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The performance of ultra long range artillery projectiles is analyzed. In the past ultra long range artillery was useless due to low accuracy, but now with relatively inexpensive guidance available for long range projectiles, such artillery is very promising. Given the projectiles' initial velocity, mass, density, and shape, the projectile's range and impact velocity are determined. For better results a MatLab program described in the paper has been used. It is shown that both ranges and impact velocities increase with mass, density, and initial velocity. For projectiles weighing 5 kg to 142 kg, and launched from a conventional cannon at initial velocity 1,700 m/s, the ranges are 127 km to 226 km and the impact velocities are 562 m/s to 1,127 m/s. For projectiles weighing 5.3 kg to 14.6 kg, and launched from an electromagnetic rail gun initial velocity 2,500 m/s, the ranges are 359 km to 499 km and the impact velocities are 1,361 m/s to 1,996 m/s. The temperature distribution on the surface of the projectile as a function of the distance from the nose tip is investigated as well. [ABSTRACT FROM AUTHOR]

We present a set of our mathematical results on a model of piezoelectric energy harvester. The model is well known and is described in engineering literature. The harvester is designed as a beam with a piezoceramic layer attached to its top face (unimorph configuration). A pair of thin perfectly conductive electrodes is covering the top and the bottom faces of the piezoceramic layer. These electrodes are connected to a resistive load. The model is governed by a system consisting of two equations. The first of them is the equation of the Euler-Bemoulli model for the transverse vibrations of a beam subject to actions of an external (viscous/fluid) damping and of an external force. The second equation represents the Kirchhoff's law for the electric circuit. Both equations are coupled due to the direct and converse piezoelectric effects. The boundary conditions for the beam equations are of clamped-free type with an additional term that is proportional to the voltage on the resistive load and is produced by the converse piezoelectric effect. The circuit equation contains an extra term that depends on the transverse displacement of the beam and is produced by the direct piezoelectric effect. We represent the system as a single operator evolution equation in a Hilbert space. The dynamics generator of this system is a non-selfadjoint operator with compact resolvent. We present precise statements and outline an approach to the proofs of the following results. 1) Asymptotic formulas for the eigenvalues and the generalized eigenvectors of the dynamics generator. 2) Completeness, minimality, and Riesz basis property of the generalized eigenvectors. 3) Solutions by the spectral decomposition method of two control problems: zero controllability problem and a version of output tracking problem. [ABSTRACT FROM AUTHOR]