Performance of time-marching techniques dedicated to nonsmooth - PowerPoint PPT Presentation

Euromech 516 — Nonsmooth contact and impact laws in mechanics Performance of time-marching techniques dedicated to nonsmooth systems A nonlinear modal analysis approach Mathias Legrand & Denis Laxalde McGill University Structural Dynamics and Vibration Laboratory July 07, 2011

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Outline 1 Introduction Context Challenges Motivation 2 Modal analysis Linear framework Nonlinear framework 3 Non-smooth application Unilateral contact without friction Invariant periodic solution Linear rod with contact Modal motions Invariant manifolds 4 Time-stepping techniques Central di ff erences – Implicit contact θ -method Comparison 5 Conclusion Summary Future work

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Outline 1 Introduction Context Challenges Motivation 2 Modal analysis Linear framework Nonlinear framework 3 Non-smooth application Unilateral contact without friction Invariant periodic solution Linear rod with contact Modal motions Invariant manifolds 4 Time-stepping techniques Central di ff erences – Implicit contact θ -method Comparison 5 Conclusion Summary Future work

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Context Context Framework • Flexible structures ◮ finite-element models, multibody ◮ possibly with many degrees-of-freedom • Dynamics and vibration ◮ transient ◮ steady-state — periodic, quasi-periodic regimes • Interface nonlinearities — contact, friction

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Context Context Framework • Flexible structures ◮ finite-element models, multibody ◮ possibly with many degrees-of-freedom • Dynamics and vibration ◮ transient ◮ steady-state — periodic, quasi-periodic regimes • Interface nonlinearities — contact, friction Nonlinear modal analysis • nonlinear modes: periodic motion of an automonous system, invariant manifold on which oscillations take place • extension of linear modal analysis, i.e. provides an essential signature of the dynamics • numerical framework • useful for engineering applications

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Challenges Challenges • Large scale engineering systems (e.g. FE models, multibody) call for (e ffi cient) numerical methods adapted to generic nonlinearities (i.e. not only polynomial) ◮ Space semi-discretization ◮ Time stepping approaches (incl. shooting for BVP) ◮ Galerkin methods in time (e.g. harmonic balance, finite-element, etc.) ◮ Space-time discretization

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Challenges Challenges • Large scale engineering systems (e.g. FE models, multibody) call for (e ffi cient) numerical methods adapted to generic nonlinearities (i.e. not only polynomial) ◮ Space semi-discretization ◮ Time stepping approaches (incl. shooting for BVP) ◮ Galerkin methods in time (e.g. harmonic balance, finite-element, etc.) ◮ Space-time discretization • Non-smooth systems (e.g. featuring interface interactions) require specific treatments and methods ◮ Contact interface laws ◮ Optimization strategies

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Challenges Challenges • Large scale engineering systems (e.g. FE models, multibody) call for (e ffi cient) numerical methods adapted to generic nonlinearities (i.e. not only polynomial) ◮ Space semi-discretization ◮ Time stepping approaches (incl. shooting for BVP) ◮ Galerkin methods in time (e.g. harmonic balance, finite-element, etc.) ◮ Space-time discretization • Non-smooth systems (e.g. featuring interface interactions) require specific treatments and methods ◮ Contact interface laws ◮ Optimization strategies • Beyond nonlinear modal analysis, the dynamic regimes of such systems have to be fully qualified including their stability ◮ equilibria ◮ periodic or quasi-periodic solutions ◮ chaos

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Motivation Time-integration algorithms Validation • Assessment of performances of time-stepping techniques dedicated to flexible mechanical systems featuring unilateral contact conditions • How are such techniques capable of finding periodic orbits of automonous (and conservative) systems with non-smooth nonlinearities? • Preservation of invariant quantities

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Motivation Time-integration algorithms Validation • Assessment of performances of time-stepping techniques dedicated to flexible mechanical systems featuring unilateral contact conditions • How are such techniques capable of finding periodic orbits of automonous (and conservative) systems with non-smooth nonlinearities? • Preservation of invariant quantities Extension • Description of the dynamics around these invariant quantities • Stability analysis

