Dynamics of Structural-Inhomogeneous Laminate and Shell Mechanical Systems with Point Constraints and Focused Masses. Part 2. Statement of the Problem of Forced Oscillations, Methods of Solution, Computational Algorithm and Numerical Results

A vibrational formulation, a technique, and an algorithm are proposed for assessing the resonance state of a package of rectangular plates and shells having point bonds and concentrated masses with different rheological properties of deformable elements under the influence of harmonic influences. The viscoelastic properties of elements are described using the linear Boltzmann-Volterra theory. An algebraic system of equations with complex coefficients is obtained, which is solved by the Gauss method. Various problems on steady-state forced vibrations for structurally inhomogeneous mechanical systems consisting of a package of plate and shell systems with concentrated masses and shock absorbers installed in it were solved. A number of new mechanical effects have been discovered associated with a decrease in the maximum resonance amplitudes of the mechanical system as a whole. The concept of “global resonance amplitude” is introduced to study the behavior of the resonance amplitudes of a mechanical system. An analysis of the numerical results showed that the interaction of resonant amplitudes is observed only in structurally inhomogeneous systems (in this case, with elastic and viscoelastic elements) and with a noticeable approximation of the natural frequencies. How to cite this paper: Mirziyod, M., Ibrokhimovich, S.I. and Khudoyberdievich, T.M. (2019) Dynamics of Structural-Inhomogeneous Laminate and Shell Mechanical Systems with Point Constraints and Focused Masses. Part 2. Statement of the Problem of Forced Oscillations, Methods of Solution, Computational Algorithm and Numerical Results. Journal of Applied Mathematics and Physics, 7, 2671-2684. https://doi.org/10.4236/jamp.2019.711182 Received: September 28, 2019 Accepted: November 2, 2019 Published: November 5, 2019 Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0). http://creativecommons.org/licenses/by/4.0/ Open Access


