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Neoton Raphson Method(전산수치해석)

Question) Employ the Newton-Raphson method to determine a real root for using initial guesses of (a) 4.52 and (b) 4.54. Discuss and use graphical and analytical methods to explain any peculiarities in your results. Perhaps the most widely used of all root-locating formular is the Newton-Raphson equation. If the initial guess at the root is , a tangent can be extended from the point . The point where this tangent crosses the x axis usually represent an improved estimate of the root. The Newton-Raphson method may be developed from the Taylor series expansion. This alternative derivation is useful in that it also provides insight into the rate of convergence of the method. The Taylor series expansion can be represented aswhere lies somewhere in the interval from to . An approximate version is obtainable by truncating the series after the first derivative term:At the intersection with the x axis, would be equal to zero, orwhich can be solved for

공학/기술| 2008.06.19| 7페이지| 1,500원| 조회(351)

수치해석, 뉴튼, 전산수치해석, 전산수치, 뉴턴랩슨

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비선형 단진자의 전산수치해석

Numerical Methods-Final Team Project-- Contents -1. Motivation2. System Introduction3. Background principle4. Induction of Equation5. Numerical coding6. Data7. Analysis8. Conclusion1. MotivationSimple pendulum vibrate non-linearly when the system is undamped system. We'll solve the problem by using RK-4 method. And then, by suing Maclaurin series, We'll compare with numerical solution by increasing the number of sin term.2 .System Introduction(1) The initial condition of degree is 0.05 rad (Approximatly 2.86°)And the system is undamped system.We'll ignore the Air resistence.(2) We've solved this kind problems by using the Maclaurin series, and also we've considered just first sin term when we solve the problem.(3) So our last goal is that we'll consider more sin term in oder of precedence and we'll compare with numerical solutions that solved by using RK-4 method.Last, we'll search the error .2. 1. Assumption- We'll ignore the Air resistence.- We'll ignore the mass of thread. .- we'll consider only the effect of gravity.2. 2. Constant- m = 1 (kg), L = 0.5 (m),= 0.05 (rad),= 1 (rad/s),= 0.01 (s)3. Background principle3. 1. The equation of motion : Newton's 2nd LawEq. (1)Observed from an inertial reference frame, the net force on a particle is proportional to the time rate of change of its linear momentum : equa1. Momentum mv is the product of mass and velocity. This law is often stated as F = m a (the net force on an object is equal to the mass of the object multiplied by its acceleration), but this approximation to the law is accurate only for speeds much less than the speed of light. An important point is that force and momentum are vector quantities and that the resultant force is found from all the forces present by vector addition.3. 2. Linearization of non-linear systemsQualitative method become particularly valuable for systems whose solution by analytic methods is difficult of impossible, as is the case for many practically important first order non linear system.Phase plane methods are qualitative methods. In this section we extend phase plane methods to nonlinear system.We call that the phase plane, a solution curve of (1) in the phase plane a trajectory.If (1) has several critical point, we discuss one after the other. Each time we move the point P0. (a,b) to be discussed to the origin. This can be done by a translationThus we can assume that p0 is the origin.(0,0) and we continue to write y1,y2. for simpicity. we also assume that p0 is isolated, that is, it is the only critical point within circular disk with center at the origin.How can we determine the type and stability property of a critical point? in most practical f1 and f2 in (1) are continuous and have continuous partial derivatives in a neighborhood of P0. then (1) and a linear system by which we shall approximate(1) near P0 will have the same kind of critical point at P0.Indeed the above assumption in the derivatives implies that h1 and h2 are small near p0:Two exceptions occur if the eigenvalues of A are equal or pure imaginary. Then, in addition to the type of critical points of the linear system. the nonlinear system may have a spiral point.3. 3. Taylor Series and Maclaurin SeriesThe Taylor series of a function f(z), the complex analog of the real Taylor series is,where,A Maclaurin series is a Taylor series with center.4. Induction of EquationSo ,assumed,with Maclaurin series,5. Numerical coding/* theta1'=theta2theta2'=-g/l*sin(theta) sin(theta)=theta로 근사 */#include#includeusing namespace std;double df1(double, double, double); //theta1 을 리턴 함수double df2(double, double, double); //theta2 을 리턴 함수double exact(double);const double g=9.81;const double l=0.5;void main(void){cout.setf(ios_base::fixed, ios_base::floatfield);//Display 6 figures as significant figuresdouble T0 = 0 , theta1_0 = 0.050 , theta2_0 = 0; //초기값double Ti, theta1_i, theta2_i; //수치해를 구하기 위한 변수double h, Tend; //시간 범위double k11, k12, k13, k14; // theta1을 구하기 위한 룬게쿠타 식double k21, k22, k23, k24; // theta2을 구하기 위한 룬게쿠타 식double T, theta1, theta2; //계산하기 위해 설정double t = 0.01; // 시간 증분double Et_Y1 = 0; //참 백분율 오차Tend = 3;h = t;Ti = T0;theta1_i = theta1_0;theta2_i = theta2_0;cout

공학/기술| 2008.06.19| 40페이지| 3,000원| 조회(313)

단진자, 수치해석, 비선형, 전산수치해석

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