Vibrations in Deformable Structures PDF

Derivation of equations of motion Free vibrations Rayleigh-Ritz method Vibrations in deformable structures Panagiotis KOTRONIS G MInstitut de Recherche en...

Derivation of equations of motion Free vibrations Rayleigh-Ritz method Vibrations in deformable structures Panagiotis KOTRONIS G MInstitut de Recherche en Génie Civil et Mécanique Institut de Recherche en Génie Civil et Mécanique Ecole Centrale de Nantes December 4, 2024 P. KOTRONIS ([email protected]) Vibrations in deformable structures 1 / 26  Derivation of equations of motion Free vibrations Rayleigh-Ritz method Bibliography 1 M. Geradin and D. Rixen. Mechanical vibrations (second edition). Theory and application to structural dynamics. John Wiley and Sons Ltd, 1997. 2 A. K. Chopra. Dynamics of Structures. Theory and Applications to Earthquake Engineering (second edition). Prentice-Hall, 2001. 3 Dynamics of Structures, Patrick Paultre, Istey, Wiley, 2011. 4 Serveur pédagogique ECN (M1 M ENG VIBRA) https://hippocampus.ec-nantes.fr/course/view.php?id=1443 5 Jurnan Schilder’ videos https://www.youtube.com/watch?v=gXitAxK28pc&list=PLMXj6GKKnHI6Lftj7CXr9WusMkXi5s9yH P. KOTRONIS ([email protected]) Vibrations in deformable structures 2 / 26  Derivation of equations of motion Free vibrations Rayleigh-Ritz method Table of contents 1 Derivation of equations of motion Hamilton’s principle for continuum systems Equations of motion 2 Free vibrations Eigenvalue problem Orthogonality, modal superposition 3 Rayleigh-Ritz method Presentation for a bar element P. KOTRONIS ([email protected]) Vibrations in deformable structures 3 / 26  Derivation of equations of motion Free vibrations Rayleigh-Ritz method Introduction Continuous system The bodies which compose the system are deformable. Each constituent possesses simultaneously inertia, stiffness and damping properties. In order to formulate the governing equations we will resort to the theory of continuum mechanics. P. KOTRONIS ([email protected]) Vibrations in deformable structures 4 / 26  Derivation of equations of motion Hamilton’s principle for continuum systems Free vibrations Equations of motion Rayleigh-Ritz method Hamilton’s principle Definition Among the feasible trajectories of the system subjected to the restrictive conditions δu(t1 ) = δu(t2 ) = 0 at the end of the considered time interval [t1 , t2 ], the real trajectory of the system is the stationary point of the mechanical action: Z t2 δ (T − V )dt = 0 t1 P. KOTRONIS ([email protected]) Vibrations in deformable structures 5 / 26  Derivation of equations of motion Hamilton’s principle for continuum systems Free vibrations Equations of motion Rayleigh-Ritz method Kinetic energy Consider an elastic body, submitted to small displacements u(x) about a reference configuration (geometric linearity ). The kinetic energy of the continuous system may be evaluated from an integration over the volume. 1 Z T = ρu̇ · u̇dV 2 V V : the volume occupied by the elastic body. ρ: the associated density of the elastic body. u̇: the velocity field observed at any point resulting from the dynamic deformation in the body. P. KOTRONIS ([email protected]) Vibrations in deformable structures 6 / 26  Derivation of equations of motion Hamilton’s principle for continuum systems Free vibrations Equations of motion Rayleigh-Ritz method Potential energy [1/3] The total potential energy is: V = Vint + Vext Vint : the stain energy of the body, Vext : the potential energy of the external forces. P. KOTRONIS ([email protected]) Vibrations in deformable structures 7 / 26  Derivation of equations of motion Hamilton’s principle for continuum systems Free vibrations Equations of motion Rayleigh-Ritz method Potential energy [2/3] The stain energy of the body Vint is: Z Vint = W (ϵ)dV V with W (ϵ) the strain energy density: Z W (ϵ) = σdϵ ϵ: the (second order) strain tensor, σ: the (second order) stress tensor. P. KOTRONIS ([email protected]) Vibrations