By Peter Proksch
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The former are the scalars; the latter, vectors. Mathematically, we define them according to the way their components change under admissible transformations. In the following discussions we consider a system whose components are defined in the general set of variables Bi and are functions of B1, B2, B3. If the variables Bi can be changed to ei by an admissible and proper transformation, then we can define new components of the system in the new variables @. The systcm will be given various names according to the way in which the new and the old components are related.
Hence the shock spectra are principally functions of wt,. See some references in the bibliography at the end of the book. The picture shown in Fig. 5:1(b) may represent, in a very crude way, a concrete floor of a steel frame building during an earthquake. A system shown in Fig. 5:1(c) may represent an astronaut in flight. A dumbbell shown in Fig. 5:1(d) has been used to model a molecule. It is clear that fairly comprehensive models of dynamic systems can be devised by this approach, see, for example, Fig.
The components of a vector in the two coordinate systems are related by %(51,32,33) =,&jvj(xliX2,x3). The components of a tensor of rank two are related by cij(3,32,z3) =Pik@js~ks(Xl,x2,~3). When only rectangular Cartesian coordinates are considered, we shall write all indices as subscripts. This convenient practice will be followed throughout this book. 9. CONTRACTION We shall now consider some operations on tensors that generate new tensors. Let Aikl be a mixed tensor so that, in a transformation from the coordinates xa to za(a = 1 , 2 , .