Mechanical oscillators described by a system of - PowerPoint PPT Presentation

Mechanical oscillators described by a system of differential-algebraic equations Kumbakonam R. Rajagopal and Dalibor Pražák Department of Mechanical Engineering Texas A&M University, College Station Department of Mathematical Analysis Charles University, Prague Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

Problem Spring �� �� �� �� x ′′ + F d + F s = F ( t ) �� �� �� �� �� �� �� �� �� �� m=1 �� �� F(t) �� �� �� �� �� �� �� �� �� �� x . . . . . . . . .displacement �� �� �� �� �� �� F d . . . . . . . dashpot force �� �� �� �� �� �� F s . . . . . . . . . spring force �� �� �� �� Dashpot �� �� F ( t ) . . . . . external force �� �� Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

Constitutive relations x ′′ + F d + F s = F ( t ) F s = f ( x ) (spring) “common” approach: F d = g ( x ′ ) (dashpot) apply the standard x ′′ + g ( x ′ ) + f ( x ) = F ( t ) ODE theory Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

“Reversed” constitutive relations x = f ( F s ) (spring) IDEA: x ′ = g ( F d ) (dashpot) what if we assume PHILOSOPHICALLY: kinematics ( x and x ′ ) are a consequence, and hence a function of the forces ( F s and F d ). x ′′ + F d + F s = F ( t ) differential-algebraic x = f ( F s ) system of equations x ′ = g ( F d ) Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

Implicit constitutive relations For some materials, it is even reasonable to assume: f ( x , F s ) = 0 (spring) g ( x ′ , F d ) = 0 (dashpot) That is to say, fully implicit constitutive relations. Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

Examples x’ x F_s F_d Bingham fluid polymer response F_d F_d x’ x’ Coulomb friction . . . with relaxation Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

Mathematical results – an overview � oscillators with reversed (monotone) constitutive relations 1 � oscillator with (generalized) Coulomb friction 2 � problem: uniqueness for 2nd order ODE’s 3 Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

Oscillators with reversed constitutive relations x ′′ + F d + F s = F ( t ) x = f ( F s ) x ′ = g ( F d ) f , g continuous, non-decreasing | f ( u ) | , | g ( u ) | ∼ | u | for | u | → ∞ F ( t ) ∈ L 2 ( 0 , T ) Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

THEOREM 1. There is at least one global solution. Proof. f k = f + k − 1 Id x = f k ( F s ) � approximation: 1 x ′ = g k ( F d ) g k = g + k − 1 Id x ′′ + � � � � − 1 ( x ′ ) � f k , g k invertible � 2 g k + f k − 1 ( x ) = F ( t ) � �� � � �� � F d F s � coercivity of f , g 3 = ⇒ k -independent estimates � limit k → ∞ (use monotonicity of f , g ). 4 Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

. . . uniqueness . . . ? x 1 , x 2 . . . solutions; F i d , F i s , i = 1 , 2 . . . the corresponding forces. ( x 1 − x 2 ) ′′ + ( F 1 / · ( x 1 − x 2 ) ′ d − F 2 d ) + ( F 1 s − F 2 s ) = 0 ( x 1 − x 2 ) ′ � 2 + ( F 1 � d )( x 1 − x 2 ) ′ s )( x 1 − x 2 ) ′ 1 d − F 2 + ( F 1 s − F 2 d = 0 2 d t � �� � � �� � ≥ 0 ??? structual properties of f assume in addition: F ( t ) ≡ F 0 . . . autonomous case = ⇒ THEOREM 2. Global (forward) uniqueness Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

Coulomb friction with relaxation x ′′ + F d + kx = F ( t ) F d = F c + g ( x ′ ) ( F c , x ′ ) ∈ A F c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coulomb-like friction force A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . monotone graph g ( · ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . relaxation function F_d F_c g x’ x’ Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

g continuous, | g ( u ) | ≤ c ( 1 + | u | ) A maximal monotone, coercive = ⇒ THEOREM 1. Global existence of solutions. moreover: g locally lipschitz = ⇒ THEOREM 2. Global (forward) uniqueness. examples of nonuniqueness: (steep relaxation) (non-monotone graph) Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

Simplification: uniqueness for ODE x ′′ + F d + F s = F ( t ) � � motivation:     x ′ x � y ′ + f ( y , t ) = 0 � neglect F s and x 1 ( y = x ′ ) � x ′′ + f ( x , t ) = 0 � neglect F d and x ′ 2 Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

Uniqueness for 1st order ODE ? y ′ + f ( y , t ) = 0 f ( · , t ) locally lipschitz: YES f ( · , t ) only Hölder: NO f ( · , t ) non-decreasing: YES (forward) Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

Uniqueness for 2nd order ODE ? x ′′ + f ( x , t ) = 0 f ( · , t ) locally lipschitz: YES f ( · , t ) only Hölder: NO f ( · , t ) non-decreasing: NO in general x ′′ + Q ( t ) x = 0, Q ( t ) ≥ 0. linear counterexample: = ⇒ uniqueness autonomous problem: x ′′ + h ( x ) = f ( t ) “quasi-autonomous” case: ???? Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators

Thank you. Kumbakonam R. Rajagopal and Dalibor Pražák Mechanical oscillators