Theoretical Study of a Chain Sliding on a Fixed Support

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 [2–7]. Delay or stochastic frame have also been investigated in [8–10].

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. (𝑚_𝑖)_{1≤𝑖≤𝑛} (with 𝑚_𝑖≥0) correspond to masses, (𝑘_𝑖)_{0≤𝑖≤𝑛} to stiffness, and (𝛼_𝑖)_{1≤𝑖≤𝑛} 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)).

(a) The graph 𝜎

(b) The graph 𝛽

Figure 1 

Two useful multivalued maximal monotone graphs.

Figure 2 

The studied model with mechanical components.

𝜎(𝑥)=⎧⎪⎨⎪⎩[−1−11,1]iiifff𝑥𝑥𝑥>=<000,.,(2.1) According to [11], this graph is maximal monotone. Therefore:

∀𝑥∈ℝ,𝜎(𝑥)=𝜕|𝑥|.(2.2)

Let us assume (see Figure 3) the following

Figure 3 

The studied model with external forces (𝐹𝑖)0≤𝑖≤𝑛, friction forces (𝑔𝑖)1≤𝑖≤𝑛, linear forces (𝑓𝑖)0≤𝑖≤𝑛, and with displacements (𝑢𝑖)0≤𝑖≤𝑛, (𝑣𝑖)1≤𝑖≤𝑛, 𝑥 and 𝜉.

(i)This mechanical system is submitted to external forces (𝐹_𝑖)_{0≤𝑖≤𝑛+1}: 𝐹₀ is exerted on the spring with stiffness 𝑘₀; For 1≤𝑖≤𝑛, 𝐹_𝑖 is exerted on material point of mass 𝑚_𝑖; 𝐹_𝑛+1 is exerted on the spring with stiffness 𝑘_𝑛+1. (ii)For 1≤𝑖≤𝑛, 𝑔_𝑖 is the friction force exerted by the support of the 𝑖th St-Venant.(iii)For 0≤𝑖≤𝑛, 𝑓_𝑖 is elastic linear force exerted by the 𝑖th spring.(iv)For 0≤𝑖≤𝑛, 𝑢_𝑖 is the displacement of the 𝑖th spring.(v)For 1≤𝑖≤𝑛, 𝑣_𝑖 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 𝑚_𝑛.

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},𝑚_𝑖𝑣̈_𝑖=𝐹_𝑖+𝑓_𝑖−1−𝑓_𝑖+𝑔_𝑖,(2.3a) by the constitutive laws of linear springs:

∀𝑖∈{0,…,𝑛},𝑓_𝑖=−𝑘_𝑖𝑢_𝑖,(2.3b) by the constitutive of laws St-Venant elements:

∀𝑖∈{1,…,𝑛},𝑔_𝑖∈−𝛼_𝑖𝜎𝑣̇_𝑖,(2.3c) by the geometrical connexions:

∀𝑖∈{0,…,𝜉𝑛+−𝑢1₀},𝜉+𝑢₀+𝑢₁+⋯+𝑢_𝑖=𝑣_𝑖+1,(2.3d)+𝑢₁+⋯+𝑢_𝑛=𝑥,(2.3e) and finally by the boundary conditions:

𝐹𝐹₀=𝑓₀,(2.3f)_𝑛+1=𝑓_𝑛.(2.3g)

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

∀𝑖∈{1,…,𝜉𝑛+−𝑢1₀}=,𝑣𝑣₁,_𝑖+1(2.4a)𝑥−𝑣_𝑛−𝑣_𝑖=𝑢_𝑖,(2.4b)=𝑢_𝑛.(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,…,𝐹_𝑛 are known.

3.1. Dynamical Case

We assume in this section that

∀𝑖∈{1,…,𝑛},𝑚_𝑖>0.(3.1)

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

∀𝑖∈{2,…,𝑛𝑚−₁𝑣1̈₁}+,𝛼𝑚₁𝜎_𝑖𝑣̇₁−𝑘₀𝜉+𝑘₀+𝑘₁𝑣₁−𝑘₁𝑣₂∋𝐹₁,(3.2a)𝑚𝑣̈_𝑖+𝛼_𝑖𝜎𝑣̇_𝑖−𝑘_𝑖−1𝑣_𝑖−1+𝑘_𝑖−1+𝑘_𝑖𝑣_𝑖−𝑘_𝑖𝑣_𝑖+1∋𝐹_𝑖,(3.2b)_𝑛𝑣̈_𝑛+𝛼_𝑛𝜎𝑣̇_𝑛−𝑘_𝑛−1𝑣_𝑛−1+𝑘_𝑛−1+𝑘_𝑛𝑣_𝑛−𝑘_𝑛𝑥∋𝐹_𝑛.(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:

andthereactio𝜉n=s𝐹0,₀𝑥=0,(3.3a)and𝐹_𝑛+1areunknown.(3.3b)

