Trouble with harmonic oscillator equation

In summary, the harmonic oscillator equation (with m=1) describes a system with b≥0 and k>0. By solving the equation of motion, we can identify regions in the bk-plane where the system will exhibit overdamped, underdamped, or critically damped motion. These regions can be determined by looking at the phase space trajectories, which differ depending on the type of motion.
  • #1
deex171
2
0
Consider the harmonic oscillator equation (with m=1),
x''+bx'+kx=0
where b≥0 and k>0. Identify the regions in the relevant portion of the bk-plane where the corresponding system has similar phase portraits.

I'm not sure exactly where to start with this one. Any ideas?
 
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  • #2
there are certain regions of b and k where the oscillator will undergo overdamped motion, underdamped motion, and critically damped motion. The phase space trajectories of overdamped motion look a lot different than those of underdamped motion. Solving the equation of motion should tell you which values of b and k correspond to which kinds of motion. I expect this is what is being asked.
 

1. What is the harmonic oscillator equation?

The harmonic oscillator equation is a mathematical formula that describes the motion of a simple harmonic oscillator, which is a system that oscillates back and forth around a stable equilibrium point. The equation is typically written as m·x'' + k·x = 0, where m is the mass of the oscillator, x is the displacement from equilibrium, x'' is the second derivative of x with respect to time, and k is the spring constant.

2. What are the applications of the harmonic oscillator equation?

The harmonic oscillator equation has many applications in physics, engineering, and other fields. It can be used to model the motion of a pendulum, the vibrations of a guitar string, the behavior of a mass on a spring, and many other systems. It is also used in quantum mechanics to describe the behavior of particles in a harmonic potential.

3. How do you solve the harmonic oscillator equation?

The harmonic oscillator equation can be solved using various mathematical techniques, depending on the specific problem at hand. One common method is to use the characteristic equation to find the general solution, which can then be modified to fit the initial conditions of the system. Another approach is to use the Lagrange method, which involves writing the equation in terms of the kinetic and potential energy of the system.

4. What is the significance of the natural frequency in the harmonic oscillator equation?

The natural frequency, denoted by ω0, is a fundamental property of a harmonic oscillator. It represents the frequency at which the oscillator would vibrate if there were no external forces acting on it. In other words, it is the frequency at which the system naturally oscillates when it is disturbed from equilibrium. The natural frequency is determined by the mass and spring constant of the oscillator and can be used to calculate the period and frequency of its motion.

5. Can the harmonic oscillator equation be used to model real-world systems?

Yes, the harmonic oscillator equation can be used to model many real-world systems, as long as they exhibit simple harmonic motion. This includes systems such as pendulums, springs, and vibrating strings. However, in more complex systems, other factors may need to be taken into account, such as damping and external forces, which can affect the behavior of the system and may require modifications to the equation.

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