Mathematical Problems in Engineering

Mathematical Problems in Engineering / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 361296 | 19 pages | https://doi.org/10.1155/2009/361296

Theoretical Study of a Chain Sliding on a Fixed Support

Academic Editor: Jerzy Warminski
Received03 Jul 2009
Accepted07 Dec 2009
Published22 Mar 2010

Abstract

A chain sliding on a fixed support, made out of some elementary rheological models (dry friction element and linear spring) can be covered by the existence and uniqueness theory for maximal monotone operators. Several behavior from quasistatic to dynamical are investigated. Moreover, classical results of numerical analysis allow to use a numerical implicit Euler scheme.

1. Introduction

This paper is the next step of a series of previous works dealing with modelling of discrete mechanical systems with finite number of degrees of freedom involving assemblies of classical smooth constitutive elements (in the mechanical point of view they correspond to linear or non linear springs, dashpots) and nonsmooth ones mainly based on St-Venant Elements. Let us cite basic rheological models [1], with different applications and developpements [27]. Delay or stochastic frame have also been investigated in [810].

In this paper we examine a new model: it can be associated with motion of a discretized beam “sliding” on soil. We do not give more details on this discretization.

This paper is organized as follows in Section 2, the model is described. In Section 3, the general model is adapted to different dynamical, semi-dynamical or quasistatic cases. In Section 4, existence and uniqueness is addressed. In Section 5, numerical scheme is described and its convergence obtained.

2. Description of the Model

We refer to previous works for description of some rheological models (see for example [1, 6]).

We consider the model of Figure 2. (𝑚𝑖)1𝑖𝑛 (with 𝑚𝑖0) correspond to masses, (𝑘𝑖)0𝑖𝑛 to stiffness, and (𝛼𝑖)1𝑖𝑛 to St-Venant elements thresholds.

The reader is referred to Appendix A.

Let 𝜎 be the multivalued graph sign defined by (see Figure 1(a)).

[]𝜎(𝑥)=1if𝑥<0,1if𝑥>0,1,1if𝑥=0.(2.1) According to [11], this graph is maximal monotone. Therefore:

𝑥,𝜎(𝑥)=𝜕|𝑥|.(2.2)

Let us assume (see Figure 3) the following

(i)This mechanical system is submitted to external forces (𝐹𝑖)0𝑖𝑛+1: 𝐹0 is exerted on the spring with stiffness 𝑘0; For 1𝑖𝑛, 𝐹𝑖 is exerted on material point of mass 𝑚𝑖; 𝐹𝑛+1 is exerted on the spring with stiffness 𝑘𝑛+1. (ii)For 1𝑖𝑛, 𝑔𝑖 is the friction force exerted by the support of the 𝑖th St-Venant.(iii)For 0𝑖𝑛, 𝑓𝑖 is elastic linear force exerted by the 𝑖th spring.(iv)For 0𝑖𝑛, 𝑢𝑖 is the displacement of the 𝑖th spring.(v)For 1𝑖𝑛, 𝑣𝑖 is the displacement of the 𝑖th St-Venant element.(vi)𝜉 is the displacement of the spring with stiffness 𝑘0.(vii)𝑥 is the displacement of the material point of mass 𝑚𝑛.

These two last notations are justified by the study of particular cases in the next sections.

The different equations of the model are successively given by the fundamental Newton law:

𝑖{1,,𝑛},𝑚𝑖̈𝑣𝑖=𝐹𝑖+𝑓𝑖1𝑓𝑖+𝑔𝑖,(2.3a) by the constitutive laws of linear springs:

𝑖{0,,𝑛},𝑓𝑖=𝑘𝑖𝑢𝑖,(2.3b) by the constitutive of laws St-Venant elements:

𝑖{1,,𝑛},𝑔𝑖𝛼𝑖𝜎̇𝑣𝑖,(2.3c) by the geometrical connexions:

𝑖{0,,𝑛1},𝜉+𝑢0+𝑢1++𝑢𝑖=𝑣𝑖+1,(2.3d)𝜉+𝑢0+𝑢1++𝑢𝑛=𝑥,(2.3e) and finally by the boundary conditions:

𝐹0=𝑓0,𝐹(2.3f)𝑛+1=𝑓𝑛.(2.3g)

We can observe that (2.3d)–(2.3e) are equivalent to

𝜉+𝑢0=𝑣1,(2.4a)𝑖{1,,𝑛1},𝑣𝑖+1𝑣𝑖=𝑢𝑖,(2.4b)𝑥𝑣𝑛=𝑢𝑛.(2.4c)

Now, we study systems (2.3a), (2.3b), (2.3c), (2.3f), (2.3g), and (2.4a)–(2.4c).