We set, for all 𝑞∈ℕ^∗,

𝐾(𝑞)=⎛⎜⎜⎜⎜⎜⎜⎝𝑘⁰−+𝑘₁𝑘₁−𝑘₁00⋯0000𝑘₁−+𝑘₂𝑘₂−𝑘₂0⋯⋯000⋮0𝑘₂⋯⋮+𝑘₃−⋯0𝑘₃⋯00−⋯𝑘⋮_𝑞−2⋯⋮⋮0𝑘_𝑞−2+𝑘_𝑞−1−𝑘_𝑞−100⋯⋯⋯0−𝑘_𝑞−1𝑘_𝑞−1+𝑘_𝑞⎞⎟⎟⎟⎟⎟⎟⎠∈ℳ_𝑞(ℝ).(3.4) Thus, by setting

𝐹𝑉==^𝑡_𝑡𝑣₁,…,𝑣_𝑛∈ℝ^𝑛,(3.5a)ℳ=𝐾⎛⎜⎜⎜⎝𝐹𝑚₁=,₁…𝐾,(𝐹𝑛)_𝑛,∈ℝ^𝑛,((33..55bc))0𝑚0₂⋯000⋯⋯⋱⋯⋯𝑚00_𝑛⎞⎟⎟⎟⎠(3.5d) and defining the maximal monotone operator 𝒜 by

𝒜𝑣₁,…,𝑣_𝑛=𝛼₁𝜎𝑣₁×⋯×𝛼_𝑛𝜎𝑣_𝑛,(3.5e) equations (3.2a)–(3.2c) imply the system of equations

𝑀𝑉̈+𝒜𝑉̇+𝐾𝑉∋𝐹.(3.6) Reactions 𝐹₀ and 𝐹_𝑛+1 can be determined thanks to (2.3f)-(2.3g) which give

𝐹𝐹₀=−𝑘₀𝑣₁,(3.7a)_𝑛+1=𝑘_𝑛𝑣_𝑛.(3.7b)

Set

𝑀=𝐼_𝑛𝑝=2𝑛,(3.8a)0ℳ0₋₁∈ℳ^2𝑛(ℝ),(3.8b) where 𝐼_𝑛 is the identity of ∈ℳ_𝑛(ℝ) and for 𝑡∈ℝ, 𝑋=(𝑉₁,𝑉₂)∈ℝ^2𝑛, with 𝑉₂=(𝑉_2,1,…,𝑉_2,𝑛),

𝒢𝑡,𝑉₁,𝑉₂=𝑉²𝜙𝑉ℳ⁻¹𝐹−ℳ⁻¹𝐾𝑉₁,(3.8c)₁,𝑉₂=^𝑛_𝑖=1𝛼_𝑖||𝑉_2,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, (𝑢_𝑖)_{0≤𝑖≤𝑛}, (𝑓_𝑖)_{0≤𝑖≤𝑛}, and (𝑔_𝑖)_{1≤𝑖≤𝑛} successivelly by

∀𝑖∈{1,…,𝑛}(,(32..34𝑔(aa2_𝑖)).--3((b23)..4,3cb)),,=𝑚_𝑖𝑣̈_𝑖−𝐹_𝑖−𝑓_𝑖−1+𝑓_𝑖.(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:

reaction𝐹𝜉₀=0,(3.10a)anddeixsptelrancaelmfeonrcti𝑥nisgius𝐹nuknn_𝑛+1konwonw,n,((33..1100bc))isknown.(3.10d)

