Время науки - The Times of Science

Время науки The Times of Science 6 №4 = ( + ) (4) Taking into account this dependence, and also taking into account the inertia of section rotation, we obtain the differential equation of transverse vibrations of the rod under the action of external forces: 4 4 + 2 2 + 0 5 4 − 4 2 2 + = ( , ) (5) where ( , ) — displacement of the neutral axis point of the rod with abscissa x, — bending stiffness in the vibration plane, – unit volume mass, – cross-sectional area, — moment of inertia of the rod cross section, 0 — coefficient characterising internal attenuation. Consider the case when the external load is given as a pulsating, continuously distributed load applied at points 1 , 2 , . . . , along the length of the load beam 1 ( ) sin + 2 ( ) cos (6) Let us assume that the concentrated masses and forces applied at points ( = 1, 2, … ) are uniformly distributed in the interval from to + according to the law of rectangle with intensity and per unit length. Thus ( , ) = − 2 2 + 1 ( ) sin + 2 ( ) cos . (7) The solution of equation (5) will be found in the form ( , ) = ( ) sin + ( ) cos . (8) Substituting ( , ) into equation (5) and introducing the variable = (9) we obtain for the determination of ( ) and ( ) the following system of linear differential equations (10): { 4 ( ) 4 + 2 2 ( ) 2 − 2 ( ) − ( 4 ( ) 4 + ( )) = 1 ( ), 4 ( ) 4 + 2 2 ( ) 2 − 2 ( ) + ( 4 ( ) 4 + ( )) = 2 ( ), Where for ≤ ≤ +1