Abstract
In this work, the energy density responses of one-dimensional structures with random properties are investigated analytically. Based on Green kernels, analytical representations of energy density for vibrating rods and beams are proposed using the superposition of energy waves. Considering random properties in rods and beams, formulations of energy density responses are obtained. Then, the mathematical expectations and variances are derived. And response intervals for random responses are developed. Finally, numerical simulations are performed to validate the proposed formulations, and characteristics of the random energy density responses of rods and beams are analysed. The main contribution of this work is that a new approach to energy density responses is proposed which facilitates the vibration analysis of structures with uncertainty parameters.
1. Introduction
Prediction of structural vibration is of interest in engineering applications. In low frequencies, the finite element method (FEM) is usually used. However, as the frequency increases, the wavelengths become smaller and more degrees of freedom are desired to obtain accurate results. FEM becomes time-consuming and even unfeasible for many problems. An approach is evaluating the responses with an averaged variable. Thus, statistical energy analysis (SEA) was proposed by Lyon et al. [1–3]. SEA, providing averaged energy for subsystems, reduces the scale of computation dramatically and has been widely studied and applied in various fields [4–8].
However, there are several limitations in SEA, and one that must be noticed is the energy distribution in subsystems, often concerned in many applications, is lost. Therefore, alternative methods become necessary. Energy flow analysis (EFA), introduced by Belov et al. [9, 10], is the representative method. In EFA, space- and frequency-averaged energy density is used as the main variable to reduce the scale of computation. Then, the dynamic behaviour of structures can be described by the control equation of energy density which is analogous to the heat conductivity equation. FEA, who desires much less computation than traditional FEM with displacement being a main variable, allows more detailed results than SEA. To provide a numerical solution for the energy responses, Nefske and Sung [11] applied FEM in solving control equations of energy density for vibrating beams. It is part of evocation of the energy finite element method (EFEM). Developing control equations for basic structures is a prerequisite for other works and is what much work is focused on. Wohlever and Bernhard [12] produced a systematic procedure to obtain energy density control equations of rods and Euler–Bernoulli beams. Bouthier et al. [13, 14] analysed Kirchhoff plates, membranes, and acoustic cavities by EFEM after constructing control equations of energy density. In-plane motions were not concerned before Park et al. [15] formulated the control equation of energy density for in-plane shear waves in coupled thin plates. To take into account the effects of rotational inertia and shear distortion, Park and Hong [16] classified the flexural waves into two kinds of waves and developed the energy control equations for Timoshenko beams. They further derived the energy control equations for three kinds of far-field propagating waves in Mindlin plates [17]. Considering lateral motion plays a more important role in high-frequency vibration, Han et al. [18] derived the energy control equation for Rayleigh–Love rods. And the energy control equation of Rayleigh–Bishop rods was formulated by taking into account both lateral motion and shear stiffness. Chen et al. [19] derived the energy density and energy intensity of an Euler–Bernoulli beam with constant axial force. Then, the energy density governing equation is established after introducing the relationship between the wavenumber and group velocity.
In practical engineering, objects to analyse consist of many basic structures. The energy transmission relationships between coupled structures become an essential problem. To predict responses for built-up structures, Cho [20] described energy relationships of coupled structures through energy reflection and transmission coefficients. Then, the global matrix can be obtained by using a joint matrix. Bitsie [21] formulated the coupled relationships between acoustic fields and structures through radiation coefficients. The coupled relationships were used in EFEM to compute responses of plates under point excitation, and the results show that energy dissipation is related to the total dissipated power. Zhang et al. [22–24] applied the energy flow method on plates with fluid load, showing that fluid load leads to a different energy density control equation for plates and influences the energy transmission between the plate and its reinforcement beams. Kwon et al. [25] analysed the wave transmission between cylindrical shells and obtained power reflection and transmission coefficients which can be used to predict energy flow for coupled cylindrical thin shells. Lin et al. [26] proposed a junction formulation for the discontinuity problem of energy density at the junction of two beams with stepped thickness. However, most works on transmission relationship neglect wave filtering effects and are only enough accurate when for weak damping effect. Liu et al. developed an alternative approach for energy transmission relationships using the coupling loss factors (CLFs) and validated the approach on coupled beams and coupled plates [27].