3. Transformations of Equations

Now, as in [1, 6], we transform system (2.3a)-(2.3b)-(2.3c)-(2.3f)-(2.3g)-(2.4b)-(2.4c) to rewrite it under the usual form (A.7) according to different kinds of problem and of boundary conditions.

Let us assume that the external forcing 𝐹1,,𝐹𝑛 are known.

3.1. Dynamical Case

We assume in this section that

𝑖{1,,𝑛},𝑚𝑖>0.(3.1)

Equations (2.3a)-(2.3b)-(2.3c)-(2.4a)–(2.4c) imply

𝑚1̈𝑣1+𝛼1𝜎̇𝑣1𝑘0𝑘𝜉+0+𝑘1𝑣1𝑘1𝑣2𝐹1,(3.2a)𝑖{2,,𝑛1},𝑚𝑖̈𝑣𝑖+𝛼𝑖𝜎̇𝑣𝑖𝑘𝑖1𝑣𝑖1+𝑘𝑖1+𝑘𝑖𝑣𝑖𝑘𝑖𝑣𝑖+1𝐹𝑖𝑚,(3.2b)𝑛̈𝑣𝑛+𝛼𝑛𝜎̇𝑣𝑛𝑘𝑛1𝑣𝑛1+𝑘𝑛1+𝑘𝑛𝑣𝑛𝑘𝑛𝑥𝐹𝑛.(3.2c)

3.1.1. Clamped Mechanical System

We assume that our mechanical system is clamped at its two extremities so that we can write the boundary conditions:

𝜉=0,𝑥=0,(3.3a)andthereactions𝐹0and𝐹𝑛+1areunknown.(3.3b)

We set, for all 𝑞,

𝑘𝐾(𝑞)=0+𝑘1𝑘100000𝑘1𝑘1+𝑘2𝑘20000𝑘2𝑘2+𝑘3𝑘3000000𝑘𝑞2𝑘𝑞2+𝑘𝑞1𝑘𝑞1000𝑘𝑞1𝑘𝑞1+𝑘𝑞𝑞().(3.4) Thus, by setting

𝑉=𝑡𝑣1,,𝑣𝑛𝑛,(3.5a)𝐹=𝑡𝐹1,,𝐹𝑛𝑛𝑚,(3.5b)𝐾=𝐾(𝑛),(3.5c)=1000𝑚20000𝑚𝑛(3.5d) and defining the maximal monotone operator 𝒜 by

𝒜𝑣1,,𝑣𝑛=𝛼1𝜎𝑣1××𝛼𝑛𝜎𝑣𝑛,(3.5e) equations (3.2a)–(3.2c) imply the system of equations

𝑀̈̇𝑉+𝒜𝑉+𝐾𝑉𝐹.(3.6) Reactions 𝐹0 and 𝐹𝑛+1 can be determined thanks to (2.3f)-(2.3g) which give

𝐹0=𝑘0𝑣1,𝐹(3.7a)𝑛+1=𝑘𝑛𝑣𝑛.(3.7b)

Set

𝐼𝑝=2𝑛,(3.8a)𝑀=𝑛0012𝑛(),(3.8b) where 𝐼𝑛 is the identity of 𝑛() and for 𝑡, 𝑋=(𝑉1,𝑉2)2𝑛, with 𝑉2=(𝑉2,1,,𝑉2,𝑛),

𝒢𝑉𝑡,1,𝑉2=𝑉21𝐹1𝐾𝑉1𝜙𝑉,(3.8c)1,𝑉2=𝑛𝑖=1𝛼𝑖||𝑉2,𝑖||.(3.8d) Then, the system (3.6) is equivalent to (A.7) (see Appendix A).

Reciprocally, if (3.6) and (3.7a)-(3.7b) hold, we define 𝑥, 𝜉, (𝑢𝑖)0𝑖𝑛, (𝑓𝑖)0𝑖𝑛, and (𝑔𝑖)1𝑖𝑛 successivelly by

(2.3b),(2.4a)-(2.4c),(3.3a)-(3.3b),𝑖{1,,𝑛},𝑔𝑖=𝑚𝑖̈𝑣𝑖𝐹𝑖𝑓𝑖1+𝑓𝑖.(3.9) Then, we can deduce (2.3a), (2.3b), (2.3c), (2.3f), (2.3g), (2.4b), and (2.4c).