As in Section 3.1.1, by setting

𝐹=_𝑡𝑉=^𝑡𝑣₁,…,𝑣_𝑛∈ℝ^𝑛,(3.11a)𝐾=⎛⎜⎜⎜⎜⎜⎜⎝𝑘₀𝐹₁,…,𝐹_𝑛−1,𝐹_𝑛−𝐹_𝑛+1∈ℝ^𝑛,(3.11b)−+𝑘₁𝑘₁−𝑘₁00⋯0000𝑘₁−+𝑘₂𝑘₂−𝑘₂0⋯⋯000⋮0𝑘₂⋯⋮+𝑘₃−⋯0𝑘₃⋯00−⋯𝑘⋮_𝑛−2⋯⋮⋮0𝑘_𝑛−2+𝑘_𝑛−1−𝑘_𝑛−100⋯⋯⋯0−𝑘_𝑛−1𝑘_𝑛−1⎞⎟⎟⎟⎟⎟⎟⎠∈ℳ_𝑛ℳand𝒜asin(3.5d)-(3.5e),(ℝ),((33..1111dc)) we can prove that equations (3.2a)–(3.2c) imply the system of equations

𝑀𝑉̈+𝒜𝑉̇+𝐾𝑉∋𝐹.(3.12) Reactions 𝐹₀ and displacement 𝑥 can be determined thanks to

𝑥=𝐹₀−=𝐹−𝑘_0𝑛+1𝑣₁,(3.13a)𝑘_𝑛+𝑣_𝑛.(3.13b)

As in Section 3.1.1, let us set

𝑀and𝜙defin𝑝e=db2y𝑛,(3.8b)-(3.8d),((33..1144ba)) and for 𝑡∈ℝ, 𝑋=(𝑉₁,𝑉₂)∈ℝ^2𝑛, with 𝑉₂=(𝑉_2,1,…,𝑉_2,𝑛),

𝒢𝑡,𝑉₁,𝑉₂=𝑉²ℳ⁻¹𝐹−ℳ⁻¹𝐾𝑉₁.(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, (𝑢_𝑖)_{0≤𝑖≤𝑛}, (𝑓_𝑖)_{0≤𝑖≤𝑛}, and (𝑔_𝑖)_{1≤𝑖≤𝑛} successivelly by

Last((e32q..3u4(aaa2)t).i--3o((bn23)..4o,3cfb))(,,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,𝑚…,𝑛−1},𝑚_𝑖=0,(3.16a)_𝑛=𝑚>0.(3.16b)

Equation (3.2b) implies

∀𝑖∈{2,…,𝑛−1},𝛼_𝑖𝜎𝑣̇_𝑖+𝑔_𝑖∋0,(3.17a) with

∀𝑖∈{2,…,𝑛−1},𝑔_𝑖=−𝐹_𝑖−𝑘_𝑖−1𝑣_𝑖−1+𝑘_𝑖−1+𝑘_𝑖𝑣_𝑖−𝑘_𝑖𝑣_𝑖+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)=⎧⎪⎪⎨⎪⎪⎩ℝ∅{0}iiff𝑥𝑥∈∈]]−−∞1,1₋,−[,1[]1,+∞[,ℝif𝑥=−1,₊if𝑥=1.(3.18) We have

∀𝑥∈ℝ,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

∀𝑖∈{2,…,𝑛−1},𝑣̇_𝑖+𝜕𝜓_{[−𝛼𝑖,𝛼𝑖]}𝑔_𝑖∋0.(3.20) Similarly, (3.2a) gives

𝑣̇₁+𝜕𝜓_{[−𝛼1,𝛼1]}𝑔₁∋0,(3.21a) with

𝑔₁=−𝐹₁−𝑘₀𝜉+𝑘₀+𝑘₁𝑣₁−𝑘₁𝑣₂.(3.21b) and (3.2c) gives

𝑚𝑣̈_𝑛+𝛼_𝑛𝜎𝑣̇_𝑛−𝑘_𝑛−1𝑣_𝑛−1+𝑘_𝑛−1+𝑘_𝑛𝑣_𝑛−𝑘_𝑛𝑥∋𝐹_𝑛.(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 𝐺𝑉==^𝑡_𝑡𝑣₁,…,𝑣_𝑛−1∈ℝ^𝑛−1,(3.23a)𝐹=_𝑡𝑔₁,…,𝑔_𝑛−1∈ℝ^𝑛−1,(3.23b)𝑍=_𝑡𝐹₁,…,𝐹_𝑛−1∈ℝ^𝑛−1,(3.23c)𝐶=𝛼₁𝐹₁,…,𝐹_𝑛−2,𝐹_𝑛−1+𝑘_𝑛−1𝑣_𝑛∈ℝ^𝑛−1,(3.23d)𝐾,𝛼₁=𝐾×(⋯𝑛−×1)−∈𝛼ℳ_𝑛−1,𝛼_{𝑛−1𝑛−1}⊂ℝ^𝑛−1,(3.23e)(ℝ),(3.23f) where 𝐾(𝑞) is defined by (3.4). Thus, according to (3.17b)–(3.21b), we have

