The differential equation of Damped Harmonic Oscillator

In summary, the solution for the differential equation of a damped harmonic oscillator can be found using the classic method if the condition b^2/m > 4*k is satisfied, where b is the damping constant, m is the mass, and k is the spring constant. Otherwise, complex numbers must be used. The solution for this equation has been proven through verification experiments and assumptions based on Hooke's law. Further information on the solution can be found in the provided link.
  • #1
Gh. Soleimani
44
1
If you consider b^2/m > 4*k, you can get the solution by using classic method (b = damping constant, m = mass and k = spring constant) otherwise you have to use complex numbers. How have the references books proved the solution for this differential equation?
 
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  • #2
I'm not sure what you mean by proving it.

If you find a solution you can always insert it into the differential equation to see if its true in the same way that the integral of some function when differentiated will produce the same function.

If you mean how did the differential equation for a damped harmonic come to be defined that would be more physics related with assumptions for the force of the spring (Hooke's law ie force proportional to spring stretch) and the motion verified by experiment.
 

1. What is a damped harmonic oscillator?

A damped harmonic oscillator is a type of system that exhibits oscillatory motion, meaning it moves back and forth between two points. It is characterized by a restoring force that is directly proportional to the displacement of the object from its equilibrium position, and a damping force that gradually decreases the amplitude of the oscillations over time.

2. What is the equation for a damped harmonic oscillator?

The equation for a damped harmonic oscillator is: m(d^2x/dt^2) + b(dx/dt) + kx = 0, where m is the mass of the object, b is the damping coefficient, k is the spring constant, and x is the displacement from equilibrium. This equation is also known as the differential equation of a damped harmonic oscillator.

3. How does damping affect the motion of a harmonic oscillator?

Damping affects the motion of a harmonic oscillator by gradually reducing the amplitude of the oscillations. This means that the object will eventually come to rest at its equilibrium position, rather than continuing to oscillate indefinitely. The amount of damping also affects the frequency and period of the oscillations.

4. How can the solution to the damped harmonic oscillator equation be represented graphically?

The solution to the damped harmonic oscillator equation can be represented graphically by plotting the displacement of the object over time. This graph will show a decaying sinusoidal curve, with the amplitude decreasing over time until it reaches zero. The frequency and period of the oscillations can also be determined from this graph.

5. What are some real-life examples of damped harmonic oscillators?

Damped harmonic oscillators can be found in many real-life systems, such as a swinging pendulum, a bouncing ball, or a car's suspension system. Other examples include a mass-spring system with friction, a guitar string, and a metronome. In these systems, the damping force can come from sources such as air resistance, friction, or internal resistance within the system itself.

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