To obtain experimental data of energy densities, Navazi et al. [28] proposed a new method for experimental measurement of energy densities of plates. Using the method, kinetic and potential energy densities were obtained by the experiment. It is shown that the experimental results agree well with that from analysis.
Since space- and frequency-averaged variables are adopted to describe dynamic response of structures, there are no mandatory requirements to decrease the size of the elements or increase the order of the interpolating functions as the frequency increases which happens in FEM for numerical computation. EFA demands much less computation power and provides a potential solution to high-frequency problems. However, most works are focused on deterministic structures whose parameters are known precisely. Due to natural uncertainties in practical parameters, deterministic models may produce unacceptable errors. Since totally avoiding uncertainties is impossible, it is meaningful to improve deterministic methods to predict uncertainties in structural vibration. There are many works dealing with uncertainties in structures [29–33]. Among them, the probabilistic method is preferable. In probabilistic analysis, parameters are supposed to satisfy a certain probabilistic distribution, and responses are provided in probabilistic form by random variables. The responses subject to a certain probabilistic distribution are supposed to be more feasible than deterministic results. In this work, new analytical formulations of energy density responses of vibrating rods and beams are developed with Green kernels. The analytical formulations facilitate the analysis of structural vibration from the point of view of energy. Energy densities can be obtained directly without solving control equations using numerical solutions. Then, random energy density responses are derived for rods and beams with probabilistic parameters. Considering the excitation location as a probabilistic variable, numerical simulations are performed to validate the derived analytical solutions and the characteristics of random response are analysed comparing with deterministic results. The analytical formulations provide helpful representations for further research on energy densities of structures. And for practical structures whose uncertainties should not be neglected, the probabilistic formulations of energy density responses provide a potential approach for the analysis of vibration in high frequencies.
2. Materials and Methods
2.1. Energy Density Governing Equation and Numerical Solution
The energy density control equation for rods and beams is as follows [12]:where is the energy density, is the group velocity of waves, is the material damping coefficient, and is the input power by the excitation with the frequency . To obtain the energy density responses, the control equation can be solved by the finite element method. The discrete control equation becomes
In equation (2), is the coefficient matrix, indicates the input power, and is the power flow in elements. The boundary of elements is denoted by , and the normal vector of the boundary is denoted by .
2.2. Analytical Formulations of Energy Density
Equation (2) provides the numerical form for the calculation of the energy density responses which is suitable for complex structures. However, it is less convenient for studying the characteristics of energy flow in structures than analytical forms. To obtain analytical solutions, one point should be noticed is that energy density responses are reverberant in high frequencies, so the vibration power of a single wave is totally reflected at free and clamped boundaries regardless of the phase. Then, the responses are the summation of a series of energy waves.
2.3. Analytical Solutions for Rods
The control equation of motion for a uniform longitudinally vibrating rod is as follows [12]:
In the equation, the coordinate system in Figure 1 is adopted, is the longitudinal displacement, is the cross-sectional area, indicates the mass density per unit length, and is the harmonic point force applied at location . The damping effect is considered by introducing the complex modulus of elasticity , where is the hysteretic damping coefficient and .
Using the wavenumber, we obtain where is a complex wavenumber. In lightly damped rods, i.e.,, the wavenumber can be approximated as where and .
The Green kernel, which indicates the displacements of the rod under a unit point force, for a longitudinally vibrating rod with infinite length is as follows [34]:where indicates the distance between the response point and the excitation.
The total energy of a vibrating rod includes two parts: the potential energy and the kinetic energy. The time-averaged kinetic () and potential () energy densities are, respectively, [12] as follows:
The averaged complex forms can be written as in which the star indicates the conjugation.
