Category Archive for: Numerical Calculus And Differential Equations

ODE Solvers in the Control System Toolbox

ODE Solvers in the Control System Toolbox Many of the functions of the Control System toolbox can be used to solve linear. time-invariant (constant-coefficient) differential equations. They are sometime more convenient to use and more powerful than the ODE solvers discussed th far, because general solutions can be found for linear, time-invariant equation Here we discuss several of…

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Velocity from an Accelerometer

Velocity from an Accelerometer An accelerometer measures acceleration and is used in aircraft, rockets, and other vehicles to estimate the vehicle’s velocity and displacement. The accelerometer integrates the  acceleration signal to produce an estimate of the velocity, and it integrates the velocity estimate to produce an estimate of displacement. Suppose the vehicle starts from rest at time t…

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Advanced Solver Syntax

Advanced Solver Syntax The complete syntax of the ODE solver is as follows and is summarized in Table 8.8-1. [t, y) = ode23(‘ydot’, tspan, yO, options, pl . p2, … ) where the options argument is created with the new odeset function, and pl , p2, … are optional parameters that can be passed to the…

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Extension to Higher-Order Equations

Extension to Higher-Order Equations To use the ODE solvers to solve an equation higher than order 2, you must first write the equation as a set of first-order equations. This is easily done. Consider the second-order equation 5y + 7y +4y = f(/) (8.6-1) Solve it for the highest derivative: J f() 4 7 . Y =…

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Trapezoidal Profile for a de Motor

Trapezoidal Profile for a de Motor ih many applications we want to accelerate the motor to a desired speed and allow it to ru at that-speed for some time before decelerating to a stop. Investigate whether an appli ‘. – voltage having a trapezoidal profile will accomplish this. Use the values R = 0.6 Q.,…

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SUMMARY

SUMMARY The solutions obtained in this section can be used to check the results of a numerical solution technique. In addition, these solutions have also pointed out the following facts that will be helpful for properly using the numerical techniques presented in the next section. For certain types of differential equations, called linear equations, the characteristic polynomial can be…

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PROBLEMS

Problems You can find the answers to problems marked with an asterisk at the end of the text. Note: Unless otherwise directed by your instructor, for each of the following problems select the easiest method (analytical or numerical) to solve the problem. Discuss the reasons for your choice. Your instructor might require you to use a numerical method…

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A Nonlinear Pendulum Medel

A Nonlinear Pendulum Medel By studying the dynamics of a pendulum like that shown in Figure 8.6-1, we can better understand the dynamics of machines such as a robot arm. The pendulum shown consists of a concentrated mass m attached to a rod whose mass is small compared to m. The rod’s length is L. The equation…

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Numerical Methods for Differential Equations

Numerical Methods for Differential Equations It is not always possible to obtain the closed-form solution of a differential equation. In this section we introduce numerical methods for solving differential equations, First we treat first-order equations, and in the next section we show how to extend the techniques to higher-order’ equations. The essence of a numerical method is to convert…

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Analytical Solutions to Differential Equations

Analytical Solutions to Differential Equations In this section we introduce some important concepts and terminology associated with differential equations, and we develop analytical solutions to some differential equations commonly found in engineering applications. These solutions will give us insight into the proper use of numerical methods for solving differential equations. They also give us some test cases to use…

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