Mathematical Formulation of Soft-Contact Problems for Various Rheological Models of Damper Wies

The paper deals with analysis of selected soft-contact problems in discrete mechanical systems. Elastic-dissipative rheological schemes representing dampers as well as the notion of unilateral constraints were used in order to model interaction between colliding bodies.Themathematical descriptions of soft-contact problems involving variational inequalities are presented.Themain finding of the paper is a method of description of soft-contact phenomenon between rigid object and deformable rheological structure by the system of explicit nonlinear differential-algebraic equations easy for numerical implementation. The results of simulations, that is, time histories of displacements and contact forces as well as hysteretic loops, are presented.


Introduction
Impact is a short-lived phenomenon of energy exchange between colliding bodies.The velocities of colliding bodies change rapidly and the reactions are impulsive in nature which means that the interactions are short-lived and reach large values.Classical theory of impact mechanics applied for rigid bodies assumes that the collision phenomenon results with discontinuous change of velocities and the reaction are modeled as impulses.However, such idealization may not be valid in many mechanical problems especially when we need to evaluate a history of reactions within a short time of collision.
In the model of impact of two elastic spheres presented by Panovko and Stronge [1,2] the local deformability of these spheres is assumed with use of nonlinear spring possessing the following property () fl  3/2 , where  denotes a parameter depending on the spheres' radii and material.The Hertz model of impact can be used for modelling of elastic collisions.
The analysis of impact and contact problems in discrete mechanical systems has received a great deal of attention in the literature [3][4][5].The mathematical description of such problems involves the notion of nonsmooth mechanics and needs a special numerical treatment [6][7][8].
In this paper we will apply rheological schemes representing deformable dampers and unilateral constraints for modeling of interaction between colliding bodies.The phenomenon of contact between rigid objects and deformable rheological structures is sometimes called "soft contact" [9].The models we analyze in this paper allow evaluation of reactions during the time of collision.
The soft-contact models analyzed in our paper involve two cases of interaction between rigid bodies.The first case concerns the impact of two rigid bodies through a deformable built-in damper.In order to simplify mathematical description we assumed that one of these bodies possesses an infinite mass.The second case of interaction applying soft-contact models describes a rigid body interaction with discrete modelling of compliance for the contact region.In such models the compliance of the contact region is modelled with use of rheological schemes (see [2]).
We will analyze both linear viscoelastic schemes and nonlinear elastoplastic and viscoelastoplastic models of dampers.Using unilateral constraints one can model contact interactions between bodies.The mathematical descriptions of softcontact problems will be presented.The notion of variational inequalities will be used in order to describe unilateral constraints and frictional properties of rheological models of dampers.
The main finding of the paper is a method we propose allowing description of the soft-contact problems with use of nonlinear explicit differential-algebraic equations.Numerical solution of such equations can be obtained using classical algorithms.We will show the results of numerical simulations in order to demonstrate the validity of the proposed formulation.

