AA242B: MECHANICAL VIBRATIONS potx

AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Direct Time-Integration Methods These slides are based on the recommended textbook: M. G´eradin and D. Rixen, “Mechanical Vibrations: Theory and Applications to Structural Dynamics,” Second Edition, Wiley, John & Sons, Incorporated, ISBN-13:9780471975465 AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Outline 1 Stability and Accuracy of Time-Integration Operators 2 Newmark’s Family of Methods 3 Explicit Time Integration Using the Central Difference Algorithm AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Multistep Time-Integration Methods Lagrange’s equations of dynamic equilibrium (p(t) = 0) M ¨ q + C ˙ q + Kq = 0 q(0) = q 0 ˙ q(0) = ˙ q 0 First-order form  C M M 0     A B  ˙ q ¨ q     ˙ u +  K 0 0 −M     −A A  q ˙ q     u =  0 0     0 =⇒ ˙ u = Au where A = A −1 B A A Direct time-integration AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Multistep Time-Integration Methods General multistep time-integration method for first-order systems of the form ˙ u = Au u n+1 = m  j=1 α j u n+1−j − h m  j=0 β j ˙ u n+1−j where h = t n+1 − t n is the computational time-step, u n = u(t n ), and u n+1 =  q n+1 ˙ q n+1  is the state-vector calculated at t n+1 from the m preceding state vectors and their derivatives as well as the derivative of the state-vector at t n+1 β 0 = 0 leads to an implicit scheme — that is, a scheme where the evaluation of u n+1 requires the solution of a system of equations β 0 = 0 corresponds to an explicit scheme — that is, a scheme where the evaluation of u n+1 does not require the solution of any system of equations and instead can be deduced directly from the results at the previous time-steps AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Multistep Time-Integration Methods General multistep integration method for first-order systems (continue) u n+1 = m  j=1 α j u n+1−j − h m  j=0 β j ˙ u n+1−j trapezoidal rule (implicit) u n+1 = u n + h 2 ( ˙ u n + ˙ u n+1 ) ⇒ ( h 2 A − I)u n+1 = −u n − h 2 ˙ u n backward Euler formula (implicit) u n+1 = u n + h ˙ u n+1 ⇒ (hA − I)u n+1 = −u n forward Euler formula (explicit) u n+1 = u n + h ˙ u n ⇒ u n+1 = (I + hA)u n AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Numerical Example: the One-Degree-of-Freedom Oscillator Consider an undamped one-degree-of-freedom oscillator ¨q + ω 2 0 q = 0 with ω 0 = π rad/s and the initial displacement q(0) = 1, ˙q(0) = 0 exact solution q(t) = cos ω 0 t associated first-order system ˙ u = Au where A = » 0 −ω 2 0 1 0 – u = [ ˙q, q] T , and initial condition u(0) = » 0 1 – AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Numerical Example: the One-Degree-of-Freedom Oscillator Numerical solution T = 3s, h = T 32 0 0.5 1 1.5 2 2.5 3 −4 −3 −2 −1 0 1 2 3 t q Exact solution Trapezoidal rule Euler backward Euler forward AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Stability Behavior of Numerical Solutions Analysis of the characteristic equation of a time-integration method consider the first-order system ˙ u = Au for this problem, the general multistep method can be written as u n+1 = m X j=1 α j u n+1−j − h m X j=0 β j ˙ u n+1−j ⇒ m X j=0 [α j I − hβ j A] u n+1−j = 0, α 0 = −1 let µ r be the eigenvalues of A and X be the matrix of associated eigenvectors the characteristic equation associated with m P j=0 [α j I − hβ j A] u n+1−j = 0 is obtained by searching for a solution of the form u n+1−m = Xa (decomposition on an eigen basis) u (n+1−m)+1 = λu n+1−m = λXa (solution form) . . . u n+1 = λu n = · · · = λ k u n+1−k = · · · = λ m Xa where λ ∈ C is called the solution amplification factor AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Stability Behavior of Numerical Solutions Analysis of the characteristic equation of a time-integration method (continue) Hence m X j=0 [α j I − hβ j A] λ m−j Xa = 0 Since X −1 AX = diag(µ r ), premultiplying the above result by X −1 leads to " m X j=0 [α j I − hβ j diag(µ r )] λ m−j # a = 0 =⇒ m X j=0 [α j − hβ j µ r ] λ m−j = 0, r = 1, 2 hence, the numerical response u n+1 = λ m Xa remains bounded if each solution of the above characteristic equation of degree m satisfies |λ k | < 1, k = 1, · · · , m AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration Operators Stability Behavior of Numerical Solutions Analysis of the characteristic equation of a time-integration method (continue) the stability limit is a circle of unit radius in the complex plane of µ r h, the stability limit is therefore given by writing λ = e iθ , 0 ≤ θ ≤ 2π =⇒ µ r h = m X j=0 α j e i(m−j )θ m X j=0 β j e i(m−j )θ one-step (m = 1) schemes forward Euler: α 1 = 1, β 0 = 0, β 1 = −1 ⇒ µ r h = e iθ − 1 the solution is unstable in the entire plane except inside the circle of unit radius and center −1 backward Euler: α 1 = 1, β 0 = −1, β 1 = 0 ⇒ µ r h = 1 − e −iθ the solution is stable in the entire plane except inside the circle of unit radius and center 1 trapezoidal rule: α 1 = 1, β 0 = − 1 2 , β 1 = − 1 2 ⇒ µ r h = 2i sin θ 1+cos θ the solution is stable in the entire left-hand plane AA242B: MECHANICAL VIBRATIONS [...]