Probabilistic Approach for Prediction of Critical Speeds
Introduction:
In rotor dynamic analysis, the first step is to compute natural frequencies ( critical speeds) and damped critical speeds. If the operation speed is close to one of the critical speed, severe vibration, resonance, may occur. Consequently, the rotor-bearing system has to be modified to shift critical speeds away from the operation speed. In general supporting bearings are redesigned or their positions are changed. Hopefully the recalculated critical speeds will not cause resonant vibration. This process is usually a time consuming trial-and-error approach. This tedious trial-and-error process is due to the conventional design method, the so-called deterministic approach where design parameters are set to be constants. Any change of one number requires recalculation of the rotor dynamic system. As opposite to the deterministic approach, the probabilistic approach considers the randomness of design variables and replaces constant design variables by random variable functions. Furthermore, a limit state function is set as the difference between the expected critical speed and the output of the modified rotor dynamic software. Then probabilistic method such as Monte Carlo Method or First Order Reliability Method, FORM can be utilized to optimize the limit state function and find and find the design variables in one calculation.
Concepts of Probabilistic Robust Design (Probabilistic Optimization):
A product design or manufacturing process can be expressed as the chart, Fig. 1, shown below. The iteration process is time consuming. Optimization schemes are desirable. The goal of optimization is to develop an actual process which meets design or manufacturing requirements. The optimization process is to modify input parameters into random variables with the use of available analytical or statistic models to come up with an output which minimizes an objective function. The objective function or the so-called limit state function is defined as the difference between the target (with required conditions) and the actual output. When the limit-state function approaches zero and its variation is minimized, the design point is reached and the set of input variables is the optimal combination of all design parameters.
Define an Objective function (called limit-state function):
Objective function = Required output – Output (from Analytical or Statistic Model)
Namely,
G(X) = Greq - Gact(X)
Process of Probabilistic Optimization:
(1) change input parameters to random variables represented by probability distribution functions. (2) calculate Gact(X). If an alytical Model is used, the model can be interfaced with the optimization program. (3) find a set of optimal design parameters which lie on an imaginary multi-dimensional surface represented by G(X) =0 with minimum gradient of G(X) indicating insignificant parameter variations.
Using these optimal parameters so-obtained to design the target product or carry out the intended operation, either the product or the operation should satisfy all required conditions.
Advantages of this approach:
consider the uncertainties of all design variables
very high accuracy
can be applied to any product and operation system
reduce costs and time
has been linked to Six Sigma process
An Example of Probability Distribution Function:
California Super Lotto Mega numbers can be used as an example to show the generation of a probability distribution function (PDF). A set of consecutive Mega numbers (50) are:
21,7,16,24,12,10,10,10,23,7,17,6,9,12,15,16,4,19,14,17,4,11,18,3,1,22,12,27, 15,22,6,2,23,18,27,3,8,12,20,10,3,21,23,13,20,8,11,16,25,22.
Taking those historic data and utilizing curve fitting techniques, a frequency map can be developed and its PDF can then be generated. The figure depicts the best fitting (Weibull distribution) curves.
The vertical columns show the occurrence frequencies of numbers. Apparently, number 10 and 12 have the highest occurrence frequency. The PDF is defined as frequency per unit gap. Therefore, the value is the relative scale for the occurrence probability of each number.
Similar process can be applied to handle collected experimental data and to create probability distribution functions for design and manufacturing parameters which are in fact random variables. Techniques of goodness of fitting can be employed to generate best forms of distribution functions. (picture here)
This example shows how to convert raw data into probability distribution functions. With available PDF's one will be able to execute above-presented probabilistic robust design process.