Introduction
Mechanical systems, consisting of a package of rectangular plates and shells with point bonds and concentrated masses with various rheological properties of deformable elements under the influence of harmonic influences, are widely used in aircraft manufacturing, mechanical engineering, shipbuilding and other industries and construction. Therefore, great importance is given to the study of forced vibrations of mechanical systems having point bonds and concentrated masses with various rheological properties of deformable elements.
Here are some of these works, for example: In [1] [2], longitudinal and torsional vibrations of a cylindrical shell with concentrated masses at its ends were studied. The work [3] [4] [5] is devoted to the calculation of the forced vibrations of shells of revolution and plates with locally attached bodies. In [6] [7] [8], development trends of the construction industry, aerospace engineering, shipbuilding, chemical and energy industries, and many other branches of modern technology are discussed, which are characterized by the increasing complexity of design decisions in the design of various objects and, in particular, are usually represented by thin-walled spatial structures and on the one hand, they raise the requirements for the reliability of these facilities in operation, and on the other hand, to reduce its weight and material consumption. Also, when designing complex structures, along with traditional metal materials, polymer materials and composites based on them are increasingly used [9] [10] [11]. Modern engineering structures made of composite materials consist of a set of elastic and viscoelastic damping elements with various rheological properties, various types of supports and forces, elastic and viscoelastic constraints with significantly different rheology. Such mechanical systems were classified in works [12] [13] as structurally heterogeneous and the task of developing reliable methods and their analysis has not yet lost its relevance.
The request for practice requires the creation of more accurate methods for solving such problems, and engineering and research organizations and higher educational institutions should focus on computer technologies widely implemented in design. These methods and the algorithms based on them should take into account the actual conditions, structures, behavior as much as possible, carry our calculations without exceeding geometric idealization, consider the real rheological properties of structural elements, the most approximate design schemes to full-scale design and, as a result of the calculations, make a scientifically sound selection of its parameters [14] [15]. Carrying out numerical experiments for complex structurally heterogeneous structures makes it possible to Journal of Applied Mathematics and Physics understand the qualitative picture of the influence of various parameters and provide reasonable recommendations, not only at the design stage, but also for a model and full-scale experiment, to significantly reduce the amount of experimental research. The development and implementation of the finite element method make it possible to solve the problem of creating a universal program for solving the statics and dynamics of an arbitrary type problem and target structures [16] [17]. The existence of such programs and software systems for a mechanical system consisting of a package of rectangular plates and shells with point bonds and concentrated masses with various rheological properties of deformable elements under the influence of harmonic influences will make judgments on publications. To analyze such structures, it is more efficient to use a model with a discrete continuum [18].
In [19], a new version of the differential-quadrature method was proposed for obtaining the vibrational characteristics of rectangular plates resting on elastic bases and bearing any number of spring-loaded masses. The accuracy of the technique is demonstrated by comparing the calculated results with published data. This work uses an uneven grid distance. The results will also demonstrate the effectiveness of the method in solving the problem of vibration of rectangular plates, bearing any number of sprung masses and resting on elastic bases.
In [20], a low computational cost developed a technique based on the eigenfunction decomposition method for studying the vibration of rectangular plates with a series of moving sprung masses. It is shown that the proposed method can significantly increase the computational efficiency of traditional methods by eliminating a large number of time-varying components in related matrices of ordinary differential equations. The dynamic behavior of the system is then investigated by conducting a comprehensive parametric study of the dynamic amplification of the coefficient of moving loads. The results show that ignoring the flexibility of the vehicle suspension system, both in the model with moving force and in the model with moving mass, can lead to a significant underestimation of the dynamic gain of the coefficient of moving loads.
These methods are effective when the mass is attached by an elastic element and has one degree of freedom. Using the method leads to insurmountable difficulties when considering a package of plates with concentrated masses and discontinuous boundary conditions.
Currently, methods, algorithms and programs for solving the problems of statics and dynamics of mechanical systems of multilayer rectangular plates and shells of elastic material [21] [22], under the influence of dynamic loads, have been sufficiently developed. Further development of plate and shell mechanical systems involves taking into account the rheological properties of the material under vibrational influences [23] [24] [25] [26]. In these works, the forced vibrations of linear viscoelastic mechanical systems under vibrational stress are mainly considered.
The brief review of the works shows that the study of the dynamic behavior of plate mechanical systems with attached masses and various discontinuous boun-Journal of Applied Mathematics and Physics dary conditions under the influence of external loads is evaluated differently, and each theory or method used has its advantages and disadvantages.
To date, the question of assessing the level of vibrational processes of plate and shell dissipative mechanical systems with attached masses with different boundary conditions remains insufficiently studied and requires extensive study.
Therefore, this problem is urgent, requiring a phased solution in the field of protection of electronic equipment from dynamic loads.

The Mathematical Formulation of the Problem of Forced Steady-State Oscillations of Viscoelastic Systems with Point Bonds
Consider a mechanical system consisting of N isotropic viscoelastic bodies (a package of rectangular plates or cylindrical shells) that occupy a volume n V and are limited by surfaces It is assumed that one linear size of each body is much smaller than the other two (class of plates and shells).
where ( ) n R t is the relaxation core of the n-the viscoelastic element or point bond, n E is the instantaneous elastic modulus.
In order for the stress to be a periodic function of time in the expression of heredity (2), the lower limit of the integral must be equal to minus infinity. The voltage will contain an aperiodic component if the lower limit is zero.
Here, in contrast to the problems of natural vibrations, the smallness of the parameters of the relaxation core is not assumed. Poisson's ratio is assumed constant (v = const). If ( ) 0 R t = , then the system is elastic. We use the principle of possible (virtual) movements, which is realized by an equation similar to (2), with an additional term-the virtual work of surface forces where , , , , where ( ) 0 nj U x is the complex amplitude of the forced oscillations. In contrast to the problem of natural oscillations, here Ω the frequency will be real. We transform the variational Equation (4) where Lis the total virtual work of the mechanical system, and Fare the kinematic conditions of rigid point connections superimposed on the system.

