# Analytical Model of Structural Damping in Friction Module of Shell Shock Absorber Connected to Spring

Authors: Ivan Shatskyi; Andrii Velychkovych
Journal: Shock and Vibration
 (2023)

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## Abstract

Vibration processes play a significant role in the modern industry. In most cases, vibration reduces strength, reliability, and durability of industrial machines, mechanisms, and structures, as well as adversely affects the health of support staff. This study presents a new dry friction shell shock absorber design. The shock absorber contains two spring sections and a friction module with an open shell and an elastic filler; at the same time, the spring sections and the friction module work in parallel. The proposed device demonstrates good damping characteristics, capable of operating under high operating loads, and at the same time has compact transverse dimensions. With a nonmonotonic loading of such a shock absorber, due to the contact interaction of the filler with an open shell, part of the energy that is supplied to the system will be dissipated. In other words, in response to the action of an external nonmonotonic loading, the phenomenon of structural hysteresis occurs in the friction module of the shock absorber. To describe the deformation of the shock absorber, a mechanical and mathematical model of a shell with a cut along the generatrix, which is the main bearing link, has been developed. By means of the technique of quasistatic analysis of structural damping in nonmobile nonconservative shell systems with a deformable filler, the hysteresis loops of the presented shock absorber are analytically described. Using them, according to the known loading history, it is possible to predict the behavior of the considered nonconservative system at any time after the start of the loading process. At each stage of the cyclic loading, the distribution of stresses and relationships between the external loading and the piston displacement was studied. Inequalities are obtained for permissible loads under which the operation of the shock absorber is safe. Such shell shock absorbers are projected to be used in the mining, oil and gas, and construction industries.


