Acoustic response of bi-directional functionally graded beam under axially varying load

Abstract

This paper investigates the effects of bi-directional gradation, length-to-height ratio, and end conditions on the acoustic behaviour of bi-directionally varying functional graded beams. The acoustic responses, including sound-power level (dB), sound-pressure level (dB), and sound-radiation efficiency, are evaluated using Rayleigh’s integral and modal superposition method. The sound power levels are presented up to the selected bandwidth, as well as the octave band center frequency. In contrast, the sound pressure levels are presented as contour plots and in directivity pattern. The buckling load, calculated for quadratically decreasing axial load, is applied in increments from 0 to its highest value. The study reveals that the highest value in the gradation indexes in both directions significantly influences the sound power levels. It is also evident from the study that thin beams have higher sound power levels compared to thick beams. The directivity pattern reveals that bi-directional functionally graded beams predict higher sound pressure levels at the critical buckling load. From the current acoustic study, it is observed that both structural and end stiffness are influential factors in sound power levels (dB) and sound pressure levels (dB).

Keywords

Rayleigh integral sound power levels octave frequency sound pressure directivity

Introduction

With the increase in the demand for customized materials, there is an upsurge in the design and fabrication of new materials. These new materials are extensively used in various engineering applications like automotive, civil, aviation, defence sectors. Functional graded structures or materials (FGMs) are among the new substances where there is an enhancement in the material property in the desired direction. The customization of FGMs led the researchers to focus on the characterization of static and dynamic analysis of structures.

Aydogdu and Taskin¹ numerically studied natural frequency characteristics of FG beams,and the effect of material grading is also presented in the study. Using Euler-Bernoulli beam hypothesis Simsek and Kocaturk² studied the effect of moving load on the vibration responses of FGM beams. Sina et al. and Sherafatnia et al.^3,4 numerically analysed vibrational responses in functional graded beams, and the studies revealed that the responses are influenced by the material gradation of the beams. Huang and Fang⁵ used Freholm integral equations in calculating dynamic responses of FG beams, where the material property significance on the irregular section of the structure is clearly studied. Thai and Vo⁶ using Hamilton’s principle analytically investigated for bending and natural frequency responses of FGM structures. From the study, the effect of material gradation and cross sectional deformation on both the responses is reported. Wattanasakulpong et al.⁷ investigated numerically and experimentally on dynamic characteristics of FGM beams by varying material grading, boundary conditions, and thickness ratio. Li et al.⁸ has developed an equation for free bending and vibration of a stepped beam using bending vibration theory. Both the CC and SS beams were analysed for natural frequency by varying the cross-sections, materials and lengths. With the help of dynamic stiffness matrix method Banerjee and Ananthapuvirajah⁹ numerically analysed dynamic characteristics of FGM beams. Zhang et al.¹⁰ proposed a mathematical model to study the effect of material gradation on the static behaviours of functional graded rectangular micro-cantilever structure. Sharma and co-authors^11,12 numerically predicted the significant influence of material gradation on the free vibrational responses of an axially inclined functional graded beam.

Bi-directional varying functional graded materials have the capability to enhance material property in two directions. The material property is defined with gradation indexes in two directions. Enrichment of material property in two directions reinforces structural stability and thereby improves the bending, buckling, vibrations and acoustic responses of beams. Wang et al.¹³ carried out a numerical study on the natural frequencies of bi-directionally varying functional graded structures. The observations from the study are helpful in designing a non-homogeneous beam vibrating at a particular frequency. Karamanli¹⁴ using Lagrange equations numerically investigated the free vibrational responses of B-FGM beams. An extensive study of dynamic behaviours of bi directionally varying functional graded beams is proposed for various sets of boundary conditions, variable axial loads, and material gradations. Pradhan et al.¹⁵ have depicted a noticeable effect on the buckling and vibrational behaviours of FG beam subjected to the variation in the material gradations along length and thickness directions. Using the continuous elements method Selmi and Mustafa¹⁶ numerically analysed dynamic behaviours of bi-directional FGM beams, where the study depicted the influence of material grading and length to height ratio on the free vibrational behaviours of the bi-directional FGM structures.

