Abstract
In the present paper, we introduce new models of pendulum motions for two cases: the first model consists of a pendulum with mass M moving at the end of a string with a suspended point moving on an ellipse and the second one consists of a pendulum with mass M moving at the end of a spring with a suspended point on an ellipse. In both models, we use the Lagrangian functions for deriving the equations of motions. The derived equations are reduced to a quasilinear system of the second order. We use a new mathematical technique named a large parameter method for solving both models’ systems. The analytical solutions are obtained in terms of the generalized coordinates. We use the numerical techniques represented by the fourth-order Runge–Kutta method to solve the autonomous system for both cases. The stabilities of the obtained solutions are studied using the phase diagram procedure. The obtained numerical solutions and analytical ones are compared to examine the accuracy of the mathematical and numerical techniques. The large parameter technique gives us the advantage to obtain the solutions at infinity in opposite with the famous Poincare’s (small parameters) method which was used by many outstanding scientists in the last two centuries.
Highlights
Nobody thought about using another technique especially the large parameter method this technique allows us to give the problem new conditions that cannot be assumed previously
Two new models have been introduced for the movement of the pendulum in the presence of new primary conditions that are not previously defined
Poincare’s method fails to solve these problems in the presence of the new condition so we must search for a new technique that matches these changes
Summary
We study a pendulum of mass M and string length l with suspended point A moving on an ellipse For this case, we take a point Q on the auxiliary circle of radius a to correspond the point A on the ellipse. When point A moves on the ellipse, the point Q moves on the circle with angular velocity ω in the plane xy. Assume the following parameters [12], b μ ≻≻1, l (3). Using Lagrange’s equation, we get the equation of motion of the pendulum in the form as follows:. E solution of this equation means that we obtain φ in terms of the large parameter and the time. E solutions of (6) are obtained in the form of power series expansions of powers of 1/μ as follows: φ(τ, μ) φ0(τ) + μ− 1φ1(τ) + μ− 2φ2(τ) + μ− 3φ3(τ) +. Substituting from (7) into (6) and equating coefficients of like powers of (1/μ) of both sides, we get a system of differential equations containing φi, i 1, 2, 3, . . ., which is solved to obtain the following:
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