with the course philosophy and the thought process required to master the Vibrations concepts. For instance the students already have knowledge of development of governing equations of motion of a typical SDOF/MDOF system, putting them in matrix form and solving them using various methodologies such as statespace technique through the course Dynamic Systems. Teaching advanced Vibrations concepts is hence possible in 22.550 Vibrations. In Vibrations, typical base excitation, isolation and force transmission are presented. The traditional multiple degree of freedom (MDOF) systems presented for two DOF is included along with modal space representations, tuned absorbers and the related material. The course then extends into very simple finite element modeling techniques with simple beam and lumped mass representations to extend the SDOF and MDOF to be able to handle more realistic modeling situations. The introduction to experimental modal analysis is the logical extension of this material to complement the finite element modeling approaches. A brief introduction into random vibrations is included to help the students be able to understand how to apply the basic SDOF and MDOF theory as a building block to understand more complicated loading scenarios. Beyond this, the introduction of base excitation of MDOF systems leads directly into structural modeling applications including seismic analysis, shock analysis and response spectrum analyses. While the theoretical foundations are presented in class, the importance of all the theory is amplified through the use of projects to solidify many different concepts. Several of the more important projects used in prior years are listed: • Concepts of response for SDOF and MDOF systems are reinforced with projects that utilize MATLAB to determine the forced response of SDOF, MDOF and mode superposition and show the interrelationship of the different approaches. Also, where possible, Simulink models are utilized to form solutions to these problems along with closed form solution if available. In addition, a LabVIEW GUI allows for the easy identification of different free and forced response characteristics for a SDOF system which is used as part of class projects to help reinforce theoretical concepts. • Development of MDOF equations by defining a project to build a seismic anchor to simulate a built-in condition is an “eye-opening assignment”. This forces the students to obtain a better grasp of the physical parameters needed to actually simulate such a severe constraint that typically is so easy to write analytically but extremely difficult to achieve in a real test situation. • A finite element modeling project is assigned and each student is given a slightly different configuration that can be modeled with simple beams and lumped masses and simplistic support boundary conditions. A general problem is posed for the students to determine how system level response will change due to a rotating sinusoidal excitation when various parameters of the system are changed (such as mass of specific components, (Center of gravity) CG shift of certain components, support stiffness changes). The effects of number of elements, distribution of mass and other items must be addressed. In many cases, the students are asked to design tuned absorbers to de-tune a troublesome mode of the system. In earlier years, the students were given configurations such as vertical pumps, muffler configurations, valve/piping systems, heat exchanges, generator turbine sets, and various other typical industrial applications. All the models are presented in class to provide some meaningful discussion of modeling scenarios deployed and alternate mechanisms to possibly model the system. • An experimental modal test is performed on a variety of simpler structures including application of tuned absorbers to see the effect of mode de-tuning. Structures have included simple beams, plates, and frames but have also included aluminum baseball bats, snowboards, tennis rackets, skis and racket ball rackets (with some equipment coming from wood, aluminum, composites and other constructions to see differences in resulting characteristics). In the evaluation of snowboards, six different commercial configurations were evaluated to observe the change characteristic shapes due to different construction configurations and resulting frequency patterns.
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