Normalization Of Dynamic Equations Of Compressed Euler Beam And Algorithm Of Critical Force

In order to solve the vibration characteristics and critical force of the compressed Euler beam under various loads and section changes,the dynamic equations of three forms（constant pressure,distributed pressure,and variable section）of the compressed Euler beam were standardized by introducing the idea of space-time scaling.At the same time,the standardized equations were solved,and the dimensionless frequency equation and modal general solution were obtained.The effects of boundary conditions,distributed pressure parameters,and variable section parameters on the vibration frequency of the system are studied by numerical analysis under three compression forms,and the critical force of the Euler beam is solved.The primary research results are as follows:（1）The results of the normalization of the dynamic equations of three kinds of compressed Euler beams show that the normalized equations in the new coordinate system represent all the dimensional dynamic equations in the normal coordinate system,no matter how the dimensional parameters change,the difference is only reflected in the reduction coefficient.After the dynamic equations of Euler beam under constant compression are normalized,the normalized equations with all coefficients equal to 1 can be obtained,but the dynamic equations with more than three constant coefficients cannot be simplified to 1.The dynamic equations of Euler beam under distributed compression and with variable cross-section can only simplify three coefficients to 1,and the other coefficients must be reduced to the corresponding dimensionless quantities in the new space-time coordinate system.（2）The results show that the frequency equation under constant pressure depends only on the generalized beam length l,while the frequency equation under distributed pressure depends on the generalized beam length l and the distributed pressure parameter g under simple boundary conditions,and the generalized spring coefficients K₁and K₂ are added to the original equations under elastic boundary conditions.The frequencies of the system decrease monotonically with the increase of the generalized beam length l,whether it is a simple boundary or an elastic boundary,and the frequencies also decrease monotonically with the influence of the distributed pressure parameter g.With the increase of the distributed pressure parameter g,the critical beam length will decrease,while with the increase of the elastic boundary,the critical beam length will increase.The influence of the distributed pressure parameter on the frequency can be expressed by the value ofλ,and the proportionλof the correction of each order frequency of the system will increase with the increase of the generalized beam length l and g.Under each boundary,the influence of g on the frequency is greater in the lower-order mode than in the higher-order mode.According to the different growth rates ofλ,a quantitative judgment can be made,that is,when the spatial change rateβis less than 2%,the influence of the distribution pressure parameter is not considered.When the frequency is 0,the critical beam length corresponding to the critical force of the Euler beam under constant pressure is a constant value,so it can be solved according to the critical beam length of the corresponding boundary,which is the same as the existing literature.For the critical force algorithm of Euler beam under distributed compression,the resolvent data l_cr-g_cr is taken when the frequency is 0,and then its application range is determined according to the error of 5%.Finally,through two groups of examples,the frequency and buckling critical force results calculated by the proposed method are compared with the existing literature and finite element method,and the results are in good agreement.（3）The results of solving the normalized equation of the Euler beam with variable compression section show that the frequency equation after solving the normalized equation only contains the generalized beam length l and the generalized variable section parameterγ,and does not depend on other specific physical parameters when considering the finite perturbation modes.Similar to the uniform Euler beam subjected to the same distributed pressure,the frequencies of the system decrease monotonically with the increase of the generalized beam length l and the generalized variable section parameterγ.The solution of the critical force is also based on the resolvent data l_cr-γ_cr when the frequency is 0.Finally,the accuracy of the algorithm is verified by examples.The universality of the resolvent data makes the critical force algorithm universal,and it does not need finite element modeling and complex numerical calculation,only needs linear interpolation and other simple calculations.