ESDU Engineer

Mobilities and Impedances

ESDU 04009 Mobilities and impedances of structures Part I: Compendium of frequency response functions

ESDU 04010 Mobilities and impedances of structures Part II: Compendium of point mobilities of infinite structures

These Items have been prepared with the aim of assisting users with the analysis of structures that are built up from basic elements. Further applications are concerned with Statistical Energy Analysis (SEA). In the first part of Item No. 04009 the background to mobility and impedance is considered, as it applies to discrete and continuous systems. The rules for obtaining the mobilities and impedances of connected structures are explained. The basic elements may be combined to form single and two degree of freedom systems. Thus a building block approach to structures may be developed. The main objective of Item No. 04009 is to provide a visual record of the mobilities of structures, particularly continuous structures such as beams, plates and cylinders. Information on both point and transfer mobilities is given as graphs of the modulus of mobility against frequency and as loci in the complex plane. Further data are provided on basic elements, such as springs, dampers and rigid bodies. The visual information on both discrete and continuous systems is provided in tables at the end of the Item. Beams and plates with single degree of freedom and other systems attached at a point are considered. For all these systems graphical records of their mobilities are provided. With an emphasis on the building block approach to the mobility of built-up structures, beams and plates with different types of dampers and springs can be easily analysed from the data provided. The Item contains a section on vibration isolation and shows how the transmission ratios can be derived from the transfer mobilities. The building block approach is used to account for the effect of foundations on systems designed to provide vibration isolation.

The graphical information allows the detail of the mobilities to be appreciated. For example, in the case of a point mobility there is an alternating sequence of resonances and antiresonances that is shown in a graph of modulus of mobility as a function of frequency. Transfer mobilities have a different sequence. Similarly, in the complex plane the point mobility may be approximated by circles with positive real values, whereas the approximate circles for transfer mobilities can have real values which are both positive and negative. Such differences are useful in assessing and checking the validity of measurements of mobility on a real structure.

Graphs of the modulus of mobility have been estimated for beams and plates with hysteretic damping and include a large number of modes of vibration. The effects of transverse shear and rotary inertia are neglected. By studying the mobilities at high frequencies it can be seen that the structures behave as though they were infinite in extent, a fact which is of particular interest in SEA. Mobilities of infinite structures are the subject of Part II. Part II, Item No. 04010, is concerned with point mobilities of infinite structures, such as beams, plates and cylinders. Point mobilities of infinite structures are directly related to the modal density of equivalent finite structures. As a structure at high frequencies may approximate to an infinite structure, the Item is applicable to SEA. The input power may be obtained from the real part of the point mobility of the infinite structure. The input power is required in SEA estimation. The information on point mobilities is mostly provided in the form of tables that list the mobilities and other data, such as the velocity field, the wave number and modal density. Point mobilities are considered for structures which include: rods with one-dimensional longitudinal and torsional wave motion; beams with normal force and moment excitation; thin isotropic and orthotropic plates with normal force excitation; rings and cylinders. In the case of a plate the force or moment has to be applied through a small, rigid disc, otherwise the stresses at the point of application are infinite. The point mobility for a plate depends on the radius of the disc. Some systems involve two types of wave motion, for example, bending and shear waves occur in thick plates. Other structures involving two wave systems, all included in the tables, are plates with in-plane excitation, plates stiffened by a beam and a ring-stiffened cylinder.