𝐺=𝐾𝑉−𝑍,(3.24) and from (3.17a)–(3.21a) we can write

𝑉̇+𝜕𝜓_𝐶(𝐺)∋0,(3.25) Under the assumption

𝑘_𝑛−1≥0∀𝑖∈{1,…,𝑛−1},𝑘_𝑖>0,(3.26) the matrix 𝐾 is symmetric definite positive (see proof in Lemma B.1 of Appendix B), so that

𝑉=𝐾⁻¹(𝐺+𝑍),(3.27) and (3.25) gives

𝐾⁻¹𝐺̇+𝑍̇+𝜕𝜓_𝐶(𝐺)∋0,(3.28) which is equivalent to

𝐺̇+𝐾𝜕𝜓_𝐶(𝐺)∋−𝑍̇.(3.29) For 𝑞 integer and 𝑢 vector of ℝ^𝑚, we denote by

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

𝑚𝑣̈_𝑛+𝛼_𝑛𝜎𝑣̇_𝑛−𝑘_𝑛−1𝑣_𝑛−1+𝑘_𝑛−1+𝑘_𝑛𝑣_𝑛∋𝐹_𝑛,(3.31) which can be rewritten under the following form:

𝑣̈_𝑛+𝛼^𝑛𝑚𝜎𝑣̇^𝑛−𝑘𝑚^𝑛−1𝐾⁻¹(𝐺+𝑍)_𝑛−1+𝑘^𝑛−1+𝑘_𝑛𝑚𝑣^𝑛∋𝐹^𝑛𝑚.(3.32) Let 𝑢 be the vector of ℝ^𝑛−1 defined by

𝑢=^𝑡(0,…,0,1).(3.33) Note that

𝑍=𝐹+𝑘_𝑛−1𝑣_𝑛𝑢.(3.34) We set 𝑀=𝐾0𝑝𝐼=0₂𝑛+1,(3.35a)∈ℳ^𝑛+1(ℝ),(3.35b) and for all 𝑡∈ℝ, 𝐺∈ℝ^𝑛−1, 𝑎,𝑏∈ℝ, 𝑋=^𝑡(𝐺,𝑎,𝑏)𝒢(𝑡,𝑋)=⎛⎜⎜⎜⎝−𝐹̇−𝑘^𝑛−1𝐹𝑏𝑏𝑢_𝑛𝑚+𝑘𝑚^𝑛−1𝐾⁻¹(𝐺+𝐹+𝑘_𝑛−1𝑎𝑢)_𝑛−1−𝑘^𝑛−1+𝑘_𝑛𝜙(𝑋)=𝜓_{[𝛼1,𝛼1]×⋯×[−𝛼𝑛−1,𝛼𝑛−1]×{0}×{0}}𝑚𝑎⎞⎟⎟⎟⎠(3.35c)(𝑋)+𝛼^𝑛𝑚||𝑏||.(3.35d) Then, system (3.29)–(3.32) is equivalent to (A.7).

Reactions 𝐹₀ and 𝐹_𝑛+1 can be determined thanks to 𝐹₀=−𝑘₀𝐾𝐹⁻¹(𝐺+𝐹+𝑘_𝑛−1𝑎𝑢)₁,(3.36a)_𝑛+1=𝑘_𝑛𝑎.(3.36b)

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

𝐺=𝑍^𝑡=[𝑘𝑋]_1𝑛−1,…,[𝑋]_𝑛−1,[𝑋]_𝑛𝑢+𝐹.(3.37) then we can calculate 𝑉 thanks to (3.27). Successively, 𝑥, 𝜉, (𝑢_𝑖)_{0≤𝑖≤𝑛}, and (𝑓_𝑖)_{0≤𝑖≤𝑛} are defined by

((32..34(aa2)).--3((b23)..4,3cb)),.(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