Buckling and vibration investigations on structures with the help of higher-order theories is proven to be accurate. The prediction is more accurate when the higher order theories, Ritz method etc., are used in the articulations of the governing expressions. Larbi et al.¹⁷ by using HSDT developed a neutral surface concept to evaluate buckling and natural frequencies of functional graded beams. It is observed from the proposed concept that inclusion of Hamilton’s principle has increased the accuracy of predicted results. Pradhan and Chakraverty¹⁸ used the Rayleigh-Ritz method in determining natural frequencies of FG beams. The study depicted that there is a significant effect of boundary conditions on vibrational behaviours. Nguyen et al.^19,20 numerically investigated on the natural frequencies of isotropic, composite, and functional graded beams with help of different higher-order theories. Vo et al., Talekar and Kotambkar^21,22 numerically investigated for the dynamic responses of composite beams using shear and normal deformation theory. The behaviour is studied for the influence of various lay-ups, Poisson’s ratio, and normal strain, the investigation is verified with finite element method. Similar investigation using Ritz method was conducted by Priyanaka and Jeyaraj²³ for evaluating the static responses of micro-laminated composite beam subjected to varying axial load.

There is great feasibility in predicting the dynamic failures of structures by using the acoustic response studies. The structures subjected to dynamic environment are acoustically analysed for sound power levels (dB), sound pressure levels (dB) and sound radiation efficiencies. Few of the researchers had concentrated on the study of sound characteristics of FGM beams. Reduction of acoustic emission from a simply supported beam with help of constrained layer damping (CLD) method is proposed by Zheng and Cai.²⁴ The study further contributed in providing the optimal size of the constrained layer patch attached to the baffled beam. Influence of truss core shapes on acoustic radiation of sandwich beams was conducted by Ruzzene.²⁵ Finite element method is used in the investigation, which observed that effect of fundamental geometry has noticeable effect on the sound radiation. Numerical investigation on natural frequency and acoustic characteristics of sandwich beams by varying the chiral core is carried out by Spadoni and Ruzzene.²⁶ The study optimized various parameters like core design, topology etc., for minimizing the intensity of natural frequency and its sound transmission of the beams. Denli and Sun²⁷ numerically optimized acoustic radiation of sandwich structures using cellular cores in a frequency band. It has been observed that the proposed numerical models show influential noise reduction in narrow and broad bands. Jeyaraj et al.²⁸ carried out a numerical investigation on the vibration and acoustic response of an isotropic structure affected by a thermal environment. From the study, it is depicted that there is significance importance of critical buckling temperature on sound radiation and sound power levels. Alshabatat and Naghshineh²⁹ investigated on optimization of the material distribution of FGM beams for altering the natural frequency and reducing sound power radiation. An elastic beam is numerically investigated by Tang and Xu³⁰ using the collocation point method for knowing the vibration and sound radiation responses. With help of spectral element method Li and Yang³¹ accurately predicted the dynamic and sound radiation behaviours of auxetic metamaterial structures.The computational results predict the dominance in acoustic response of metamaterials with increasing negative poison’s ratio. Gunasekaran et al.^32–34 analytically investigated the effect of non-uniform edge loadings on the dynamic responses of various structures. It is evident from the study that there is a phenomenal influence of compression load on the acoustic responses of structures. Vibration and acoustic characteristics of anisotropic structures were numerical investigated by Vaiduriyam et al.³⁵ From the study it was observed that there is a raise in the sound power levels in the lower octave bands.

It is observed from the literature that acoustic response study motivated several researchers in predicting the dynamic behaviour of structures. The dynamic responses are proven to be more accurate when the study is evaluated using higher order beam theories. The literature clearly highlighted the significance of variable axial loads in determining the exact dynamic behaviour of the structures. Few of the researchers have focussed for vibrational and acoustic responses from a bi-directional functionally graded beams. But, it is more evident that there is a great need in analysing the acoustic responses of bi directional FG beams by employing higher-order theories. Hence, the present study emphasizes on numerically analysis using Rayleigh’s integral and super position method for determining the acoustic responses of B-FGM beams. A detailed methodology for determining various acoustic responses of B-FGM beams is detailed in the next sections.

Methodology

Figure 1 illustrates diagram representation of the problem to be studied on the B-FGM beam. CC, SS and CF are the three end conditions for which the B-FGM beam is investigated. The bi-directional FG beam subjected to a quadratically decreasing type of VAL as shown in Figure 2, where the load is applied along the length of the beam. It can be observed that the variation in the in-plane axial load is defined with the below equation.

F_{x}^{e} (X) = F_{0} {χ_{1} {(X + \frac{L}{2})}^{2} + χ_{2} (X + \frac{L}{2}) + χ_{3}}

(1)

F_{x}^{e} (X) = F_{0} P (X)

(1a)

Figure 1.

Bi-directional FG beam geometry and co-ordinates.

Figure 2.

