Distributed system simulation using infinite product expansions
Linear distributed dynamic system simulation using infinite product expansions for transcendental terms in transfer functions
Goodson, R. E.↗
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A High Frequency Model of Cascade Noise
Closed form asymptotic expressions for computing high frequency noise generated by an annular cascade in an infinite duct containing a uniform flow are presented. There are two new elements in this work. First, the annular duct mode representation does not rely on the often-used Bessel function expansion resulting in simpler expressions for both the radial eigenvalues and eigenfunctions of the duct. In particular, the new representation provides an explicit approximate formula for the radial eigenvalues obviating the need for solutions of the transcendental annular duct eigenvalue equation. Also, the radial eigenfunctions are represented in terms of exponentials eliminating the numerical problems associated with generating the Bessel functions on a computer. The second new element is the construction of an unsteady response model for an annular cascade. The new construction satisfies the boundary conditions on both the cascade and duct walls simultaneously adding a new level of realism to the noise calculations. Preliminary results which demonstrate the effectiveness of the new elements are presented. A discussion of the utility of the asymptotic formulas for calculating cascade discrete tone as well as broadband noise is also included.
Envia, Edmane↗
Updating Critical Temperature Calculations for Several Secondary High Explosives
In designing experiments where high explosives (HEs) are heated, it is important to have an understanding of where thermal runaway may occur. This determination is often done by using the Frank-Kamenetskii (FK) equation. For several highly-studied HEs, the parameters necessary for FK calculations are typically referenced from decades-old literature, and more recent experimental data have shown that these values require adjustment. For example, some of the legacy parameters dangerously overpredict the critical temperature by tens of degrees relative to more recent observed values. Herein, we first summarize historical results and highlight the importance of insulation effects when estimating critical temperatures for small samples, which is especially relevant to small-scale aging, compatibility, and characterization experiments. Here, we then present updated FK parameters for several common secondary CHNO HEs like RDX, HMX, PETN, and TATB, as well as the first reported values for 2,6-Diamino-3,5-dinitropyrazine-1-oxide (LLM-105). Our updated parameters produce critical-temperature curves that are consistent with both mm-scale differentical scanning calorimetry (DSC) measurements and cm-scale data from the Lawrence Livermore National Laboratory one-dimensional time-to-explosion (ODTX) experiment. In analyzing the critical temperature versus sample size curves derived from the transcendental FK equation, we discovered that these curves are remarkably well described by a simple power-law function with a universal power of 0.142 and a HE-specific scaling factor.
45 MILITARY TECHNOLOGY, WEAPONRY, AND NATIONAL DEF↗
Boundary-layer effects in composite laminates: Free-edge stress singularities, part 6
A rigorous mathematical model was obtained for the boundary-layer free-edge stress singularity in angleplied and crossplied fiber composite laminates. The solution was obtained using a method consisting of complex-variable stress function potentials and eigenfunction expansions. The required order of the boundary-layer stress singularity is determined by solving the transcendental characteristic equation obtained from the homogeneous solution of the partial differential equations. Numerical results obtained show that the boundary-layer stress singularity depends only upon material elastic constants and fiber orientation of the adjacent plies. For angleplied and crossplied laminates the order of the singularity is weak in general.
Wanag, S. S.↗
Theory of the optimally coupled Q-switched laser
The general equations describing Q-switched laser operation are transcendental in nature and require numerical solutions, which greatly complicates the optimization of real devices. Here, it is shown that, using the mathematical technique of Lagrange multipliers, one can derive simple analytic expressions for all of the key parameters of the optimally coupled laser, i.e., one which uses an optimum reflector to obtain maximum laser efficiency for a given pump level. These parameters can all be expressed as functions of a single dimensionless variable z, defined as the ratio of the unsaturated small-signal gain to the dissipative (nonuseful) optical loss, multiplied by a few simple constants. Laser design tradeoff studies and performance projections can be accomplished quickly with the help of several graphs and a simple hand calculator. Sample calculations for a high-gain Nd:YAG and a low-gain alexandrite laser are presented as illustrations of the technique.
Degnan, John J.↗