3.1.2. Clamped-Free Mechanical System

We assume that our mechanical system is clamped at its left extremity and free at its right extremity so that we can write the boundary condition:

𝜉=0,(3.10a)reaction𝐹0isunknown,(3.10b)displacement𝑥isunknown,(3.10c)andexternalforcing𝐹𝑛+1isknown.(3.10d)

As in Section 3.1.1, by setting

𝑉=𝑡𝑣1,,𝑣𝑛𝑛,(3.11a)𝐹=𝑡𝐹1,,𝐹𝑛1,𝐹𝑛𝐹𝑛+1𝑛𝑘,(3.11b)𝐾=0+𝑘1𝑘100000𝑘1𝑘1+𝑘2𝑘20000𝑘2𝑘2+𝑘3𝑘3000000𝑘𝑛2𝑘𝑛2+𝑘𝑛1𝑘𝑛1000𝑘𝑛1𝑘𝑛1𝑛(),(3.11c)and𝒜asin(3.5d)-(3.5e),(3.11d) we can prove that equations (3.2a)–(3.2c) imply the system of equations

𝑀̈̇𝑉+𝒜𝑉+𝐾𝑉𝐹.(3.12) Reactions 𝐹0 and displacement 𝑥 can be determined thanks to

𝐹0=𝑘0𝑣1,𝐹(3.13a)𝑥=𝑛+1𝑘𝑛+𝑣𝑛.(3.13b)

As in Section 3.1.1, let us set

𝑝=2𝑛,(3.14a)𝑀and𝜙denedby(3.8b)-(3.8d),(3.14b) and for 𝑡, 𝑋=(𝑉1,𝑉2)2𝑛, with 𝑉2=(𝑉2,1,,𝑉2,𝑛),

𝒢𝑉𝑡,1,𝑉2=𝑉21𝐹1𝐾𝑉1.(3.14c) Then, system (3.12) is equivalent to (A.7).

As in Section 3.1.1, reciprocally, if (3.12) and (3.13a)-(3.13b) hold, we define 𝑥, 𝜉, (𝑢𝑖)0𝑖𝑛, (𝑓𝑖)0𝑖𝑛, and (𝑔𝑖)1𝑖𝑛 successivelly by

(2.3b),(2.4a)-(2.4c),(3.3a)-(3.3b),Lastequationof(3.9).(3.15) Then, we can deduce (2.3a), (2.3b), (2.3c), (2.3f), (2.3g), (2.4b), and (2.4c).

3.2. Semi-Dynamical Case

In this section, we assume that 𝑖{1,,𝑛1},𝑚𝑖𝑚=0,(3.16a)𝑛=𝑚>0.(3.16b)

Equation (3.2b) implies

𝑖{2,,𝑛1},𝛼𝑖𝜎̇𝑣𝑖+𝑔𝑖0,(3.17a) with

𝑖{2,,𝑛1},𝑔𝑖=𝐹𝑖𝑘𝑖1𝑣𝑖1+𝑘𝑖1+𝑘𝑖𝑣𝑖𝑘𝑖𝑣𝑖+1.(3.17b) As in [6, 7], we introduce 𝛽, the inverse graph of 𝜎 (in the sens of [11], see Figure 1(b)):

][][,][,𝛽(𝑥)=if𝑥,11,+{0}if𝑥1,1if𝑥=1,+if𝑥=1.(3.18) We have

𝑥,𝛽(𝑥)=𝜕𝜓[1,1](𝑥),(3.19) where 𝜕𝜓[1,1] is the convex indicatrix function of the convex domain [1,1]. Thus, (3.17a) is equivalent to

̇𝑣𝑖{2,,𝑛1},𝑖+𝜕𝜓[𝛼𝑖,𝛼𝑖]𝑔𝑖0.(3.20) Similarly, (3.2a) gives

̇𝑣1+𝜕𝜓[𝛼1,𝛼1]𝑔10,(3.21a) with

𝑔1=𝐹1𝑘0𝑘𝜉+0+𝑘1𝑣1𝑘1𝑣2.(3.21b) and (3.2c) gives

𝑚̈𝑣𝑛+𝛼𝑛𝜎̇𝑣𝑛𝑘𝑛1𝑣𝑛1+𝑘𝑛1+𝑘𝑛𝑣𝑛𝑘𝑛𝑥𝐹𝑛.(3.22)