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## Body

## 1. Introduction

Vibration processes that occur in the operation of almost all, without exception, modern machines, and mechanisms, typically lead to undesirable consequences. In the vast majority of cases, vibration decreases the strength, reliability, and durability of industrial machines, mechanisms, and structures, as well as affects the health of personnel. This is why the research and design works and theoretical investigations in the field of development of new means of vibration protection and methods for their numerical analyses are of crucial importance [1].Today, the problems of vibration isolation of objects and vibration damping remain relevant for a number of branches of mechanical engineering, instrumentation, and construction [2, 3]. In particular, in oil and gas drilling techniques, a drill string is a long structure connecting downhole tools and a drilling rig. Its dynamic characteristics are one of the key factors affecting the efficiency and cost of drilling [4]. Innovative mining technologies involve the use of a standard tool for complex operations, often associated with abnormal dynamic loadings [5, 6]. The specifics of the operation of drill strings [7] and sucker rod strings [8] lead to vibration loadings of downhole equipment, which can lead to emergency situations [9]. Particular attention should be paid to the problems associated with premature failure, loss of tightness, and unintentional self-loosening of threaded connections due to exposure to intense vibrations [10]. This phenomenon can lead to significant financial losses due to the need for regular maintenance of threads and cause emergency situations [11, 12]. In this regard, the task of ensuring effective vibration protection of objects, as well as ensuring their isolation from shock impact, has always been and remains extremely relevant.The global direction of solving the formulated problem is the use of highly specialized vibration protection devices–shock absorbers [13], dampers [14], elastic couplings, and dynamic absorbers [15]. As for traditional vibration dampers (hydraulic, metal, rubber), the market offers a wide variety of technical solutions, and their properties have been studied at a level sufficient for engineering practice. Due to the development of new technologies, attempts to introduce magnetorheological fluid dampers [16], regenerative shock absorbers [17], and vibration dampers with electromagnetic damping [18] are becoming more frequent. The energy-harvesting technology using the vibration energy can be divided into piezoelectric, electrostatic, and electromagnetic methods [19]. In paper [20], a mechatronic shock absorber based on ball screw pair is designed, which can produce a large damping force. The study [21] proposes a novel design of a regenerative shock absorber for the in-wheel motor to capture the vibration energy from the rough road surface while driving an electric vehicle. A novel hydraulic energy harvesting interconnected shock absorber is proposed in paper [22] to improve the efficiency of energy harvesting, ride comfort, and road handling performance. New methods as use of the magnetorheological elastomer (a smart material with elastic property variable in the external magnetic field) were developed in the last period to absorb the vibration energy [23].Separately, it is necessary to pay attention to vibration-dampening systems in the construction industry. High-rise structures are vulnerable to structural vibrations induced by wind and earthquake loadings [24]. Passive-tuned mass and liquid dampers are a popular choice among structural vibration control systems for being reliable, economical, and easy to implement [25]. Passive vibration control devices like tuned liquid column dampers not only significantly reduce buildings’ vibrations but also can serve as a water storage facility [26]. Particular interest is aroused by the dynamic damper designs based on the effect of collisions [27, 28]. The dynamic conditions of the object connected with a vibration-impact damper are changed both by a redistribution of vibration energy from the object to the damper and by an increase in the vibration energy dissipation [29]. There exists an optimum filling ratio and mass ratio in which the damper can reach the best damping state [27, 30].Due to the simplicity of designs and inexpensive maintenance, engineers often prefer shock absorbers and dampers with dry friction when designing modern vibration protection systems [31, 32]. Friction dampers are devices that use dry friction to dissipate the energy of a system in order to limit its vibratory response. They work by keeping in contact wih two surfaces that move relative to each other in order to generate friction [33]. The use of friction dampers has been widely proposed for a variety of mechanical systems for which applying viscoelastic materials, fluid-based dampers, or other viscous dampers is impossible. An important example is the application of friction dampers in aircraft engines to reduce the blades’ vibration amplitudes [34]. Recurrent impact protection devices usually need to dissipate large amounts of energy to prevent damage to the infrastructure they protect. The device proposed in article [35] is composed of rigid parts with articulated joints, an elastic element that allows the recovery of its shape, and an element that dissipates energy by friction. Dry friction damping technology is an effective means of cardan drives of helicopter tail rotors [36]. Friction effects in the so-called Coulomb damper are widely used in material mixing equipment and automatic washing machines [37]. A number of designs of friction dampers have been developed to improve the seismic characteristics of industrial and civil structures [38]. When compared to other means to attenuate vibration, friction dampers stand out by their advantages. To name a few, they work in harsh environments and in the absence of electric or hydraulic power, and they adapt to a wide excitation bandwidth without tuning [33].Minimum overall dimensions and weight, stability of vibration protection properties, manufacturability, operational reliability, and low cost are well-known requirements for a modern vibration protection system. In addition, when designing vibration protection devices for systems experiencing extreme loadings, it is necessary to strive for a combination of the high bearing capacity of the structure with relatively low rigidity and the required level of damping. In the case of a strict limitation imposed on the transverse dimension of the vibration isolator, for vibration protection of heavily loaded objects in the mining, metallurgical, construction, and oil and gas industries, we propose to use the original design of the shell shock absorber (Figure1(a)). The main feature of the proposed design is the use of an open cylindrical shell as the main working link, which provides excellent damping characteristics for the device. In the shock absorber, together with the shell module, two additional spring sections operate in parallel, which makes it possible to provide the required load-bearing capacity of the shock absorber.Figure 1
Friction shock absorber with open shell: (a) full-scale shock absorber samples; (b) a circuit diagram of the shock absorber; 1: shell with a cut along the generatrix; 2: piston; 3: elastic filler; 4: helical compression spring.
(a)(b)In Figure1(b), the shock absorber works as follows. External axial loading is perceived by pistons 2, which compress filler 3 and compression springs 4. Filler transforms longitudinal displacements of pistons 2 into radial deformations of shell 1. The open shell 1 and the compression springs 4, deforming within the limits of elasticity, accumulate energy and, when the external loading disappears (or decreases), return pistons 2 to their original (or intermediate) position. In this case, part of the energy supplied to the shock absorber is dissipated mainly due to mutual slippage with the friction of the filler and the shell. It is important that all three bearing links of the elastic element (shell with a cut 1 and two compression springs 4) will work in parallel; that is, in general, their holding capacity is summed up, and each of these links will carry a share of the external loading proportional to its own rigidity. The tribological properties of the filler-shell pair are selected for reasons of providing the level of structural damping required in a particular operational situation.From the point of view of mechanics, the proposed shock absorber is a nonconservative deformable system with dry friction [39, 40]. When performing mathematical modeling of the behavior of such systems under the influence of a nonmonotonic loading, nonlinear contact problems arise on the frictional interaction of shells with elastic bodies [41, 42].The contact interaction of real bodies is one of the most common and complex natural phenomena. The problems of contact mechanics are of great interest to researchers as a result of their practical application in the field of mechanical engineering [43, 44], the construction industry [45, 46], biomechanics, etc. [47, 48]. In general, when solving contact problems, researchers pay considerable attention to the analysis of the strength, rigidity, and stability of the contacting elements [49, 50]. Numerical-analytical models of the contact interaction of deformable systems with the environment are considered in studies [51–53]. A class of problems related to the effect of contact interaction of cut faces during the bending of thin shells is topical [54–58]. Similar problems, in which simultaneous loadings of plates and shells by tension (compression) and bending are considered, were solved in [59–61]. The elastic and boundary equilibrium of thin-walled structures on an elastic basis, in which cuts are present, was studied in publications [62, 63]. The works [64–67] are devoted to the study of dynamical systems with different laws of friction.With a nonmonotonic loading of the friction shock absorber (Figure1), as a result of the contact interaction of the filler with an open shell, the part of the energy supplied to the system will be dissipated. In other words, in response to the action of an external nonmonotonic loading, the phenomenon of structural hysteresis occurs in the friction shock absorber. Systems of passive and active damping of vibrations in mechanical engineering and construction were considered in the works [68–70]. The studies [71, 72] considered the phenomena of magnetostriction and internal damping in metals and alloys. A detailed review of the hysteresis properties of composites with a polymer matrix was carried out in [73], and an analysis of damping structures fabricated using modern additive technologies was carried out in [74]. Various models of structural damping and practical approaches to quantifying the energy dissipation of vibrations of structures equipped with dampers are presented in articles [75–77].Practical experience and experimental studies [78–80] have shown that the use of open shells in shock absorbers can significantly reduce their rigidity compared to continuous shells [81, 82] and, as a result, reduce the resonant frequencies of a dynamic system. It is predicted that such shock absorbers will be especially useful when used in vibration machines of superresonance action in the foundry for vibration protection of drill strings and sucker rod strings. Such a conducive forecast for a wide range of applications of the developed shock absorber became the main motivator of our research.The aim of our study is to obtain an analytical description of the deformation diagram “external loading-draft” for the presented friction shock absorber. With the help of such a diagram, according to the known history of the loading, it is possible to predict the behavior of the considered nonconservative system at any time after the start of the loading process, as well as calculate the amount of energy dissipated in this case. At the same time, it is planned to investigate the distribution of stresses in the friction module of the shock absorber at each stage of the cyclic loading. To achieve the aim, the following tasks must be solved: to develop a mechanical-mathematical model of an open cylindrical shell, to perform the formulation, to obtain a solution to the problem of structural damping during the contact interaction of the deformable filler and the open shell, taking into account the presence of friction between them, and analytically to describe the process of drawing a deformation diagram of the shell shock absorber taking into account the parallel connection of the friction module and springs.


