Modal superposition

The second section deals with the analysis methods that are used in fuzzy finite element analysis. An overview of different approaches is presented first the transformation method, affine analysis, and global optimisation. The application to dynamic analysis is considered. The core of this section is the presentation of a consistent analysis approach to predict the effect of parameter uncertainties on different characteristics of dynamic behaviour of structures, resonance frequencies and dynamic response levels. A hybrid approach is developed, based on modal superposition and optimisation. 

Moens (2002) has developed a hybrid approach for fuzzy frequency response functions (FRF) FE analysis. The approach uses the classical concept of modal superposition. This section briefly presents the modal snperposition principle, first in a deterministic formulation and then in a fuzzy formulation. 

The deterministic modal superposition concept states that, considering the first n modes, this FRF equals ... 

Figure 4(a) gives a graphical overview of this deterministic modal superposition procedure. It introduces the function T>(co) = (ki — aP-ihi) to express the modal response denominator as a function of the frequency. 

For undamped strnctnres with one or more interval inputs, the total envelope FRF can be calculated using a step by step interval translation of the deterministic modal superposition procedure, as shown in fignre 4(b).The full mathematical description of this method, developed by Moens, can be fonnd in Moens Vandepitte (2004). It consists of three major steps ... 

Figure 4 Deterministic and interval modal superposition algorithm.

It is noted that Equation 6 can be interpreted as the modal superposition formula for the impulse response functions. Similar expressions can be derived if other response functions are considered. For example, the structural response rAt) is written as... 

However, the right hand side decouples only approximately with respect to in- wind or lateral wind forces and with respect to the control forces ofthe absorbers as well. The floor displacements (and rotations) in Cartesian coordinates can subsequently be determined by modal superposition, x =. ... 

For a mathematical solution of the system governed by the above equations, suitable initial conditions are also needed. For example, the initial conditions may be given by the static solutions with zero initial velocities. The standard modal superposition method may be used to solve the problem. 

S. H. Swanger and D. M. Boore, Simulation of strong motion displacements using surface wave modal superposition . Bull. Seism. Soc. Am., 68, 907-922 (1978). 

Lateral force method Modal analysis Modal response spectrum analysis Modal superposition Response spectrum Seismic performance assessment... 

The most probable total response shall be obtained as the square-root-sum-of-the-squares (SRSS) of the response from individual modes, when using the response spectrum method. Use either modal superposition or direct integration approach for time history analysis. 

Use either modal superposition of three components of earthquake motion or direct integration approach for time history analysis. The integration time step shall be justified if the time history direct integration method of analysis is used. 

By means of various methods (Ewins 1984), the respective frequency response Xij a)) can be approximated by the superposition of several (n) adapted harmonic oscillators (so-called modal harmonic oscillators). Their parameters, the modal eigenfrequency co , the associated eigenvector as well as the modal damping 6 , correspond to the modal matrices Mm, Dm, and Cm (Gawronski 2004), but they are not comparable with a real mass, damping, or stiffness. Those modal harmonic oscillators only describe an... 

The fundamental works of P. Medgyessy on the decomposition of superpositions of density functions and discrete distributions (Medgyessy 1977) has been applied to set up conditions for determining the modality (i.e. the number of maxima) of density functions and distributions. 

Figure 26.15 TOF spectra of desorbed NO from photodissodation of NO2 adsorbed on Pd(lll), taken with a quadrupole mass spectrometer. Different photon energies for desorption are used, as indicated in the top right corner of the individual panels. Notice the bi modal distribution is more pronounced at lower photon energies. The modified Maxwell-Boltzmann distributions, used to fit the data by their superposition, are represented by lines. Reproduced from Hasselbrink etot J. Chem. Phys., 1990,92 3154, with permission oftheAmerican Institute of Physics...

The simulations are carried out for the first 20 s of the first three, and the first 100 s of the last three earthquakes. The major response is seen in this time frame, and it also allows for more detail in the illustrations. A direct integration method for the solution of the equation of motion in Eq. 18.2 is used. Superposition of modal responses is not possible for systems with non-proportional damping, as is the case with the current structure with added dampers. The Newmark p method (by using the unconditionally stable average acceleration method) is used as the solver for aU simulations in this study. The function that implements this ordinary differential equation (ODE) solver makes sure that the simulation time step is 20 times smaller than the smallest period of the structure. If this is not the case, it interpolates the excitation data for a smaller time step, and later outputs the response at a 0.01 s. In this work, the building type structure has a minimum period of 0.108 s. Thus, the simulation takes place at 0.108/20 = 0.0054 s. 

The dynamic response U can be computed by modal analysis, as the superposition of the modal contributions... 

The complete spatial variation of each radiation mode is given by Eq. (25-1), and the range of values of the propagation constant satisfies 0 p < kn. For a particular value of j8, the modal fields can be regarded as a Fourier superposition of the fields of a family of plane waves, all inclined at angle 6 to the waveguide axis. In the uniform cladding, 6 is related to j8 by... 

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