Direct substitution method

The direct-substitution method is a simple and useful method for finding the values of parameters m the characteristic equation that put the system just at the limit of stability. 

In Chap. 12 we will show that we can convert from the Laplace domain (Russian) into the frequency domain (Chinese) by merely substituting ia for s in the transfer function of the process. This is similar to the direct substitution method, but keep in mind that these two operations are different. In one we use the transfer function. In the other we use the characteristic equation. 

We could make a root locus plot in the U) plane. Or we could use the direct-substitution method (let U) = iv) to find the maximum stable value of. Let us use the Routh stability criterion. This criterion cannot be applied in the z plane because it gives the number of positive roots, not the number of roots outside the unit circle. The Routh array is... 

Both processes have been used successfully in field trials. The 1978 trial in Brazos County, Texas [35] used both on the same job for comparison purposes. After three years no significant difference in performance has been noted. Only 2 of the last 18 field trials placed in the United States during 1979 utilized the preblending method, exclusively [33] indicating a perference for the direct substitution method. 

Direct Substitution Method. Sulfur and asphalt were poured together at 300°F and were added to the heated aggregate at 300°-320°F. The three materials were mixed in the Hobart mixer for 2 min. Standard Marshall test samples for Marshall and dynamic stiffness tests were prepared as in the emulsification method. Bar samples for fatigue testing were prepared by kneading the mix into 3 X 3 X 12-in. bars to obtain compaction equal to the Marshall compacts. 

Sand-Sulfur-Asphalt Method. The direct substitution method was used to evaluate the substitution of sulfur for asphalt in preparing paving materials with improved properties. Sulfur and asphalt were combined and added to graded sand in the Hobart mixer and were mixed for 2 min. Test samples were prepared as described for the direct substitution method. 

Evaluation of sulfur-asphalt binders for paving materials showed no differences in materials prepared using either the emulsified or the direct substitution method in preparing the binder. Therefore, the simpler direct substitution method was used in all subsequent testing. 

The number of performed Picard iterations is denoted by 1. Because of the way the two subspaces U and / are constructed, this scheme will converge much faster than the direct substitution method applied to the original system of equations. We refer to [4] for more details about the convergence analysis. 

The acceleration over the direct substitution method is apparent since convergence is achieved in one iteration from both initial guesses. 

Use the direct substitution method to determine Kcm for the system with the characteristic equation given by Eq. 11-99. 

The direct-substitution method is related to the Routh stability criterion in Section 11.4.2. If the characteristic equation has a pair of roots on the imaginary axis, equidistant from the origin, and all other roots are in the left-hand plane, the single element in the next-to-last row of the Routh array will be zero. Then the location of the two imaginary roots can be obtained from the solution of the equation. 

The direct-substitution method is also related to the frequency response approach of Chapter 14, because both techniques are based on the substitution S = jid. 

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