## 2. Materials and Methods

### 2.1. Mechanical and Mathematical Model of an Open Shell

To describe the deformation of a friction shock absorber, we first construct a mechanical-mathematical model of a shell with a cut along the generatrix, which is the main bearing link of the shock absorber (Figure1(b)). Let us take into account the main feature of the design, the rigidity of the cut shell in the tangential direction is significantly less than along the generatrix. The main contribution to the compliance of the system will be made by changing the shape of the filler due to the bending deformation of the cut shell (reducing the curvature of the open ring in the cross section). Therefore, the following assumptions are logical. The filler material is assumed to be incompressible. To a cut isotropic shell that bends under conditions of a nonaxisymmetric contact loading, we put in correspondence a strongly closed orthotropic cylindrical shell under the action of an axisymmetric contact loading (Figure 2). We keep the thicknesses and radii of the shells the same. We choose the elastic modulus for an equivalent orthotropic shell such that it identifies, on average, the properties of the cut shell and its continuous model.Figure 2
Scheme of a thought experiment for constructing an open shell model: (a) isotropic open shell; (b) equivalent orthotropic closed shell.
(a)(b)To do this, we carry out a thought experiment. LetE0 be Young’s modulus for the material of the cut shell (Figure 2(a)). Let us denote by Ee the modulus of elasticity of the equivalent model to be determined (Figure 2(b)). Let us subject both shells to the action of internal pressure q and find deformations and displacements. We will use the cylindrical coordinate system r,ϑ,z (axis z in Figure 2 is not shown, and it is directed along the axis of the shell). As a result of the predominant effect of bending deformation, we will additionally assume that the middle surface of the cut shell is inextensible, and the middle surface of the orthotropic shell is stretched only in the tangential direction.Taking into account the accepted assumptions, we will solve the plane problem of the loading of an open shell by internal pressure. The equilibrium equations have the following form:(1)dNϑdϑ+Qϑ=0,dQϑdϑ−Nϑ=−qR,dMϑdϑ−QϑR=0,ϑ∈0,2π,where Nϑ is a tangential force, and Mϑ and Qϑ denote bending moment and transverse force.Having integrated the system of equation (1) under the boundary conditions:(2)Nϑ=0,Mϑ=0,Qϑ=0whenϑ=0,2π.We obtain expressions for forces and moments:(3)Nϑ=2qRsin2ϑ2,Qϑ=−qRsinϑ,Mϑ=−2qR2sin2ϑ2.We find the deflection function by the differential equation(4)d2wdϑ2+w=−12R2E0h3Mϑ,ϑ∈0,2π.Integrating expression (4), we have(5)wϑ=Acosϑ+Bsinϑ+12qR4E0h31−12ϑsinϑ.Here, the first two summands describe the movement of the section of an open shell as a rigid body. Now, we find the average value of the bending(6)w1R=12π∫02πwϑRdϑ=18qR3E0h3.In an orthotropic momentless shell, a tangential force arises under the action of internal pressureNϑ=qR. For tangential deformation, we will have the following expression:(7)w2R=qEeRh.Requiring that the average deflections in the left parts of expressions (6) and (7) be the same, we find the equivalent elastic constant of the model continuous shell:(8)Ee=118h2R2E0.


### 2.2. Analytical Model of Structural Damping in a Friction Shock Absorber

Let us consider structural damping in a model strongly orthotropic shell with a deformable filler. In view of the symmetry of the structure with respect to a plane equidistant from the pistons, we will study half of the system (Figure3), considering the section belonging to the plane of symmetry to be supported by a rigid barrier. Therefore, we have an elastic incompressible deformable cylinder (filler), with the radius R and length a, which fills a strongly orthotropic shell of thickness h. A linear compression spring is installed between the piston collar and the end face of the shell. An external cyclic loading Q is applied to an absolutely rigid smooth piston. The nature of the contact interaction of the shell and filler is determined by the law of dry friction. Let us study the energy dissipation in a frictional contact.Figure 3
Calculation scheme of a friction shock absorber.The frequency-dependent energy dissipation that occurs due to internal friction in the filler is negligible compared to amplitude-dependent structural damping during contact sliding. Therefore, the viscosity of the filler material is not taken into account.The initial model relations for the filler will be the equation of equilibrium, equation of incompressibility, and Hooke’s law for incompressible material averaged over the cross-sectional area(9)dσζdζ+2aRτ=0,dudζ+2aRw=0,dudζ=aEσζ−σ,ζ∈0,1,where σζ,u are axial stresses and filler displacement, σ and τ are normal and tangential contact stresses, w denotes radial displacement on the surface r=R, E denotes Young’s modulus of the filler material, and ζ=z/a is a dimensionless coordinate.For a model orthotropic shell, describing its elastic equilibrium by the relations of the momentless theory, and assuming that its middle surface is stretched only in the tangential direction, we have(10)w0=−σR2Eeh,u0=0,ζ∈0,1,where w0,u0 are radial and axial displacement of the shell; Ee is an equivalent elastic constant determined by the formula (8).The frictional interaction of the shell and filler is described by the relations of one-sided contact and Coulomb’s law of dry friction in a quasi-static interpretation:(11)w0=w,σ<0,τ=fσsgn∂u∂QQ˙,∂u∂Q≠0,τ≤−fσ,∂u∂Q=0,where f is a coefficient of friction of the shell-filler contact pair.To find the coordinates of the point of delimitation of the areas of adhesion and sliding,α it is necessary to use the condition of continuity of axial stresses in the filler:(12)σζα−0=σζα+0.Let us now consider the situation at the end of the filler. With parallel connection, the external loading on the piston is divided into two parts(13)Q=Q1+Q2,where Q1 is a filler loading, and Q2 denotes a spring loading.The first of them is proportional to the unknown pressurep under the piston, and the second is proportional to the displacement of the extreme coil of the spring u20:(14)Q1=pπR2,Q2=c2u20.Here,c2 is a coefficient of linear rigidity of the spring.Given that under the piston(15)p=−σζ0,u20=u0.From formulas (13)–(15), we obtain the boundary condition at the lower end of the filler:(16)Q=−σζ0πR2+c2u0.The upper end of the cylinder is stationary:(17)u1=0.Therefore, the boundary value problem of the contact interaction of the shell and filler, taking into account the forces of dry friction, is formulated. We built its solution separately for each of the sections–sliding and ideal adhesion at each stage of the cyclic loading.