3.2.1. Clamped Mechanical System

We assume that our mechanical system is clamped at its two extremities so that we can write the boundary conditions (3.3a)-(3.3b). As in [6, 7], let us set 𝑉=𝑡𝑣1,,𝑣𝑛1𝑛1,(3.23a)𝐺=𝑡𝑔1,,𝑔𝑛1𝑛1,(3.23b)𝐹=𝑡𝐹1,,𝐹𝑛1𝑛1,(3.23c)𝑍=𝑡𝐹1,,𝐹𝑛2,𝐹𝑛1+𝑘𝑛1𝑣𝑛𝑛1𝛼,(3.23d)𝐶=1,𝛼1××𝛼𝑛1,𝛼𝑛1𝑛1,(3.23e)𝐾=𝐾(𝑛1)𝑛1(),(3.23f) where 𝐾(𝑞) is defined by (3.4). Thus, according to (3.17b)–(3.21b), we have

𝐺=𝐾𝑉𝑍,(3.24) and from (3.17a)–(3.21a) we can write

̇𝑉+𝜕𝜓𝐶(𝐺)0,(3.25) Under the assumption

𝑘𝑛10𝑖{1,,𝑛1},𝑘𝑖>0,(3.26) the matrix 𝐾 is symmetric definite positive (see proof in Lemma B.1 of Appendix B), so that

𝐾𝑉=1(𝐺+𝑍),(3.27) and (3.25) gives

𝐾1̇̇𝑍𝐺++𝜕𝜓𝐶(𝐺)0,(3.28) which is equivalent to

̇𝐺+𝐾𝜕𝜓𝐶̇(𝐺)𝑍.(3.29) For 𝑞 integer and 𝑢 vector of 𝑚, we denote by

[𝑢]𝑞(3.30) the  𝑞th  component  of  𝑢. Equation (3.22) gives

𝑚̈𝑣𝑛+𝛼𝑛𝜎̇𝑣𝑛𝑘𝑛1𝑣𝑛1+𝑘𝑛1+𝑘𝑛𝑣𝑛𝐹𝑛,(3.31) which can be rewritten under the following form:

̈𝑣𝑛+𝛼𝑛𝑚𝜎̇𝑣𝑛𝑘𝑛1𝑚𝐾1(𝐺+𝑍)𝑛1+𝑘𝑛1+𝑘𝑛𝑚𝑣𝑛𝐹𝑛𝑚.(3.32) Let 𝑢 be the vector of 𝑛1 defined by

𝑢=𝑡(0,,0,1).(3.33) Note that

𝑍=𝐹+𝑘𝑛1𝑣𝑛𝑢.(3.34) We set 𝑝=𝑛+1,(3.35a)𝑀=𝐾00𝐼2𝑛+1(),(3.35b) and for all 𝑡, 𝐺𝑛1, 𝑎,𝑏, 𝑋=𝑡(𝐺,𝑎,𝑏)̇𝒢(𝑡,𝑋)=𝐹𝑘𝑛1𝑏𝐹𝑏𝑢𝑛𝑚+𝑘𝑛1𝑚𝐾1(𝐺+𝐹+𝑘𝑛1𝑎𝑢)𝑛1𝑘𝑛1+𝑘𝑛𝑚𝑎(3.35c)𝜙(𝑋)=𝜓[𝛼1,𝛼1]××[𝛼𝑛1,𝛼𝑛1]×{0}×{0}𝛼(𝑋)+𝑛𝑚||𝑏||.(3.35d) Then, system (3.29)–(3.32) is equivalent to (A.7).

Reactions 𝐹0 and 𝐹𝑛+1 can be determined thanks to 𝐹0=𝑘0𝐾1(𝐺+𝐹+𝑘𝑛1𝑎𝑢)1,𝐹(3.36a)𝑛+1=𝑘𝑛𝑎.(3.36b)

Reciprocally, as in Section 3.1.1, if (3.29)–(3.32) hold, we can determine 𝐺 and 𝑍 thanks to

𝐺=𝑡[𝑋]1[𝑋],,𝑛1,𝑍=𝑘𝑛1[𝑋]𝑛𝑢+𝐹.(3.37) then we can calculate 𝑉 thanks to (3.27). Successively, 𝑥, 𝜉, (𝑢𝑖)0𝑖𝑛, and (𝑓𝑖)0𝑖𝑛 are defined by

(2.3b),(2.4a)-(2.4c),(3.3a)-(3.3b).(3.38) Then, we can deduce (2.3a), (2.3b), (2.3c), (2.3f), (2.3g), (2.4b), and (2.4c).

3.2.2. Clamped-Free Mechanical System

We assume that our mechanical system is clamped at its left extremity and free at its right extremity so that we can write boundary condition (3.10a)–(3.10d).

