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Figure 8. Ratio of dispersion measures for different uncertainty models; damping model (a). (a) Member 3; (b) member 5; —, case C (with ¢7=0-05)/case B; -.-.-, case C (with &7 = 0-01)/case B; ---, case A/case C (with ¢7 = 0-05).

Figure 8 Ratio of dispersion measures for different uncertainty models; damping model (a). (a) Member 3; (b) member 5; —, case C (with ¢7=0-05)/case B; -.-.-, case C (with &7 = 0-01)/case B; ---, case A/case C (with ¢7 = 0-05).

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Here AEF = axial stiffness, m = mass per unit length, 7, = strain rate dependent viscous damping coefficient, 4, = strain rate dependent hysteretic damping coefficient, c,; = velocity dependent viscous damping coefficient, c, = velocity dependent hysteretic damping coefficient, F = magnitude of the harmonic concentrated force, x) = distance at which the force is acting and 6(x — x) represents the Dirac delta function centered at x = xo. In the presence of element harmonic forces and nodal displacements, the displacement within the rod element can be taken to be given by
Here AEF = axial stiffness, m = mass per unit length, 7, = strain rate dependent viscous damping coefficient, 4, = strain rate dependent hysteretic damping coefficient, c,; = velocity dependent viscous damping coefficient, c, = velocity dependent hysteretic damping coefficient, F = magnitude of the harmonic concentrated force, x) = distance at which the force is acting and 6(x — x) represents the Dirac delta function centered at x = xo. In the presence of element harmonic forces and nodal displacements, the displacement within the rod element can be taken to be given by
A useful feature which emerges from the study of the expressions for energy spectra is that the energy dissipated can be expressed as a fraction of the kinetic energy as follows
A useful feature which emerges from the study of the expressions for energy spectra is that the energy dissipated can be expressed as a fraction of the kinetic energy as follows
It may be noted that the presence of damping makes the dynamic stiffness coefficients and equivalent nodal forces to be complex valued. where L is the length of the member. Notice that only the elements of the upper triangular matrix are listed above. The presence of force within the element results in the equivalent nodal forces. Thus, corresponding to the concentrated force F exp [if] acting at x = xo, see Figure 1, the amplitude of the equivalent nodal force in global co-ordinates can be shown to be given by
It may be noted that the presence of damping makes the dynamic stiffness coefficients and equivalent nodal forces to be complex valued. where L is the length of the member. Notice that only the elements of the upper triangular matrix are listed above. The presence of force within the element results in the equivalent nodal forces. Thus, corresponding to the concentrated force F exp [if] acting at x = xo, see Figure 1, the amplitude of the equivalent nodal force in global co-ordinates can be shown to be given by
Figure 3. The example truss considered; for all members: AE = 23-75 x 10°N; m = 0-882 kg/m. The value of c; in model (a) is selected on the basis that the damping coefficient in the first mode is 1:5%. For this value of damping, the dissipated energy as a fraction of T(@) is computed using equation (10). The values of the other damping model parameters, ©, 4; and 7 in models (b—d) are then selected such that in all hese models, at the first natural frequency, the same fraction of 7T(@) gets dissipated as in model (a). Figure 4 shows E(w) for the top chord members, 2, 3 and 4, for the above damping models. It has been verified that in all these cases he power input is numerically equal to the sum of energy dissipated in all the truss members; this serves as a check on the correctness of the calculations made. The behavior of E(w) for large frequencies is seen to be strongly governed by the choice of damping model. From equation (15) it follows that for the first four damping ypes listed above, (a—d), the bandwidth parameter, B,, varies as m°, w~', w? and @, respectively. It can be expected that the behavior of E(w) at large frequencies
Figure 3. The example truss considered; for all members: AE = 23-75 x 10°N; m = 0-882 kg/m. The value of c; in model (a) is selected on the basis that the damping coefficient in the first mode is 1:5%. For this value of damping, the dissipated energy as a fraction of T(@) is computed using equation (10). The values of the other damping model parameters, ©, 4; and 7 in models (b—d) are then selected such that in all hese models, at the first natural frequency, the same fraction of 7T(@) gets dissipated as in model (a). Figure 4 shows E(w) for the top chord members, 2, 3 and 4, for the above damping models. It has been verified that in all these cases he power input is numerically equal to the sum of energy dissipated in all the truss members; this serves as a check on the correctness of the calculations made. The behavior of E(w) for large frequencies is seen to be strongly governed by the choice of damping model. From equation (15) it follows that for the first four damping ypes listed above, (a—d), the bandwidth parameter, B,, varies as m°, w~', w? and @, respectively. It can be expected that the behavior of E(w) at large frequencies
Figure 4. Energy spectra for top chord of the nominal truss; - . - . -, Member 2; - - -, member 3; —, member 4. (a) Damping type (a); (b) damping type (b); (c) damping type (c); (d) damping type (d); (e) damping type (e).
