There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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Formulas For Natural Frequency And Mode Shape

The Viscosity can be imagined as the force that should be applied to a layer of fluid belonging to the plane fixed to the velocity of the layer placed at a fixed distance. For example a fluid flowing in a tube at different speeds: the minimum speed is in the edge of the section (due to friction) and the maximum speed is at the center. The viscosity in this example is that pressure which is exerted on the wall allows a constant speed on the whole section.

Non-newtonian fluids.

Explicit formula and graphs of few special functions are derived in the paper on the basis of various definitions of various fractional derivatives and various fractional integrals. Their applications are also reviewed in the paper.

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Applications of Fractional Calculus

Linear fractional order controllers; A survey in the frequency domain

Structural vibration attenuation using a fractional order PD controller designed for a fractional order process

: Following the paradigm shift in the pharmaceutical industry from batch to continuous production, additional instrumentation and revision of control strategies to optimize material ﬂow throughout the downstream processes are required. Tableting manufacturing is one of the most productive in terms of turnover and investment into new sensor technologies is an important decision-making step. This paper proposes a continuous solution to detect changes in material properties, and a control algorithm to aid in minimizing risk at the end-product line. Some of the sub-processes involved in tableting manufacturing perform changes in powder and liquid mixtures, granulation, density, therefore changing ﬂow conditions of the raw material. Using impedance spectroscopy in a continuous sensing and monitoring context, it is possible to perform online identiﬁcation of generalized (fractional) order parametric models where the coefﬁcients are correlated to changes in material properties. The model parameters are then included in a self-tuning control gain used in ratio control as part of the local process control loop. The solution proposed here is easy to implement and poses a signiﬁcant added value to the current state of art in pharmaceutical manufacturing technologies.

Impedance Spectroscopy Sensing Material Properties for Self-Tuning Ratio Control in Pharmaceutical Industry

Bioelectrical impedance analysis of thermal-induced cutaneous nociception

This paper provides an overview of the distributed parameter properties of non-Newtonian fluids and a proposal for identifying them. A low cost setup is described along with a proposed methodology protocol. The paper introduces the problem of moving from a linear framework of fluid properties towards a nonlinear one and motivates the choice for lumped nonlinear parameter model structures. It follows identification using nonlinear least squares in various liquids. The results obtained suggest that parameters of the proposed fractional order impedance model are susceptible to changes in density as being one important feature of non-Newtonian fluids. Further use of these findings across disciplines is given in the conclusion section of the paper.

Experiment Design and Estimation Methodology of Varying Properties for Non-Newtonian Fluids

The control community recognizes the improved performance, stability and intrinsic robustness brought by fractional orders of differentiation and integration. However, the full potential of Fractional Order Proportional Integral Derivative (FOPID) controllers is yet to be grasped in the modern industrial setting. Mass adoption is hindered by implementation difficulties associated to fractional order terms. Available approximation methods usually obtain high order equivalent integer order transfer functions that could be difficult to implement. The present study introduces a novel frequency-domain approximation strategy based on non-linear least squares optimization routine. The result is an alternative approximation strategy that obtains an accurate frequency domain approximation with a reduced order transfer function. Validations are performed on various numerical examples, also highlighting the efficiency of the method for fractional orders greater than 1.

Isabela Roxana Birs

Papers

Structural vibration attenuation using a fractional order PD controller designed for a fractional order process

Impedance Spectroscopy Sensing Material Properties for Self-Tuning Ratio Control in Pharmaceutical Industry

Bioelectrical impedance analysis of thermal-induced cutaneous nociception

Experiment Design and Estimation Methodology of Varying Properties for Non-Newtonian Fluids

A low-order approximation method for fractional order PID controllers