Models of dynamics of complex heterostructures under pulsed force effects

. Relevance is due to insufficient knowledge and development of models of dynamics of complex heterostructures. Modern mechatronic systems include hybrid components consisting of complex heterogeneous structures: mechanical, electrical, electronic, etc. Heterostructures function in extreme conditions due to kinematic, dynamic, temperature, vibration and other external factors and are exposed to external influences. To create a reliable system for protecting complex heterostructures, it is necessary to build appropriate mathematical models that allow you to adequately describe the processes occurring in complex heterostructures. The purpose of the work is to study and develop heterostructural models based on the equations of point dynamics, plate and multilayer structures under the action of force and impulse loads in the presence or absence of internal friction in heterostructures. Particular attention is paid to the study of models of heterostructures with dissipation, with many degrees of freedom under the action of impulse loads.


Introduction
Complex hybrid heterostructures are constantly exposed to external influences, among which vibrational and shock influences are most powerful.They subject the system to significant overloads.Overloads of several tens of g can cause irreversible degradation processes in heterostructure materials, leading to the formation of cracks [1], lamination of printed circuit boards, violation of solder contacts, destruction of welds, etc.For example, many mechatronic systems of Western manufacturers are vulnerable in climatic conditions of the Russian Federation, or do not withstand extreme loads.The lack of knowledge of complex hybrid heterostructures and, as a result, the problem of developing models for studying the behavior of heterostructures under the influence of extreme influences makes the work relevant.
A lot of articles by various authors are devoted to improving the reliability of complex heterostructures.To protect against external influences, various methods of protecting heterostructures from shock and vibration overloads [2][3] are used, including in the form of multilayer fills with high dissipative properties and allowing to transfer the energy of shock loads or periodic force into thermal radiation [4].Heterogeneous layers should have not only high dissipative properties, but also have a significant resource for wear resistance, be resistant to thermal influences, and have a decent fatigue strength.

Materials and methods
To develop the protection of complex heterostructures in modern conditions, it is necessary to use mathematical models to conduct computational experiments.When developing technologies for vibration shock protection of heterostructures, fundamental models of heterostructures [5] can be used without taking into account and taking into account internal friction, which allow analyzing the results of effects of impulse loads on technical systems.
In most cases, heterostructures can be represented as rod, plate and multilayer structures.In the simplest case, these are heterostructures with one, and in general with many degrees of freedom, for which it is necessary to compose systems of linear or nonlinear differential equations, the latter of which have no analytical solutions.
To describe the processes occurring under the action of shock loads, a numerical and analytical modeling apparatus [6,7] is used, which involves the use of numerical methods and computer technology.
The purpose of this study is to study and develop heterostructural models of mechatronics under the influence of impulse loads.

Fundamental models of heterostructure dynamics
When a load of mass  moving horizontally at a speed  0 is hit by a spring with mass   = 0, the process of joint movement of the load and the spring is described by the equation According to the energy balance condition [8], we equate the kinetic energy of the moving load  to the potential energy of the compressed spring .We get When a load of mass , moving vertically at a speed , hits a spring with mass   = 0, the spring receives a dynamic deflection  д .
or  д =   χ, where   =   -static deflection of the spring under the action of the applied force.
Internal forces and stresses change in the same ratio on impact σ д = χσ с .
The coefficient of dynamism depends on the rigidity of the system and the kinetic energy of the falling load.In particular, with the instantaneous lowering of the load, the kinetic energy  = 0 and χ = 2. Then When a load of mass , moving vertically at a speed  0 , is hit by a spring with a buffer of mass  1 on it, both masses after impact move at a common speed .
Expression (1) will take the form The figure 1 shows the dependence of the dynamic coefficient of the χ on the initial speed of the  0 load at different mass ratios  =

Model of pulsed effects on heterostructures
The differential equation of the motion of the system under the force () without taking into account internal friction is where  -moving the system;  -its mass;  -stiffness.
Solving the equation under zero initial conditions (0) = 0, (0) = 0 for the time interval 0 ≤  ≤ τ has the form for where  0 -maximum power;  -the moment of time for which the movement is determined; τ -duration of force action;  0 -the circular frequency of free undamped oscillations of the system.
We will substitute expressions for () describing the most often used forms of pulse loadings [9,10] (4) in ( 2) and ( 3), performance of integration and the designation  0  = ,  0  = ,  0  =   turn out formulas for definition of ()/  for four forms of impulses.
Here  is the static movement of the system under the influence of the  0 force.
Maximum values of displacements - д , expressed depending on the ratio )   , where χ -coefficient of dynamism.
To do this, we calculate statically the deflection of   by the formula  0  =   , and we determine the coefficient of dynamism χ from the graphs in the figure 2, for some forms of pulses given by the equations (for 0 ≤  ≤ ): Displacement maxima range from 0.5 to 1.0 Τ = τ  1 ratio.
Тhe coefficient of dynamism χ for some form pulses (4)

Exposure to periodic pulses
When determining the motion of the system under the action of periodic pulses, it is necessary to take into account the internal friction in the system [11].Consideration of frequency-independent internal friction in problems of free oscillations of dissipative systems is realized using the hypothesis of complex rigidity [12]  () + ( + )() = 0, where ()complex system movement; imaginary unit, where γthe coefficient of internal friction associated with the coefficient of loss η = γ/(1 − γ 2 /4).