Variation of quadratically decreasing axial load along the length of the B-FGM beam Karmanli et al. (2019).

The distributed external load will be obtained when the coefficients $χ_{1} = 3$ , $χ_{2} = - 6$ and, $χ_{3} = 3$ are substituted in equation (1). Figure 2 represents the axial load variation along the length of the beam. Displacement field was defined according to the Karamanli¹⁴ shear and normal deformation theory. Mass density (ρ), modulus of elasticity (E), modulus of rigidity (G)and Poissons’s ratio (µ) will differ as defined in the below mentioned relations,

E (x, z) = E_{m} e^{{p_{x} (\frac{X}{L} + \frac{1}{2}) + p_{z} (\frac{z}{h} + \frac{1}{2})}}

(2)

G (x, z) = \frac{E (x, z)}{2 (1 + μ_{m})}

(3)

ρ (x, z) = ρ_{m} e^{p_{x} (\frac{X}{L} + \frac{1}{2}) + p_{z} (\frac{z}{h} + \frac{1}{2})}

(4)

The material properties Em, µm and ρm are determined by the point $(\frac{- L}{2}, \frac{- h}{2})$ and gradation indexes px and pz. px and pz defines the material property increment in x and z directions which are defined in length (L) and thickness (t) directions respectively. If gradation indexes px and pz are set at 0, material will be isotropic in nature.

In accordance with Reddy’s beam hypothesis,

U (x, z, t) = u (x, t) + z ϕ (x, t) - α z^{3} (\emptyset (x, t) + \frac{\partial ω (x, t)}{\partial x})

(5)

W (x, z, t) = w (x, t)

(5a)

in the above equations axial displacement u, transverse displacement w, rotation component φ and, α = 4/3 h² constitutes to the B-FGM beams cross-section. Eventually strain displacement equation from equation (5) can be presented as,

ε_{x x} = \frac{\partial U}{\partial x} = \frac{\partial u}{\partial x} + z \frac{\partial_{\emptyset}}{\partial x} - α z^{3} (\frac{\partial_{\emptyset}}{\partial x} + \frac{\partial_{ω}^{2}}{\partial x^{2}})

(6a)

ϒ_{x z} = \frac{\partial U}{\partial z} + \frac{\partial W}{\partial x} = \emptyset + \frac{\partial W}{\partial x} - β z^{2} (\emptyset + \frac{\partial ω}{\partial x})

(6b)

and, β = 3α, Now, the stress strain equation is defined by,

{\begin{cases} σ_{x x} \\ σ_{x z} \end{cases}} = [\begin{array}{l} E (x, z) & 0 \\ 0 & G (x, z) \end{array}] {{\begin{cases} ε_{x x} \\ γ_{x z} \end{cases}}}

(7)

The strain energy,

U = \frac{1}{2} \int_{v} (σ_{xx} ϵ_{xx} + σ_{xz} γ_{xz}) dV

(8)

Volume V of the bi directional FG beam and it’s strain component can be calculated by keeping equations (6) and (7) in the equation (8).

U = \frac{1}{2} \int_{v} [E (x, z) {(\frac{\partial u}{\partial x})}^{2} + (z^{2} - 2 α z^{4} + α^{2} z^{6}) {(\frac{\partial ϕ}{\partial x})}^{2} + α^{2} z^{6} {(\frac{\partial^{2} w}{\partial x^{2}})}^{2} + 2 (z - α z^{3}) \frac{\partial u}{\partial x} \frac{\partial ϕ}{\partial x} - 2 α z^{3} \frac{\partial u}{\partial x} \frac{d^{2} w}{d x^{2}} + 2 (α^{2} z^{6} - α z^{4}) \frac{\partial ϕ}{\partial x} \frac{d^{2} w}{d x^{2}}} + G (x, z) {(1 - 2 β z^{2} + β^{2} z^{4}) ϕ^{2} + (1 - 2 β z^{2} + β^{2} z^{4}) {(\frac{\partial w}{\partial x})}^{2} + (1 - 2 β z^{2} + β^{2} z^{4}) ϕ (\frac{\partial w}{\partial x})}] dV .