## 2.1. Mechanical and Mathematical Model of an Open Shell

To describe the deformation of a friction shock absorber, we first construct a mechanical-mathematical model of a shell with a cut along the generatrix, which is the main bearing link of the shock absorber (Figure1(b)). Let us take into account the main feature of the design, the rigidity of the cut shell in the tangential direction is significantly less than along the generatrix. The main contribution to the compliance of the system will be made by changing the shape of the filler due to the bending deformation of the cut shell (reducing the curvature of the open ring in the cross section). Therefore, the following assumptions are logical. The filler material is assumed to be incompressible. To a cut isotropic shell that bends under conditions of a nonaxisymmetric contact loading, we put in correspondence a strongly closed orthotropic cylindrical shell under the action of an axisymmetric contact loading (Figure 2). We keep the thicknesses and radii of the shells the same. We choose the elastic modulus for an equivalent orthotropic shell such that it identifies, on average, the properties of the cut shell and its continuous model.Figure 2
Scheme of a thought experiment for constructing an open shell model: (a) isotropic open shell; (b) equivalent orthotropic closed shell.
(a)(b)To do this, we carry out a thought experiment. LetE0 be Young’s modulus for the material of the cut shell (Figure 2(a)). Let us denote by Ee the modulus of elasticity of the equivalent model to be determined (Figure 2(b)). Let us subject both shells to the action of internal pressure q and find deformations and displacements. We will use the cylindrical coordinate system r,ϑ,z (axis z in Figure 2 is not shown, and it is directed along the axis of the shell). As a result of the predominant effect of bending deformation, we will additionally assume that the middle surface of the cut shell is inextensible, and the middle surface of the orthotropic shell is stretched only in the tangential direction.Taking into account the accepted assumptions, we will solve the plane problem of the loading of an open shell by internal pressure. The equilibrium equations have the following form:(1)dNϑdϑ+Qϑ=0,dQϑdϑ−Nϑ=−qR,dMϑdϑ−QϑR=0,ϑ∈0,2π,where Nϑ is a tangential force, and Mϑ and Qϑ denote bending moment and transverse force.Having integrated the system of equation (1) under the boundary conditions:(2)Nϑ=0,Mϑ=0,Qϑ=0whenϑ=0,2π.We obtain expressions for forces and moments:(3)Nϑ=2qRsin2ϑ2,Qϑ=−qRsinϑ,Mϑ=−2qR2sin2ϑ2.We find the deflection function by the differential equation(4)d2wdϑ2+w=−12R2E0h3Mϑ,ϑ∈0,2π.Integrating expression (4), we have(5)wϑ=Acosϑ+Bsinϑ+12qR4E0h31−12ϑsinϑ.Here, the first two summands describe the movement of the section of an open shell as a rigid body. Now, we find the average value of the bending(6)w1R=12π∫02πwϑRdϑ=18qR3E0h3.In an orthotropic momentless shell, a tangential force arises under the action of internal pressureNϑ=qR. For tangential deformation, we will have the following expression:(7)w2R=qEeRh.Requiring that the average deflections in the left parts of expressions (6) and (7) be the same, we find the equivalent elastic constant of the model continuous shell:(8)Ee=118h2R2E0.


## 2.2. Analytical Model of Structural Damping in a Friction Shock Absorber

Let us consider structural damping in a model strongly orthotropic shell with a deformable filler. In view of the symmetry of the structure with respect to a plane equidistant from the pistons, we will study half of the system (Figure3), considering the section belonging to the plane of symmetry to be supported by a rigid barrier. Therefore, we have an elastic incompressible deformable cylinder (filler), with the radius R and length a, which fills a strongly orthotropic shell of thickness h. A linear compression spring is installed between the piston collar and the end face of the shell. An external cyclic loading Q is applied to an absolutely rigid smooth piston. The nature of the contact interaction of the shell and filler is determined by the law of dry friction. Let us study the energy dissipation in a frictional contact.Figure 3
Calculation scheme of a friction shock absorber.The frequency-dependent energy dissipation that occurs due to internal friction in the filler is negligible compared to amplitude-dependent structural damping during contact sliding. Therefore, the viscosity of the filler material is not taken into account.The initial model relations for the filler will be the equation of equilibrium, equation of incompressibility, and Hooke’s law for incompressible material averaged over the cross-sectional area(9)dσζdζ+2aRτ=0,dudζ+2aRw=0,dudζ=aEσζ−σ,ζ∈0,1,where σζ,u are axial stresses and filler displacement, σ and τ are normal and tangential contact stresses, w denotes radial displacement on the surface r=R, E denotes Young’s modulus of the filler material, and ζ=z/a is a dimensionless coordinate.For a model orthotropic shell, describing its elastic equilibrium by the relations of the momentless theory, and assuming that its middle surface is stretched only in the tangential direction, we have(10)w0=−σR2Eeh,u0=0,ζ∈0,1,where w0,u0 are radial and axial displacement of the shell; Ee is an equivalent elastic constant determined by the formula (8).The frictional interaction of the shell and filler is described by the relations of one-sided contact and Coulomb’s law of dry friction in a quasi-static interpretation:(11)w0=w,σ<0,τ=fσsgn∂u∂QQ˙,∂u∂Q≠0,τ≤−fσ,∂u∂Q=0,where f is a coefficient of friction of the shell-filler contact pair.To find the coordinates of the point of delimitation of the areas of adhesion and sliding,α it is necessary to use the condition of continuity of axial stresses in the filler:(12)σζα−0=σζα+0.Let us now consider the situation at the end of the filler. With parallel connection, the external loading on the piston is divided into two parts(13)Q=Q1+Q2,where Q1 is a filler loading, and Q2 denotes a spring loading.The first of them is proportional to the unknown pressurep under the piston, and the second is proportional to the displacement of the extreme coil of the spring u20:(14)Q1=pπR2,Q2=c2u20.Here,c2 is a coefficient of linear rigidity of the spring.Given that under the piston(15)p=−σζ0,u20=u0.From formulas (13)–(15), we obtain the boundary condition at the lower end of the filler:(16)Q=−σζ0πR2+c2u0.The upper end of the cylinder is stationary:(17)u1=0.Therefore, the boundary value problem of the contact interaction of the shell and filler, taking into account the forces of dry friction, is formulated. We built its solution separately for each of the sections–sliding and ideal adhesion at each stage of the cyclic loading.


## 3. Results and Discussion

First, let us consider the cyclic deformation of the “filler-shell” subsystem under some loadingQ1 with cycle asymmetry coefficient s=Q1min/Q1max≡pmin/pmax∈0,1. The process of nonmonotonic loading was divided into stages where the loading is monotonic. As a result of solving problem (9)–(12), (16), and (17) according to the method started in [41, 70], we obtained expressions for contact σ, τ and axial σζ stresses and piston displacements δ at all stages of loading. Let’s present the solutions of the problem for each stage separately.For clarity of notation, we introduce an additional superscript Roman index. The number I will indicate the stage of active (initial) loading, the number II will indicate the stage of unloading, and the number III will indicate the stage of repeated loading.


