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Unformatted text preview: LAB #7 Resonance Goal : Observe the phenomenon of resonance; find numerical approximations of solutions to non-autonomous systems of differential equations. Required tools : Matlab routines pplane , ode45 ; m-files; systems of differential equations. Discussion Assume that we have a box of mass 1g on a table and attached to a spring. Initially, the spring is unstretched. We pull the box 3 cm to the right and give it an initial speed of 1 cm/sec to the left. From Newtons 2 nd Law F = ma , we have seen that the motion of the spring is governed by the equation my 00 + y + ky = 0 ( * ) where m is the mass of the box, is the coefficient of friction and k is the spring constant. y(t) Assignment (1) Assume that m = 1 , = 0, and k = 0 . 25 dynes/cm. Thus, it takes 0 . 25 dynes of force to stretch the spring 1 cm. Convert equation ( * ) to a system, enter the system into pplane . Use the Keyboard Input option to plot the phase plane portrait corresponding to a 3 cm stretching with initial velocity- 1 cm/sec. Under the Graph pull down menu select Plot y vs t. Use this graph to estimate the period of the motion. Would you consider this slow or fast oscillation? (The time is in seconds.) Use your graph to estimate the amplitude of the motion. (2) Find the general solution to ( * ), with the values of and k from (1). Find a formula for the solution that satisfies y (0) = 3 and y (0) =- 1. Use your answer to find the exact value for the period and amplitude of the motion you estimated in (1). (3) The period of the oscillations is determined by the stiffness of the spring. (A stiff spring is one which takes a large force to stretch it.) How do you guess stiffness should relate the period of the motion. i.e., should stiff springs oscillate faster or slower ? Test your guess by graphing the solution curve for a stiff spring and a non-stiff spring. Note: This will require that you change k in equation( * ). How should you change it to model a stiffer spring?...
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