(9)

where the stiffness coefficients are described in equations (26) and (27) of Appendix A. The strain energy is defined from,

U = \frac{1}{2} \int_{- L / 2}^{L / 2} e^{p_{x} (\frac{X}{L} + \frac{1}{2})} [A {(\frac{\partial u}{\partial x})}^{2} + (D + α^{2} H - 2 α F) {(\frac{\partial ϕ}{\partial x})}^{2} + α^{2} H {(\frac{\partial^{2} w}{\partial x^{2}})}^{2} + 2 (B - α C) \frac{\partial u}{\partial x} \frac{\partial ϕ}{\partial x} - 2 α C \frac{\partial u}{\partial x} \frac{d^{2} w}{d x^{2}} + 2 (α^{2} H - α F) \frac{\partial ϕ}{\partial x} \frac{d^{2} w}{d x^{2}} + (A_{s} + β^{2} F_{s} - 2 β D_{s}) (ϕ^{2} + {(\frac{\partial w}{\partial x})}^{2} + 2 ϕ \frac{\partial w}{\partial x})] dx

(10)

where kinetic energy K is,

K = \frac{1}{2} \int_{v} ρ ({\dot{u}}_{i} δ \dot{{\dot{u}}_{i})} dv = \frac{1}{2} \int_{- L / 2}^{L / 2} e^{p_{x} (\frac{X}{L} + \frac{1}{2})} {I_{0} {(\frac{\partial u}{\partial t})}^{2} + (I_{2} + α^{2} K_{1} - 2 α J_{2}) {(\frac{\partial ϕ}{\partial t})}^{2} + I_{0} {(\frac{\partial w}{\partial t})}^{2} + 2 (I_{1} - α J_{1}) \frac{\partial u}{\partial t} \frac{\partial ϕ}{\partial t} + 2 (α^{2} K_{1} - α J_{2}) \frac{\partial ϕ}{\partial t} \frac{\partial^{2} w}{\partial x \partial t} - 2 α J_{1} \frac{\partial u}{\partial t} \frac{\partial^{2} w}{\partial x \partial t} + α^{2} K_{1} {(\frac{\partial^{2} w}{\partial x \partial t})}^{2}} dx

(11)

in the above equation t represents time and, the inertial coefficients are described as,

(I_{0}, I_{1}, I_{2}, J_{1}, J_{2}, K_{1}) = b \int_{- h / 2}^{h / 2} ρ_{m} e^{p_{z} (\frac{z}{h} + \frac{1}{2})} (1, z, z^{2}, z^{3}, z^{4}, z^{6}) dz

(12)

where as the work done is written as,

V = - \frac{1}{2} \int_{- L / 2}^{L / 2} F_{x}^{e} (x) [\int_{- \frac{L}{2}}^{x} {{(\frac{\partial ϕ}{\partial x})}^{2} + 2 \frac{\partial ϕ}{\partial x} \frac{\partial w}{\partial x} + {(\frac{\partial w}{\partial x})}^{2}} dx] dx

(13)

and the total potential energy is written as,

\prod = U + V - K

(14)

then

0 = \int_{t_{1}}^{t_{2}} (K - U - V) dt

0 = \int_{t_{1}}^{t_{2}} \int_{0}^{L} \frac{1}{2} \int_{- L / 2}^{L / 2} e^{p_{x} (\frac{X}{L} + \frac{1}{2})} [A {(\frac{\partial u}{\partial x})}^{2} + (D + α^{2 H} - 2 α F) {(\frac{\partial ϕ}{\partial x})}^{2} + α^{2} H {(\frac{\partial^{2} w}{\partial x^{2}})}^{2} + 2 (B - α C) \frac{\partial u}{\partial x} \frac{\partial ϕ}{\partial x} - 2 α C \frac{\partial u}{\partial x} \frac{d^{2} w}{d x^{2}} + 2 (α^{2} H - α F) \frac{\partial ϕ}{\partial x} \frac{d^{2} w}{d x^{2}} + (A_{s} + β^{2} F_{s} - 2 β D_{s}) (ϕ^{2} + {(\frac{\partial w}{\partial x})}^{2} + 2 ϕ \frac{\partial w}{\partial x})] d x - \frac{1}{2} \int_{- L / 2}^{L / 2} e^{p_{x} (\frac{X}{L} + \frac{1}{2})} {I_{0} {(\frac{\partial u}{\partial t})}^{2} + (I_{2} + α^{2} K_{1} - 2 α J_{2}) {(\frac{\partial ϕ}{\partial t})}^{2} + I_{0} {(\frac{\partial w}{\partial t})}^{2} + 2 (I_{1} - α J_{1}) \frac{\partial u}{\partial t} \frac{\partial ϕ}{\partial t} + 2 (α^{2} K_{1} - α J_{2}) \frac{\partial ϕ}{\partial t} \frac{\partial^{2} w}{\partial x \partial t} - 2 α J_{1} \frac{\partial u}{\partial t} \frac{\partial^{2} w}{\partial x \partial t} + α^{2} K_{1} {(\frac{\partial^{2} w}{\partial x \partial t})}^{2}} d x - \frac{1}{2} \int_{- L / 2}^{L / 2} F_{x}^{e} (x) [\int_{- \frac{L}{2}}^{x} {{(\frac{\partial ϕ}{\partial x})}^{2} + 2 \frac{\partial ϕ}{\partial x} \frac{\partial w}{\partial x} + {(\frac{\partial w}{\partial x})}^{2}} d x] d x

