Mit2 092 f09_lec18
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The document covers dynamic analysis modeling in finite element analysis, detailing methods such as mode superposition, explicit methods (like central difference), and implicit methods (like trapezoidal rule) for numerical solutions. It emphasizes the importance of stability and accuracy in time step selection, with explicit methods being conditionally stable and implicit methods unconditionally stable. Additionally, the document explores Rayleigh damping for constructing damping matrices in dynamic systems, providing equations for determining damping ratios.
Mit2 092 f09_lec18
- 1. 2.092/2.093 — Finite Element Analysis of Solids Fluids I Fall ‘09 Lecture 18 -Modeling for Dynamic Analysis Solution Prof. K. J. Bathe MIT OpenCourseWare From last lecture, MU ¨+ CU˙ + KU = R(t); 0U , 0U˙ (1) KU = FI , the internal force calculated from the element stresses. Mode Superposition The mode superposition method transforms the displacements so as to decouple the governing equation (1). Thus, consider: n U =Σ φixi (2) i=1 We start with the general solution, where φi is an eigenvector. Then Eq. (1) becomes x¨i +2ξiωix˙i + ωi 2 xi = φiT R = ri (i =1,...,n) (3) For damping, assume a diagonal C matrix: ⎤ ⎡ ΦT CΦ = ⎢⎢⎣ .. . zeros 2ξiωi . zeros .. ⎥⎥⎦ Φ = φ1 ... φn 0 The initial conditions are xi = φiT M 0U , 0x˙i = φTi M 0U˙. We consider and solve n such single-DOF systems: The mass m is 1, and the stiffness is ωi 2 . In Eq. (1) we have fully coupled equations. By performing the transformation, we obtain n decoupled equations. ri can be a complicated function of time. Direct Integration In direct integration, no transformation is performed. I. Explicit Method: Central Difference Method The explicit method evaluates Eq. (1) at time t to obtain the solution at time t +Δt. Assume we 1
- 2. Lecture 18 Modeling for Dynamic Analysis Solution 2.092/2.093, Fall ‘09 already have the values for tU , t−ΔtU, t−2ΔtU , ... Consider the following three linearly independent equations: M tU ¨+ C tU˙ + K tU = tR (4) tU ¨= Δ1 t2 t+ΔtU − 2 tU + t−ΔtU (5) tU˙ = 1 t+ΔtU − t−ΔtU (6) (2Δt) These equations can be solved for t+ΔtU . Assume C = 0 and M = diagonal mass matrix Ml, 1 Mlt+ΔtU = tRˆ (7) (Δt)2 All known quantities go to the right-hand side into tRˆ . Ml is a diagonal matrix, hence we have t+ΔtUi (Δt)2 t ˆ = Ri(Ml)ii for every ith component. If (Ml)ii is zero, the equation can not be solved. This corresponds to an infinite frequency. For the method to be stable, we must have The Condition of Stability: Δt ≤ Δtcr = Tπ n = ω2 n (8) If C is diagonal as well, the method still works in the same way! Note that K only appears in the right-hand side of the equation. II. Implicit Method: Trapezoidal Rule An implicit method evaluates Eq. (1) at time t +Δt to obtain the solution at time t +Δt. M t+ΔtU ¨+ C t+ΔtU˙ + K t+ΔtU = t+ΔtR (9) t+ΔtU˙ = tU˙ + 12 t+ΔtU ¨+ tU ¨Δt (10) The last term in Eq. (10) tells why the trapezoidal rule is also called the constant average acceleration method. We need one more linearly independent equation to solve the system. t+ΔtU = tU +Δt tU˙ + 1 t+ΔtU ¨+ tU ¨ (Δt)2 (11) 4 Here, the last two terms are incremental displacements. Substituting Eqs. (10) and (11) into (9): (c1M + c2C + K) t+ΔtU = t+ΔtRˆ (12) where c1 and c2 are constants, and are given in the textbook (see Sections 9.1-9.3 for more information). t+ΔtRˆ is obtained from known quantities. The larger Δt is, the smaller c1M + c2C becomes. 2
- 3. Lecture 18 Modeling for Dynamic Analysis Solution 2.092/2.093, Fall ‘09 This method is unconditionally stable. In other words, there is no condition on the time step size to have stability. (Not all implicit methods are unconditionally stable.) In numerical analysis, stability and accuracy are distinct requirements. Stability is the first fundamental requirement. But even if the scheme is stable, the result will not be accurate unless a sufficiently small time step has been used. For conditionally stable explicit methods The time step Δt is chosen for stability and accuracy. • → For unconditionally stable implicit methods The time step Δt is chosen for accuracy. • → How to Construct C Rayleigh damping is widely used. For the C matrix, assume C = αM + βK where α and β are constants to be selected. φTi Cφj =2ξiωiδij if i = j, δij = 1 (13) if i =6j, δij =0 For two values of Eq. (13) we obtain φTi (αM + βK) φi =2ξiωi α + βωi 2 =2ξiωi (14) Let’s use i = 1 and i = 2. We get two independent equations: α + βω12 =2ξ1ω1 (15) α + βω22 =2ξ2ω2 which we can solve for α and β. (Obviously, we must have ω1 =6ω2.) Then we use Eq. (14) to estimate what damping ratios are implicitly assumed in the remaining frequencies. 1 ξi = α + βωi 2 2ωi αβ =+ ωi 2ωi 2 Hence, αβ ξi =+ ωi, i =3, 4,...,n (16) 2ωi 2 where 2αωi is the (low-frequency) mass-proportional damping, and β 2 ωi is the (high-frequency) stiffness- proportional damping. See the textbook for examples on how this method may be applied when more than two damping ratios need to be matched approximately. 3
- 4. MIT OpenCourseWare http://ocw.mit.edu 2.092 / 2.093 Finite Element Analysis of Solids and Fluids I Fall 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.