The calculus are similar to those of Section 3.2.1; Equation (3.29) holds and (3.31) is replaced by

𝑚̈𝑣𝑛+𝛼𝑛𝜎̇𝑣𝑛𝑘𝑛1𝑣𝑛1+𝑘𝑛1𝑣𝑛+𝐹𝑛+1𝐹𝑛.(3.39) Using notations (3.23a)–(3.23f), we obtain the system (A.7), where we set 𝑝=𝑛+1,(3.40a)𝑀and𝜙aredenedby(3.35b)and(3.35d),(3.40b) and for all 𝑡, 𝐺𝑛1, 𝑎,𝑏, 𝑋=𝑡(𝐺,𝑎,𝑏),

̇𝒢(𝑡,𝑋)=𝐹𝑘𝑛1𝑏𝐹𝑏𝑢𝑛𝐹𝑛+1𝑚+𝑘𝑛1𝑚𝐾1(𝐺+𝐹+𝑘𝑛1𝑎𝑢)𝑛1𝑘𝑛1𝑚𝑎𝑛+1().(3.40c) The reaction 𝐹0 and the displacement 𝑥 can be determined thanks to (3.36a) and

𝐹𝑥=𝑛+1𝑘𝑛+𝑎.(3.41)

3.3. Quasistatic Case

In this section, we assume that

𝑖{1,,𝑛},𝑚𝑖=0.(3.42)

As it has been previously noticed, (3.17a)-(3.17b) and (3.21a)-(3.21b) are not modified, and (3.22) gives

̇𝑣𝑛+𝜕𝜓[𝛼𝑛,𝛼𝑛]𝑔𝑛0,(3.43a) with

𝑔𝑛=𝐹𝑛𝑘𝑛1𝑣𝑛1+𝑘𝑛1+𝑘𝑛𝑣𝑛𝑘𝑛𝑥.(3.43b)

3.3.1. Clamped Mechanical System

We assume that our mechanical system is clamped at its two extremities so that we can write the boundary conditions (3.3a)-(3.3b).

As in Section 3.2.1, following [6, 7], we set 𝑉=𝑡𝑣1,,𝑣𝑛𝑛,(3.44a)𝐺=𝑡𝑔1,,𝑔𝑛𝑛,(3.44b)𝐹=𝑡𝐹1,,𝐹𝑛𝑛𝛼,(3.44c)𝐶=1,𝛼1××𝛼𝑛,𝛼𝑛𝑛,(3.44d)𝐾=𝐾(𝑛)𝑛(),(3.44e)where 𝐾(𝑞) is defined by (3.4).

Thus, we have ̇𝐺=𝐾𝑉𝐹,𝑉+𝜕𝜓𝐶(𝐺)0(3.45) Under assumption

𝑘𝑛0,𝑖{1,,𝑛1},𝑘𝑖>0,(3.46) the matrix 𝐾 is symmetric definite positive (see proof in Lemma B.1), so that

𝑉=𝐾1̇(𝐺+𝐹),(3.47)𝐺+𝐾𝜕𝜓𝐶̇(𝐺)𝐹.(3.48) We set 𝑝=𝑛,(3.49a)𝑀=𝐾𝑛(),(3.49b) and, for all 𝑡, for all 𝑋𝑛̇𝒢(𝑡,𝑋)=𝐹,(3.49c)𝜙(𝑋)=𝜓𝐶(𝑋).(3.49d) Then, the system (3.48) is equivalent to (A.7).

Reactions 𝐹0 and 𝐹𝑛+1 can be determined thanks to 𝐹0=𝑘0𝐾1(𝐺+𝐹)1,𝐹(3.50a)𝑛+1=𝑘𝑛𝐾1(𝐺+𝐹)𝑛.(3.50b)

3.3.2. Clamped-Free Mechanical System

We assume that our mechanical system is clamped at its left extremity so that we can write the boundary condition (3.10a) and (3.10b). Boundary conditions for its right extremity is given later.

The calculus is similar to those of [6, 7].

