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Some features of the forced vibrations modelling

Passing to the analysis of the methods to calculate the forced vibrations in an elastic system, we should mark first of all that the matrix methods can be divided into two basic ones: the direct solution method (see e.g. [7, pp.296-297]) and the normal co-ordinates method (see e.g. [5], [8, pp.539-560]).

In accord to this first, one seeks the solution of a system of a following type:

in the form

''This system of the linear heterogeneous equations has the following solution:

where (_e)  is the characteristic determinant of (20) taken at the value _e, and ^k(_e) is the determinant obtained from the characteristic equation by way of substituting the kth column elements composed of  Q₁^e, Q₂^e, ... , Q_s^e '' [7, p.296].

In (22) the same problems are inherent as in the above method for the free vibrations calculation. For elastic systems having a large number of elements, one has first calculate numerically all _e , and then, again numerically, to calculate the determinants in (22). It means that in the essence this method is not analytical too, because it does not indicate in analytical form the pattern of the vibration process dependence on the acting force frequency and the system parameters.

The second matrix method bases on the modification of the initial generalised co-ordinates of a system to the normal co-ordinates:

where _a   are the normal co-ordinates). In these co-ordinates, the generalised force has a form

.

''Using the expression for the kinematic and potential energy in the normal co-ordinates, find

[8, p.539].

The equation (25) is similar to a modelling differential equation for a single body having ideal elastic constraints. ''We can integrate (25) using the symbolic method of integration. If P(t) was the integrable function with respect to time t₁, we can present the integral of (25) in the following form:

,

where C_1a and C_2a are the integration constants, _a is the free vibration frequency and t₀ is the initial moment of time. The term containing P_a(t) is a particular solution of (25)'' [8, p.539].

Using this technique, A.N. Krylov [6, pp.161-173] has considered the problem of forced vibrations of a shaft with n fitted gears. In the case ''when at the points for which the variable x values being a₁,a₂ , a₃, ..., a_n  there have been applied the forces Q₁, Q₂, Q₃, ..., Q_n and the pairs whose moments are M₁, M₂, M₃, ..., M_n'' [6, p.166], ''the elastic line equation, when the end of the shaft x=0 is supported, will be

But when this end is embedded,

and when it is free,

(where  S(x) = (1/2)(cosh x + cos x); T(x) = (1/2)(sinh x + sin x); U(x) = (1/2)(cosh x - cos x); V(x) = (1/2)(sinh x - sin x) are so-called unit matrixes, which Caushy used in many works), and (x)   is in all these cases one and the same function determined as

The constant arbitrary values are determined by the boundary conditions for the shaft end corresponding to the value x=1'' [6, p.166].