Construction of Solving Equations of Linear Problems on Forced Steady-State Oscillations of Dissipative Mechanical Systems
The solution to variational Equation (6) where k nj γ are the unknown coefficients.
In the equations obtained after substituting the sum (7) into relation (6) (1) it is assumed that the external perturbing force has a distributed character.
The task, in fact, will not change if the applied force is taken concentrated. Then the virtual work on the area of the element will be replaced by the virtual work of the concentrated force, and only the amplitude vector 0 nj P  will change, which, in addition to the amplitude of the concentrated force, will be filled with zeros. We solve the system (8) by the Gauss method with the selection of the main element in columns and also in rows. The initial system of Equations (8) has complex coefficients; therefore, a program that implements the algorithm was written for systems with complex coefficients and complex unknowns.
The right side of the system, i.e. vector 0 nj P  , set in the form  (7). Substituting the coordinates of any point in the system into the last formula, we obtain the amplitude of its oscillations for a given driving frequency Ω .

Numerical Implementation of the Algorithm
In this section, several problems will be solved in which the amplitude-frequency characteristics for the displacements of individual points of structurally inhomogeneous viscoelastic systems will be obtained and analyzed. The relaxation core for viscoelastic shock absorbers is selected as the Rzhanitsyn-Koltunov core, i.e.: ( ) where , , A α β are the kernel parameters [25]. The viscosity of the shock absorber is adopted such that its creep deformation during the quasistatic process is a small fraction (~12%) of the total deformation.
The parameters of the relaxation core (10)    In this case, in specific calculations, the kernel parameters are as follows:. The coefficient of stiffness moments C 0 ranged from 10 −4 to 10 −1 .
As a harmonically exciting load, a driving harmonic load P(t) uniformly distributed over the area of the upper plate is considered. The amplitude vector of this load is equal to a unit vector. Journal of Applied Mathematics and Physics One can introduce the concept of "global resonance amplitude" to study the behavior of the resonance amplitude of a mechanical system depending on the geometric and physico-mechanical parameters of mechanical systems: where max А k k A δ = is the k-number of the resonant peak. Figure 5 shows a graph of the global coefficient of resonance amplitude А δ versus the instantaneous stiffness of the C 0 shock absorber for a structurally inhomogeneous mechanical system of the central point of the bottom plate. Depending on the value of C 0 , the amplitude А δ is determined either by the first, second, or third resonance peak. The minimum А δ value corresponds to the value of C 0 at which the low frequencies (first and second) of the mechanical system get as close as possible.
The role of the global resonance amplitude depending on C 0 for homogeneous mechanical systems is played by the resonance amplitude corresponding to the first frequency ( Figure 5). For dissipative inhomogeneous mechanical systems, the resonance amplitude is corresponding to both the first and second frequencies. "Role change" occurs at the point where the real parts of the natural frequencies are closest to each other ( Figure 6).      ison with other points, the energy intensity will be the highest (intense). Thus, the results obtained for the structurally inhomogeneous viscoelastic structure under consideration are completely consistent with the solutions of the problem of free damped vibrations and confirm the fact of a sharp increase in the intensity of dissipative processes when the fundamental frequencies approach each other in inhomogeneous viscoelastic systems.

Conclusions
1) Based on the principle of possible displacements, a mathematical formulation and methods for solving the problem of forced vibrations of structurally homogeneous and inhomogeneous mechanical systems consisting of a package of plates (or shells) with point supports and attached masses under the influence of harmonic loads are formulated.
2) The problems of forced oscillations of dissipative homogeneous and inhomogeneous plate systems with concentrated supports and mass attachments are solved.
3) To describe the dissipative properties of the system as a whole, the concept of "global resonance amplitude" is introduced. In the case of a structurally homogeneous mechanical system, the global resonant amplitude is entirely determined by the resonant amplitude corresponding to the first natural frequency. 4) In the case of a structurally inhomogeneous mechanical system, the global resonance amplitude, depending on the stiffness of the shock absorber, is played by the resonance amplitudes corresponding to the first natural frequency (up to C = C 0 ), and then the resonant amplitude corresponding to the second natural frequency.

Recommendations
In some cases, the real structure working under vibrational influences can be represented as a combination of plate dissipatively inhomogeneous mechanical systems with an attached finite number of rigid masses, and others can have a complex structure due to the use of various materials with different rheological properties. Therefore, when designing such structures, it is constantly necessary to solve the problems of calculating the strength of the underlying change in the global resonance amplitude from the geometric and physico-mechanical parameters of the system's elements. Journal of Applied Mathematics and Physics