Laplace Transform is often used in solving the free vibration problems of structural beams. In existing research, there are two types of simplified models of continuous beam placement. The first is to regard the continuous beam as a single-span beam, the middle bearing of which is replaced by the bearing reaction force; the second is to divide the continuous beam into several simply supported beams, with the bending moment of the continuous beam at the middle bearing considered as the external force. Research shows that the second simplified model is incorrect, and the frequency equation derived from the first simplified model contains multiple expressions which might not be equivalent to each other. This paper specifies the application method of Laplace Transform in solving the free vibration problems of continuous beams, having great significance in the proper use of the transform method.
In this paper, a new simple method for determination of flow parameters, axial dispersion coefficients DL and Péclet numbers Pe was presented. This method is based on an accurate measurement model considering pulse tracer response. Our method makes it possible to test the character of gas flow motion and precisely measure flow parameters for different pressures and temperatures. The idea of combining the transfer function, numerical inversion of the Laplace transform and optimisation method gives many benefits like a simple and effective way of finding solution of inverse problem and model coefficients. The calculated values of flow parameters (DL and/or Pe) suggest that in the considered case the gas flow is neither plug flow nor perfect mixing under operation condition. The obtained outcomes agree with the gas flow theory. Calculations were performed using the CAS program type, Maple®.