SELF

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S.B. Karavashkin, O.N. Karavashkina

4. The solutions change when transforming the model

We can much complicate the basic model using the standard procedures. Using the superposition, we can account the external force affecting few line elements simultaneously. Using the spectral expansion, we can model the elastic line response to the external unharmonic affection. We can complicate the vibration, extending (2)- (4) for the case of inclined force affection. In this case the solution will describe in the implicit form the inclined waves propagating from the external force application point. The main regularities of vibration process will correspond to the considered in the items 3.1–3.4. When passing to a distributed line, the inclined pattern of wave process will remain in full correspondence with results presented in [4].

A heterogeneous infinite elastic line can serve as the basic model also for other types of similar-structure elastic lines. We can yield for them also the exact analytic solutions by way of related transformation of solutions (2)–(4). For example, using the conventional analogy in linear and rotary motion of a solid body, we can yield solutions for a system of elastically constrained disks. Extending additionally the limiting process to a distributed line, we can study the torsion vibrations of heterogeneous shafts, cables and so on. Using the dynamical electromechanical analogy DEMA described in [10], we can yield a full set of solutions for complex electric filters. By a simple modification of modelling system (1), we can account the resistance of constraints, remaining the solutions complete, analytical and exact, etc.

As an example, consider few simple and visual models.

4.1. m₁ = m₂

The transformation of a heterogeneous elastic line into that homogeneous will equalise the parameters characterising both parts of a heterogeneous line. It means that in this case

Noting (24), the basic system (2)- (4) will take the following form:

for i k 

for k i n

and for i n + 1

Solutions (25)–(28) fully coincide with the results presented in [1] where the similar solutions were obtained by direct use of the presented method to yield exact analytical solutions. This coincidence of results serves to a definite extent as the check of, whether the solutions given here are correct.

4.2.  m₂ = 0

In this case we can consider the elastic line as a semi-infinite line with a free end. With it the equalities

will be true. With regard to (28), the solutions (2)–(4) will take the following form:

for i k

for k i n

and for i n + 1

At i = n , (31) will coincide with (30) and will not depend on the index i . Thus, we can ignore it in the general system of solutions (29)–(30). The rest two expressions for the first and second sections fully coincide with the related results yielded in [3]. If we go on with this transformation and take additionally k = n , we will yield the solutions for a homogeneous semi-finite elastic line under external force affecting its free end:

This result coincides with the solution presented in [1].