Mathematical model of air damper with permeable fabric shell
One of the phase of landing process for cargo dropped system with air dampers (System) is when the cargo moves on the damper during its compression. It may lead to cargo bounce and sometimes to its overturn. The System trajectory on this phase depends on experiment conditions: landing velocity, mass that comes to the damper, environment conditions and regime of gas expelling from air damper camera that is controlled with valves. Generally, materials of dampers shells are not hermetic - air permeability may have an influence on System landing dynamics. In this article we describe the mathematical model of the air damper with air permeability fabric shell for its further using in the tasks of System landing parameters prediction with variable experiment conditions. Considering dynamic model consists of cargo (mass) and cylindrical air damper with a valve. The law of the mass moving describes by one-dimensional equilibrium equation, where reaction from the damper is defined with equation of state for ideal gas. Forth integration method is used to solutions on each step. The model of gas expelling through the shell has been developed. Analysis of existing permeability theories shows that Rysev’s law is the the most appropriate for the considering task: this theory is based on Rahmatulin’s non-linear law of pressure gradient to gas velocity, but transformed with similarity theory to nondimensional parameters, related only to fabric characteristics - it makes possible not to consider experiment conditions if we knew them. Based on assumption that gas conditions inside the camera keep constant during the integration step of dynamic mass-to-damper model, we find the value of flow rate through the fabric shell on this step (with formulas of Rysev’s law) and add it to the value of valve flow rate. There are two experiment-calculated tasks are presented to evaluate the influence of permeability on System dynamics: 1 - we put the mass on air damper and find fabric characteristics; 2 - we make a drop test with the same mass and damper. Analysis shows well correlation of experimental results and calculated results received from the developed model. Comparison of the solutions with and without permeable shells of the air damper shows the essential difference on the phase of mass bounce. Thus, we can conclude that the influence of gas permeability through the damper shell has to be taking into account to define the dynamics of mass-to-air damper system, especially in the tasks of bounce and overturn prediction.