Soft-Contact Problem Formulation with Use of Viscoelastic Rheological Scheme
Let us consider the description of the soft-contact phenomenon in the system visualized in Figure 1.This system is composed of a rigid body (material point) possessing a mass  and of a mass-less linear viscoelastic rheological model of damper having stiffness parameters  1 and  2 and viscosity .The unilateral constraints are visualized in Figure 1 by two horizontal bold lines representing mass-less bumpers.We also assume the gravity load acting where  denotes gravity constant.
The coordinates , , and  shown in Figure 1 determine the displacement of the body and deformation of the rheological structure. denotes the force of mutual interaction between the body and the structure.The configuration shown in Figure 1 represents such a time instant when  = 0 and  = 0 and  = 0.In case of this configuration both bumpers are in contact but the springs' and dashpot's forces as well as the reaction  are equal to zero.
The description of the soft-contact problem for the analyzed system may be obtained by formulation of the equation of the body's motion and evolution of the structure's deformation.
The equations describing the soft-contact problem are as follows:  ż +  2  =  (1c) where (1a) describes the motion of the material point  while (1b) and (1c) represent the evolution of the structure's deformation.The system of equations (1d) defines the relationships between a coordinate  fl  −  and the reaction .It should be emphasized that despite the linearity of viscoelastic scheme to be analyzed, the system of equations ( 1) is nonlinear because of the form of constraints defined in (1d) involving variational inequality.We will prove that (1d) can be visualized via a mapping shown in Figure 2(a).Let us divide the set { ≥ 0} into two subsets { = 0} ∪ { > 0}.If  > 0, then the inequality (ũ − ) ≥ 0 ∀ũ ≥ 0 is satisfied only for  = 0.If  = 0, the inequality transforms to the following form ũ ≥ 0 ∀ũ ≥ 0 being satisfied for  ≥ 0. The result of the proof is shown in Figure 2(a).
Our objective is to transform (1a), (1b), (1c), and (1d) to the system of explicit nonlinear differential-algebraic equations.The expected system of equations should have the following form: where (2a) and (2b) can be easily obtained using (1a) and (1c), respectively.
The fundamental problem is to find the functions   and   describing time histories of displacement  and reaction , respectively.Let us note that (1b) can be rewritten in the following equivalent form: Equation ( 3) is a linear relation between  and  fl  − .Thus, using (3) and the mapping shown in Figure 2(a) leads to the solution as follows: Graphical solution leading to (4b) is visualized in Figure 2(b) where the dashed lines represent two possible locations of the function expressed by (3) for  −  > 0 and  −  < 0.

Elastic-Viscoplastic Rheological Scheme
This section is devoted to analysis of a soft-contact problem with use of elastic-viscoplastic rheological scheme shown in Figure 3.The scheme in Figure 3 was obtained using Figure 1 and replacing the spring  2 by the slider modelling dryfriction or plasticity phenomena [10].We assume that the slider force is denoted by  while the limit force is   .
The equations defining the soft-contact problem have the following form: The relationships expressed via (5e) describe constitutive characteristics of the slider.A graphical visualization of (5e) is shown in Figure 4 [8,11].In order to prove the equivalence of (5d) and the mapping in Figure 4 We will demonstrate that the system of equations (5a), (5b), (5c), (5d), and (5e) can be replaced by the following differential-algebraic equations: Let us note that the functions   and   were formulated in previous section (see (4a) and (4b)).Our objective in this section is to find the mappings   and   describing time history of velocity ż and friction force , respectively.Substituting (5b) into (5c) gives Using Figure 4 the left-hand side of ( 7) can be visualized in a graph as it was shown in Figure 5.The multivalued function shown in Figure 5 is a constitutive relationship of the system of two elements joined in parallel, the slider and the dashpot (see Figure 3).

Shock and Vibration
The unknown functions   and   can be determined using (7) and the graph shown in Figure 5.The results are as follows:

