V.2 No 1

75

Some features of the forced vibrations modelling

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

In accordance with this first, the solution for the system of a type

is sought 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 yielded from the characteristic equation by way of substituting the elements of kth column 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 free vibrations calculation. For elastic systems with a large number of elements, we have first to 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 show analytically the pattern of the vibration process dependence on the affecting force frequency and the system parameters.

The second matrix method bases on the transformation of the initial generalised coordinates of a system to the normal coordinates:

where _a are the normal coordinates. In these coordinates, the generalised force has a form

.

“Using the expression for the kinematic and potential energy in the normal coordinates, find

[8, p.539].

The equation (25) is similar to a modelling differential equation for a single body with ideal elastic constraints. “We can integrate (25) using the symbolic method of integration. If P(t)   was the integrable function of time t₁ , we can present the integral of (25) as

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] considered the problem of forced vibrations of a shaft with n fitted gears, as we mentioned it above. In the case “when at the points for which the values of variable x 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 shaft end x = 0  is supported, will be

When this end was embedded,

and when it was 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)  in all these cases is 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].