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

In summary, the conversation discusses a mass-spring oscillator affected by friction. The equation for the friction force is given, and the differential equation for the system is derived. However, due to the complexity of the equation, it is not possible to solve it analytically unless a small amplitude approximation is made. Numerical values for initial conditions and parameters are provided, making it possible to solve the equation numerically.
  • #1
bolzano95
89
7
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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  • #2
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##.
 
  • #3
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?
 

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

1. What is a differential equation?

A differential equation is a mathematical equation that relates a function with its derivatives. It is commonly used to model physical systems and their behavior over time.

2. What is a mass-spring oscillator?

A mass-spring oscillator is a physical system that consists of a mass attached to a spring. The mass is able to move back and forth due to the restoring force of the spring.

3. How do you set up a differential equation for a mass-spring oscillator?

To set up a differential equation for a mass-spring oscillator, we use Newton's second law of motion which states that the sum of all forces acting on an object is equal to its mass multiplied by its acceleration. In this case, the forces acting on the mass are the force of gravity and the force of the spring, which is proportional to the displacement of the mass from its equilibrium position.

4. What are the steps to solve a differential equation for a mass-spring oscillator?

The steps to solve a differential equation for a mass-spring oscillator are as follows:

  1. Set up the differential equation using Newton's second law.
  2. Solve for the second derivative of the position function.
  3. Substitute the second derivative into the differential equation.
  4. Solve the resulting differential equation using techniques such as separation of variables or integrating factors.
  5. Use the initial conditions (position and velocity at time t=0) to find the values of the constants in the solution.

5. What are the applications of solving a differential equation for a mass-spring oscillator?

Solving a differential equation for a mass-spring oscillator has many applications, including modeling the motion of a pendulum, a swinging door, or a vibrating guitar string. It is also used in engineering and physics to analyze and design systems such as shock absorbers, car suspensions, and earthquake-resistant buildings.

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