Figure 4. Energy spectra for top chord of the nominal truss; - . - . -, Member 2; - - -, member 3; —, member 4. (a) Damping type (a); (b) damping type (b); (c) damping type (c); (d) damping type (d); (e) damping type (e).
Figure 5. (a) Scatter of nodal co-ordinates of truss with random geometry; —, nominal truss. (b) Sample realizations of truss with random geometry; *, nominal location of the nodes.
Figure 5. (a) Scatter of nodal co-ordinates of truss with random geometry; —, nominal truss. (b) Sample realizations of truss with random geometry; *, nominal location of the nodes.
These equations have been listed in the order in which they are solved in the numerical algorithm. A careful inspection of these equations reveals that these equations can be solved sequentially such that, at any given step, only one non-linear equation needs to be handled. Figure 5(b) shows a few sample realizations of the truss geometries in which the randomness parameters, &17, (i=1,..., 13), associated with the member lengths, are taken to be 0-05.
These equations have been listed in the order in which they are solved in the numerical algorithm. A careful inspection of these equations reveals that these equations can be solved sequentially such that, at any given step, only one non-linear equation needs to be handled. Figure 5(b) shows a few sample realizations of the truss geometries in which the randomness parameters, &17, (i=1,..., 13), associated with the member lengths, are taken to be 0-05.
Figure 6. Ensemble averages of E(w) for member 3; sample size = 500; uncertainty model A; ...., 5 percentage points; -.-.-, 50 percentage points; -*-*-, 95 percentage points; -O-O-, mean; - x - x -, mean + standard deviation. (a) Damping type (a); (b) damping type (b); (c) damping type (c); (d) damping type (d); (e) damping type (e).
Figure 6. Ensemble averages of E(w) for member 3; sample size = 500; uncertainty model A; ...., 5 percentage points; -.-.-, 50 percentage points; -*-*-, 95 percentage points; -O-O-, mean; - x - x -, mean + standard deviation. (a) Damping type (a); (b) damping type (b); (c) damping type (c); (d) damping type (d); (e) damping type (e).
where V,,(@)=amplitude of vellocity at x, and *=conjugation. These calculations are repeated for five randomly chosen points of harmonic load application. The estimates of E;(@), the energy in the jth subsystem due to the power input at the ith subsystem, and JT‘ (@), the power input to the ith subsystem are evaluated using the expressions
where V,,(@)=amplitude of vellocity at x, and *=conjugation. These calculations are repeated for five randomly chosen points of harmonic load application. The estimates of E;(@), the energy in the jth subsystem due to the power input at the ith subsystem, and JT‘ (@), the power input to the ith subsystem are evaluated using the expressions
Figure 9. SEA model for the truss example. Here JT denotes the vector of power input with I7T;(@) as given by equation (22) and IT;(@) = 0 for all i 4 5. Numerical work has been carried out for the damping models listed in section 4 and using the two methods of finding C(@) described in the previous section. Figures 12 and 13 show the spectra of E(@) for members 5 and 3 for the damping model (a). These spectra are now averaged over frequency bands, as is done in the traditional SEA. The 1/3 octave bands with center frequencies of 20, 31:5, 50, 80, 125, 200, 315, 500, 800, 1250, 2000, 5000 and 8000 Hz are selected [5]. This way of averaging amounts to assuming that the spectra consist of segments of random processes which are ergodic within each of the frequency bands considered. The frequency band averages of E(w) obtained using methods I and II, are compared with the ensemble statistics in Figures 14 and 15. The ensemble averages correspond to the randomness model A (see section 5). It can be observed that the ensemble mean and frequency average using method I compare reasonably well with each other, especially for driving frequencies beyond | kHz, for all the five damping models studied. This lends credence to the computation of the C-matrix using the approach described in section 6.1.1. On the other hand, the results of method II tend to overestimate the mean energy especially for member 3 which is away from the point of driving. This feature arises due to the effect of indirect coupling between near and far subsystems, which method II does not take cognizance of.
