Perturbation of periodic equilibrium by Arnaud Lazarus A spectral method to solve linear periodically time-varying systems 1
A few history Late 19 th century Emile Léonard Mathieu: Wave equation for an elliptical membrane moving through a fluid (1868) Achille Marie Gaston Floquet: «Sur les équations différentielles linéaires à coefficients périodiques», 1881, Comptes rendus de l académie des sciences. George William Hill [Hill, 1886]: «On the part of the motion of the lunar perigee which is a function of the mean motions of the sun and the moon», 1886, Acta Mathematica. 2
A few history 20 th century Schrödinger equation with periodic potential. Parametric pendulum. A hamonic-based method for computing the stability of periodic solutions of dynamical systems. Modelling the vibration of cracked rotors in the frequency domain. 3
Stability of dynamical systems Fixed points: x 0 (t) = x 0 Governing equation in the state space: a. Beam under conservative compressive load 4
Stability of dynamical systems Fixed points: x 0 (t) = x 0 Governing equation in the state space: a. Beam under conservative compressive load 5
Stability of dynamical systems Fixed points: x 0 (t) = x 0 Governing equation: λ control parameter Bifurcation diagram with the ANM continuation method x 0 constant solution (static equilibrium) 6
Stability of dynamical systems Perturbed solution Governing equation: Perturbation: Fundamental solutions Linear system: α n : eigenvalues of J 7
Stability of dynamical systems Perturbed solution Governing equation: Perturbation: Fundamental solutions Linear system: α n : eigenvalues of J 8
Stability of dynamical systems Perturbed solution Governing equation: Perturbation: Fundamental solutions Linear system: α n : eigenvalues of J 9
Stability of dynamical systems Periodic equilibrium state: x 0 (t) = x 0 (t+t) Governing equation in the state space: b. Beam under compressive following force 10
Stability of dynamical systems Periodic equilibrium state: x 0 (t) = x 0 (t+t) Governing equation in the state space: b. Beam under compressive following force 11
Stability of dynamical systems Periodic equilibrium state: x 0 (t) = x 0 (t+t) Governing equation in the state space: λ control parameter x 0 (t) T-periodic solution T = 2π/ω Bifurcation diagram with to the ANM continuation method + harmonic balance 12
Stability of dynamical systems Perturbed solution Governing equation in the state space: λ control parameter Perturbation: x 0 (t) T-periodic solution T = 2π/ω Linear T-periodic system: J(t) T-periodic 13
Stability of dynamical systems Floquet theory N linearly independent solution y n (t): Floquet form: Poincaré stability p n (t) T-periodic: T = 2π/ω 14
Stability of dynamical systems Floquet theory N linearly independent solution y n (t): Floquet form: Poincaré stability p n (t) T-periodic: T = 2π/ω 15
Stability of dynamical systems Floquet theory N linearly independent solution y n (t): Floquet form: Poincaré stability p n (t) T-periodic: T = 2π/ω 16
Stability of dynamical systems Hill s method Fourier series expansions: Floquet form: Hill Linear T-periodic system: Hill 17
Stability of dynamical systems Hill s method Eigenproblem: H, infinite dimensional Hill Matrix To the infinity limit: N families of eigenvalues and eigenvectors Finite H: Most convergent members associated with l = 0 [Lazarus & Thomas, 2010] 18
Stability of dynamical systems The forced Guitar string Two polarizations: Modal expansion: 19
Stability of dynamical systems The forced Guitar string Harmonic based continuation method for quadratic dynamical systems (MANlab): Continuation of the first two modes: Dynamical system: 20
Stability of dynamical systems The forced Guitar string Resonance curves with H = 5: + Hill method Orbits in the p-q plane a. Amplitude of the first harmonic of p 1 and q 1 b. Phase 21
Stability of dynamical systems The forced Guitar string Evolution of Floquet s exponents against control parameter λ: a. Re(s) = fct(ω) for H = 5 b. Im(s) = fct(ω) for H = 3 22
Vibration of cracked rotors Synopsis Influence of imperfections on the global vibratory behaviour of large Rotating Machines? a. Turbo alternator b. GT-MHR project 23
Influence of imperfections Healthy rotor S 0 a. Simple model with 2 dofs b. Physical fundamental solutions (eigenmodes) 24
Influence of imperfections Healthy rotor S 0 a. Simple model with 2 dofs b. Physical fundamental solutions (eigenmodes) 25
Influence of imperfections Open crack S 1 T = π/ω a. Simple model with 2 dofs b. Stiffness function k(φ) in the fixed frame 26
Influence of imperfections Breathing crack S 2 [Andrieux & Varé, 2002] T = 2π/Ω a. Simple model with 2 dofs b. Stiffness function k(φ) in the fixed frame 27
Non-axisymmetrical rotating oscillators Floquet-Hill method Fourier series: + Hill matrix S 1 case: Open crack with 28
Non-axisymmetrical rotating oscillators Floquet-Hill method Fourier series: + Hill matrix S 2 case: Breathing crack with 29
Non-axisymmetrical rotating oscillators Fundamental solutions 4 independent fundamental solutions: Complex, coupling x- and y- directions. Poly-harmonic: (± ω x ±2jΩ)t or (± ω y ±2jΩ)t. a. In x- for Ω * = 0.34 and j max = 5 b. In y- for Ω * = 0.34 and j max = 5 30
Non-axisymmetrical rotating oscillators Stability analysis Evolution of Floquet exponents α n against rotating speed: a. Re(s) = fct(ω) for j max = 3 b. Im(s) = fct(ω) for j max = 3 31
Non-axisymmetrical rotating oscillators Stability analysis Evolution of Floquet exponents α n against rotating speed: a. Re(s) = fct(ω) for j max = 3 b. Im(s) = fct(ω) for j max = 3 32