(i)  First Case: Displacement-Force Model
We assume that the displacement 𝑥 is known and that the force 𝐹𝑛+1 is unknown.
We introduce 𝑉, 𝐺, 𝐶 and matrix 𝐾 defined by (3.44a), (3.44b), (3.44d), and (3.44e) and 𝐹 defined by
𝐹=𝑡𝐹1,,𝐹𝑛1,𝐹𝑛+𝑘𝑛𝑥𝑛,(3.51) and we obtain, as in Section 3.3.1, ̇𝐺=𝐾𝑉𝐹,(3.52)𝐺+𝐾𝜕𝜓𝐶̇(𝐺)𝐹.(3.53) By setting 𝑝=𝑛,(3.54a)𝑀and𝜙aredenedby(3.49b)and(3.49d),(3.54b) and, for all 𝑡, for all 𝑋𝑛, ̇𝒢(𝑡,𝑋)=𝐹,(3.54c) we remark that system (3.53) is equivalent to (A.7).
Reactions 𝐹0 and 𝐹𝑛+1 can be determined thanks to
𝐹𝑛0=𝑘0𝐾1(𝐺+𝐹)1,𝐹(3.55a)𝑛+1=𝑘𝑛𝑥+𝑘𝑛𝐾1(𝐺+𝐹)𝑛.(3.55b)

(ii)  Second Case: Force-Displacement Model
We assume that external forcing 𝐹𝑛+1 are known and displacement 𝑥 is unknown.
The calculus are similar to the previous case.
Equation (3.43b) is replaced by
𝑔𝑛=𝐹𝑛+1𝐹𝑛𝑘𝑛1𝑣𝑛1+𝑘𝑛1𝑣𝑛.(3.56) Following the same method, we introduce 𝑉, 𝐺, and 𝐶 defined by (3.44a)-(3.44b)-(3.44d), and matrix 𝐾 defined by (3.11c). Vector 𝐹 is defined by 𝐹=𝑡𝐹1,,𝐹𝑛1,𝐹𝑛𝐹𝑛+1.(3.57)
So, (3.52) is replaced by
𝐺=𝐾𝑉𝐹,(3.58) and (3.48) is replaced by ̇𝐺+𝐾𝜕𝜓𝐶̇(𝐺)𝐹.(3.59)

Remark 3.1. As in [6], let us notice that matrix 𝐾 defined by (3.11c) for force-displacement model corresponds to matrix 𝐾(𝑛) for displacement-force model defined by (3.4) with 𝑘𝑛=0.(3.60)

According to previous remark, assumption

𝑖{1,,𝑛1},𝑘𝑖>0(3.61) and Lemma B.1 ensure that matrix 𝐾 is symmetric definite positive. Thus, like previously, the system is equivalent to

̇𝐺+𝐾𝜕𝜓𝐶̇(𝐺)𝐹.(3.62) By giving 𝑝, 𝜙 defined by (3.49a)–(3.49d), 𝒢 defined by for all 𝑡, for all 𝑋𝑛,

̇𝒢(𝑡,𝑋)=𝐹,(3.63a) and 𝑀 defined by

𝑀=𝐾𝑛(),(3.63b) we remark that system (3.62) is equivalent to (A.7). Reactions 𝐹0 and displacement 𝑥 can be determined thanks to

𝐹0=𝑘0𝐾1(𝐺+𝐹)1,𝐹(3.64a)𝑥=𝑛+1𝑘𝑛+𝐾1(𝐺+𝐹)𝑛.(3.64b)

4. Existence of Uniqueness Results

Thus, as proved in [1], all the systems of Section 3 can be written under the form (A.7) and, according to Proposition A.1 (see Appendix A), have a unique solution. For all systems, Table 1 provides the corresponding integer 𝑝, function 𝜙, and matrix 𝑀. It is easy to prove that 𝜙 is convex proper and lower semi-continuous function on 𝑝 and that 𝑀 is symmetric positive definite.