𝑚𝑣̈_𝑛+𝛼_𝑛𝜎𝑣̇_𝑛−𝑘_𝑛−1𝑣_𝑛−1+𝑘_𝑛−1𝑣_𝑛+𝐹_𝑛+1∋𝐹_𝑛.(3.39) Using notations (3.23a)–(3.23f), we obtain the system (A.7), where we set 𝑀and𝜙aredefin𝑝ed=b𝑛y+(31.,35b)and(3.35d),((33..4400ba)) and for all 𝑡∈ℝ, 𝐺∈ℝ^𝑛−1, 𝑎,𝑏∈ℝ, 𝑋=^𝑡(𝐺,𝑎,𝑏),

𝒢(𝑡,𝑋)=⎛⎜⎜⎜⎝−𝐹̇−𝑘^𝑛−1𝐹𝑏𝑏𝑢_𝑛−𝐹_𝑛+1𝑚+𝑘𝑚^𝑛−1𝐾⁻¹(𝐺+𝐹+𝑘_𝑛−1𝑎𝑢)_𝑛−1−𝑘^𝑛−1𝑚𝑎⎞⎟⎟⎟⎠∈ℳ^𝑛+1(ℝ).(3.40c) The reaction 𝐹₀ and the displacement 𝑥 can be determined thanks to (3.36a) and

𝑥=−𝐹^𝑛+1𝑘_𝑛+𝑎.(3.41)

3.3. Quasistatic Case

In this section, we assume that

∀𝑖∈{1,…,𝑛},𝑚_𝑖=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

𝑣̇_𝑛+𝜕𝜓_{[−𝛼𝑛,𝛼𝑛]}𝑔_𝑛∋0,(3.43a) with

𝑔_𝑛=𝐹_𝑛−𝑘_𝑛−1𝑣_𝑛−1+𝑘_𝑛−1+𝑘_𝑛𝑣_𝑛−𝑘_𝑛𝑥.(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 𝐺𝑉==^𝑡_𝑡𝑣₁,…,𝑣_𝑛∈ℝ^𝑛,(3.44a)𝐹=_𝑡𝑔₁,…,𝑔_𝑛∈ℝ^𝑛,(3.44b)𝐶=𝛼₁𝐹₁,…,𝐹_𝑛∈ℝ^𝑛,(3.44c)𝐾,𝛼=₁𝐾×(⋯𝑛)×∈ℳ−_𝑛𝛼_𝑛,𝛼_𝑛⊂ℝ^𝑛,(3.44d)(ℝ),(3.44e)where 𝐾(𝑞) is defined by (3.4).

Thus, we have 𝑉̇𝐺+=𝜕𝜓𝐾_𝐶𝑉−𝐹,(𝐺)∋0(3.45) Under assumption

𝑘_𝑛≥0,∀𝑖∈{1,…,𝑛−1},𝑘_𝑖>0,(3.46) the matrix 𝐾 is symmetric definite positive (see proof in Lemma B.1), so that

𝐺̇+𝑉=𝐾𝜕𝐾𝜓⁻¹_𝐶(𝐺+𝐹),(3.47)(𝐺)∋−𝐹̇.(3.48) We set 𝑀=𝐾𝑝∈=ℳ𝑛,_𝑛(3.49a)(ℝ),(3.49b) and, for all 𝑡∈ℝ, for all 𝑋∈ℝ^𝑛𝜙𝒢(𝑋(𝑡),𝑋=)𝜓=_𝐶−𝐹̇,(3.49c)(𝑋).(3.49d) Then, the system (3.48) is equivalent to (A.7).

Reactions 𝐹₀ and 𝐹_𝑛+1 can be determined thanks to 𝐹𝐹₀=−𝑘₀𝐾⁻¹(𝐺+𝐹)₁,(3.50a)_𝑛+1=𝑘_𝑛𝐾⁻¹(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 𝐹_𝑛+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=^𝑡𝐹₁,…,𝐹_𝑛−1,𝐹_𝑛+𝑘_𝑛𝑥∈ℝ^𝑛,(3.51) and we obtain, as in Section 3.3.1, 𝐺̇+𝐾𝐺𝜕=𝜓_𝐶𝐾𝑉−𝐹,(3.52)(𝐺)∋−𝐹̇.(3.53) By setting 𝑀and𝜙aredefined𝑝b=y𝑛(,3.49b)and(3.49d),((33..5544ba)) and, for all 𝑡∈ℝ, for all 𝑋∈ℝ^𝑛, 𝒢(𝑡,𝑋)=−𝐹̇,(3.54c) we remark that system (3.53) is equivalent to (A.7).
Reactions 𝐹₀ and 𝐹_𝑛+1 can be determined thanks to
𝐹𝐹_𝑛0=−𝑘₀𝐾⁻¹(𝐺+𝐹)₁,(3.55a)_𝑛+1=−𝑘_𝑛𝑥+𝑘_𝑛𝐾⁻¹(𝐺+𝐹)_𝑛.(3.55b)