(15)

The reader has referred Priyanaka and Jeyaraj²³ for the details of boundary exponents used for different end conditions and, the displacement relations u (x, t), w (x, t) and the rotation relations φ (x, t) are defined below.¹⁴

u (x, t) = \sum_{j = 1}^{m} A_{j} Θ_{j} (x) e^{i ω t}, Θ_{j} (x) = {(x + \frac{L}{2})}^{p_{u}} {(x - \frac{L}{2})}^{q_{u}} x^{m - 1}

(16a)

w (x, t) = \sum_{j = 1}^{m} B_{j} φ_{j} (x) e^{i ω t}, φ_{j} (x) = {(x + \frac{L}{2})}^{p_{w}} {(x - \frac{L}{2})}^{q_{w}} x^{m - 1}

(16b)

ϕ (x, t) = \sum_{j = 1}^{m} C_{j} ψ_{j} (x) e^{i ω t}, ψ_{j} (x) = {(x + \frac{L}{2})}^{p_{ϕ}} {(x - \frac{L}{2})}^{q_{ϕ}} x^{m - 1}

(16c)

in the equation (16), A_j, B_j & C_j are undetermined coefficients and Θ_j (x), ϕ_j (x) and ψ_j (x) are the trial relations and p_ζ and q_ζ are the boundary exponents where, ζ = u, w, φ.

By keeping equation (16) into equation (15) governing differential equations in motions are calculated, where the eigenvalue’s are determined from,

([K] - F_{0} [S]) {Δ} = 0

(17)

where stiffness matrix [K] and geometric matrix [S] are shown in Appendix equations (29) and (30).

The natural frequency ω can be determined from,

([K] - ω^{2} [M]) {Δ} = 0

(18)

where, mass matrix [M] is defined in equation (31) of Appendix.

Modal superposition method is used in determining the forced vibration frequencies of the B-FGM beams.

[M] {\ddot{Δ}} + [C] {\dot{Δ}} + [K] {Δ} = [H] e^{j ω t}

(19)

where, [C] is damping matrix, ∆ is displacement function while [H] is harmonic load vector.

Rayleigh integral is used in determining the nature of acoustic radiation.³²

p (r) = \frac{j ω ρ_{0}}{2 π} \int_{S} \dot{W} (r_{s}) \frac{e^{- jk | r - r_{s} |}}{| r - r_{s} |} ds

(20)

where, pressure

p (r)

, surface velocity

ω (r)

, wave number k, fluid density ρ0 and distance among the surface and field

| r - rs |

are determined and sound intensity calculated by.

I (r) = \frac{1}{2} R e (p (r) \dot{w^{*}} (r_{s}))

(21)

The sound power radiated.

\bar{W} = \oint I (r) n (r_{s}) ds

(22)

where,

n (r_{s})

is surface normal.

\bar{W} = \frac{1}{2} Re (\oint p (r) \dot{W^{*}} (r) ds)

(23)

with above relation and by solving equation (24) sound power level for a bi-directional functionally graded beam is determined by.

SWL = 10 \log \frac{\bar{W}}{W_{ref}}

(24)

here, reference sound power Wref is 10–12 Watts, similarly acoustic radiation efficiency (σ) is determined by,

σ = \frac{\bar{W}}{ρ_{0} C_{0} S 〈 \bar{{\dot{W}}^{2}} 〉}

(25)

From the above relation, sound speed C₀ presented in m/s, $〈 \bar{{\dot{W}}^{2}} 〉 = {\dot{W}}_{n}^{H} N {\dot{W}}_{n}$ and N = (1/2) NI. unit matrix I, N elements number. W_n and normal velocity with its conjugate velocity denoted by ${\dot{W}}_{n}^{H}$ .

Validation study

Using earlier studies, the present investigation on acoustic responses of bi directionally varying functional graded beams is validated.