System 𝑝 function 𝜙 matrix 𝑀

(3.6) 2 𝑛 𝜙 ( 𝑉 1 , 𝑉 2 ) = 𝑛 𝑖 = 1 𝛼 𝑖 | 𝑉 2 , 𝑖 | ( 𝐼 𝑛 0 0 1 )
(3.12) 2 𝑛 𝜙 ( 𝑉 1 , 𝑉 2 ) = 𝑛 𝑖 = 1 𝛼 𝑖 | 𝑉 2 , 𝑖 | ( 𝐼 𝑛 0 0 1 )
(3.29)–(3.32) 𝑛 + 1 𝜙 ( 𝑡 , ( 𝐺 , 𝑎 , 𝑏 ) ) = 𝜓 [ 𝛼 1 , 𝛼 1 ] × × [ 𝛼 𝑛 1 , 𝛼 𝑛 1 ] × { 0 } × { 0 } 𝛼 ( 𝐺 , 𝑎 , 𝑏 ) + 𝑛 𝑚 | 𝑏 | ( 𝐾 0 0 𝐼 2 )
(3.29)–(3.39) 𝑛 + 1 𝜙 ( 𝑡 , ( 𝐺 , 𝑎 , 𝑏 ) ) = 𝜓 [ 𝛼 1 , 𝛼 1 ] × × [ 𝛼 𝑛 1 , 𝛼 𝑛 1 ] × { 0 } × { 0 } 𝛼 ( 𝐺 , 𝑎 , 𝑏 ) + 𝑛 𝑚 | 𝑏 | ( 𝐾 0 0 𝐼 2 )
(3.48) 𝑛 𝜙 ( 𝑋 ) = 𝜓 [ 𝛼 1 , 𝛼 1 ] × × [ 𝛼 𝑛 , 𝛼 𝑛 ] ( 𝑋 ) 𝐾
(3.53) 𝑛 𝜙 ( 𝑋 ) = 𝜓 [ 𝛼 1 , 𝛼 1 ] × × [ 𝛼 𝑛 , 𝛼 𝑛 ] ( 𝑋 ) 𝐾
(3.62) 𝑛 𝜙 ( 𝑋 ) = 𝜓 [ 𝛼 1 , 𝛼 1 ] × × [ 𝛼 𝑛 , 𝛼 𝑛 ] ( 𝑋 ) 𝐾

5. Convergence of Numerical Scheme

All the models examined here can be written under the form (A.7). Based on [1, 12], general writing of the implicit Euler scheme corresponds to

𝑋𝑛{0,,𝑁1},𝑛+1𝑋𝑛𝑋+𝑀𝜕𝜙𝑛𝑡𝒢𝑛,𝑋𝑛,𝑋0=𝜉.(5.1) with time step , discretized time 𝑡𝑛=𝑛, and approximations 𝑋0,,𝑋𝑁 of the exact solution provided by the numerical scheme. Previous studies [12] ensure that this numerical scheme is convergent with order 1/2 (systems (3.6), (3.12), (3.29)–(3.32), and (3.29)–(3.39)) or 1 (systems (3.48), (3.53), and (3.62)).

In practice for computation of solutions, three cases can be distinguished, based on further expression of 𝑋𝑛+1:

𝑋𝑛+1=[]𝐼+𝑀𝜕𝜙1𝑋𝑛𝑡+𝒢𝑛,𝑋𝑛,(5.2) where 𝐼 is the identity and [𝐼+𝑀𝜕𝜙]1 is the inverse of the graph 𝐼+𝑀𝜕𝜙 (see [11]). According to [11], [𝐼+𝑀𝜕𝜙]1 is a monovalued operator, providing a unique solution 𝑋𝑛+1𝑝. In the first case, effective computations of 𝑋𝑛+1 associated with diagonal matrix 𝑀 is explicit: this situation corresponds to systems (3.6) and (3.12). In the second case, 𝜙 is defined as the indicatrix function of a closed convex set: this situation corresponds to systems (3.48), (3.53), and (3.62). Effective computation of 𝑋𝑛1 is given by the projection of a given vector on a closed convex set (see [6]). In the third case (for systems (3.29)–(3.32) and (3.29)–(3.39)), 𝜙 is involving indicatrix function of a closed convex set and a norm function. In such case, computation of 𝑋𝑛+1 leads to the following problem: according to [11], 𝑋𝑛+1 is the solution of minimization problem: considering 𝑀 the norm define by the inner product given by (A.4)

𝑥𝑀=𝑡𝑥𝑀1𝑍𝑥,(5.3a)𝑛=𝑋𝑛𝑡+𝒢𝑛,𝑋𝑛(5.3b) solve

min𝑥𝐷(𝜕𝜙)1𝜙(𝑥)+2𝑥𝑍𝑛2𝑀(5.3c) and such problem can be solved in practice following efficient algorithms [13].

6. Conclusion

In this paper, a mechanical system involving finite degrees of freedom and nonsmooth terms have been investigated from the mechanical point of view. Dynamical, semi-dynamical, and quasistatic modeling have been established. The main results are theoretical ones:

(i)all the problems are well posed;(ii)it has been explained how a numerical approximation of solutions can be effectively computed.

All the mechanical systems have been considered in a deterministic frame. Theoretical results and corresponding effective computations could be extended to the stochastic frame.

