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How to solve a differential equation for a mass-spring oscillator?
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[QUOTE="bolzano95, post: 6483591, member: 638687"] [B]Homework Statement:[/B] There is an mass-spring oscillator made of a spring with stiffness k and a block of mass m. The block is affected by a friction. At the time ##t=0s## the block is pulled from the equilibrium position for ##x=5cm## in the right direction and released. Derive and solve the equation for a displacement x, velocity v and power P. [B]Relevant Equations:[/B] II. Newton's Law There is an mass-spring oscillator made of a spring with stiffness [B]k[/B] and a block of mass [B]m[/B]. The block is affected by a friction given by the equation: $$F_f = -k_f N tanh(\frac{v}{v_c})$$ ##k_f## - friction coefficient N - normal force ##v_c## - velocity tolerance. At the time ##t=0s## the block is pulled from the equilibrium position for ##x=5cm## in the right direction and released. 1. Derive the differential equation. 2. Solve it for a displacement x. 3. Solve it for a velocity v. 4. Solve it for the power loss P, because of the friction. [ATTACH type="full" width="625px"]281850[/ATTACH] 1. Deriving the necessary equation: In the vertical direction: N=mg. Because the only forces acting on the block in horizontal direction are the spring force and friction we can write: $$-kx + k_f\, N\, tanh(\frac{\dot{x}}{v_c})=m\ddot{x}$$ $$m\ddot{x} - k_f\: mg\: tanh(\frac{\dot{x}}{v_c}) +kx=0$$ Here my solving stops, because I'm not sure how to implement the standard solution ##x(t)= Ce^{\lambda t}## because velocity is inside the function ##tanh(\frac{\dot{x}}{v_c})##. [/QUOTE]
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How to solve a differential equation for a mass-spring oscillator?
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