Research On Dropping Shock Dynamics Performance Of Nonlinear Packaging System Based On Newton-Harmonic Balancing Method

The economy is developing at a rapid pace and the circulation of goods continues.Although the logistics system is gradually improving,the phenomenon of falling and damage of goods during transportation,loading and unloading,etc.still occurs.In the packaging dynamics,due to the nonlinearity of the cushion material,the cushion packaging system is mostly a nonlinear system.The non-linear dynamic equation analysis method is used to study the nonlinear dropping shock dynamics problem.The analytical expression of the system response can be obtained,which has certain theoretical guiding significance for the dynamic performance analysis,evaluation and design of the nonlinear cushioning packaging system.Newton-harmonic balancing method is an approximate analysis method for solving the vibration problem of nonlinear conservative systems,which can obtain satisfactory accuracy for strong nonlinear vibration problems.The outstanding advantage of this method is to avoid the problem of solving complex nonlinear algebraic equations in the harmonic balance method,and it does not depend on small parameters.For the dropping shock problem of product circulation,Newton-harmonic balancing method is introduced to establish a suitable approximate analytical analysis method for it,and first-order,second-order,third-order approximate solutions are obtained.For the typical nonlinear system such as cubic,tangent,hyperbolic tangent and suspension system,the dropping shock dynamics equations are established.Then the dynamics equation is dimensionlessly processed by introducing the relative dimensionless parameters.For the analysis of force characteristics,the dimensionless dynamic equations of typical systems are uniformly classified into a nonlinear conservative system dynamic equation with cubic term or cubic-five term.Taking the cubic non-linear non-dimensional dropping shock dynamics equation as the research object,the first-order,second-order and third-order approximate solutions of the system and the dimensionless displacement maximum response,the dimensionless acceleration maximum response and the dropping shock time of the system were obtained by applying Newton-harmonic balancing method.The cubic cushioning packaging system(hard spring characteristics)and the suspension system with three terms(soft spring characteristics)are used as examples to verify the analytical solution accuracy of Newton-harmonic balancing method.Compared the NHB approximate solution results with the fourth-order Runge-Kutta numerical results and the variable iteration method results,the analysis of the example is shown that the relative errors of the second-order and third-order approximate analytical solutions of Newton-harmonic balancing method are controlled within 2.5%,the accuracy meets the engineering requirement,and the NHB third-order approximate solution has the highest precision.Based on the analytical expression of the maximum dimensionless acceleration response obtained by NHB third-order approximate solution,the system's dropping damage evaluation algebraic equation was established,and the system dropping damage evaluation curve(surface)was obtained.Taking the nonlinear non-linear non-dimensional dropping shock dynamics equation with cubic-five terms as the research object,the first-order,second-order and third-order approximate solutions of the system and the dimensionless displacement maximum response,the dimensionless acceleration maximum response and the dropping shock time of the system were obtained by applying Newton-harmonic balancing method.The tangential system(hard spring characteristic)and the suspension system(soft spring characteristic)with cubic-five terms are used as examples to verify the analytical solution accuracy of Newton-harmonic balancing method.Compared the NHB approximate solution results with the approximate solution of the variable iterative method results and the fourth-order Runge-Kutta numerical results,The comparison of the analysis results is shown that the second-order and third-order approximate solution errors of Newton-harmonic balancing method are controlled within 2%,the accuracy can meet the engineering requirement,and the third-order analytical solution accuracy is higher than the second-order analytical solution.Based on the analytical expression of the maximum dimensionless acceleration response of NHB third-order approximate solution,the system's dropping damage evaluation algebraic equation was established and the dropping damage evaluation boundary curve(surface)was obtained.