Uncertain Spring Vibration Equation

In the application of classical differential equation,spring vibration equation plays an important role to describe the phenomenon of spring vibration,whose external force is a given function.External force is often affected by the interference of noise in reality due to air or material friction.Some scholars use Wiener process to describe the disturbance term,and then design random spring vibration equation to describe the phenomenon of spring vibration.But is it really reasonable to apply random spring vibration equation in describing forced vibration phenomenon of the spring in reality?In this dissertation,a counter example of random spring vibration equation is giv-en to illustrate that it is unreasonable to describe the phenomenon of spring vibration by using random spring vibration equation.We first try to describe the noise of ex-ternal force by Liu process and derive an uncertain spring vibration equation.For a special type of uncertain spring vibration equations,this dissertation obtains its ana-lytical solution.For general uncertain spring vibration equation,we give an existence and uniqueness condition and some stability concepts,and discuss their relationships of stability.However,for the general uncertain spring vibration equation,it is difficult to find its analytic solution.In order to solve this problem,this dissertation shows that the solution of an uncertain spring vibration equation can be represented by a family of solutions of spring vibration equations,which provides a possibility for the numerical solution of the general uncertain spring vibration equationIn general,the mainly innovations of this dissertation are as follows:This dissertation first employs uncertain differential equation to investigate the phenomenon of spring vibration,and derives an uncertain spring vibration equa-tion.In addition,for a special type of uncertain spring vibration equations,this dissertation obtains its analytical solution·For general uncertain spring vibration equation,we proved an existence and u-niqueness theorem and some stability theorems·This dissertation shows that the solution of an uncertain spring vibration equation can be expressed by a family of solutions of spring vibration equations.Further-more,this dissertation derives the inverse uncertainty distribution,expected val-ue,extreme values,first hitting time,and time integral of the solution for general uncertain spring vibration equation.