### 3.1. The Stage of the Initial (Active) Load of the Shock Absorber

Shock absorber load parameters are0≤p≤pmax,p˙>0,∂u/∂p>0.At the initial stage of loading, contact and axial stresses are described by the following formulas:(18)σIζ=−p1+2εe−λζ,τIζ=−fp1+2εe−λζ,σζIζ=−pe−λζ,ζ∈0,1,where λ=2fa/R1/1+2ε is parameter of exponential decay of axial and contact stresses; ε=18E/E0R3/h3 is auxiliary dimensionless rigidity parameter.Graphically, these dependencies are shown in Figure4. Here and below, to illustrate the results obtained, as an example, a system with parameters f=0.5, a/R=3, h/R=0.18, E/E0=10−4. Then, ε≈0.31, λ≈1.86. The selected numerical parameters correspond to real samples of shock absorbers for the elastic suspension of the sucker rod string [8].Figure 4
Stress distribution at the active loading stage.If we trace the behavior of contact and axial stresses in the friction module of the shock absorber (Figure1), then they reach their maximum absolute values in the planes of the ends of the filler (at the edges of the contact zone), decreasing with distance from these planes.The displacement of the piston at this stage is a linear function ofp and is calculated by the following formula:(19)δI=uζI0=paE2ε1+2ε1−e−λλ.During the operation of the shock absorber, structural energy dissipation is present only in the areas of mutual sliding of the filler and the shell. Relation (19) describes the portion of the loading diagram (or structural hysteresis loop) corresponding to the initial loading. The linearity of this expression means that at this stage the loading of the shock absorber, the adhesion in the contact pair is not achieved.


### 3.2. The Stage of Shock Absorber Unloading

At this stage, the shock absorber load parameters changepmax≥p≥0, p˙<0. At the beginning of this stage, the unloading of the structure occurs in the mode of partial reverse sliding. The segment ζ∈0,1 is divided into two areas: ζ∈0,α–the reverse sliding area, where ∂uζ/∂p<0; ζ∈α,1–adhesion area, where ∂uζ/∂p=0. The coordinate of the point of delimitation of the areas is unknown.In the adhesion zone, there is an old pattern of stress and displacement distribution, which was achieved at the end of the previous stage:(20)σIIζ=σIζp=pmax,τIIζ=τIζp=pmax,σζIIζ=σIζp=pmax,ζ∈α,1.In the sliding zone, shear stresses change their direction to the opposite. Changing a parameterλ, we got:(21)σIIζ=−p1+2εeλζ,τIIζ=fp1+2εeλζ,σζIIζ=−peλζ,ζ∈0,α.Having ensured the fulfillment of condition (12), we found the coordinate of the point of delimitation of the areas of reverse sliding and adhesion(22)α=1λlnpmaxp1/2.Stress graphs are shown in Figure5 (hereinafter, the solid lines will represent the results of the current stage of loading or unloading, and the dashed lines will represent the results of the previous stages).Figure 5
Stress distribution at the unloading stage (mode of partial sliding).The described unloading process takes place until the pointα reaches the upper limit ζ=1. From the condition 0≤α≤1, we find the corresponding range of pressure change: e−2λpmax≤p≤pmax, where e−2λ is a characteristic value of the cycle asymmetry coefficient p. Piston displacement in this loading range(23)δII=δmaxI−aE2ε1+2ε1λpmax−p2ate−2λpmax≤p≤pmax.When the pressure under the piston drops top=e−2λpmax, reverse sliding conditions will be satisfied on the entire contact surface ζ∈0,1. In this case, in the unloading range 0≤p≤e−2λpmax, formula (21) will be true. The characteristic stress distribution corresponding to this episode of unloading is shown in Figure 6. We will find the piston draft by the following formula:(24)δII=paE2ε1+2εeλ−1λat0≤p≤e−2λpmax.Figure 6
Stress distribution at the unloading stage (full sliding mode).Consequently, the beginning of the unloading stage is characterized by the fact that nonunilateral sliding conditions are satisfied on the contact surfaces (Figure5). In other words, the contact surface is divided into areas in which the velocity of the mutual displacement of the contacting bodies and, consequently, the contact stresses have opposite symbols. Formula (23) gives us a nonlinear relationship between piston displacement and loading. Toward the end of the unloading stage, one-sided sliding is achieved on the contact surfaces (Figure 6), and formula (24) indicates that the unloading section of the deformation diagram will always end in a straight line.


### 3.3. The Stage of Repeated Loading of the Shock Absorber

Shock absorber load parameterspmin≤p≤pmax, p˙>0. Depending on the value pmin, two cases are possible: s≥e−2λ and s≤e−2λ. Let’s consider each of them in turn.Let1≥s≥e−2λ. In this case, at the beginning of the repeated loading stage, we have direct sliding in the piston zone ζ∈0,β and the situation of the end of the preliminary stage in the region ζ∈β,1. The coordinate of the point of delimitation of the areas β is unknown.In the adhesion zone∂uζ/∂p=0,ζ∈β,1, stress is obtained from expressions (21) at p=pmin:(25)σIIIζ=σIIζp=pmin=−pmin1+2εeλζ,ζ∈β,α,−pmax1+2εe−λζ,ζ∈α,1,τIIIζ=τIIζp=pmin=fpmin1+2εeλζ,ζ∈β,α,−fpmax1+2εe−λζ,ζ∈α,1,σζIIIζ=σζIIζp=pmin=−pmineλζ,ζ∈β,α,−pmaxe−λζ,ζ∈α,1.In the sliding zone∂uζ/∂p=0,ζ∈0,β, stress distributions are described by formula (18):(26)σIIIζ=σIζ,τIIIζ=τIζ,σζIIIζ=σζIζ,ζ∈0,β.From the condition of continuity of axial stresses, we find the boundary of regions with opposite symbols of contact stresses(27)β=1λlnppmin1/2.Results (20) and (21) are graphically presented in Figure 7.Figure 7
Stress distribution at the repeated loading stage at1≥s≥e−2λ.It is noteworthy that there are three regions at once on the contact surface. The difference between them lies in the fact that in each two adjacent areas, the tangential contact stresses have opposite symbols. It should be noted that the number of such areas for an arbitrary number of loading cycles with an arbitrary amplitude can be greater (this depends on the history of the loading).The described process takes place until the pointβ reaches the point α (until the curves III are aligned with the curves І). This moment corresponds to the end of the cycle p=pmax.Finally, the displacement of the piston at1≥s≥e−2λ is determined by the following formula:(28)δIII=δminII+aE2ε1+2ε1λp−pmin2.Let now0≤s≤e−2λ. Then, in the zone of direct sliding ζ∈0,β, dependencies (26) take place, and in the adhesion region ζ∈β,1, stress state (21) recorded at the end of unloading:(29)σIIIζ=σIIζp=pmin=−pmin1+2εeλζ,τIIIζ=τIIζp=pmin=fpmin1+2εeλζ,σζIIIζ=σζIIζp=pmin=−pmineλζ,ζ∈β,1.To calculate the parameterβ and piston draft at this stage, formulas (27) and (28) are preserved, respectively. Figure 8 illustrates the distribution of stresses characteristic of the situation under consideration.Figure 8
Stress distribution at the repeated loading stage at0≤s≤e−2λ.This case is realized until the pointβ reaches the upper edge of the gap 0,1, which in terms of loading means pmin≤p≤e2λpmin. When p=e2λpmin again, we enter the loading mode with full direct sliding on the entire contact surface.


