Introduction
In our case, the displacement field of two interacting distributed parameter non-linear mechanical systems is considered, when their interaction is vibration-shock in nature, and there is a pre-interval Δ > 0 between them (Δ is the interval between the displacements of non-linear mechanical systems), and the interaction angle changes in the interval 0 < α < 900, and the vibration shock process has a periodic character. In this case, the interaction of the mentioned systems is described by the function
where T is the interaction period; x1 and x2 - are the coordinates of the longitudinal movement of the system, and either
(where
and
are the corresponding geometric dimensions of the system). Obviously, the vibration displacement of both systems has a random character and is completely described by the function
during vibration shock processes, the equations of motion of the interacting systems in our case will have the form (j=1, 2)
where δ is the Dirac functionaries; δ1,j - Corner symbol; aj - propagation speed of elastic waves in systems; bj - viscous resistance coefficient of vibration shock systems [1];
mj - mass of interacting systems;
- disturbing power; x22 - the distance from the point of concern to the anti-vibration means;
- relative coordinate;
- rate of damping (harmful energy absorption) in the first system;
- power function transferred to the second system. These functions characterize the vibration shock force between the systems. In this case, the initial and boundary conditions for finding the displacement field of interacting systems are zero. The action of gravity is not taken into account in the equations of motion. Vibration shock occurs during the rotation of the first system about its 0Z axis, with angular velocity ω, the first system is a completely rigid body [2]. The mechanical model of interacting systems is presented in the first figure, where the velocity vector field is determined by the following relationship Figure 1.
Figure 1: Model of two interacting mechanical systems.
- the rotor of the speed vector field characterizes the system winding process, which in our case is determined by the following formula
as we can see, in this case, rot
is a constant vector and is directed in the direction of the OZ axis, and its modulus is equal to double the angular velocity of rotation
.
From the first equation, let's move on to integration-differential equations of interacting systems, where
and
and are non-linear functions, and we are looking for a solution considering the periodicity of T. Periodic modes that satisfy the first equation will also satisfy the following equations
where the periodic Green's function for nonlinear mechanical systems with distributed parameters has the form
The last function characterizes the dynamic operation of the given system with superimposed frequency kω from point yj to point xj.
The function
in this case is defined by the following equation
It can be seen from equation (2) that the displacements defined in one period of vibration shock during interaction take the form of a parametric representation
where the momentum of the force and the phase of the oscillations satisfy the conditions
, U is the relative coordinate; R[0,1] - system state recovery coefficient. Accordingly, we will have
- which is the sum of the loading and unloading stresses
- is the reduced mass [3].
The goal of our research is to determine
and
, and from equation (4) determine the field of corresponding displacements
and
.
If we look for a general solution in the form of an expansion of the own forms of free oscillations, then we can present the periodic Green's function in the following form,
where
are the roots of the following equation
the
are frequencies of the given system's own oscillations, in addition
and
µ is the coefficient of absorption of harmful energies, ωj - are the frequencies of forced oscillations of vibration shock processes,
when
then
.
Methods
In this case, an elastic-damping ring is included in both interacting systems, which is characterized by parabolic type hysteresis absorption ability, so all members of the equation (5) contain a multiplier with exponential suppression of oscillations, therefore, over time, the values of the second terms in the equation (4) approach zero, and the impulsive forced oscillations are defined only with the first members. To find the general solution U0j (xj,t), in equation (3) it is allowed that U0j = 0, as a result of which we get
where the hysteresis losses of harmful energies are described by the equation [4],
K2 - is the average dynamic stiffness of the anti-vibration agent; λ - is the rate of absorption of harmful energies of forced vibration shock oscillations [5]. Accordingly, from equation (5), we have
If we insert the equations (7) and (8) into the equation (6) and integrate, we get
where
is a phase shift [6],
arg A1 is the principal value of the A1 amplitude argument, and the modulus is defined by the equations [7]
The analysis of equations (4), (5), and (9) shows that even in the case of increased hysteresis losses of harmful energies in interacting vibration shock systems [8], it is impossible to completely exclude the impulsive loads acting on the systems and the corresponding critical and resonance events, which are accompanied by periodic vibration shock processes. Obviously, the mentioned events are more pronounced when the self-oscillation frequency of one or both systems momentarily approaches the frequency of forced vibration shock processes. In addition, critical moments are fixed during the phase shifts of forced oscillations of oscillatory systems, in this case, the frequencies of forced oscillations approach mutually opposing phase moments. It can be seen from equation (9) that by increasing the damping capacity, the amplitudes of both the current and resonant modes decrease, accordingly, the impulse loads acting on the systems decrease. It is not excluded that sub-harmonic resonance modes may also develop in systems during forced vibration shock processes. In this case, the work of dissipative forces in one period of vibration shock is determined by the display
and the work of the disturbing force takes into account the orthogonal of the harmonic function
where
, (ϕ2 - is the new phase shift indicator). The equation for balancing the work of dissipative and disturbing forces will take the form in this case
It can be seen from equation (12) that by selecting the optimal parameters of hysteresis losses,
it is possible to almost exclude sub-harmonic modes superimposed on the main resonant modes in vibration shock processes. The optimal values of impulses of forced oscillations, their duration and phases, which ensure minimum vibration loads of vibration shock systems, are determined by the following formulas
where
- is the critical value of displacement when
, then
, i.e.
will take its maximum value when U01 = 0, then
and under the conditions mentioned above it will take its maximum value.
Conclusion
From the analysis of the inequalities (13) it can be seen that, in the case of parabolic type hysteresis losses, the value of µ changes automatically in relation to impulsive loads, which will allow us to transfer vibration processes to automatic modes and, accordingly, practically safe operation of the mentioned systems. The obtained results are of particular importance in heavy machinery construction, and their consideration and implementation will increase the safe and long-lasting operation of manufactured products.