.21Transient ResponseAnalysis ofStructural Systems21–1Chapter 21: TRANSIENT RESPONSE ANALYSIS OF STRUCTURAL SYSTEMS 21–2§21.1 MODAL APPROACH TO TRANSIENT ANALYSISConsider the following large-order finite element model equations of motion for linear structures:M¨u(t) + D˙u(t) + Ku(t) = f(t), D = (αM + βK), (α, β) are constant. (21.1)where the size of the displacement vector, n, ranges from several thousands to several millions.Now suppose we would like to obtain the displacement response, u(t), for expected applied force,f(t). There are two approaches: direct time integration and modal superposition. We will differdirect time integration techniques for transient response analysis to the latter part of the course, andconcentrate on modal superposition techniques. To this end, we first decompose the displacementvector, u(t), in terms of its modal components byu(t) = Ψq(t)(21.2)where Ψ is the mode shapes of the free-vibration modes, and q(t) is the generalized modal displace-ment. The mode shape matrix Ψ has the property of simultaneously diagonalizing both the massand stiffness matrices. That is, it is obtained from the following eigenvalue problem:K Ψ = M ΨΛΛ = diag(ω21, ..., ω2N)(21.3)where ωjis the j-th undamped frequency component of (21.1).21–221–3 §21.1 MODAL APPROACH TO TRANSIENT ANALYSISIn structural dynamics, one often employs the following special form of mode shapes (eigenvector):ΨTK Ψ = λ, ΨTM Ψ = I = diag(1, 1, ..., 1)(21.4)Substituting (21.2) into (21.1) and pre-multiplying the resulting equation by ΨTresults in the fol-lowing uncoupled modal equation:¨qi(t) + (α + βω2i) ˙qi(t) + ω2iqi(t) = pi(t), i = 1, 2, 3, ..., n.pi(t) = Ψ(i, :)Tf(t)(21.5)The above equation can be cast into a canonical form˙xi(t) = Aixi(t) + b pi(t), xi=[qi(t) ˙qi(t)]TAi=01−ω2i−(α + βω2i) , b =01 (21.6)whose solution is given byxi(t) = eAitxi(0) +t0eAi(t−τ)b pi(τ ) dτ(21.7)It should be noted that the above solution provides only for one of the n-vector generalized modalcoordinates, q(t)(n ×1).21–3Chapter 21: TRANSIENT RESPONSE ANALYSIS OF STRUCTURAL SYSTEMS 21–4Carrying out for the entire n-modal vector, the physical displacement, u(t)(n × 1), can be obtainedfrom (21.2) byu(t) = Ψ(1:n, 1:n) q(t)(1:n, 1), q =[q1(t), q2(t), ..., qn(t)]T(21.8)While the solution method described in (21.2) - (21.8) appears to be straightfowrad, its practicalimplementation needs to overcome several computational challenges, which include:(a) When the sizeof discrete finiteelement model increases, the taskfor obtaining alarge number ofmodes (m), if not all, m << n, becomes computationally expensive. In practice, it is customaryto truncate only part of the modes and obtain an approximate solution of the formu(t) ≈ Ψ(1:n, 1:m) q(t)(1:m, 1), q =[q1(t), q2(t), ..., qm(t)]T, m << n (21.9)It is not uncommon to have m/n <(1/100 − 1/1000) .(b) A typical vehicle consists of many substructures whose structural characteristics are distinctlydifferent from one to another. For example, a fuselage has different structural characteristicsfrom wing structures. Likewise, engine blocks are considerably stiffer than the car framestructure. The impact of stiffness differences on the computed modes and mode shapes can leadto accuracy loss, and frequently to an unacceptable level.(c) In modern manufacturing arrangements, rarely an aerospace company or automobile companydesigns, manufactures, assembles and tests the entire vehicle system. This means, except for21–421–5 §21.2 SOLUTION BY DIRECT INTEGRATION METHODSthe final performance evaluation, each substructure can be modeled, analyzed and tested beforeit can be assembled, as a separate and independent structure.An alternative approach is to numerically integrate the equations of motion (21.1). We will discusscomputationalproceduresoftwodirectionalgorithmsinthenextsection. Theiralgorithmicpropertieswill be examined later in the course.§21.2 SOLUTION BY DIRECT INTEGRATION METHODSThere are two distinct direct time integration methods: explicit and implicit integration formulas.We summarize their computational sequences below.§21.2.1 Central Difference Method for Undamped Case (D = 0)First, we express the acceleration vector¨u from(21.1) as¨u(t) = M−1(f − Ku(t)) (21.10)Hence, it is clear that if the mass matrix is diagonal, the computation for obtaining the accelerationvector would be greatly simplified. We now describe direction time integration by the centraldifference method:Initial step:21–5Chapter 21: TRANSIENT RESPONSE ANALYSIS OF STRUCTURAL SYSTEMS 21–6Given the initial conditions, {˙u(0), u(0), f(t)}, obtain the velocity at the half step {t = h, h =t}by¨u(0) = M−1(f(0) − Ku(0))˙u(12h) =˙u(0) +12h¨u(0)u(h) = u(0) + h˙u(12h)(21.11)Subsequent stepsTtotal= hnmaxfor n = 1:nmax¨u(n) = M−1(f(n) − Ku(n))˙u(n +12) =˙u(n −12) + h¨u(n)u(n +1) = u(n) + h˙u(n +12)end(21.12)§21.2.2 The Trapezoidal Rule for Undamped Case (D = 0)This method is also referred to Newmark’s implicit rule with its free parameter chosen to be (α =12,β = 1/4). Among several ways of implementing the trapezoidal integration rule, we will employa summed form or half-interval rule given as follows.21–621–7 §21.2 SOLUTION BY DIRECT INTEGRATION METHODS˙u(n +12) =˙u(n) +12h¨u(n +12)u(n +12) = u(n) +12h˙u(n +12)˙u(n +1) = 2˙u(n +12) −˙u(n)u(n +1) = 2u(n +12) − u(n)(21.13)In using the preceding formula, one multiply the first of (21.13) by M to yieldM˙u(n +12) = M˙u(n) +12h M¨u(n +12)(21.14)The term M¨u(n +12) in the above equation is obtained from (21.1) asM¨u(n +12) = f(n +12) − D˙u(n +12) − Ku(n +12)(21.15)which, when substituted into (21.14), results inM˙u(n +12) = M˙u(n) +12h {f(n +12) − D˙u(n +12) − Ku(n +12)}⇓[M +12hD]˙u(n +12) = M˙u(n) +12h {f(n +12) − Ku(n +12)}(21.16)21–7Chapter 21: TRANSIENT RESPONSE ANALYSIS OF STRUCTURAL SYSTEMS 21–8Now multiply the second of (21.13) by [M +12hD] to obtain[M +12hD] u(n +12) = [M +12hD] u(n) +12h [M +12hD]˙u(n +12)(21.17)third, substitute the second term in the righthand side of (21.17) by (21.16), one obtains[M +12hD] u(n +12) = [M +12hD] u(n) +12h {M˙u(n) +12h {f(n +12) − Ku(n +12)}⇓[M +12hD + (12h)2K] u(n +12) = M {u(n) +12h˙u(n)}+12h Du(n) + (12h)2f(n +12)(21.18)Implicit integration stepsAssemble: A = [M +12hD + (12h)2K]Factor: A = LU21–821–9 §21.3 DISCRETE APPROXIMATION OF MODAL SOLUTIONfor n = 0:nmaxb(n) = M