Elastoplastic Rheological Scheme
Let us analyze nonlinear elastoplastic rheological scheme shown in Figure 4.The model contains two elastic elements (springs  1 and  2 ) and one friction element (slider).We will demonstrate that the solution of this soft-contact problem is more complicated that it was in case of the systems analyzed in previous sections.
The system of equations describing the problem is similar to (5a), (5b), (5c), (5d), and (5e).The only difference is that we should replace (5c) by the following relation: Moreover, we will formulate a differential-algebraic system of equations similar to (6a), (6b), (6c), (6d), and (6e).The fundamental issue is to find the values of mappings describing time history of velocity ż and friction force .
Let us note that using (9) and the graph shown in Figure 4 representing constitutive relations of frictional element (slider), it is possible to evaluate ż and  if |− 2 | <   .The solution is as follows: It should be emphasized that evaluation of ż and  if | −  2 | =   is complicated and needs a special treatment.First of all one should establish additional relationships constituting so-called differential successions of relationships described in (5d) and (5e).The notion of differential successions was used by authors in various mechanical problems [12][13][14].Using differential successions we will define additional relations being satisfied by the time derivatives of variables in (5d) and (5e).
We will try to establish relationships between the rates u and Ṙ based on (5d) and the graph shown in Figure 2. We will formulate such relations in case of three subsets being defined below: Let us note that subsets  and  correspond to horizontal and vertical branches of the graph in Figure 2. The subset  corresponds to the origin of the graph in Figure 2.
It is easy to note that at points (, ) ∈ , the value  = 0 is constant; thus Ṙ = 0. On the other hand at the same points (, ) ∈ , the value of u can be arbitrary which gives u ∈ (−∞, +∞).A similar analysis leads to the conclusion that if (, ) ∈  then u = 0 and Ṙ ∈ (−∞, +∞).At points (, ) ∈  there exist two options.If  does not vary in time then u = 0 and Ṙ ≥ 0. The second case we should consider concerns a situation when  does not vary over time then Ṙ = 0 and u ≥ 0.
Summarizing the above analysis leads to the following relationships: The relationships described in ( 12) constitute time differential successions of the relations in (5d).The graphs of ( 12) are shown in Figure 7.
A similar analysis can be used in order to determine relationships between variables ż and Ṙ based on (5e) and the graph shown in Figure 4. We assume the following decomposition of (5e) into five subsets: If { ż , } ∈  1 then the value of ż = 0 does not change in time but the velocity of the slider force  is arbitrary; thus Ṫ ∈ (−∞, +∞).If { ż , } ∈  2 then one should consider two cases ż = 0 and Ṫ ≤ 0 or ż ≥ 0 and Ṫ = 0.If { ż , } ∈  3 then ż ≥ 0 and Ṫ = 0. Similarly, we can define the relationships { ż , Ṫ } in case of subsets  4 and  5 .
Summarizing all cases gives the following formula: Let us move back to the problem of evaluation of ż and  if | −  2 | =   .We will begin with the case  −  2  =   which is equivalent to the condition  =   .We will use the relationships given in (12) along with the following equation resulting from (9): Additionally, let us note that using (13b) gives ż ≥ 0 for  =   (subsets  2 and  3 ).Moreover, for subsets  2 and  3 the differential successions expressed in ( 14) can be visualized in one graph shown in Figure 8.If (, ) ∈  (see (11b)) then using (12) gives Ṙ = 0. Applying (15) along with the graph shown in Figure 8 gives ż = 0.If (, ) ∈  (see (11b)) then using (12) gives u = 0 ⇒ ẋ = ẏ .Applying (5b) leads to the following equation: Substituting ( 16) into (15) gives the following relationship: Using (17) and the graph shown in Figure 8 leads to the solution It can be proved that the solution expressed via (18) is valid also if (, ) ∈ .
Using a similar procedure it is possible to evaluate ż if  −  2  = −  which is equivalent to the condition  = −  .
Finally, the total set of differential-algebraic equations defining the problem of a material point  impacting on elastoplastic rheological structure is as follows:  where functions   and   are given in (4a) and (4b), respectively, while functions   and   have the following definitions:

