The Theory Of Power Flow And Its Applications In Vibration Analysis Of Rod And Beam Structures

It is different from the traditional methods of structural dynamics that the theory of power flow is from a point of energy for structure vibration analysis.The rod and beam structures can not only exist as independent structures,but also as important components of the plane or space frame and grid structures,it is of great significance to apply the theory of power flow to vibration analysis on them.In this paper,we systematically discussed the theoretical basis of power flow and applied it to vibration analysis of rod and beam structures,and the main contents include:(1)The general system composed of elastic media is as the basic model,the expressions of power flow density and transient power flow are derived from the physical equations,geometric equations,and motion equations,then the steady state power flow is obtained from using the energy theorem.For the important harmonic case,the expression of complex power flow is deduced,and then,the concept of mechanical impedance/admittance is introduced,and the relationship between it and complex power flow is derived.(2)Using the theory of Love rod and classical rod,the theory of Timoshenko beam and Euler-Bernoulli beam as the theoretical models,respectively.The complex power flow at any section of the rod/beam element was deduced by the governing equations of the rod/beam element,when the power flow theory was introduced.Taking the two-phase rod/beam structure with concentrated mass lump,spring and damping support at the joint as a model,the method of reverberation-ray matrix(MRRM)was used to solve the wave amplitude solution of the rod/beam structure,and the complex power flow,active power flow and reactive power flow at any section of the rod/beam structure was derived.And then,the structures are used as numerical examples to analyze the influence of the structural stiffness ratio,mass lump,spring stiffness,damping coefficient,and material damping on the power flow.These results show that:(1)The power flow represents the vibration energy of the structure,in which the active power flow corresponds to the energy transmitted by the traveling wave and can be transmitted to a distant place,and the reactive power flow corresponds to the oscillating energy of the standing wave and cannot be transmitted far away.The frequency corresponding to the peak of active power flow and reactive power flow is the natural frequency of the structure.When the structure is not damped,the active power flow is zero,and the reactive power flow can reflect the vibration characteristics of the structure.(2)Compared to the classical rod theory and the Euler-Bernoulli beam theory,the Love rod theory and the Timoshenko beam theory are more applicable to the higher frequency of power flow analysis of rod and beam structures.(3)For the rod structure,the Poisson's ratio has less influence on the low-frequency power flow,but has a greater impact on the high-frequency power flow.In the Love rod theory,the energy transmitted by the longitudinal wave is cut off from the cutoff frequency of the rod.At this time,the longitudinal wave does not transmit energy,and all the energy is absorbed by the transverse vibration.(4)For the rod and beam structure,the structural stiffness ratio,the concentrated mass lump and the spring stiffness of joint and other factors all have different effects on the power flow in different frequency ranges by changing the natural frequency of the structure.The change of the joint damping and material damping does not change the natural frequency of the structure,but it has an influence on the magnitude of the peak(or reverse peak)of the power flow.With the increase of joint damping coefficient or material damping factor,the power flow peak decreases,and outside the peak area,active power flow increases,while reactive power flow is hardly affected.