76 D. Day et al. In this work, data input errors are used as perturbations to introduce acceleration error and evaluate the limits of the error in stress. Three different example cases are analyzed. The first case uses the magnitude of the error in acceleration to determine a bound on the modal von Mises stress for a free-free beam. In the second and third cases, two different beams are studied in a random vibration setting. For these problems, the sources of error are perturbations in the modulus, density, damping, and length. The random vibration solution uses the method presented in [2], where the VRMS stress is calculated using modal stress and displacement amplitudes. An in-depth theoretical development is discussed and the stress-acceleration error relation for the example problem is studied first, followed by results from the other two cases in a random vibration context. 8.2 Theory The equation of motion for a damped, multi-degree of freedom (MDOF) system under load F(t) can be expressed as [m] ¨u(t) +[c] ˙u(t) +[k] u(t) =F(t) (8.1) where [m], [c], [k] are the system mass, damping, and stiffness matrices, respectively. In a direct solution, the displacements, u(t), are computed by numerically solving the coupled partial differential equations of motion. Modal superposition, used in modal-based methods such as random vibration, takes advantage of the modal degrees of freedom, or modal coordinates, qn(t), to uncouple the equations using the system mode shapes, φn, as given in [3]: un(t) =φnqn(t) (8.2) [M] ¨q(t) +[ ]T [c] [ ] ˙q(t) +[K] q(t) =Q(t) (8.3) Here, [M] =[ ] T[m][ ] is the modal mass matrix, [K] =[ ] T[k][ ] is the modal stiffness matrix, andQ(t) =[ ]TF(t) is the modal force vector. The displacements are calculated by solving the n uncoupled equations for each qn(t), which are used to determine the strains and stresses at a given time. In random vibration problems, the quantities of interest are statistical in nature, so a metric such as the VRMS is used to assess failure. For random vibration of MDOF systems in the time domain, the mean square von Mises stress can be calculated using the modal coordinates and the stress mode shapes of the structure [2], σ 2 VRMS =E p 2 (t) = i,j ΓijTij = i,j E qi(t)qj(t) Ψ σ i TAΨσ j (8.4) Here, E[p2(t)] is the expected value, or mean, of the square of the von Mises stress, p(t) = σ(t) TAσ(t), where σ(t) T =[σxxσyy σzz σyzσxz σxy]. The VRMS is calculated using the modal covariance, Γij =E[qi(t)qj(t)], the stress modes σ i T, σ j , and matrixA, defined in [2]. In the frequency domain, the modal coordinate, qi(ω), is related to the input loads, fj(ω), by the transfer function, Hij(ω) [2]. qi (ω) =Hij (ω)fj (ω) (8.5) From [4], the spectral density matrix of the response, [SX(ω)], is calculated from the transfer function and the given spectral density matrix, [SF(ω)], [SX(ω)] =[H (ω)] [SF (ω)] [H (ω)] † (8.6) This reduces in the case of a single input to [SX(ω)] = |H (ω)| 2 [SF (ω)] (8.7)
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