### 3.4. Analytical Method of Drawing the Deformation Diagram of the Shell Shock Absorber

Let us summarize the study by bringing together expressions that describe all stages of the loading cycle. Let us denotec1=πR2E/a1+2ε/2ε as linear rigidity of the conservative subsystem “shell-filler.”Then, based on formulas (14), (15), (17), (23), (24), and (28), we obtain the hysteresis loop equation in implicit form (dependences of δ on Q1=Q−c2δ).For active loading,(30)δI=Q−c2δIc11−e−λλ,0≤Q−c2δI≤Qmax−c2δmaxI.For unloading,(31)δII=δmaxI−1c1λQmax−c2δmaxI−Q−c2δII2,Qmax−c2δmaxI≥Q−c2δII≥e−2λQmax−c2δmaxI,Q−c2δIIc1eλ−1λ,e−2λQmax−c2δmaxI≥Q−c2δII≥0.For repeated loading,(32)δIII=δminII+1c1λQ−c2δIII−Qmin−c2δminII2,Qmin−c2δminII≤Q−c2δIII≤minQmin−c2δminIIe−2λ,Qmax−c2δmaxI,Q−c2δIIIc11−e−λλ,minQmin−c2δminIIe−2λ,Qmax−c2δmaxI≤Q−c2δIII≤Qmax−c2δmax.ISolving equations (30)–(32) with respect to Q, we found explicit expressions of Q by means of δ at all stages of the cycle:For active loading,(33)QI=c1λ1−e−λ+c2δ,0≤δ≤δmax.For unloading,(34)QII=QmaxI−c2δmax−c1λδmax−δ2+c2δ,δmax≥δ≥e−λδmax,c1λeλ−1+c2δ,e−λδmax≥δ≥0.For repeated loading,(35)QIII=QminII−c2δmin+c1λδ−δmin2+c2δ,δmin≤δ≤mineλδmin,δmax,c1λ1−e−λ+c2δ,mineλδmin,δmax≤δ≤δmax.In contrast to previous works [41, 42, 83], in which the phenomenon of structural hysteresis in shell dampers is described in terms of the power cycle (Qmin,Qmax), the equations (33)–(35) obtained here present the nonconservative properties of the damper in terms of the kinematic cycle (δmin,δmax). This approach actually made it possible to study energy dissipation in the parallel connection of the nonlinear and the linear element.According to expressions (33)–(35), a structural damping loop was built (Figure 9). Still here, λ≈1.86, in addition, value c1/c2=5 was recorded.Figure 9
Diagram of the cyclic loading of the shock absorber: ОА: active (initial) loading; АСО: unloading (here, AC is a nonlinear section, and CO is a linear section); ВА or DEA: repeated loading.Let us move on to the assessment of permissible loadings. LetQ=Qmax. The analysis showed that the most dangerous place in the open shell is a point with coordinates R,π,0. Indeed, here the maximum hoop stress from bending and the greatest modulus axial stress from compression are achieved:(36)σϑmaxshell=maxϑ∈0,2π1hNϑ−6hMϑ=2pmax1+2εRh1+6Rh≈Q1maxπR2121+2εRh2,σζminshell=−Q2max2πRh.The strength of the bearing links of the structure will be ensured if the irregularities are true:(37)maxσeqshell≡σϑmaxshell−σζminshell≤σshell,maxQ2spring≡Q2max≤Qspring.Here,σeqshell is Tresca equivalent stress, σshell is an allowable stress for the shell material, and Qspring is an allowable compression force for the spring.From formulas (13), (14), and (33), it follows that the maximum loadings on the filler and on the spring are divided in proportion to the corresponding rigidities of the subsystems:(38)Q1max=c1λ/1−e−λc1λ/1−e−λ+c2Qmax,Q2max=c2c1λ/1−e−λ+c2Qmax.Then, from relation (36)–(38), we obtain the conditions for the safe operation of the shock absorber in the form of restrictions on the external loading:(39)Qmax≤σshellπR2c1λ/1−e−λc1λ/1−e−λ+c2121+2εRh2+c2c1λ/1−e−λ+c2R2h−1,Qmax≤Qspringc1λ/1−e−λ+c2c2.Which of the inequalities (39) is stronger depends on the values of the geometric and mechanical parameters of all structural elements of the shock absorber.Therefore, formulas (33)–(35) and (39) are the main ones for the engineering calculation of a shock absorber and make it possible to construct a deformation diagram and determine the holding capacity of the device for any coefficient of loading cycle asymmetry.


## 3.1. The Stage of the Initial (Active) Load of the Shock Absorber

Shock absorber load parameters are0≤p≤pmax,p˙>0,∂u/∂p>0.At the initial stage of loading, contact and axial stresses are described by the following formulas:(18)σIζ=−p1+2εe−λζ,τIζ=−fp1+2εe−λζ,σζIζ=−pe−λζ,ζ∈0,1,where λ=2fa/R1/1+2ε is parameter of exponential decay of axial and contact stresses; ε=18E/E0R3/h3 is auxiliary dimensionless rigidity parameter.Graphically, these dependencies are shown in Figure4. Here and below, to illustrate the results obtained, as an example, a system with parameters f=0.5, a/R=3, h/R=0.18, E/E0=10−4. Then, ε≈0.31, λ≈1.86. The selected numerical parameters correspond to real samples of shock absorbers for the elastic suspension of the sucker rod string [8].Figure 4
Stress distribution at the active loading stage.If we trace the behavior of contact and axial stresses in the friction module of the shock absorber (Figure1), then they reach their maximum absolute values in the planes of the ends of the filler (at the edges of the contact zone), decreasing with distance from these planes.The displacement of the piston at this stage is a linear function ofp and is calculated by the following formula:(19)δI=uζI0=paE2ε1+2ε1−e−λλ.During the operation of the shock absorber, structural energy dissipation is present only in the areas of mutual sliding of the filler and the shell. Relation (19) describes the portion of the loading diagram (or structural hysteresis loop) corresponding to the initial loading. The linearity of this expression means that at this stage the loading of the shock absorber, the adhesion in the contact pair is not achieved.