Figure 9. SEA model for the truss example. Here JT denotes the vector of power input with I7T;(@) as given by equation (22) and IT;(@) = 0 for all i 4 5. Numerical work has been carried out for the damping models listed in section 4 and using the two methods of finding C(@) described in the previous section. Figures 12 and 13 show the spectra of E(@) for members 5 and 3 for the damping model (a). These spectra are now averaged over frequency bands, as is done in the traditional SEA. The 1/3 octave bands with center frequencies of 20, 31:5, 50, 80, 125, 200, 315, 500, 800, 1250, 2000, 5000 and 8000 Hz are selected [5]. This way of averaging amounts to assuming that the spectra consist of segments of random processes which are ergodic within each of the frequency bands considered. The frequency band averages of E(w) obtained using methods I and II, are compared with the ensemble statistics in Figures 14 and 15. The ensemble averages correspond to the randomness model A (see section 5). It can be observed that the ensemble mean and frequency average using method I compare reasonably well with each other, especially for driving frequencies beyond | kHz, for all the five damping models studied. This lends credence to the computation of the C-matrix using the approach described in section 6.1.1. On the other hand, the results of method II tend to overestimate the mean energy especially for member 3 which is away from the point of driving. This feature arises due to the effect of indirect coupling between near and far subsystems, which method II does not take cognizance of.
Figure 10. Substructure for computation of the C-matrix.
Figure 10. Substructure for computation of the C-matrix.
Figure 11. SEA model for the substructure. The relationship between the random variables randomness and the spectra of energy is highly Cc | | and lognormal distributions to the simulated data u a of randomness (types A and B, see section 5) and of E(w) for members 5 (close to the point of driving of driving) were studied. A limited amount of results U; describing the systen non-linear with no clearh discernible patterns. Consequently, determining the probability distribution E(@ by rules of transformation of random variables or by invoking any of the limi theorems of probability is difficult. In view of this, it is of interest to investigat the digitally simulated data on E(q@) can be described by empirical distribution: 2]. Given that E(@) > 0, the usefulness of fitting exponential, Rayleigh, gamm« has been investigated. The sefulness of Gaussian distribution with non-zero mean in describing the data wa so considered. The parameter estimation capabilities of MATLAB statistic: toolbox was employed in this investigation. The range of system paramete: variations surveyed included: damping models (types a—e, see section 4), source: driving frequency (centra frequencies of 1/3 octave bands up to 8 kHz). The probability distribution functior1 and 3 (away from the poin are shown in Figures 16-18 In each of these figures, the probability distribution of E(@), estimated using 500( samples (marked in the caption as simulation results empirically fitted probability distribution function , is shown together with th (marked in the caption a
Figure 11. SEA model for the substructure. The relationship between the random variables randomness and the spectra of energy is highly Cc | | and lognormal distributions to the simulated data u a of randomness (types A and B, see section 5) and of E(w) for members 5 (close to the point of driving of driving) were studied. A limited amount of results U; describing the systen non-linear with no clearh discernible patterns. Consequently, determining the probability distribution E(@ by rules of transformation of random variables or by invoking any of the limi theorems of probability is difficult. In view of this, it is of interest to investigat the digitally simulated data on E(q@) can be described by empirical distribution: 2]. Given that E(@) > 0, the usefulness of fitting exponential, Rayleigh, gamm« has been investigated. The sefulness of Gaussian distribution with non-zero mean in describing the data wa so considered. The parameter estimation capabilities of MATLAB statistic: toolbox was employed in this investigation. The range of system paramete: variations surveyed included: damping models (types a—e, see section 4), source: driving frequency (centra frequencies of 1/3 octave bands up to 8 kHz). The probability distribution functior1 and 3 (away from the poin are shown in Figures 16-18 In each of these figures, the probability distribution of E(@), estimated using 500( samples (marked in the caption as simulation results empirically fitted probability distribution function , is shown together with th (marked in the caption a
Figure 12. Energy spectra for member 5 before * Frequency band averaging computed using SEA methods; damping model (a); —, method I; -.-.-, method II.
