Physics
Linear Analysis
Linear analysis is a mathematical method used to study the behavior of physical systems. It involves approximating the system's behavior as a linear function of its inputs and outputs, allowing for the use of linear algebra to solve equations and make predictions. Linear analysis is particularly useful for studying systems that exhibit small deviations from equilibrium.
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4 Key excerpts on "Linear Analysis"
- S. Graham Kelly(Author)
- 2008(Publication Date)
- CRC Press(Publisher)
65 Chapter 2 Linear algebra 2.1 Introduction Linear algebra is the algebra used for analysis of linear systems. It provides a foundation on which solutions to mathematical problems can be developed. Success in obtaining a solution to a mathematical problem requires finding the specific solution among a possible set of solutions, the solution space. An understanding of the solution space and the properties of elements of the solution space leads to the development of solution techniques. It is in this spirit that this review of linear algebra is presented. An exact solution of a problem is a solution for the dependent variables which satisfies without error the mathematical problem for all possible values of the independent variables. An exact solution, while desirable, is not always possible. Approximate solutions are sought when an exact solution is not available. Approximate solutions are of two types. Variational methods are used to determine continuous functions of the independent variables which provide in some sense the “best approximation,” chosen from a specified set, to the exact solution. Numerical solutions provide an approximation to the exact solution only at discrete values of independent variables. Linear algebra provides a framework in which these approximate solutions can be devel-oped and in which the error between the exact solution and an approximate solution can be estimated. Linear algebra provides a framework for developing solutions to linear prob-lems. Modeling of engineering systems often leads to nonlinear mathematical problems. Exact solutions exist for only a few nonlinear problems. Often assump-tions are made such that the nonlinear problem can be approximated by a linear problem. Even if the assumptions that linearize the problem are not valid, some understanding of the solution can be obtained by studying the linearized prob-lem.
eBook - PDF- Arthur Akers, Max Gassman, Richard Smith(Authors)
- 2006(Publication Date)
- CRC Press(Publisher)
These block diagrams, however, are much more generic than fluid power circuits, yet considerable quantitative and qualitative insight may be gained from their analysis. These block diagrams will be manipulated so a network of simple 101 102 LINEAR SYSTEMS ANALYSIS blocks can be consolidated into a single function, the transfer function, that may be said to transform the input signal into the output. These consolidated functions will be used to determine the stability of a system. Two fundamental entities, the spring-mass-damper and the single fluid volume with differential flow rates, are building blocks for fluid power sys-tems. These two entities will be analyzed to introduce ideas concerning damping, resonant frequency, and a system time constant. 5.2 LINEAR SYSTEMS The description linear has been encountered several times in this text. For dynamic systems, a linear system is one in which the equations of motion are ordinary differential equations with constant coefficients. A corollary of this statement is the fact that if two or more inputs are applied to a linear system concurrently , then the output will be the sum of the outputs that would occur if the inputs acted on the system individually. At the level at which control theory will be introduced in this chapter, a system will either be linear or approximations will have been made so the system can be treated as linear. It should be stated categorically that approximating nonlinear systems by linear ones is not generally a valid approach for simulation. The linearized system can be investigated for stability properties near a specific operating point and this information can then be used in a more general nonlinear simulation. The term stability will be explained in more analytical terms as the mathematical developments are continued. At this stage, consider a stable system as one that will have bounded response when subject to a bounded input.
eBook - PDF- S. Graham Kelly(Author)
- 2006(Publication Date)
- CRC Press(Publisher)
3 Linear Algebra 3.1 INTRODUCTION Application of Lagrange’s equations to the lagrangian of a mechanical system results in a differential equation, or a set of differential equations whose solution is the system response. Application of Lagrange’s equations to a discrete system leads to a set of ordinary differential equations, while application of Lagrange’s equations to a distributed parameter system leads to partial differential equations. Mathematical methods are employed to determine the solution of the equations, and hence the system response. Linear algebra is the algebra of linear systems. It provides a foundation in which solutions to mathematical problems are developed. Success in obtaining a solution to a mathematical problem requires finding the specific solution from a possible set of solutions, the solution space. An understanding of the solution space and properties of elements of the solution space leads to the development of solution techniques. It is the spirit in which this review of linear algebra is presented. An exact solution of a problem is a solution for the dependent variables that satisfies, without error, the mathematical problem for all possible values of the independent variables. An exact solution, while desirable, is not always possible. Approximate solutions are sought when an exact solution is not available. Approximate solutions are of two types. Variational methods are used to determine continuous functions of the independent variables which provide in some sense the “best approximation”, chosen from a specified set, to the exact solution. Numerical solutions provide an approximation to the exact solution only at discrete values of independent variables. Linear algebra provides a framework in which these approximate solutions are developed, and a framework in which the error between the exact solution and an approximate solution is estimated. Linear algebra provides a framework for developing solutions to linear problems.
eBook - PDF- Robert F. Stengel(Author)
- 2015(Publication Date)
- Princeton University Press(Publisher)
Chapter 4 Methods of Analysis and Design With the ready availability of digital computation, it is increasingly easy to analyze the dynamics of aircraft by integrating their nonlinear equa-tions of motion. The numerical results are as good as the data that are used for computation, and even subtle differences in response that arise from changing parameters or conditions can be stored, printed, and plot-ted for interpretation. What is missing in this direct approach is an under-lying framework for simplifying, unifying, and generalizing the results, for learning the underlying lessons that the numbers only imply. In a tra-dition that predates digital computation by several decades, we can gain considerable insight by referring to linearized models that are derived from the nonlinear equations and by investigating the probabilistic ef-fects of system uncertainty. Small perturbations from a nominal flight path can be simulated and evaluated using the linearized equations of motion. Linear models pro-vide particular advantages for analysis because the solutions are additive and underlying modes of motion can be identified. These perturbations may arise from small variations in initial conditions, control inputs, or disturbance effects. The nominal path may represent a changing flight con-dition, but more often it is a steady or quasisteady path. In the first case, linear, time-varying (LTV) models are appropriate, while in the second, even simpler linear, time-invariant (LTI) models are satisfactory. LTI models receive emphasis as the chapter evolves, as they form the basis for classical stability-and-control analysis. Methods for deriving and analyzing linear models of aircraft dynam-ics are presented here.
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