How to solve a differential equation for a mass-spring oscillator?

AI Thread Summary
The discussion centers on solving a differential equation for a mass-spring oscillator affected by friction. The derived equation incorporates spring force and friction, leading to a complex form that includes the hyperbolic tangent function. Participants express uncertainty about solving it analytically due to the velocity term in the friction equation, suggesting that a small amplitude approximation may be necessary. The conversation highlights the need for numerical values for parameters like stiffness, friction coefficient, and mass to proceed further. Ultimately, the consensus leans towards numerical methods as the likely solution approach.
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Homework Statement
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.
Relevant Equations
II. Newton's Law
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 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.

image1.png

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})##.
 
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You are given a numerical value for the initial displacement. Are you given numerical values for any of the other parameters such as ##k, k_f, v_c## and ##m##?

At first glance, the differential equation looks difficult to solve unless you can make a small amplitude approximation such that ##\dot x## is always much less than ##v_c##.
 
TSny said:
You are given a numerical value for the initial displacement. Are you given numerical values for any of the other parameters such as ##k, k_f, v_c## and ##m##?

At first glance, the differential equation looks difficult to solve unless you can make a small amplitude approximation such that ##\dot x## is always much less than ##v_c##.
Oh yes, the initial conditions and parameters of the system given.

So there is no chance of solving this analytically, only numerically?
 
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