Substituting the Green kernel into equations (12) and (13), the averaged kinetic and potential energy densities become by taking . Thus, noticing , the energy density responses of an infinite rod under a unit point force are
The representation for an infinite rod under a sinusoidal force with amplitude is then written as
To obtain the representation for a finite rod, the boundary effect should be taken into account. Since what exists in a vibrating rod is a reverberant field, power flow is considered to be reflected totally at boundaries regardless of the phase. Then, the response is the summation of a series of energy waves, which can be written aswhere is the order of reflection, is the length of the rod, and indicates the location of the concerned location. As the order of reflection tends to infinite, the representation of the energy density responses is
Then, the energy density Green kernel for a finite rod can be written as
2.4. Analytical Solutions for Beams
The control equation of motion for a uniform Euler–Bernoulli beam excited by a transverse harmonic point force is as follows [12]:
In the equation, is the transverse displacement, is the cross-sectional area, is the mass density per unit length, is the harmonic point force applied at location , the complex modulus of elasticity is , and indicates the flexural rigidity.
The wavenumber of the bending beam is
In lightly damped beams, the wavenumber can be written aswhere and .
The Green kernel for an infinite beam is as follows [35]:
Similar to a rod, the average total energy density in a beam is the sum of the kinetic and potential energy density:where the star indicates the conjugation.
Substituting the displacement with the Green kernel, the energy densities become
Based on the approximation , the energy density representation for an infinite Euler–Bernoulli beamcan be adopted without introducing unacceptable errors by noticing that .
The energy density responses for a finite beam are calculated by the superposition of a series of bending waves:
Then, the energy density Green kernel for a finite rod can be written as
It can be observed that excitation terms are introduced in equations (18) and (27). The effect of boundaries is taken into account by the sum of reflected waves. The expressions become analytical forms which have no unknown constants, and it is convenient for analysing the energy density characteristics of rods and beams.
3. Energy Densities for One-Dimensional Structures with Random Uncertainties
3.1. Energy Densities for Rods with Random Uncertainties
According to equation (18), the energy density formulation of vibrating rods with random parameters can be written as where denotes an independent random variable with the mathematic expectation and the standard deviation .
Then, the expectation of the response is in which means expectation, and indicates the function which possesses the same form as the deterministic formulation of energy densities:
The variance of response is obtained as And the standard deviation is
As an example, the energy density of vibrating rods experiencing a random-location excitation can be written as
The mathematic expectation and variance of the random variable are
Equation (34) can be expanded as a Taylor series at , and the first two terms are retained as
Then, assuming the random location as a Gaussian random parameter, the expectation of energy density for rods is as with
It is the same expression as that of rods experiencing a deterministic-location excitation at . Thus, the standard deviation of the energy density responses is
Within standard deviations, the corresponding energy density interval isin which at least of the values of response are covered. The maximum and minimum values in the interval are
3.2. Energy Density for Beams with Random Uncertainties
For an Euler–Bernoulli vibrating beam with random parameters, the energy density representation is
Then, the expectation of responses is similarlyin which
The variance of response is obtained as
And the standard deviation is
Then, for a vibrating beam experiencing random-location excitation, the energy density is
The mathematic expectation and the variance of the random variable are given as
The first two terms of the Taylor series at are retained. The energy density formulation is approximately written as
Then, the expectation and the standard deviation of the energy densities can be obtained by assuming the location as a Gaussian random parameter:
Within standard deviations, the corresponding energy density interval is in which at least of the values of response are covered. Maximum and minimum energy densities in the interval for beams are
4. Results and Discussion
4.1. A Simulation for the Rod
For the rod cases, the uniform clamped-free rod shown in Figure 2 with , , , and is driven by a harmonic point force with . Figure 3 shows the analytical energy density distribution in frequencies at location with that obtained from EFEM and wave methods. And the distributions in space at are presented in Figure 4.