Sound radiation validation

For validating the current investigation, the sound power level investigation by Zheng and Cai²⁴ on an isotropic beam using Rayleigh’s integral is considered. An aluminium beam subjected to SS boundary condition and having dimensions of 0.4 × 0.03 × 0.004 m³ is numerically analysed. Figure 3 corresponds to the calculated sound power levels with the current and Zheng and Cai²⁴ studies. From Figure 3 it is evident that the sound power levels determined with present approach is having a good agreement with that of Zheng and Cai²⁴ results.

Figure 3.

Comparison of sound power level with Zheng and Cai²⁴ study with the validation it is obvious that current research could be employed for further investigations.

Results and discussion

In current numerical investigation sound power level (dB) and sound pressure level (dB) are calculated for various aspect ratios (L/h), end conditions, and gradation indexes of the bidirectional FG beams. The acoustic responses are studied by considering two aspect ratios thick beam (L/h = 5) & thin beam (L/h = 20), three different boundary conditions and, for specific gradation indexes varied along (x) and (z) directions. A quadratically decreasing load as shown in Figure 2 is considered for the analysis, its buckling load intensity is varied from 0 to its critical value. A bi-directional functionally graded beam with length 1 m, and having b × h as square cross-section used in the analysis, where height can be determined based on the aspect ratio L/h. The B-FGM beam has the material properties as Em = 210 GPa; μ_m = 0.3; ρ_m = 2700 kg·m³. With the outlined parameters a detailed investigation is carried out and presented in the below sections.

Effect on sound power level

This section details the effect of boundary conditions, aspect ratios and gradation indexes on the sound power level (dB) of B-FGM beams.

Effect of gradation index on sound power level

Figure 4, illustrates the significance of increment in the gradation indexes on the CC B-FGM beam’s sound power level (dB). From Figure 4, and for various gradation indexes, it is observed that the nature of variation in the sound power level within the calculated frequency band follows a similar trend. As anticipated, the fundamental frequencies are decreasing for an increase in the intensity of buckling load. All three plots illustrate that the fundamental frequency is significantly lower at 0.99 P_cr than compared with 0 and 0.5 P_cr load intensities. It can be observed from the plots that an increase in the gradation index leads to reduction in sound power levels. Plots in Figure 4(b) and (c) show a significant increase in the excitation frequency with an enhancement in the gradation index in the P_z direction.

Figure 4.

Influence of both gradation indexes on sound power level of CC bi-directional FG beam with $\frac{L}{h} = 5$ . (a) Varying P_x. (b) Varying P_z. (c) Varying P_x and P_z.

Similarly, from, Figure 4(a) there is a slight decrease in the excitation frequency observed with an increase in the gradation index along the P_x direction. Plots in Figure 4 indicate that as the natural frequency increases, there is a decreasing trend in sound power levels. It can be predicted that the influential of lateral stiffness of the B-FGM beams significantly affects the decreasing trend of sound power levels with an increment in the gradation indexes. From Figure 5 it is observed that gradation indexes (P_x) and (P_z) variation will have noticeable in fluence on the sound power levels. In the case of variations in both the gradation indexes, the sound power levels exhibit marginal influence, where the influence is more pronounced in the higher modes. Plots in Figure 5 illustrate, that as the buckling load intensity increases, the natural frequency exhibits a decreasing trend. The decreasing trend is more evident in the lower frequency band. From the plots in Figures 4(c) and 5(c), it is evident that there is a significant variation in the sound power levels with an increase in the gradation indexes (P_x) and (P_z). This variation is observed to be less significant in the gradation indexes in either the x and z direction.

Figure 5.

Influence of both gradation indexes on sound power level of CC bi-directional FG beam with $\frac{L}{h} = 20$ . (a) Varying P_x. (b) Varying P_z. (c) Varying P_x and P_z.

Effect of aspect ratio on sound power level

The plots in Figure 6 depict that there is a noticeable influence on the sound power levels (dB) due to the change in the aspect ratios of CC and SS bi-directional FG beams. For better comprehension, the sound power levels are plotted for a constant gradation indexes P_x = 1 and P_z = 1. For a change in aspect ratio from thick beam (L/h = 5) to thin beam (L/h = 20), there is a significant variation observed in the sound power levels (dB). While at selected frequency bands, the sound power levels of thin beams are slightly higher compared to those of thick beam levels. It can be predicted that the sound power level (dB) decrease with an increase in the stiffness of the beams and vice versa. The variation in the sound power levels due to changes in the aspect ratio is significant in higher frequency band. A significant variation is observed in Figure 6(b) of the SS B-FGM beam compared to the CC B-FGM beam.

Figure 6.