(ii)  Second Case: Force-Displacement Model
We assume that external forcing 𝐹_𝑛+1 are known and displacement 𝑥 is unknown.
The calculus are similar to the previous case.
Equation (3.43b) is replaced by
𝑔_𝑛=𝐹_𝑛+1−𝐹_𝑛−𝑘_𝑛−1𝑣_𝑛−1+𝑘_𝑛−1𝑣_𝑛.(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=^𝑡𝐹₁,…,𝐹_𝑛−1,𝐹_𝑛−𝐹_𝑛+1.(3.57)
So, (3.52) is replaced by
𝐺=𝐾𝑉−𝐹,(3.58) and (3.48) is replaced by 𝐺̇+𝐾𝜕𝜓_𝐶(𝐺)∋−𝐹̇.(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 𝑘_𝑛=0.(3.60)

According to previous remark, assumption

∀𝑖∈{1,…,𝑛−1},𝑘_𝑖>0(3.61) and Lemma B.1 ensure that matrix 𝐾 is symmetric definite positive. Thus, like previously, the system is equivalent to

𝐺̇+𝐾𝜕𝜓_𝐶(𝐺)∋−𝐹̇.(3.62) By giving 𝑝, 𝜙 defined by (3.49a)–(3.49d), 𝒢 defined by for all 𝑡∈ℝ, for all 𝑋∈ℝ^𝑛,

𝒢(𝑡,𝑋)=−𝐹̇,(3.63a) and 𝑀 defined by

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

𝑥=𝐹₀−=𝐹−𝑘_0𝑛+1𝐾⁻¹(𝐺+𝐹)₁,(3.64a)𝑘_𝑛+𝐾⁻¹(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 ℝ^𝑝 and that 𝑀 is symmetric positive definite.

Table 1 

The dimension of the system, the convex function and the symmetric positive definite matrix used for the above described mechanical models.

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

∀𝑛∈{0,…,𝑁−1},𝑋^𝑛+1−𝑋_𝑛ℎ+𝑀𝜕𝜙𝑋^𝑛∋𝒢𝑡_𝑛𝑋₀,𝑋_𝑛,=𝜉.(5.1) with time step ℎ, discretized time 𝑡_𝑛=ℎ𝑛, and approximations 𝑋₀,…,𝑋_𝑁 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 𝑋_𝑛+1:

𝑋_𝑛+1=[𝐼+ℎ𝑀𝜕𝜙]⁻¹𝑋_𝑛+ℎ𝒢𝑡_𝑛,𝑋_𝑛,(5.2) where 𝐼 is the identity and [𝐼+ℎ𝑀𝜕𝜙]⁻¹ is the inverse of the graph 𝐼+ℎ𝑀𝜕𝜙 (see [11]). According to [11], [𝐼+ℎ𝑀𝜕𝜙]⁻¹ is a monovalued operator, providing a unique solution 𝑋_𝑛+1∈ℝ^𝑝. In the first case, effective computations of 𝑋_𝑛+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 𝑋_𝑛1 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 𝑋_𝑛+1 leads to the following problem: according to [11], 𝑋_𝑛+1 is the solution of minimization problem: considering ‖⋅‖_𝑀 the norm define by the inner product given by (A.4)

𝑍‖𝑥‖_𝑀=^𝑡𝑥𝑀⁻¹𝑥,(5.3a)_𝑛=𝑋_𝑛+ℎ𝒢𝑡_𝑛,𝑋_𝑛(5.3b) solve

min_{𝑥∈𝐷(𝜕𝜙)}𝜙(𝑥)+12ℎ‖‖𝑥−𝑍^𝑛‖‖^2𝑀(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 ℝ^𝑝. If 𝜙 is a convex proper and lower semi-continuous function from ℝ^𝑝 to ]−∞,+∞], we can define its subdifferential 𝜕𝜙 by

𝑦∈𝜕𝜙(𝑥)⟺𝐷(∀𝜕ℎ𝜙)∈=ℝ{^𝑝𝑥,∶𝜙𝜕(𝜙𝑥(+𝑥)ℎ≠)−∅}𝜙.(𝑥)≥⟨𝑦,ℎ⟩,(A.1) Moreover, 𝜕𝜙 is a maximal monotone graph in ℝ^𝑝×ℝ^𝑝.

