30 An Efficient Method for the Quantification of the Frequency Domain. . . 313 m1 m2 m3 x1 x2 x3 k1 = 4· 10 4 N m k3 = 4· 10 4 N m k6 =12 · 10 4 N m k5 = 4· 10 4 N m k4 = 4· 10 4 N m k2 = 4· 10 4 N m Fig. 30.1 Three-degree-of-freedom mass-spring system respectively, and modal damping values D 0:01 0:015 0:02 T : Figure 30.1 depicts the system, where the masses m1, m2, andm3 correspond to the diagonal of the mass matrix. By solving the generalized eigenvalue problem, the undamped eigenvalues D 2:274 104 9:100 104 2:5286 105 T rad2 s2 and the corresponding circular peak frequencies !p D 150:80 301:59 502:65 T rad s of the system can be obtained with .!p/l Dq. /l 1 2. / 2 l 8 l D1;2;3. The peak frequency is the frequency, where the magnitude of the frequency response function according to Eq. (30.19) of a single-degree-of-freedom system has its maximum. In this example, the chosen system properties and time length result in discrete frequency values exactly at the peak frequency positions of each mode for all considered configurations. This explains the uncommon values within the mass matrix. The system is excited by a random excitation at the first and second degree of freedom, where the independent and identically distributed (i.i.d.) random variables with respect to time and space follow a normal distribution with mean value zero and variance 224 N2. The unit of the (co)variances of excitation is Newton to the power of two N2 . Therefore, the random excitation can be described by Nf N E.Nf/; C.Nf; Nf/ with E.Nf/ i D0 and C.Nf; Nf/ i;j Dıi;j2 24 N2 8i;j (30.27) using the Kronecker delta ıi;j D 0 W i ¤j 1 W i Dj : (30.28) Generally, the matrix of linear combination coefficients Acan be arbitrarily chosen fromRmg m . In this representative study, it was assumed that only the relative displacements and not the total displacements can be measured. In addition, the linear combination coefficients have been designed to act as a modal filter. More details about the design of modal filters can be found in [2, 11]. Finally, the matrix of linear combination coefficients is defined as AD 2:9837 25:2322 22:2485 0:9912 1:2659 0:2746 ; (30.29) where the first and second row correspond to the coefficients of the first linear combiner (lc1) and second linear combiner (lc2), respectively.
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