Appendices

A. A Few Theoretical Reminders about the Class of Maximal Monotone Differential Equations Used

The reader is referred to [11]. Let , be scalar product on 𝑝. If 𝜙 is a convex proper and lower semi-continuous function from 𝑝 to ],+], we can define its subdifferential 𝜕𝜙 by

𝑦𝜕𝜙(𝑥)𝑝,𝜙(𝑥+)𝜙(𝑥)𝑦,,𝐷(𝜕𝜙)={𝑥𝜕𝜙(𝑥)}.(A.1) Moreover, 𝜕𝜙 is a maximal monotone graph in 𝑝×𝑝.

If 𝐶 is a closed convex nonempty subset of 𝑝, we denote by 𝜓𝐶 the indicatrix of 𝐶 defined by

𝑥𝐶,𝜓𝐶(𝑥)=0,if𝑥𝐶,+,if𝑥𝐶.(A.2) In this particular case, 𝜕𝜓𝐶, which is the subdifferential of 𝜓𝐶, is given by

(𝑥,𝑦)𝐶×𝑝,𝑦𝜕𝜓𝐶(𝑥)𝑧𝐶,𝑦,𝑥𝑧0,(A.3a)𝑥𝐶,𝜕𝜓𝐶(𝑥)=.(A.3b) The domain of the maximal monotone operator 𝜕𝜓𝐶 is equal to 𝐶.

We observe that if 𝑝 is equipped with its canonical scalar product ,, and with another scalar product,

𝑥,𝑦𝑀=𝑡𝑥𝑀1𝑦,(A.4) where 𝑀 is symmetric positive definite, then we can relate the subdifferential 𝜕𝜙 of 𝜙 relatively to the canonical scalar product , and the subdifferential 𝜕𝑀𝜙 relatively to ,𝑀 by

𝜕𝑀𝜙(𝑥)=𝑀𝜕𝜙(𝑥).(A.5)

We give now the general mathematical formulation of our problem. We assume that 𝑇 is strictly positive and that 𝒢 is a function from [0,𝑇]×𝑝 to 𝑝 which is Lipschitz continuous with respect to its second argument, that is, there exists 𝜔0 such that

[]𝑡0,𝑇,𝑋1,𝑋2𝑝,𝒢𝑡,𝑋1𝒢𝑡,𝑋2𝑋𝜔1𝑋2.(A.6a) Moreover, we assume that

𝑌𝑝,𝒢(,𝑌)𝐿(0,𝑇;𝑝).(A.6b)

Proposition A.1. If the matrix 𝑀 is symmetric positive definite and 𝜙 is convex proper and lower semicontinuous on 𝑝, under assumptions (A.6a)-(A.6b), for all 𝜉𝐷(𝜕𝜙), there exists a unique function 𝑋 in 𝑊1,1(0,𝑇;𝑝) such that ̇][,𝑋(𝑡)+𝑀𝜕𝜙(𝑋(𝑡))𝒢(𝑡,𝑋(𝑡))a.e.on0,𝑇𝑋(0)=𝜉,(A.7) where the differential inclusion can be written as an inequality: for almost every 𝑡 in ]0,𝑇[, 𝑝𝐺̇𝑋,𝜙(𝑋(𝑡)+)𝜙(𝑋(𝑡))(𝑡,𝑋(𝑡))(𝑡),𝑀.(A.8)

Proof of this result can be found in [1, Proposition 3.1], based on [11, Proposition 3.13, page 107] and (A.5).

B. 𝐾(𝑞) Defined by (3.4) Is Symmetric Definite Positive

Lemma B.1. Under assumption 𝑘𝑞0,𝑖{1,,𝑞1},𝑘𝑖>0,(B.1) matrix 𝐾(𝑞) defined by (3.4) is symmetric definite positive.

Proof. We have, for all 𝑋=(𝑥1,,𝑥𝑞)𝑞, 𝑡𝑋𝐾(𝑞)𝑋=𝑞𝑖=1𝑘𝑖1+𝑘𝑖𝑥2𝑖2𝑞𝑖=1𝑘𝑖𝑥𝑖𝑥𝑖+1,=𝑘0𝑥21+𝑞𝑖=2𝑘𝑖1𝑥2𝑖+𝑞1𝑖=1𝑘𝑖𝑥2𝑖+𝑘𝑞1𝑥2𝑞12𝑞1𝑖=1𝑘𝑖𝑘𝑖+1𝑥𝑖+1,=𝑘0𝑥21+𝑞1𝑖=1𝑘𝑖𝑥𝑖+1𝑥𝑖2+𝑘𝑞𝑥2𝑞.(B.2) Under assumption (B.1), 𝑡𝑋𝐾(𝑞)𝑋=0 then implies 𝑋=0.

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Copyright © 2009 Jérôme Bastien and Claude-Henri Lamarque. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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