Figure 12. Energy spectra for member 5 before * Frequency band averaging computed using SEA methods; damping model (a); —, method I; -.-.-, method II.
Figure 13. Energy spectra for member 3 before eee band averaging computed using SEA methods; damping model (a); —, method I; -.-.-, method II. empirical most (bu J J 200 Hz, results). The following were the observations made: (1) lognormal distribution was observed to fit the data better than the other models considered; the goodness of the fit was examined using the Kolmogorov—Smirnov test and in not all) of the cases, it was observed that at 5% significance level, the ata could be taken as being lognormal distributed; the cases in which the data id not pass the test includes the cases shown in Figures 16(b), 16(c) and 18(b); (2) the effect of randomness in geometry of the truss was to produce probability distributions which were highly non-Gaussian and had long upper tails, especially in frequency ranges above about 200 Hz and (3) in frequency ranges below about he data could be well described by Gaussian, gamma or lognormal distributions.
Figure 13. Energy spectra for member 3 before eee band averaging computed using SEA methods; damping model (a); —, method I; -.-.-, method II. empirical most (bu J J 200 Hz, results). The following were the observations made: (1) lognormal distribution was observed to fit the data better than the other models considered; the goodness of the fit was examined using the Kolmogorov—Smirnov test and in not all) of the cases, it was observed that at 5% significance level, the ata could be taken as being lognormal distributed; the cases in which the data id not pass the test includes the cases shown in Figures 16(b), 16(c) and 18(b); (2) the effect of randomness in geometry of the truss was to produce probability distributions which were highly non-Gaussian and had long upper tails, especially in frequency ranges above about 200 Hz and (3) in frequency ranges below about he data could be well described by Gaussian, gamma or lognormal distributions.
Figure 14. Comparison of ensemble averages and SEA results for member 5; -O-O-, ensemble mean; —, 50 percentage point; - x - x -, SEA: method J; -*-*-, SEA: method II. (a) Damping type (a); (b) damping type (b); (c) damping type (c); (d) damping type (d); (e) damping type (e).
Figure 14. Comparison of ensemble averages and SEA results for member 5; -O-O-, ensemble mean; —, 50 percentage point; - x - x -, SEA: method J; -*-*-, SEA: method II. (a) Damping type (a); (b) damping type (b); (c) damping type (c); (d) damping type (d); (e) damping type (e).
Figure 15. Comparison of ensemble averages and SEA results for member 3; -O-O-, ensemble mean; —, 50 percentage point; - x - x -, SEA: method I; -*-*-, SEA: method II. (a) Damping type (a); (b) damping type (b); (c) damping type (c); (d) damping type (d); (e) damping type (e).
Figure 15. Comparison of ensemble averages and SEA results for member 3; -O-O-, ensemble mean; —, 50 percentage point; - x - x -, SEA: method I; -*-*-, SEA: method II. (a) Damping type (a); (b) damping type (b); (c) damping type (c); (d) damping type (d); (e) damping type (e).
Figure 18. Probability distribution function of E(w) for member 3; damping type (e); —, uncertainty model A, simulation; ..... , uncertainty model A, empirical; - - , uncertainty model B, simulation; -..-, uncertainty model B, empirical; *, extreme values, uncertainty model A; O, extreme values, unvertainty model B. (a) Driving frequency = = 80 Hz; (b) driving frequency = 500 Hz; (c) driving frequency = 2000 Hz.
Figure 18. Probability distribution function of E(w) for member 3; damping type (e); —, uncertainty model A, simulation; ..... , uncertainty model A, empirical; - - , uncertainty model B, simulation; -..-, uncertainty model B, empirical; *, extreme values, uncertainty model A; O, extreme values, unvertainty model B. (a) Driving frequency = = 80 Hz; (b) driving frequency = 500 Hz; (c) driving frequency = 2000 Hz.
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