Numerical Simulations
Computer simulations were carried out for the systems analysed in previous sections.The parameters of these systems are presented in Table 1 (compare Figures 1, 3, and 6).We assume that a material point of mass  falling freely in the earth's gravitational field impacts a rheological structure.The initial position of the material point equals   = −3 m while its initial velocity is V  = 5 m/sec.The solutions of nonlinear differential equations defining the soft-contact problems were obtained applying the classical 4th-order fixed-time step Runge-Kutta algorithm.
Figure 7: Graphs of mappings expressed via (12).Let us begin with the results obtained in case of viscoelastic system shown in Figure 1.Time history graphs of displacements , , and  are presented in Figure 9, while Figure 10 visualizes the reaction.These figures represent the solutions of (2a), (2b), (2c), and (2d) with initial conditions (0) =   , ẋ (0) = V  , and (0) = 0. Let us note that for time sections [0, 0.4 sec] and [1.4 sec, 2.4 sec] the reaction equals zero,  = 0 and () < (), which means that the bumpers are not in contact.In other time instants, the reaction is bigger than zero,  > 0 and () = ().Thus, the bumpers are in contact.
We can also note analyzing Figures 9 and 10 that the variables , , , and  tend to the following values: Graphs shown in Figure 11 represent the solutions (hysteretic loops) on planes (, ) and (, ).Analysing both loops one can see limit points (, ) = (981 N, 0.75 m) and (, ) = (981 N, 0.5 m).
In case of elastic-viscoplastic model shown in Figure 3 the solutions of (6a), (6b), (6c), (6d), and (6e) are presented in Figures 12,13,and 14.Let us note that for  > 1 sec the variable  does not change over time but the structure vibrates.During these vibrations there are short time instants when  = 0 (see Figure 13).During such time instants the bumpers are not in contact.
The first time period of loading for  < 1 sec is represented in hysteretic loops shown in Figure 14 by a thin line.The thick line in Figure 14 represents the second period of vibrations for  > 1 sec.
The solutions of (19a), (19b), (19c), (19d), and (19e) describing elastoplastic system shown in Figure 6 are presented in Figure 15 (displacements), Figure 16 (reaction), and Figure 17 (hysteretic loops).Let us note that previously analyzed graphs shown in Figure 12 are similar to that of Figure 15.After a short transient time period, the coordinate  is constant while  and  change over time.For such time periods when  <  (see Figure 15) and  = 0 (see Figure 16) the bumpers are not in contact.
Analyzing hysteretic loops shown in Figure 17 leads to the conclusion that the process of energy dissipation is formed in the first period of impact when  changes over time.This time period is represented in Figure 17 by broken line OABCD.The next period of vibrations when the structure deforms elastically is represented by section DE in Figure 17.Numerical examples analyzed in this section illustrate the solutions of the soft-contact problems defined via differentialalgebraic equations derived in Sections 2, 3, and 4. Based on these equations it is possible to carry out an optimization procedure in order to find an optimal set of parameters for rheological structures modelling energy dissipators.

Final Remarks
Discrete rheological schemes analyzed in this paper are used to model energy-absorbing devices in machinery, vehicles, and buildings.The scheme visualized in Figure 6 was considered in [12] as a model of rail bumper while a viscoplastic scheme, similar to that shown in Figure 3, was analyzed in [15] as a model of magnetorheological fluid damper for seismic retrofit of buildings.Finding the response of rheological models exhibiting elasticity, viscosity, and plasticity phenomena against impact loading seems to be very important in order to investigate dissipative capabilities of these systems.
The fundamental characteristics of each damper model are its hysteretic loop (see Figures 11(a), 14(a), and 17(a)).For example the hysteretic loop visualized in Figure 11(a) is typical for viscoelastic rheological models.The loops shown in Figures 14(a also used for the modelling of the contact region compliance between elastoplastic deformable bodies.Thanks to the method proposed in this paper the three analyzed soft-contact problems were formulated with use of nonlinear explicit differential-algebraic equations.As it was demonstrated numerical solution of such equations can be obtained using classical Runge-Kutta algorithm.Thus, there was no need to use more advanced solvers.Moreover, we did not apply any regularization procedure as it is sometimes suggested in the literature where multivalued mappings describing friction and plasticity phenomena (see Figure 4) are arbitrarily replaced by regular functions (see [16] and cited literature).
In case of viscoelastic and elastic-viscoplastic systems analyzed in Sections 2 and 3, respectively, explicit forms of equations defining the problems were obtained directly.In case of elastoplastic system presented in Section 4 explicit form of equations was obtained after formulation of additional relationships being satisfied by the time derivatives of variables defining the problem.
The rheological schemes analyzed in this paper were composed of finite number of elastic, viscous, and plastic elements.Because of strongly nonlinear nature of the analyzed problems, the method proposed herein cannot be simply generalized to any number of elements.Each scheme should be treated individually as it was shown analyzing three separate soft-contact problems in this paper.
It is worth mentioning that the method presented in this paper can also be applied in case of soft-contact viscoelastic problems described by fractional derivatives (see [17]).

Figure 10 :
Figure 10: Time history of reaction  in viscoelastic system shown in Figure 1.

Figure 12 :
Figure 12: Time history of displacements in elastic-viscoplastic system shown in Figure 3.

Figure 13 :Figure 14 :
Figure 13: Time history of reaction  in elastic-viscoplastic system shown in Figure 3.

Figure 15 :
Figure 15: Time history of displacements in elastoplastic system shown in Figure 6.

Figure 16 :
Figure 16: Time history of reaction  in elastoplastic system shown in Figure 6.