The transient dynamic analysis of constrained mechanical systems may require the solution of a mixed set of algebraic and differential equations of motion. The usual formulation of these equations is expressed in terms of the accelerations of the system components. A canonical form of the equations of motion in terms of the system velocities and the time derivative of the system momenta may be used instead. This is a natural form of the equations in which all the state variables are explicitly expressed, and have the same physical importance. The numerical solution obtained from the canonical equations shows more accuracy and stability, specifically for systems with large and fluctuating forces. For the mechanical systems that undergo an impact, the usual numerical solution of the equations of motion is not valid. Two different methods of analysis of impact problems are presented. In one method, the variations of the impulsive force during the contact period are directly added to the vector of forces in the canonical equations of motion. In the second method, based on the assumption of instantaneous nature of impact, a set of momentum balance-impulse equations is derived by explicitly integrating the canonical equations. These equations are solved at the time of impact for the jump in the system momenta right after impact. Necessary parameters are evaluated for the performance of the two methods of analysis. These parameters include the maximum relative indentation, the maximum contact force, and the coefficient of restitution. The parameters are determined for the collision between two bodies in a system with any general geometric or material properties. The influence of friction modeling in the magnitude and the direction of the total force at the contact surfaces is discussed. The dynamics of a vehicle collision is studied in order to illustrate the efficiency of obtaining a solution to the canonical equations, the simplicity of solving the momentum balance-impulse equations.

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The objective of this study is to address the gap that exists in the literature regarding quantifiable guidelines, benchmarks and consistency of applications. During the research acceptable benchmarks for the calculation or quantification of the elements linked to materiality and audit risk were found. The benchmarks are in compliance with the practices and the requirements of the ISAs and regulations. Models and benchmarks based on literature were used as a basis and modified for application in the auditing environment. The combination of literature, responses from public practitioners and experience based on best practices resulted in the development of a modified risk-based assessment model. The conclusion from the empirical study indicated that there are no defined rules or basis for calculating materiality and audit risk. The inconsistencies in responses indicate that audit firms and developer of key concepts interpret and apply the above-mentioned term different in practice. The interpretations of the relevant ISAs, appear to be conceptually correct as no major non-compliances were identified. Various instances indicated that there is a lack of guidance with regard to the quantification or qualification of benchmarks. The implementation of the Sarbanes-Oxley Act (2002) was an event that leads to the consideration of more conservative benchmarks. The most consistent benchmark that stood the test of time was Discussion paper 6 (1984). The 30 years since the development of these benchmarks indicate that little attention has been given to one of the most complex issues in auditing. Companies within different industries are not generic and exceptions will occur where the auditor needs to apply professional judgment to accommodate the deviations. Further research is required to assist the audit professionals and students in the development of consistent benchmarks to increase the reputation of the profession. The conclusion drawn from this study is that audit materiality and audit risk has a significant impact on the audit process as even the audit report is influenced by proper audit planning and guidelines to support the auditor in audits. / MCom (Accountancy), North-West University, Vaal Triangle Campus, 2013