Viscoelastic Fractional Derivative Model And Its Applycation On Solid Rocket Motor

Solid rocket motor (SRM) is an important propulsive component of solid missile and astronautical system. Its dynamic analysis plays a very important role in system stability and vibration environment forecast. In this paper, viscoelastic fractional derivative (VFD) model is used for the first time in dynamic analysis of SRM, and various verification tests are carried out. Theoretical developments and experiment results are used to SMR dynamic response analysis. Through the theoretical study and experiments, some achievements are obtained as follows:Firstly, solid propellant relaxation module is fitted with three parameter VFD model. They are in good agreement from 10^-3 to 10⁵ seconds. Three parameter VFD model's accuracy can compare with 19 parameter Prony series model. Long period large fixed strain tension test shows that the stress also meet VFD model until the test specimens are going to be broken with the phenomenon of surface macrocrackls, but the model parameters are different from those in relaxation module test. The VFD model can be used not only in short time vibration analysis but also in long period storage relaxation calculation so long as different parameters are adopted. Vibration test of pole-shaped grain is conducted under various frequencies. The experiment results of acceleration frequency response is in agreement with theoretical data, which shows VFD model can also be used in SMR frequency domain analysis.Secondly, in theoretical aspect, equivalent viscous damping method for VFD model is proposed and it is used in finite element calculation. The method is applicable to small damping system and little derivative order with narrow band response. It can simplify the response calculation of some kind rubbers and solid propellants, whose calculation time is only about one tenth of original VFD solution. By means of Laplace transformation and its inverse transformation, analytical solutions are deduced for one-end-fixed pole, cantilever and uncompressible tube composed of VFD material. The solutions is used to the response analysis of SRM grains. Finite element program is compiled and dynamic substructure method for VFD materials is developed. Modal truncation and phase space enlargement are adopted for reduction of the substructure degree of freedoms. Integrated system can be uncoupled directly so that the calculation is simplified.Finally, finite element and dynamic substructure method for VFD model are applied to dynamic response calculation of SMR. Power spectrum density of SMR grain response is obtained under random excitation of trapezium acceleration power spectrum density. It centered in a narrow frequency band, and forms a high resonant peek. Through the system damping, high frequency excitation component give little contribution to the response, therefore, the response of resonant frequency domain must be considered seriously in SRM vibration analysis. Analytical solution is obtained for tube shaped SRM grain under step jump inner pressure, which shows that the stress forms a slow peak little higher than inner pressure in a short time. Atthe beginning, the hoop stress is tension and the grain is dangerous, but later on the stress become compress in three dimension and equal to one another, so the grain is much safer.As mentioned above, three parameter VFD model can characterize the viscoelasticity of solid propellant. Applied it to SRM dynamic analysis, time and frequency domain response can be calculated.