Vibration of Plates and Shells with Added Masses
This chapter is devoted to analysis of plates and shells with added elements (discrete additives). In the analysis a complex system will be simplified allowing for an efficient investigation of the influence of added elements’ on the dynamic characteristi
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1.1 Introduction This chapter is devoted to analysis of plates and shells with added elements (discrete additives). In the analysis a complex system will be simplified allowing for an efficient investigation of the influence of added elements’ on the dynamic characteristics of plates’ and shells’. “Added elements” are defined either as concentrated masses [53, 129] or as continuous masses (lying on a small surface) and absolutely stiffly attached to the plates and shells. In addition, they can be lumped oscillators or lumped (continuous) joint elements. One of the first works focused on investigation of a plate with added masses vibration, belongs to Gershgorin [82]. The problem has been reduced to the consideration of forced vibration of plate jointly supported with the masses. The equations of plates and masses have been integrated separately (similar approaches have been often applied in many dynamical problems of constructions with added masses). The added element is considered to be “unjoined” during analysis and therefore its’ and plate’s vibrations are analysed separately. The dynamic influence on the plate has been reduced to analyzing harmonic vibrations. Using physical model of “plate (shell) - added mass” one finds many interesting dynamical phenomena. This classical approach has been step by step expanded by including many different properties such as: a) Geometry of plates and shells (circle plates and shells [75, 76], ring plates and shells, trapezoidal plates and shells, triangle plates and shells [200]); b) Real properties and material structure (orthotropy, anisotropy [28], elasticity and damping [33], composits [96], elastic support [220], ribs [124]), and so on. Now a brief review of the research oriented on investigation of different lumped-continuous constructions with an emphasis on the results of free plates and shells vibration with discrete masses in the frame of the linear theory will be given. This review includes only the works that are relevant to this chapter subject rather than the totality of them. It should be pointed out that during analysis of the added masses’ influence on constructions with plates and shells, a few simple and fundamental
J. Awrejcewicz et al., Nonlinear Dynamics of Continuous Elastic Systems © Springer-Verlag Berlin Heidelberg 2004
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1 Vibration of Plates and Shells with Added Masses
methods have been used, leading to analysis of “shell - mass” or “plate mass” models. The applied calculation models can be divided into the following groups. 1. Continuous and absolutely stiff models. It has been assumed that mass joints on plates or shells occupy small surfaces, which is typical for technical situations. It has also been assumed that a contact surface is either square or rectangular (in a case of a shell a curvature is negligible). This model has been used by Christienko [54, 53] during the vibration analysis of shells with discrete added elements and by Palamarˇcuk [179] during the vibration analysis of a system consisting of a rib cylindrical shell and an absolutely stiff
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