... τ 2 6 AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Newmark’s Family of Methods The Newmark Method In summary tn+1 ¨ q(τ )dτ = (1 − γ)h¨n + γh¨n+1 + rn q q tn tn+1 ¨ q(τ )(tn+1 − τ )dτ = tn 1 ¨ ¨ − β h2 qn + βh2 qn+1 + rn 2 where rn = rn = 1 − γ h2 q(3) (˜) + O(h3 q(4) ) τ 2 1 − β h3 q(3) (˜) + O(h4 q(4) ) τ 6 and tn < τ < tn+1 ˜ AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS. .. scheme AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Newmark’s Family of Methods The Newmark Method Particular values of the parameters γ and β γ= γ= 1 1 ¨ and β = leads to linearly interpolating q(τ ) in [tn , tn+1 ] 2 6 « „ ¨ ¨ qn+1 − qn ¨ ¨ qln (τ ) = qn + (τ − tn ) h 1 1 ¨ and β = leads to averaging q(τ ) in [tn , tn+1 ] 2 4 ¨ ¨ qn+1 + qn ¨ qav (τ ) = 2 AA242B: MECHANICAL VIBRATIONS AA242B:. .. , N AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Newmark’s Family of Methods Stability of a Time-Integration Method Undamped case (continue) the algorithm is stable if γ≥ 1 2 furthermore, the algorithm is unconditionally stable if 1 β≥ 4 „ 1 γ+ 2 «2 1 1 the choice γ = and β = leads to an unconditionally stable 2 4 time-integration operator of maximum accuracy AA242B: MECHANICAL VIBRATIONS. .. unconditionally stable 2 4 time-integration operator of maximum accuracy AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Newmark’s Family of Methods Stability of a Time-Integration Method Undamped case (continue) Stability of the Newmark scheme AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Newmark’s Family of Methods Stability of a Time-Integration Method Damped case (C = 0) consider... 2 AA242B: MECHANICAL VIBRATIONS ξ2 = ω 2 h2 1 + βω 2 h2 AA242B: MECHANICAL VIBRATIONS Newmark’s Family of Methods Amplitude and Periodicity Errors Free-vibration of an undamped linear oscillator (continue) amplitude error ρ − ρex = ρ − 1 = − 1 2 „ « 1 γ− ω 2 h2 + O(h4 ) 2 relative periodicity error 1 ∆φ ∆T = 1 = T φ 1 φ − 1 φex 1 φex = 1 ωh −1= φ 2 „ β− 1 12 « ω 2 h2 + O(h3 ) AA242B: MECHANICAL VIBRATIONS. .. preserved AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Newmark’s Family of Methods The Newmark Method Taylor’s expansion of a function f f (tn + h) = f (tn ) + hf (tn ) + h2 hs (s) 1 f (tn ) + · · · + f (tn ) + 2 s! s! Z tn +h f (s+1) (τ )(tn + h − τ )s dτ tn Application to the velocities and displacements tn+1 ˙ qn+1 = ¨ q(τ )dτ ˙ qn + tn tn+1 qn+1 ˙ = q n + h qn + ¨ q(τ )(tn+1 − τ )dτ tn AA242B:. .. that for any h ∈ [0, h0 ], a finite variation of the state vector at time tn induces only a non-increasing variation of the state-vector un+j calculated at a subsequent time tn+j AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Newmark’s Family of Methods Stability of a Time-Integration Method ˙ The application of the Newmark scheme to M¨ + Cq + Kq = p(t) q can be put under the form un+1 =... (h) 1 1   (1 − γ)hpn + γhpn+1 γhK  , H1 = M +2γhC bn+1 =  1 βh C M + βh2 K − β h2 pn + βh2 pn+1 2   −M + (1 − γ)hC (1 − γ)hK  1 H0 = −  1 − β h2 C − hM −M + − β h2 K 2 2 AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Newmark’s Family of Methods Stability of a Time-Integration Method Effect of an initial disturbance δu0 = u0 − u0 =⇒ δun+1 = A(h)δun = A2 (h)δun−1 = · · · = A(h)n+1 δu0... moduli of an eigenvalue of A(h) is greater than unity =⇒ δun+1 will not be amplified by the time-integration operator if all moduli of all eigenvalues of A(h) are less than unity AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Newmark’s Family of Methods Stability of a Time-Integration Method Undamped case decouple the equations of equilibrium by writing them (for the purpose of analysis)... h2 1+βωi2 h2 1 )ξ 2 2 ´ 1 + 1 − (γ − 2 )ξ 2 = 0 where ξ 2 = characteristic equation has a pair of conjugate roots λ1 and λ2 if „ «2 1 4 γ+ − 4β ≤ 2 2 , i = 1, · · · , N 2 ωi h AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Newmark’s Family of Methods Stability of a Time-Integration Method Undamped case (continue) the eigenvalues λ1 and λ2 can be written as = ρe ±i φ ρ = s φ = λ1,2 where . AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Direct Time-Integration Methods These slides. Time Integration Using the Central Difference Algorithm AA242B: MECHANICAL VIBRATIONS AA242B: MECHANICAL VIBRATIONS Stability and Accuracy of Time-Integration