This is typically the case for laser arrays that are injection locked to a single master source. In some situations, the harmonic oscillators are driven by identical sources. Where I 1 and I 2 are the source intensities. The harmonic oscillator model is widely used to simulate reality with no need to perform physical experiments. The accuracy of such predictions has been verified by many physical experiments. It is a prediction of the motion of the mass-spring system with the specified data. According to ODE theory, once the parameters and initial data are specified, the ODE has a unique solution. To predict the outcome of an experiment by using this model, the parameters in the model must be identified and the initial position and velocity ( x 0 and x. In fact, the spring model is accurate enough to make useful predictions for many physical phenomena. This parameter is not zero when there is an external periodic force with circular frequency Ω and phase shift ρ, a case that is prevalent in many applications (for example in electrical engineering). With the linear damping force, we recover the basic model with A = 0. 2, but these latter choices lead to nonlinear ODEs that are more difficult to analyze. The simplest model for the damping force assumes that it is proportional to the velocity of the mass and acts in the direction opposite to the motion: − ϵ x. Could predictions be made from such a model? The usual procedure for modeling macroscopic mechanical systems is to mimic fundamental forces with constitutive laws.
Also, imagine the complexity of a model that took into account molecular forces. But the dynamical behavior of the system, which depends on forces at the molecular level, is so complicated that a fundamental model of damping forces is beyond current understanding. At a fundamental level, electromagnetic forces are acting. Why does the spring stop oscillating? The mass moves through air, the spring warms up, energy is radiated away due to heat, and perhaps other internal mechanisms are active. Thus, our model does not take into account at least one force acting on the mass. But, a real spring will eventually stop oscillating and return to equilibrium. By observation of springs, this is the correct qualitative behavior, at least for a short amount of time. Thus the displacement x is a periodic function of time. What does it predict?įrom Exercise 2.2 part (a), the general solution of this ODE involves periodic functions, which in this case are sines and cosines. This model is an approximation to reality. Where a dot over a variable denotes differentiation with respect to time, two dots denote the second derivative with respect to time, and so on. Under the assumption of constant mass, a spring model is given by There is a constant k, depending on the material properties of the spring that are ultimately due to electromagnetism, such that F = − kx. The usual force law is Hooke’s law: the restoring force is proportional to the displacement and in the direction toward the equilibrium position of the mass. Instead, the force may be modeled by a constitutive law, which is meant to be a good approximation of reality. To obtain a fundamental model, the laws of electromagnetism (Maxwell’s laws) and the Lorentz force law would have to be used to determine the restoring force of the spring.
The restoring force of the spring is clearly electromagnetic. What is this force? Recall that there are four fundamental forces: the weak and strong nuclear forces, gravity, and electromagnetism. By the nature of a spring, the force acts in the direction that would restore the spring to its equilibrium position. The force F acting on the mass is due to the elasticity of the material used to make the spring.
Also, choose a horizontal coordinate system with its origin at the equilibrium position of the mass oriented so that the coordinate, say x, measures displacement with positive values corresponding to the stretched spring. Imagine a spring attached to a wall and to a mass in a horizontal configuration so that the force of gravity does not affect the motion (see Fig. 2.1).
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