Row normal upper Phi EndMatrix minus omega squared Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix Superscript upper T Baseline Start 1 By 1 Matrix 1st Row upper M EndMatrix Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix right-parenthesis Start 1 By 1 Matrix 1st Row bold-italic q EndMatrix prime 2nd Column Blank 2nd Row 1st Column equals left-parenthesis Start 1 By 1 Matrix 1st Row upper K EndMatrix prime minus omega squared Start 1 By 1 Matrix 1st Row upper M EndMatrix prime right-parenthesis Start 1 By 1 Matrix 1st Row bold-italic q EndMatrix Superscript prime Baseline 2nd Column equals Start 1 By 1 Matrix 1st Row normal upper Phi EndMatrix Superscript upper T Baseline Start 1 By 1 Matrix 1st Row bold-italic upper F EndMatrix equals Start 1 By 1 Matrix 1st Row bold-italic upper F EndMatrix prime EndLayout"/> (1.124)
Fn′ are the components of the modal force vector F′. Using the above defined orthogonality and normalisation we get:
or in another form:
The response shape is reconstructed using (1.122). Thus, with modal coordinates expressed by independent degrees of freedom – the modes – the system decouples. This can by seen by the diagonal form of the matrix. For a system with N degrees of freedom, the modal solution is exact. The similarity to Equation (1.28) underlines that every mode can be interpreted as a single resonator. Thus, every dynamical system can be interpreted as a group of independent resonators of modal mass mn and stiffness ωn2mn In accordance with Equation (1.63) we may introduce a modal damping ηn for every mode. In that case the response in modal coordinates is:
In computational dynamics the solution of the homogeneous equation for normal modes and the calculation of the system response using the diagonal form is called modal frequency response. Finite element methods are frequently applying this method to simplify the solution and to condense the system equations to a reduced coordinate system. In addition modes are an excellent option to exchange model information or simulation results between different solvers. Usually, the modal base is truncated and not all modes are considered. In that case the modal solution is only an approximation of the direct solution. The denominator ωn2−ω2 reveals that the participation of every mode to the global solution decreases with the distance from the modal frequency. Thus, a reasonable set of modes near the frequency of interest can by sufficient.
Finally, we conclude that the change in the coordinate base to modal coordinates keeps the general format of the equation of motion.
1.5 Random Process
As clean harmonic signals are rare in technical systems we need a mathematical toolbox to deal with nonharmonic signals. Due to the Fourier theory (Nelson and Elliott, 1993) every deterministic time signal can be synthesized by a sum of harmonic components. This provides a theoretical link between periodic and harmonic but also transient signals. See appendix A.1 for a brief introduction.
The same theory can also be applied to stochastic or random signals but in a slightly different manner. Fourier analysis and methods for investigating the random processes and the description of mechanical systems by impulse response or frequency response functions is an important toolset for the description of vibroacoustic systems. In addition and due to digitalisation most signals or spectra are given as a discrete, digital set of values. This discrete formulation creates some pitfalls that may also lead to misinterpretations. Even though the signal analysis is not a major subject in vibroacoustic simulation it is a very important especially when acoustic experiments are performed.
Signals from technical processes are often not predictable and are generated randomly like pressure fluctuations in a turbulent flow, the impact of raindrops on a roof, or the stochastic combustion in a jet. For dealing with such signals the above described methods must be adapted. In addition we need formulations that allow for the definition of the statistics of random processes as there is no deterministic functionality between time or frequency and the physical quantity, for example force or displacement.
1.5.1 Probability Function
Imagine a random process creating signal sequences as shown in Figure 1.19. At each time the signal value f(t) may be different and has a certain continuous value. One option to characterize this signal is to define the probability that the signal value is less or equal to a specific value fk. Thus we define a probability for f(t) to be less than or equal to fk
Figure 1.19 Stochastic fluctuation with time. Source: Alexander Peiffer.
Next, we are interested in the probability that the value of f(t) is in a range defined by Δf=f2−f1 meaning the probability Prob[f1<f≤f2]. Now we can define the probability density function as
Consequently