in deformable structures 8 / 26  Derivation of equations of motion Hamilton’s principle for continuum systems Free vibrations Equations of motion Rayleigh-Ritz method Potential energy [3/3] The potential energy of the external forces Vext is: Z Z Vext = − f imp · udV − timp · udS V Sσ S = Sσ + Su : the total surface. Su : the portion of the surface on which displacements are imposed. Sσ : the portion on which surface tractions are imposed. f imp : the applied body forces. timp : the surface tractions imposed on Sσ. P. KOTRONIS ([email protected]) Vibrations in deformable structures 9 / 26  Derivation of equations of motion Hamilton’s principle for continuum systems Free vibrations Equations of motion Rayleigh-Ritz method General equations of motion [1/2] Starting from the Hamilton’s principle and by integration by parts we derive the general equations of motion (see Ref. 1, TO DO). ∇ · σ + f̄ = ρü σ · n = t̄ on Sσ ∂W (ϵ) n: the outward normal to the surface Sσ and σ = ∂ϵ P. KOTRONIS ([email protected]) Vibrations in deformable structures 10 / 26  Derivation of equations of motion Hamilton’s principle for continuum systems Free vibrations Equations of motion Rayleigh-Ritz method General equations of motion [2/2] Remarks The equilibrium conditions are natural conditions to the principle in the sense that the principle takes care of their best possible enforcement by the trial functions adopted for u. The essential compatibility conditions are (geometric linearity, HPP): 1 ∂uj ∂ui ϵ = ∇s u ϵij = ( + ) in V 2 ∂xi ∂xj u = uimp on Su (kinematic conditions). P. KOTRONIS ([email protected]) Vibrations in deformable structures 11 / 26  Derivation of equations of motion Eigenvalue problem Free vibrations Orthogonality, modal superposition Rayleigh-Ritz method Eigenvalues [1/5] Consider a body free of applied forces assuming harmonic motion: u(x, t) = ū(x)cosωt For a linear elastic material (quadratic strain energy) it is easy to show that: T = Tmax sin2 ωt V = Vmax cos 2 ωt with the kinetic and potential energy amplitudes: 1 Z Tmax = ω 2 ρū(x) · ū(x)dV 2 V 1 Z Vmax = ∇s ū(x) : C : ∇s ū(x)dV 2 V P. KOTRONIS ([email protected]) Vibrations in deformable structures 12 / 26  Derivation of equations of motion Eigenvalue problem Free vibrations Orthogonality, modal superposition Rayleigh-Ritz method Eigenvalues [2/5] We choose a priori a time interval in such a way that δu(x, t0 ) = δu(x, t1 ) = 0, for example: π π   [t0 , t1 ] = − , 2ω 2ω we then get π π π Z Z 2ω 2ω sin2 ωtdt = cos 2 ωtdt = −π 2ω −π 2ω 2ω P. KOTRONIS ([email protected]) Vibrations in deformable structures 13 / 26  Derivation of equations of motion Eigenvalue problem Free vibrations Orthogonality, modal superposition Rayleigh-Ritz method Eigenvalues [3/5] - Rayleigh principle Hamilton’s principle for a free vibration problem therefore becomes the Rayleigh principle: δ(Tmax − Vmax ) = 0 In other words, because the total energy of the system is constant, a variation of the maximum kinetic energy is compensated by a variation of the maximale potential energy. P. KOTRONIS ([email protected]) Vibrations in deformable structures 14 / 26  Derivation of equations of motion Eigenvalue problem Free vibrations Orthogonality, modal superposition Rayleigh-Ritz method Eigenvalues [4/5] After some developments we have (see Ref I. and Ref III. TO DO): ∇ · (C : ∇s ū(x)) + ω 2 ρū(x) = 0 (C : ∇s ū(x)) · n = 0 on Sσ This is an eigenvalue problem of the Sturm-Liouville type as it can take the form: L x = λx with L a differential operator and λ a scalar. P. KOTRONIS ([email protected]) Vibrations in deformable structures 15 / 26  Derivation of equations of motion Eigenvalue problem Free vibrations Orthogonality, modal superposition Rayleigh-Ritz method Eigenvalues [5/5] The (infinite) eigenvalues and eigensolutions take the form 0 ≤ ω12 ≤ ω22 ≤ ω32 ≤... ū1 (x), ū2 (x), ū3 (x),... P. KOTRONIS ([email protected]) Vibrations in deformable structures 16 / 26  