Non-axisymmetrical rotating oscillators Strutt diagram Parametric instability of the rotating system: a. Stability map in the S 1 case a. Influence of fixed damping (S 1 case) 33
Non-axisymmetrical rotating oscillators Strutt diagram Parametric instability of the rotating system: a. Stability map in the S 2 case b. Influence of fixed damping (S 2 case) 34
Non-axisymmetrical rotating oscillators Time integration Free whirling is a linear combination of the fundamental solutions: a. Free whirling in the x-direction for Ω * = 0.3 b. Power Spectral Density 35
Extension to the Finite Element Method Finite Element Method (open cracks) Hypothesis: Constant spin speed Ω. Linear governing equations. Gyroscopic coupling allowed. Three-dimensional modelling for taking imperfections into account: Rotor modelled in the rotating frame R. Stator modelled in the fixed frame R (working frame). Mesh of a rotating machine 36
Extension to the Finite Element Method Finite Element Method Partial equilibrium equations of the rotating part in R : [1] Partial equilibrium equations of the fixed part in R: [2] Coupling between rotor and stator [3] [Combescure & Lazarus, 2008] 37
Extension to the Finite Element Method Finite Element Method The whole machine is a parametric oscillator with a period T: Global mesh in time domain = Relevant number of substructures in the frequency domain. 38
Extension to the Finite Element Method Finite Element Method Coupling between harmonic contributions considered through the rotor-stator condition [3]: Γ and Γ are chosen so that conditions could be imposed in the modal basis. Rotor-stator coupling in the time domain [3] = interface conditions between boundary nodes U j and U j. Representation of the Γ and Γ boundary in the frequency domain 39
Extension to the Finite Element Method Finite Element Method The modal basis Φ SS is calculated, at rest, in the frequency domain, with the component synthesis method: [Craig and Bampton, 1968] Hill s determinant for high dimensional systems Rotordynamic analysis of the whole system is performed by projected the classical equilibrium equations on the previous modal basis: 40
Validation on a 3D axi-symmetrical case Description Linear dynamic behaviour of «healthy» shaft: Small displacements around the equilibrium position. Balanced system rotates at a constant spin speed Ω. a. Axi-symmetrical shaft on non-isotropic bearings in the physical space b. 3D- Finite Element Modelling 41
Validation on a 3D axi-symmetrical case Real modes at rest Modal basis computed at rest for the four first modes: a. First beam mode in the x- direction b. First disc mode in the y-direction 42
Validation on a 3D axi-symmetrical case Complex modes in rotation 3D modelling: rotor in the rotating frame and stator in the fixed frame. Free whirling projected on Φ SS with j max = 0 Ω Ω a. Backward beam modes for Ω = 5 rpm Campbell diagram for flexible disc 43
Application to the ROTEC test bench Description Study of the free parametric oscillations of a «simple» test rig: Ω constant and negligible gyroscopic coupling. No damping consideration. The rotating beam is rectangular: Length: 0.70 m Cross-section: 0.034 x 0.024 m Four flexible rectangular struts: Length: 0.21 m Cross-section: 0.0119 x 0.007 m Ω Two different configurations: Particular case: the rigid support plate is fixed. General case: the rigid support plate is free. 44
Application to the ROTEC test bench Description dbx dby a. Test bench general view b. Non-contacting sensors and excitation 45
Application to the ROTEC test bench 3D modelling (particular case) The modal basis is constituted by the two first beam modes: ω x = 20 Hz ω y = 23 Hz 2 first modes shapes at standstill 46
Application to the ROTEC test bench 3D modelling (particular case) 3D modelling: rotor considered in the rotating frame Free whirling projected on the Φ SS modal basis with j max = 1. Two parametric modes in the backward and forward direction. Ω First parametric mode for the spin speed Ω = 60 rpm 47
Application to the ROTEC test bench Experimental results (particular case) For each 1 Hz: Power Spectrum Density of the sensors responses dbx and dby in the [0-17] Hz range. ω x -2Ω ω x b. Experimental frequency spectra b. Comparison between the numerical and experimental Campbell diagrams 48
Application to the ROTEC test bench 3D modelling (general case) The modal basis is constituted by the four first beam modes: ω 1x = 15 Hz ω 1y = 20 Hz 2 first in-phase modes shapes at standstill 49
Application to the ROTEC test bench 3D modelling (general case) The modal basis is constituted by the four first beam modes: 2 first out of phase modes shapes at standstill ω 2x = 24 Hz ω 2y = 31 Hz 50
Application to the ROTEC test bench 3D modelling (general case) 3D modelling: rotor in the rotating frame and stator in the fixed frame Four parametric modes in the x- and y- direction: Ω j max = 2 First parametric modes in the x- direction for the spin speed Ω = 420 rpm 51
Application to the ROTEC test bench Experimental results (general case) For each 1 Hz: Power Spectrum Density of the sensors responses dbx and dby in the [0-14] Hz range. Ω Ω a. PSD of the db sensors for Ω = 78 Hz b. PSD of the parametric modes for Ω = 8 Hz 52
Taking home message Perturbation Classiclineartheory Eigenmodes Dynamical systems Constant state Periodic state Perturbation Floquet linear theory Parametric instabilities 53
Perturbation of periodic equilibrium Bibliography 54