## 3.2. The Stage of Shock Absorber Unloading

At this stage, the shock absorber load parameters changepmax≥p≥0, p˙<0. At the beginning of this stage, the unloading of the structure occurs in the mode of partial reverse sliding. The segment ζ∈0,1 is divided into two areas: ζ∈0,α–the reverse sliding area, where ∂uζ/∂p<0; ζ∈α,1–adhesion area, where ∂uζ/∂p=0. The coordinate of the point of delimitation of the areas is unknown.In the adhesion zone, there is an old pattern of stress and displacement distribution, which was achieved at the end of the previous stage:(20)σIIζ=σIζp=pmax,τIIζ=τIζp=pmax,σζIIζ=σIζp=pmax,ζ∈α,1.In the sliding zone, shear stresses change their direction to the opposite. Changing a parameterλ, we got:(21)σIIζ=−p1+2εeλζ,τIIζ=fp1+2εeλζ,σζIIζ=−peλζ,ζ∈0,α.Having ensured the fulfillment of condition (12), we found the coordinate of the point of delimitation of the areas of reverse sliding and adhesion(22)α=1λlnpmaxp1/2.Stress graphs are shown in Figure5 (hereinafter, the solid lines will represent the results of the current stage of loading or unloading, and the dashed lines will represent the results of the previous stages).Figure 5
Stress distribution at the unloading stage (mode of partial sliding).The described unloading process takes place until the pointα reaches the upper limit ζ=1. From the condition 0≤α≤1, we find the corresponding range of pressure change: e−2λpmax≤p≤pmax, where e−2λ is a characteristic value of the cycle asymmetry coefficient p. Piston displacement in this loading range(23)δII=δmaxI−aE2ε1+2ε1λpmax−p2ate−2λpmax≤p≤pmax.When the pressure under the piston drops top=e−2λpmax, reverse sliding conditions will be satisfied on the entire contact surface ζ∈0,1. In this case, in the unloading range 0≤p≤e−2λpmax, formula (21) will be true. The characteristic stress distribution corresponding to this episode of unloading is shown in Figure 6. We will find the piston draft by the following formula:(24)δII=paE2ε1+2εeλ−1λat0≤p≤e−2λpmax.Figure 6
Stress distribution at the unloading stage (full sliding mode).Consequently, the beginning of the unloading stage is characterized by the fact that nonunilateral sliding conditions are satisfied on the contact surfaces (Figure5). In other words, the contact surface is divided into areas in which the velocity of the mutual displacement of the contacting bodies and, consequently, the contact stresses have opposite symbols. Formula (23) gives us a nonlinear relationship between piston displacement and loading. Toward the end of the unloading stage, one-sided sliding is achieved on the contact surfaces (Figure 6), and formula (24) indicates that the unloading section of the deformation diagram will always end in a straight line.


## 3.3. The Stage of Repeated Loading of the Shock Absorber

Shock absorber load parameterspmin≤p≤pmax, p˙>0. Depending on the value pmin, two cases are possible: s≥e−2λ and s≤e−2λ. Let’s consider each of them in turn.Let1≥s≥e−2λ. In this case, at the beginning of the repeated loading stage, we have direct sliding in the piston zone ζ∈0,β and the situation of the end of the preliminary stage in the region ζ∈β,1. The coordinate of the point of delimitation of the areas β is unknown.In the adhesion zone∂uζ/∂p=0,ζ∈β,1, stress is obtained from expressions (21) at p=pmin:(25)σIIIζ=σIIζp=pmin=−pmin1+2εeλζ,ζ∈β,α,−pmax1+2εe−λζ,ζ∈α,1,τIIIζ=τIIζp=pmin=fpmin1+2εeλζ,ζ∈β,α,−fpmax1+2εe−λζ,ζ∈α,1,σζIIIζ=σζIIζp=pmin=−pmineλζ,ζ∈β,α,−pmaxe−λζ,ζ∈α,1.In the sliding zone∂uζ/∂p=0,ζ∈0,β, stress distributions are described by formula (18):(26)σIIIζ=σIζ,τIIIζ=τIζ,σζIIIζ=σζIζ,ζ∈0,β.From the condition of continuity of axial stresses, we find the boundary of regions with opposite symbols of contact stresses(27)β=1λlnppmin1/2.Results (20) and (21) are graphically presented in Figure 7.Figure 7
Stress distribution at the repeated loading stage at1≥s≥e−2λ.It is noteworthy that there are three regions at once on the contact surface. The difference between them lies in the fact that in each two adjacent areas, the tangential contact stresses have opposite symbols. It should be noted that the number of such areas for an arbitrary number of loading cycles with an arbitrary amplitude can be greater (this depends on the history of the loading).The described process takes place until the pointβ reaches the point α (until the curves III are aligned with the curves І). This moment corresponds to the end of the cycle p=pmax.Finally, the displacement of the piston at1≥s≥e−2λ is determined by the following formula:(28)δIII=δminII+aE2ε1+2ε1λp−pmin2.Let now0≤s≤e−2λ. Then, in the zone of direct sliding ζ∈0,β, dependencies (26) take place, and in the adhesion region ζ∈β,1, stress state (21) recorded at the end of unloading:(29)σIIIζ=σIIζp=pmin=−pmin1+2εeλζ,τIIIζ=τIIζp=pmin=fpmin1+2εeλζ,σζIIIζ=σζIIζp=pmin=−pmineλζ,ζ∈β,1.To calculate the parameterβ and piston draft at this stage, formulas (27) and (28) are preserved, respectively. Figure 8 illustrates the distribution of stresses characteristic of the situation under consideration.Figure 8
Stress distribution at the repeated loading stage at0≤s≤e−2λ.This case is realized until the pointβ reaches the upper edge of the gap 0,1, which in terms of loading means pmin≤p≤e2λpmin. When p=e2λpmin again, we enter the loading mode with full direct sliding on the entire contact surface.