Derivation of equations of motion Eigenvalue problem Free vibrations Orthogonality, modal superposition Rayleigh-Ritz method Orthogonality Orthogonality properties (where eigensolutions have been normalized according to modal masses): Z ρūi · ūj dV = δij V Z ∇s ūi : C : ∇s ūj dV = ωi2 δij V Modal superposition The modal superposition is applied as well (TO DO). P. KOTRONIS ([email protected]) Vibrations in deformable structures 17 / 26  Derivation of equations of motion Eigenvalue problem Free vibrations Orthogonality, modal superposition Rayleigh-Ritz method Conclusions Remarks Discrete systems with an ever increasing number of degrees of freedom will tend to continuous systems. Inversely, continuous systems can be modeled by discrete approximations, such as finite elements. Continuous systems have an infinite number of eigenfrequencies and eigenmodes. Orthogonality has to be understood in terms of a scalar products between continuous functions, involving a volume integral. P. KOTRONIS ([email protected]) Vibrations in deformable structures 18 / 26  Derivation of equations of motion Eigenvalue problem Free vibrations Orthogonality, modal superposition Rayleigh-Ritz method Case studies [1/3] Examples (from Ref. II) P. KOTRONIS ([email protected]) Vibrations in deformable structures 19 / 26  Derivation of equations of motion Eigenvalue problem Free vibrations Orthogonality, modal superposition Rayleigh-Ritz method Case studies [2/3] Examples (from Ref. II) P. KOTRONIS ([email protected]) Vibrations in deformable structures 20 / 26  Derivation of equations of motion Eigenvalue problem Free vibrations Orthogonality, modal superposition Rayleigh-Ritz method Case studies [3/3] Example P. KOTRONIS ([email protected]) Vibrations in deformable structures 21 / 26  Derivation of equations of motion Free vibrations Presentation for a bar element Rayleigh-Ritz method Eigenvalues with the Rayleigh-Ritz method [1/5] Principle The method is used to calculate the eigenvalue problem. We approximate the solution by superimposing admissible functions wi (x ) extracted from a complete base. wi (x ) are multiplied by time dependent amplitudes qi that play the role of the generalized coordinates. The n approximate solutions named ω̃i2 verify: ω̃i2 ≥ ωi2 i = 1,... n with ωi2 the n first eigenfrequencies of the continuous system. P. KOTRONIS ([email protected]) Vibrations in deformable structures 22 / 26  Derivation of equations of motion Free vibrations Presentation for a bar element Rayleigh-Ritz method Eigenvalues with the Rayleigh-Ritz method [2/5] So we write: n X u(x , t) = wi (x )qi (t) i=1 and thus n X δu(x , t) = wi (x )δqi (t) i=1 P. KOTRONIS ([email protected]) Vibrations in deformable structures 23 / 26  Derivation of equations of motion Free vibrations Presentation for a bar element Rayleigh-Ritz method Eigenvalues with the Rayleigh-Ritz method [3/5] We then have (presentation is done for a bar element): ω2 ω2 X X Z l Tmax = mu 2 dx = mij qi qj 2 0 2 i j Z l 1 du 2 1 XX Vmax = EA( ) dx = kij qi qj 2 0 dx 2 i j with: Z l Z l dwi dwj mij = mwi wj dx kij = EA dx 0 0 dx dx P. KOTRONIS ([email protected]) Vibrations in deformable structures 24 / 26  Derivation of equations of motion Free vibrations Presentation for a bar element Rayleigh-Ritz method Eigenvalues with the Rayleigh-Ritz method [4/5] The Rayleigh principle becomes:   ω2 X X 1 XX δ(Tmax − Vmax ) = δ  mij qi qj − kij qi qj  = 0 2 i j 2 i j In matrix form it is: " # ω2 T 1 δ(Tmax − Vmax ) = δ q M q − qT K q = 0 2 2 with: K = [kij ] M = [mij ] qT = {q1 q2... qn } ω2 T 1 Tmax = q Mq Vmax = qT K q 2 2 P. KOTRONIS ([email protected]) X X Vibrations in deformable structures 25 / 26  Derivation of equations of motion Free vibrations Presentation for a bar element Rayleigh-Ritz method Eigenvalues with the Rayleigh-Ritz method [5/5] X X ω2 mij qj − kij qj = 0 i = 1,... , n j j K q = ω2M q P. KOTRONIS ([email protected]) Vibrations in deformable structures 26 / 26