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Outline 1 Introduction Context Challenges Motivation 2 Modal analysis Linear framework Nonlinear framework 3 Non-smooth application Unilateral contact without friction Invariant periodic solution Linear rod with contact Modal motions Invariant manifolds 4 Time-stepping techniques Central di ff erences – Implicit contact θ -method Comparison 5 Conclusion Summary Future work

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Linear framework Linear framework (1) • Linear 1 st -order ODE with constant coe ffi cients z + Dz = 0 ˙ (1) • Assumed solution z ( t ) = Z e λ t ⇒ ( D − λ I ) Z = 0 (2) ◮ generally complex eigen elements ( λ i , Z i ) ◮ uncouped equations ≡ invariant subsets ◮ superposition principle ≡ eigen modes span the configuration space • Invariant set formulation ◮ Master coordinates ( x 1 , y 1 ) = ( u 1 , v 1 ) ◮ Functional dependence x i ( t ) = a 1 i u 1 ( t ) + a 2 i v 1 ( t ) i = 1 , 2 ,..., N (3) y i ( t ) = b 1 i u 1 ( t ) + b 2 i v 1 ( t ) ◮ Substitution into equation of motion (1) to get a ik ’s and b ik ’s

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Linear framework Linear framework (2) • Equations of motion in real modal space �   1 − ξ 2 − ξ k ω k ω k � � � � ˙ u k   u k  k  =   k = 1 , 2 ,..., N (4)   ˙  �  v k v k   1 − ξ 2  ω k − ξ k ω k    k • Reconstruction: superposition principle • System with two degrees-of-freedom: first linear mode 2 1 2 0 1 −1 0 −2 −2 −1.5 −1 −1 −0.5 0 0.5 1 1.5 −2 2

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Nonlinear framework Nonlinear framework (1) • Theoretical tools ◮ Central manifold theorem ◮ Lyapunov theorem ◮ Normal form theorem • Assumed solution � Z n e n λ t z ( t ) = ⇒ F ( λ, Z ) = 0 (5) n ◮ Z = ( Z 1 , Z 2 ,..., Z n ) ◮ continuation technique: Z ( λ ) ◮ geometry of the invariant manifold • Invariant set formulation ◮ Master coordinates ( x 1 , y 1 ) = ( u , v ) ◮ (Nonlinear) functional dependence x i = X i ( u , v ) i = 1 , 2 ,..., N (6) y i = Y i ( u , v )

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Nonlinear framework Nonlinear framework (2) ◮ Time di ff erentiation x i = ∂ X i u + ∂ X i ˙ ∂ u ˙ ∂ v ˙ v i = 1 , 2 ,..., N (7) y i = ∂ Y i u + ∂ Y i ˙ ∂ u ˙ ∂ v ˙ v ◮ Substitution into equation of motion (1) — i=1,2,...,N Y i = ∂ X i ∂ u v + ∂ X i ∂ v f 1 ( u , X 1 ,..., X n , v , Y 2 ,..., Y N ) (8) f i ( u , X 1 ,..., X n , v , y 2 ,..., Y N ) = ∂ Y i ∂ u v + ∂ Y i ∂ v f 1 ( u , X 1 ,..., X n , v , Y 2 ,..., Y N ) ◮ Power series expansions, nonlinear Galerkin technique... ◮ Reduced-dynamics on the invariant manifold u = v ˙ (9) v = f 1 ( u , X 1 ,..., X n , v , Y 2 ,..., Y N ) ˙

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Nonlinear framework Nonlinear framework (3) ◮ energy-dependent frequency ◮ perturbation techniques ◮ stability analysis ◮ Cubic nonlinearity: first nonlinear mode 5 2 0 1 0 −5 −2 −1.5 −1 −1 −0.5 0 0.5 1 1.5 −2 2

Introduction Modal analysis Non-smooth application Time-stepping techniques Conclusion Outline 1 Introduction Context Challenges Motivation 2 Modal analysis Linear framework Nonlinear framework 3 Non-smooth application Unilateral contact without friction Invariant periodic solution Linear rod with contact Modal motions Invariant manifolds 4 Time-stepping techniques Central di ff erences – Implicit contact θ -method Comparison 5 Conclusion Summary Future work