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

∀𝑥∈𝐶,𝜓_𝐶(𝑥)=0+,∞,iiff𝑥𝑥∈∉𝐶𝐶,.(A.2) In this particular case, 𝜕𝜓_𝐶, which is the subdifferential of 𝜓_𝐶, is given by

∀(𝑥,𝑦)∈𝐶×ℝ^𝑝,∀𝑦𝑥∈∉𝜕𝐶𝜓,_𝐶(𝜕𝑥𝜓)_𝐶⟺∀𝑧∈𝐶,⟨𝑦,𝑥−𝑧⟩≥0,(A.3a)(𝑥)=∅.(A.3b) The domain of the maximal monotone operator 𝜕𝜓_𝐶 is equal to 𝐶.

We observe that if ℝ^𝑝 is equipped with its canonical scalar product ⟨,⟩, and with another scalar product,

⟨𝑥,𝑦⟩_𝑀=^𝑡𝑥𝑀⁻¹𝑦,(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]×ℝ^𝑝 to ℝ^𝑝 which is Lipschitz continuous with respect to its second argument, that is, there exists 𝜔≥0 such that

∀𝑡∈[0,𝑇],∀𝑋₁,𝑋₂∈ℝ^𝑝,‖‖𝒢𝑡,𝑋₁−𝒢𝑡,𝑋₂‖‖≤𝜔‖‖𝑋₁−𝑋₂‖‖.(A.6a) Moreover, we assume that

∀𝑌∈ℝ^𝑝,𝒢(⋅,𝑌)∈𝐿^∞(0,𝑇;ℝ^𝑝).(A.6b)

Proposition A.1. If the matrix 𝑀 is symmetric positive definite and 𝜙 is convex proper and lower semicontinuous on ℝ^𝑝, under assumptions (A.6a)-(A.6b), for all 𝜉∈𝐷(𝜕𝜙), there exists a unique function 𝑋 in 𝑊^1,1(0,𝑇;ℝ^𝑝) such that 𝑋̇(𝑡)+𝑀𝜕𝜙(𝑋(𝑡))𝑋∋(𝒢0)(𝑡=,𝑋𝜉,(𝑡))a.e.on]0,𝑇[,(A.7) where the differential inclusion can be written as an inequality: for almost every 𝑡 in ]0,𝑇[, ∀ℎ∈ℝ^𝑝,𝜙(𝑋(𝑡)+ℎ)−𝜙(𝑋(𝑡))≥𝐺(𝑡,𝑋(𝑡))−𝑋̇(𝑡),ℎ_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 𝑘_𝑞≥0,∀𝑖∈{1,…,𝑞−1},𝑘_𝑖>0,(B.1) matrix 𝐾(𝑞) defined by (3.4) is symmetric definite positive.

Proof. We have, for all 𝑋=(𝑥₁,…,𝑥_𝑞)∈ℝ^𝑞, ^𝑡𝑋𝐾(𝑞)𝑋=^𝑞_𝑖=1𝑘_𝑖−1+𝑘_𝑖𝑥^2𝑖−2^𝑞_𝑖=1=𝑘₀𝑘_𝑖𝑥_𝑖𝑥_𝑖+1,𝑥²¹+^𝑞_𝑖=2𝑘_𝑖−1𝑥^2𝑖+^𝑞−1_𝑖=1𝑘_𝑖𝑥^2𝑖+𝑘_𝑞−1𝑥^2𝑞−1−2^𝑞−1_𝑖=1𝑘_𝑖=𝑘₀𝑘_𝑖+1𝑥_𝑖+1,𝑥²¹+^𝑞−1_𝑖=1𝑘_𝑖𝑥_𝑖+1−𝑥_𝑖²+𝑘_𝑞𝑥^2𝑞.(B.2) Under assumption (B.1), ^𝑡𝑋𝐾(𝑞)𝑋=0 then implies 𝑋=0.