Significance of aspect ratios on the sound power level of CC & SS B-FGM beams with P_x = 1 & P_z = 1. (a) CC beam. (b) SS beam.

This can be attributed to the dominant stiffness in the CC B-FGM beams. The fundamental frequencies are in the lower frequency band for the change in the aspect ratio of the SS B-FGM beams. From Figure 6(b) it is also evident that the sound power levels of thick beams are significantly higher than those of thin beams. From both plots, it evident that the sound power levels of thin beams decrease with an increase in the mode number. At the fundamental frequency, the sound power levels of thick beams are noted to be higher.

Effect of end condition on sound power level

The influence of end conditions on the sound power level (dB) of bi directional FG beams is represented in Figure 7 for a gradation index P_x = 1 & P_z = 1 and aspect ratios L/h = 5 & 20. From the plots, it is evident that the boundary conditions have a remarkable influence on the sound power level of bi-directional FG beams. As expected, the fundamental frequencies increase in the order of CF, SS and CC bi-directional FG beams due to the increasing end stiffness. It is apparent from the two plots in Figure 7, that influence of boundary conditions on sound power levels is more significant for thin beams compared to thick beams.

Figure 7.

Influence of end conditions on sound power level for B-FGM beams with P_x = 1 & P_z = 1 and $\frac{L}{h} = 5 & 20$ . (a) For $\frac{L}{h} = 5$ . (b) For $\frac{L}{h} = 20$ .

The variation in sound power levels for an increase in the mode number was observed to be more significant for the low stiffness SS & CF B-FGM beams compared to the high stiffness CC bi-directional functional graded beams. It can be observed that the behaviour of sound power levels for an increase in the natural frequency can be attributed to the significance of boundary stiffness in the B-FGM beams’ boundary conditions.

Effect of end condition on sound power level

Apparently from Figure 8, it is understood that the nature of variation in sound radiation efficiency is insignificant for increment in the buckling load magnitude. Though there is a marginal variation in the levels of sound radiation efficiency observed between the thick beam in Figure 8(a) and the thin beam in Figure 8(b). The marginal change is due to the change in the structural stiffness which is effected due to the variation in aspect ratio.

Figure 8.

Effect of aspect ratios on radiation efficiency of CC bi directional FGM beams with P_x = 1 & P_z = 1. (a) $\frac{L}{h} = 5$ . (b) $\frac{L}{h} = 20$ .

Continuing from the earlier studies, the effect of buckling load intensity on sound power levels (dB) is presented as octave band center frequency (Hz) in Figure 9(a) and (b) for thin CC and SS B-FGM beams. These figures depict that the octave bands exhibit an initial rise, followed by drop, and then another rise over the selected frequency band. It is due to the effect of increased material property and enhancement of gradation indexes on the structural stiffness. It is also observed that effect of buckling load intensity is negligible at higher band frequencies. The overall sound power levels of CC, SS and CF bi-directional FG beams for the increment in the buckling load magnitude are presented in Figure 9(c). The curve is almost linear for the CF B-FGM beam, decreasing linearly for the SS B-FGM beam and, decreasing quadratically for the CC B-FGM beam.

Figure 9.

Octave band and overall sound power level representation of various B-FGM. (a) Octave band for CCL/h = 20. (b) Octave band for SSL/h = 20. (c) Overall sound power level.

It is apparent that the nature of the decrement is highly sensitive to the boundary stiffness of the B-FGM beams. The higher the boundary stiffness, the higher the rate of decrement of overall sound power level with an increase in load intensity.

