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Any real-time diagnosis of machinery systems is imperative to prevent their malfunctions. One of such promising diagnoses is first to regard any signal taken from the machinery operation as the time series data. We then think of the series itself expressing the displacement under the influence of an external force F. The time rate variables of the series are velocity V and acceleration A that is proportional to F. Under this force F, if the system makes any changes, its kinetic energy KE also changes. The time rate change of KE is the power PW, which is proportional to the product of V and A. The PW is very sensitive to any subtle change in F that controls the system dynamics. We generalize the method to obtain these physical variables from the time series with physical wavelets. The wavelets are the windows through which we selectively view the series and its rate variables in the frequency region confined within each wavelet width. Any subtle change in F will appear on the selective correlation of V and A. The change in F that may lead to the malfunction of the systems will sometimes start at the order of 0.01%, which is often comparable to adverse noise. Even with this noise, however, the selective correlation will identify the small change leading to the malfunction. It can also predict the abnormal operating condition to control the system adoptively. We further make the physical wavelet orthogonal to each other in order to reconstruct the state space. In the space, we then draw a trajectory from the time series, which will reveal the physical dynamics of the operating conditions. This reconstruction is similar to so called time delay embedding in Chaos analysis. Their bases chosen to reconstruct the state space, however, have no direct physical meaning. An example of the diagnoses and prognoses on a drilling machine tool by using the physical wavelets is discussed in detail.
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