Substitute
Substituting in (6) the initial  0 = 0, υ 0 =  0  −1 corresponding to the application of the  0 pulse to the stationary system at the moment  = 0, we obtain For a finite number of  + 1 periodic pulses with a  0 period, the solution is built by superimposing functions (7) with different time starts here   -moving after  periods  0 , for time within  0 ≤ ( + 1) 0 from the moment of application of the first pulse,  -is the number of pulse repetitions,  = 0 for a single pulse.
Introducing relative time At small , the expression (8) describes the unsteady fluctuations of the system.
where  0 * -minimum positive value derived from the expression With low dissipation γ ≤ 0,1 The global maximum is set from the analysis of  -values   .
The largest of the    at  = 1.With a large , the fluctuations will be practically steady.Then in case of  → ∞ , at γ ≤ 0,1, .

At
0  =  impulsive resonance occurs.The resonance amplitude can be determined by the formula

Multiple Degrees of Freedom Model
The movement of a system with  degrees of freedom is described by  -differential equations of the form where   -complex mass  movements.Real movements   denote   under initial conditions Expression ( 11) is a decomposition of solutions of the dissipative system of equations ( 9) by the forms of natural oscillations.Imagine   as a decomposition of   = с  φ  , where φ  -coefficients of the forms of natural oscillations determined from ( − 1) equations .

Shock processes in rod systems
The equation of vibrations of a homogeneous rod [13] μ Solution ( 12) in a complex form is represented by moving The real part  =  satisfies the conditions (, 0) = 0, μ(, 0) = − 0 (), where ⋅   0 - − th natural frequency of the rod with internal friction, where  -is the length of the rod, λ  -are the roots of the characteristic equation.Constants   ,   ,   are determined from the conditions of fixing the rod,   is determined from the conditions of normalization of the beam function: The real solution for an instantaneous pulse has the form , where с  = (1/μ  ) ∫  0 ()  ()  0 , and its maximum value is determined by the formula For a pulse with a finite value of the pulse duration, formula ( 14) is applicable depending on the type of pulse load and depends on the coefficient   : -for a short-term pulse -for a short-term pulse of constant intensity along the entire length of the rod с  = (ε  /μ  ) ∫   ()  0 ; -for a concentrated pulse  applied at point  0 The normalized beam functions   ( 0 ) by  for the values  = 1, … 5 and various conditions for fixing the edges can be found in the reference books on the calculation of floors for impulse loads [14].

Shock processes in rectangular plate systems
The equation of natural vibrations of plates taking into account internal friction for an instantaneous pulse has the form The solution of the problem of a rectangular plate on a two-parameter basis is sought in the form of a double row of beam functions and is proposed in [15] (, , ) = ∑     (, ) − γ 2    sin    ∞ =1 ,) where   (, ) -the th form of the plate's own oscillations, equal to where  and  are the dimensions of the plate along the x , ;   ,   ,   ,   ,   are determined from the normalization conditions where ,  -indexes of beam functions   () and   () corresponding to the rods-strips cut from the plate along the ,  axes and having the appropriate fastening conditions.
The correspondence between the number of the th oscillation frequency of the plate and the indices ,  are established according to Table 2. .
The values of λ  4 are tabulated for various conditions of fixing the plates, and can be found in reference books on the calculation of overlaps for pulse loads.
The parameter   depends on the type of pulse and is determined by: -in the general case ,   ( 0 ) =       . (17)

Conclusion
The developed models have a place to find wide application in technology and technology.One of the applications is building beam and rod structures, mast structures.Important is the ability to simulate the behavior of dissipative systems taking into account friction (9).
Models allow you to investigate dynamic processes that occur under the influence of impulse loads.
The theory of building models of heterostructures of complex systems has been developed.Fundamental mathematical models of heterostructures are proposed.Models of heterostructures were built in the presence of exposure in the form of periodic and impulse loads.The results of modeling heterostructures under the influence of dynamic loads of various types are presented.General model of multilayer heterostructures and with multiple degrees of freedom are described.

.
=  1 sin 0  +  2 cos 0 , where  0 = √   -the frequency of natural vibrations of the load attached to the spring, spring stiffness.When  = 0,  = 0 and ̇=  0 are constant The largest value of the force   , compressing the spring (dynamic load) is determined by the expression   =    =  0   0 .

Fig. 1 .
Fig. 1.Dependence of the dynamic coefficient of the χ on the initial speed of the  0

Table 1 .
Some values of  and ε are given in Table1.Some values of  and ε.

Table 2 .
The correspondence between the number of the th oscillation frequency of the plate and the indices ,