Effect on sound pressure level

It is evident from Figures 4–6 that thick B-FGM beams emit lower sound power levels (dB) compared to thin B-FGM beams. Therefore, a detailed investigation on the effect of sound pressure level (dB) for an increment in the magnitude of buckling load and change in boundary conditions of thick B-FGM beams is carried out in this section. The sound pressure levels (dB) are calculated along the length, up to a height of 1 m and at 0.6 m distance from the fixed support of the B-FGM beams. Tables 1 and 2 detail the sound pressure level (dB) radiated as contour plots for various bi directional functional graded beams under different boundary conditions. Table 1 corresponds to the fundamental frequency of SS and CF B-FGM beams for variation in the buckling load intensity. At critical buckling load, there is a shift in the sound pressure radiation towards the right, while for the other two load intensities, the sound pressure is concentrated at the center for the SS B-FGM beam. But, the sound pressure radiation of CF bi-directional functionally graded beam is shifting towards the left. The shift is obvious, as the boundary stiffness will be higher at clamped support. One can predict the nature of sound radiation at the fundamental frequency of CC bi-directional FG beam in Table 2 to be concentric at the center, it is due to the equal end stiffness at both ends. Sound pressure radiation for the third mode of CC B-FGM beam is significant at the critical load intensity and is predicting similar to the earlier sound power level plots. The behaviour of sound pressure radiation as contour plots in this section coincides with the behaviour of sound power levels depicted in the earlier section. Similar to the contour plot representation, the sound pressure level (dB) is presented as a directivity pattern in Figure 10 for CC and SS B-FGM beams. The CC B-FGM beam in Figure 10(a) was observed to exhibit a significant variation in the sound pressure directivity compared to the SS B-FGM beam in Figure 10(b). Where there is no such variation observed when both the B-FGM beams are subjected to critical buckling load. While the sound pressure radiation was observed to be sensitive to the 0 Pcr and 0.5 Pcr buckling load intensities. It can be anticipated that the influence of structural stiffness along with end stiffness, has a remarkable effect on sound pressure radiation of B-FGM beams. The directivity nature of sound pressure for the third mode of the CC bi directional FG beam presented in Figure 10(c) depicts an irregular pattern, while coincides with the vibrational pattern at the third mode. This observation is due to the non-homogeneous material composition of bi-directional functionally graded beams compared to isotropic material beams. As a result, there will be variations in the structural stiffness of the bi-directional functional graded beams.

Table 1.

Contour representation of significance in increasing the in-plane axial load on the sound pressure level radiated for 1st mode of SS and CF B-FGM beams.

Table 2.

Contour representation of effect of increment in in-plane axial load on the sound pressure level radiated for first and third mode of CC B-FGM beams.

Figure 10.

Directivity pattern representation for first and third mode of various B-FGM beams. (a) First mode of CC beam (b) First mode of SS beam. (c) Third mode of CC beam.

Conclusion

The acoustic responses of bi-directionally varying functional graded beams using the Rayleigh integral and super position method are investigated. The acoustic responses sound power level (dB) and sound pressure level (dB) are evaluated for various aspect ratios, boundary conditions, and gradation indexes in the x and z directions. The study highlights the following observations:

• Sound power levels (dB) are significantly influenced by an increase in both the gradation indexes Px & Pz. An increase in the material gradation influences the material stiffness, which, in turn, affects the sound power levels.

• As anticipated, thin beams have a great impact on sound power levels compared to thick beams. This is due to the lower material mass of thin beams, which reduces stiffness and allows for a longer duration of vibration.

• It is noteworthy that end stiffness has a greater impact on the natural frequencies of the B-FGM beams compared to sound power levels, as marginal variation in the sound power levels is observed for the change in end stiffness.

• The behaviours of the sound pressure levels, presented as directivity and contour plots, closely correlate with the sound power levels. Whereas, the directivity pattern clearly indicates a significant variation in the sound pressure levels with an increase in the buckling load intensity. With the reduction in the structural stiffness at the critical buckling load, the structures vibrate more critically, resulting in an increase in the acoustic response.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

B. Somi Naidu

Jeyaraj Pitchaimani

Appendix

(26)

(A, B, D, C, F, H) = b \int_{- h / 2}^{h / 2} E_{m} e^{p_{z} (\frac{Z}{h} + \frac{1}{2})} (1, z, z^{2}, z^{3}, z^{4}, z^{6}) dz

(27)

(A_{s}, D_{s}, F_{s}) = b \int_{- h / 2}^{h / 2} {G_{m} e}^{p_{z} (\frac{Z}{h} + \frac{1}{2})} (1, z, z^{2}, z^{3}, z^{4}, z^{6}) d z

(28)

[∆] = {\begin{cases} {A} \\ {B} \\ {C} \end{cases}}

(29)

[K] = [\begin{array}{l} [k_{11}] & [k_{12}] & [k_{13}] \\ [k_{12}] & [k_{22}] & [k_{23}] \\ [k_{13}] & [k_{23}] & [k_{33}] \end{array}]

(30)

[S] = [\begin{array}{l} [0] & [0] & [0] \\ [0] & [S_{22}] & [0] \\ [0] & [0] & [0] \end{array}]

(31)

[M] = [\begin{array}{l} [M_{11}] & [M_{12}] & [M_{13}] \\ {[M_{12}]}^{⊤} & [M_{22}] & [M_{23}] \\ {[M_{13}]}^{⊤} & {[M_{23}]}^{⊤} & [M_{33}] \end{array}]

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