In Figures 3 and 4, it can be observed that the proposed analytical solution to the rod coincides well with the numerical (EFEM) solution of the energy density governing equation. Both of them represent well the global variation in the classical wave solution regardless of the excitation frequency.
Figure 5 shows the energy density interval at point in frequencies by assuming the excitation coordinate as a Gaussian random variable and the standard deviation . Due to the random effect of the excitation location, the energy densities are no longer deterministic. A confidence response interval is generated by potential energy densities for both frequency and space responses. The practical responses can be any value in the interval. However, the mean energy densities are equal to the results from deterministic parameters without deviations. As the excitation frequency increases, the energy density interval width increases. And it is indicated that the uncertainty of energy densities for the rod becomes more significant in high frequencies than in low frequencies.
The random energy densities in space at are shown in Figure 6. The mean energy densities are identical to the results from deterministic parameters without uncertainties. The energy density interval width remains with increasing distance from the driving point which indicates that energy densities of the rod in different locations possesses the same uncertainty at a fixed frequency.
The standard deviations and the relative standard deviations of energy densities are presented, respectively, in Figures 7 and 8. It is shown that the standard deviation at the free end of the rod increases while that at the clamped end decreases as the frequency increases. The relative standard deviation of energy density increases as the frequency increases, and it stays the same with increasing distance from the driving point. It is indicated that, although the absolute uncertainty changes in various ways in different locations, the relative uncertainty of energy densities for the rod increases in high frequencies and remains at a steady level at a fixed frequency regardless of the coordination.
4.2. A Simulation for the Beam
The beam used for the validation is shown in Figure 9 with , , , and . Figure 10 provides the energy density distributions in frequencies at from the proposed analytical, the numerical (EFEM), and the wave methods. And the energy density distributions in space at are shown in Figure 11. It is observed that, for the beam, results from the proposed analytical method and the numerical method coincide well and represent well the global variation in the classical wave solution regardless of the excitation frequency.
Similar to the previous example for the rod, Figure 12 shows the energy density responses of the beam at point in the case of the excitation location standard deviation . Due to the random effect of the excitation location, the energy densities of the beam become random. And the potential responses cover an interval in the frequency domain and spatial domain. And the mean energy densities are equal to the results from deterministic parameters without deviations. As the excitation frequency increases, the uncertainty of energy density for the beam becomes more significant. Figure 13 shows the energy densities in space at whose potential values form an interval too. The mean energy densities are identical to the results from deterministic parameters without uncertainties. And the energy densities of the beam in different locations possess the same uncertainty at a fixed frequency.
The standard deviation and the relative standard deviation of energy densities are presented, respectively, in Figures 14 and 15. It is observed that the standard deviation is shown to decrease with the increasing frequency at all locations of the beam. The relative standard deviation increases with the increasing frequency at all locations of the beam and stays the same with increasing distance from the driving point. It is indicated although the absolute uncertainty goes down, the relative uncertainty of energy densities for the beam is more significant in higher frequencies and remains at a steady level in space at a fixed frequency. And the rate of growth becomes smaller in higher frequencies which differ from that for the rod.
5. Conclusions
Vibration prediction of many practical structures demands more on vibration research due to structural complexity and high-frequency analysis. As an approach to high-frequency responses, EFA is now widely applied to many engineering fields. In this work, a general method was presented for predicting the energy density responses. Based on the Green kernels of vibrating rods and beams, the method provided new formulations for rods and beams using the summation of propagating waves. The advantage of the proposed method is that it provides a systematic way to analyse energy responses of structures with analytical formulations and enabled direct solutions of energy density for vibrating structures which promotes the analysis of vibration in energy view. With the proposed method, formulations of energy densities are derived for rods and beams with probabilistic parameters. Simulations for a random-location excitation case were performed to validate the formulations. The proposed method provides an approach to predict energy density responses analytically which facilitates vibration energy analysis of structures. And the extension of the method to structures with an uncertain parameter of other types is the subject of future research.
Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was funded by Ph. D funding of NCIAE (BKY-2020-12).