## 3.4. Analytical Method of Drawing the Deformation Diagram of the Shell Shock Absorber

Let us summarize the study by bringing together expressions that describe all stages of the loading cycle. Let us denotec1=πR2E/a1+2ε/2ε as linear rigidity of the conservative subsystem “shell-filler.”Then, based on formulas (14), (15), (17), (23), (24), and (28), we obtain the hysteresis loop equation in implicit form (dependences of δ on Q1=Q−c2δ).For active loading,(30)δI=Q−c2δIc11−e−λλ,0≤Q−c2δI≤Qmax−c2δmaxI.For unloading,(31)δII=δmaxI−1c1λQmax−c2δmaxI−Q−c2δII2,Qmax−c2δmaxI≥Q−c2δII≥e−2λQmax−c2δmaxI,Q−c2δIIc1eλ−1λ,e−2λQmax−c2δmaxI≥Q−c2δII≥0.For repeated loading,(32)δIII=δminII+1c1λQ−c2δIII−Qmin−c2δminII2,Qmin−c2δminII≤Q−c2δIII≤minQmin−c2δminIIe−2λ,Qmax−c2δmaxI,Q−c2δIIIc11−e−λλ,minQmin−c2δminIIe−2λ,Qmax−c2δmaxI≤Q−c2δIII≤Qmax−c2δmax.ISolving equations (30)–(32) with respect to Q, we found explicit expressions of Q by means of δ at all stages of the cycle:For active loading,(33)QI=c1λ1−e−λ+c2δ,0≤δ≤δmax.For unloading,(34)QII=QmaxI−c2δmax−c1λδmax−δ2+c2δ,δmax≥δ≥e−λδmax,c1λeλ−1+c2δ,e−λδmax≥δ≥0.For repeated loading,(35)QIII=QminII−c2δmin+c1λδ−δmin2+c2δ,δmin≤δ≤mineλδmin,δmax,c1λ1−e−λ+c2δ,mineλδmin,δmax≤δ≤δmax.In contrast to previous works [41, 42, 83], in which the phenomenon of structural hysteresis in shell dampers is described in terms of the power cycle (Qmin,Qmax), the equations (33)–(35) obtained here present the nonconservative properties of the damper in terms of the kinematic cycle (δmin,δmax). This approach actually made it possible to study energy dissipation in the parallel connection of the nonlinear and the linear element.According to expressions (33)–(35), a structural damping loop was built (Figure 9). Still here, λ≈1.86, in addition, value c1/c2=5 was recorded.Figure 9
Diagram of the cyclic loading of the shock absorber: ОА: active (initial) loading; АСО: unloading (here, AC is a nonlinear section, and CO is a linear section); ВА or DEA: repeated loading.Let us move on to the assessment of permissible loadings. LetQ=Qmax. The analysis showed that the most dangerous place in the open shell is a point with coordinates R,π,0. Indeed, here the maximum hoop stress from bending and the greatest modulus axial stress from compression are achieved:(36)σϑmaxshell=maxϑ∈0,2π1hNϑ−6hMϑ=2pmax1+2εRh1+6Rh≈Q1maxπR2121+2εRh2,σζminshell=−Q2max2πRh.The strength of the bearing links of the structure will be ensured if the irregularities are true:(37)maxσeqshell≡σϑmaxshell−σζminshell≤σshell,maxQ2spring≡Q2max≤Qspring.Here,σeqshell is Tresca equivalent stress, σshell is an allowable stress for the shell material, and Qspring is an allowable compression force for the spring.From formulas (13), (14), and (33), it follows that the maximum loadings on the filler and on the spring are divided in proportion to the corresponding rigidities of the subsystems:(38)Q1max=c1λ/1−e−λc1λ/1−e−λ+c2Qmax,Q2max=c2c1λ/1−e−λ+c2Qmax.Then, from relation (36)–(38), we obtain the conditions for the safe operation of the shock absorber in the form of restrictions on the external loading:(39)Qmax≤σshellπR2c1λ/1−e−λc1λ/1−e−λ+c2121+2εRh2+c2c1λ/1−e−λ+c2R2h−1,Qmax≤Qspringc1λ/1−e−λ+c2c2.Which of the inequalities (39) is stronger depends on the values of the geometric and mechanical parameters of all structural elements of the shock absorber.Therefore, formulas (33)–(35) and (39) are the main ones for the engineering calculation of a shock absorber and make it possible to construct a deformation diagram and determine the holding capacity of the device for any coefficient of loading cycle asymmetry.


## 4. Conclusions

The idea of using the bending strain of load-bearingthin-walled links for shell dampers and shock absorbers is quite fruitful and will form the basis of a number of technical solutions. It should be noted that the shell shock absorber has both damping and shock-absorbing properties. However, for the qualitative implementation of the theoretical concepts on which the principle of the operation of the shock absorber is based, its constituent elements must have certain characteristic properties. Open shell requires a combination of low rigidity with a sufficient level of strength and durability. For the directional transformation of displacements, the filler must easily change its shape. On the other hand, in order to cause the open shell to deform under contact pressure, the filler material must have a high bulk modulus. These requirements are met by elastomers, which are also capable of frequency-dependent energy dissipation. The tribological characteristics of the filler-shell contact pair are selected in order to ensure the level of structural damping required in a particular operational situation.This study has three main innovation outcomes.(1)
A new design of the dry friction shock absorber is introduced. The device contains two spring sections and a friction module (an open shell with an elastic filler). The spring sections and the friction module work in parallel. The proposed shock absorber with compact transverse dimensions demonstrates good performance.(2)
To describe the deformation of a friction shock absorber, a mechanical-mathematical model of a shell with a cut along the generatrix, which is the main bearing link of the device under consideration, has been developed. An open isotropic shell was modeled by an equivalent continuous strongly orthotropic momentless shell. The elasticity characteristics of such an equivalent shell were determined on the basis of solving an additional problem of loading a cut shell with internal pressure.(3)
A technique for the quasistatic analysis of structural damping in nonmobile nonconservative shell systems with an elastic filler has been developed. The hysteresis loop for an open cylindrical shell with a deforming filler is described analytically. The distribution of stresses at each stage of the cyclic loading was studied, and the permissible loading was determined at which the operation of the shock absorber would be safe.Thus, the purpose of the study has been achieved. The results obtained make it possible to evaluate the behavior of friction shock absorbers with a working link in the form of an open cylindrical shell with an elastic filler and a spring connected in parallel under cyclic loading conditions and to carry out an engineering calculation of their rigidity and holding capacity, focused on the practical needs of use and efficient operation.In future studies, it is advisable to study the effect of frictional heat generation and wear in the contact interface on the properties of the shock absorber during long-term operation.The authors see good prospects for using the presented friction shock absorber in the mining, oil and gas, and construction industries.



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