Note that other cases exist which may lead to nonlinear equations which go beyond the scope of this example. Consider an object of mass m attached to a spring of constant k. We simply add a term describing the damping force to our already familiar equation describing a simple harmonic oscillator to describe the general case of damped harmonic motion. We solve this differential equation for our equation of motion of the system, x t. We assume a solution in the form of an exponential, where a is a constant value which we will solve for.
Plugging this into the differential equation we find that there are three results for a, which will dictate the motion of our system. We can solve for a by using the quadratic equation. The physical situation has three possible results depending on the value of a, which depends on the value of what is under our radical.
This expression can be positive, negative, or equal to zero which will result in overdamping, underdamping, and critical damping, respectively. In this case, the system returns to equilibrium by exponentially decaying towards zero. The system will not pass the equilibrium position more than once. In this case, the system oscillates as it slowly returns to equilibrium and the amplitude decreases over time. Figure 1 depicts an underdamped case. In this case, the system returns to equilibrium very quickly without oscillating and without passing the equilibrium position at all.
Driven harmonic oscillators are damped oscillators further affected by an externally applied force. If a frictional force damping proportional to the velocity is also present, the harmonic oscillator is described as a damped oscillator. Driven harmonic oscillators are damped oscillators further affected by an externally applied force F t. This type of system appears in AC driven RLC circuits resistor-inductor-capacitor and driven spring systems having internal mechanical resistance or external air resistance.
For strongly underdamped systems the value of the amplitude can become quite large near the resonance frequency see. Resonance : Steady state variation of amplitude with frequency and damping of a driven simple harmonic oscillator. It only takes a minute to sign up.
Connect and share knowledge within a single location that is structured and easy to search. We use source. But at higher velocity, flow becomes turbulent and inertial forces acting on the flowing fluid have to be taken into account. Damped harmonic oscillations are an extremely broad paradigm, and there are many physically dissimilar examples for which the force behaves in completely different ways as a function of velocity.
As pointed out by Gert and Kyle Kanos, for low velocities as determined by the Reynolds number the flow of air will be laminar. At high velocities the quadratic dependence comes about due to turbulence making the flow far away from the object independent of the flow in the immediate neighborhood of the object. One can then consider the change in momentum of the air flow that is intercepted by the object. Clearly the change in momentum if a given quantity of intercepted air is proportional to the velocity and the amount of intercepted air per unit time is also proportional to the velocity, therefore the friction force must be quadratic in the velocity.
At low velocities this reasoning becomes invalid, the flow of the air perturbed by the moving object is no longer local. However, this is only true for an object moving at a uniform velocity; precisely the long range effect of the moving object on the fluid will cause the friction force to depend on the entire history of the object's trajectory.
The second term in the brackets yields the familiar Stokes formula for the friction force. The first term is the effect of the inertia of fluid, if the object accelerates then part of the fluid will accelerate with it due to the no-slip boundary conditions. The last term yields the effect of the history of the object's motion on the friction force. This is discussed on the Wikipedia entry on Drag emphasis theirs :. The equation for viscous resistance or linear drag is appropriate for objects or particles moving through a fluid at relatively slow speeds where there is no turbulence i.
In this case, the force of drag is approximately proportional to velocity, but opposite in direction. So because the author is assuming a laminar flow for the air around the oscillating mass, it uses the linear form Stokes' limit for drag. Sign up to join this community. The best answers are voted up and rise to the top.
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