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It is the non-uniform linear equation of the second order with constant coefficients, and the uniform corresponding to the equation (is (. Therefore; it is necessary to find x. If to that, the private decision x, it is necessary to look for in a look, where M and N — the which are subject to definition. So,

As in balance position balance force force of a tension of a spring is counterbalanced by the body weight, P = . Let's substitute in the differential equation expression P and we will replace - through x, the equation in a look will turn out:

The private decision (characterizing actually fluctuations, it was received in the assumption that, i.e. that the frequency of external force does not coincide with the frequency of own fluctuations. If, business is absolutely differently. Really, the equation (it is possible to copy in a look now

Nature of the movement entirely is defined by these. Three various cases are possible. Let's consider a a case, when. This inequality takes place when resistance of the environment is small. If to, roots (have an appearance. Then the common decision can be written down in a look

The received equation defines so-called free fluctuations of freight. It is called as the equation of the harmonious oscillator. This linear differential equation of the second order with constant coefficients. Its characteristic equation:

The uniform equation corresponding (1, is the equation (with roots of the characteristic equation (. Let's assume that resistance of the environment is small, t. e. Thus the common decision of a uniform has an appearance (:

Let's direct an axis Oh down the vertical straight line passing through a freight subweight point. The beginning of coordinates About we will choose in situation balance freight, that is in a point in which the weight of freight is counterbalanced by spring tension force.