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The energy flow analysis (EFA) method is developed to predict the energy density of a high damping beam with constant axial force in the high-frequency range. The energy density and intensity of the beam are associated with high structural damping loss factor and axial force and introduced to derive the energy transmission equation. For high damping situation, the energy loss equation is derived by considering the relationship between potential energy and total energy. Then, the energy density governing equation is obtained. Finally, the feasibility of the EFA approach is validated by comparing the EFA results with the modal solutions for various frequencies and structural damping loss factors. The effects of structural damping loss factor and axial force on the energy density distribution are also discussed in detail.

With the development of high-speed aircrafts and transportation vehicles, high-frequency vibration of components is of great concern for both academic and industrial communities in recent years. As one of the typical structural components, beams with high damping treatment for vibrational reduction are extensively used in mechanical and aerospace engineering. Moreover, these beam-type structures sometimes experience axial forces arising from initial stress and temperature variation, which can significantly influence the dynamic characteristics of structures. Consequently, high-frequency dynamic response prediction of a high damping beam with axial force is of great significance for a beam-type structure design.

Both the statistical energy analysis (SEA) [

Wohlever et al. [

For structures with initial stress, Zhang et al. [

For a beam with constant axial force, driven by a harmonic point force as depicted in Figure

A pined-pinned transversely vibrating beam with constant axial force.

Substituting the general solution

Then the complex wavenumber,

In terms of equation (

In a lightly damped system, from equations (

As a result, the general solution of equation (

The right first two terms of equation (

The energy density in a vibrating beam is the sum of the kinetic and potential energy densities which are associated with bending strain and constant axial force, respectively. The time averaged energy density can be represented as [

Substituting equation (

From equations (

In the case of hysteric damping, the dissipated power of per volume smoothed by time and space averaging is proportional to the potential energy density smoothed by time and space averaging, for an elastic system harmonically vibrating with frequency

In a vibrating beam with constant axial force, the time averaged potential energy density can be expressed as [

Substituting equation (

Combining equations (

For the elastic medium, the power balance equation at the steady state is expressed as

Substituting equations (

Unlike the lightly damping system in [

As shown in Figure ^{th} region of the beam. The constants

From equations (

Because there are no energy outflow and inflow at both ends of the pinned-pinned beam, the following energy intensity boundary conditions can be determined as

At the interface between the regions ① and ②, the energy density is continuous and the energy intensity is subject to conservation of energy. The resulting boundary condition can be written as

The input power of the vibrating beam is obtained by the impedance method and expressed as [

Substituting equation (

According to equation (

To validate the proposed EFA formulation and investigate the effects of structural damping loss factor and axial force, various energy flow analyses are performed for a pinned-pinned beam at both ends shown in Figure

The beam is made of aluminum alloy with density

Figure _{0} = 1 N, and the structural damping loss factor _{0} = 0.1, 1, 10 kN), as shown in Figure

Energy density distribution of pinned-pinned beams with _{0} = 1 N, and

Energy density distribution of pinned-pinned beams with _{0} = 0.1, 1, 10 kN.

The exact energy density distribution obtained by modal analysis method oscillates spatially without space averaging. At high-frequency ranges, the variance of classical solutions is uniform resulting from short wavelength. Therefore, the developed EFA results, obtained with space averaging, are analogous to the classical solutions as the analysis frequency increases.

Figures _{0} = 1 N, and

Energy density distribution of pinned-pinned beams with _{0} = 1 N, and

Energy density distribution of pinned-pinned beams with _{0} = 1 N, and

Figure _{0} = 1 N, and structural damping loss factor

Energy density distribution of pinned-pinned beams with _{0} = 1 N, and

Total energy of pinned-pinned beams _{0} = 1 N,

The effects of constant axial force can be explained by the input power and group velocity. Figures

Input power of pinned-pinned beams _{0} = 1 N,

Group velocity

Figure _{0} = 1 N, and axial force

Energy density distribution of pinned-pinned beams with _{0} = 1 N, and

Total energy of pinned-pinned beams _{0} = 1 N,

Figures

Input power of pinned-pinned beams _{0} = 1 N,

Group velocity

The analysis model consists of a beam fixed at both ends as shown in Figure

A fixed-fixed beam with constant axial force.

A fixed-free beam with constant axial force.

Similarly the impedance of fixed-free beam at driving point is determined by substituting the equation (

Figure _{0} = 1 N,

Energy density distribution of fixed-fixed beams with _{0} = 1 N,

Energy density distribution of fixed-free beams with _{0} = 1 N,

An energy flow model for a beam with axial force that includes high damping effect is established to evaluate the energy density in the high-frequency range. The wavenumber is expressed as the function of the damping loss factor and the axial force. Particularly, the energy transmission equation and the energy dissipation equation are derived from the wavenumber without approximation of its real and imaginary term. Then, the energy density governing equation is obtained for the high damping beam with axial force.

To verify the developed energy flow model, numerical analyses are performed for the simply supported beam with various damping loss factor and axial force. The EFA solutions are greatly consistent with the modal solutions in all cases. Furthermore, it is found that both the high damping factor and large axial force can significantly alter the level of energy density as well as the distribution. Due to the resulting deceasing input power, the increasing damping loss factor makes the energy density decay more dramatically along axial position. However, that is the contrary case for the axial force. The proposed method is expected to be useful for the prediction of the high-frequency vibration of high damping structures with initial stress.

When a high damping beam with axial force is excited at _{0} by a transverse harmonic force, the governing equation of motion can be expressed as

The steady-state solution of equation (

By substituting equation (

The forced vibration response of the beam can be expressed as the linear combination of the normal modes

The impedance at the driving point of the finite beam with axial force can be obtained from its definition:

The time averaged energy density is given by

Substituting equation (

The transverse displacement and the bending moment are zero at a simply supported end. Hence, the boundary conditions can be stated as

The general solution of equation (

The vibration mode of the beam is given by

At a fixed end, the transverse displacement and the slope of the displacement are zero. Hence, the boundary conditions are given by

By virtue of general solution in equation (

The vibration mode of the beam is expressed as

If the beam is fixed at

According to the general solution in equation (

The vibration mode of the beam is written as

The data used to support the finding of this study are available from the corresponding author upon request.

The authors declare that there no conflicts of interest.

The presented work was supported by National Natural Science Foundation of China (Grant no. 51505096) and Natural Science Foundation of Heilongjiang Province of China (Grant no. QC2016056).