Dynamic and sound radiation characteristics of a non-uniformly heated isotropic plate

Abstract

The influence of non-uniform heating on the anisotropic plate’s dynamic and acoustic response characteristics is investigated numerically. The effect of non-uniform heating on the plate is accounted in the form of pre-stress developed on the plate due to non-uniform heating of the plate. The influence of structural boundary conditions, nature of temperature variation, and level of temperature on vibration and acoustic response characteristics are investigated in detail. Thermal buckling strength and fundamental buckling mode are highly sensitive to the nature of temperature variation across the plate. Free vibration mode shapes of the plates with high aspect ratio and free edges are changing their pattern with an increase in temperature. The resonant amplitude of sound power radiated associated with a particular mode is dominated by vibration nodes and anti-nodes shifting relative to the excitation location. Further, an increase in sound power level with an increase in temperature is observed clearly in the lower octave frequency bands.

Keywords

Non-uniform heating thermal buckling vibration response sound radiation finite element method Rayleigh integral

Introduction

Structural members used in aerospace and automobile industries are thin-walled to have the advantage of reduced weight. These members are exposed to heat during their service and subjected to thermal stress whenever the free expansion of the structure is restricted. Thermal stress alters structural stiffness and influences the static and dynamic behavior of the structures. Thin-walled structural members under compressive load are subjected to buckling mode of failure. The main problem with thermal loading is that even a small temperature rise above the ambient temperature will produce a significant amount of thermal stress. Prediction of static and dynamic characteristics of these structures over a range of elevated temperatures is essential to complete the design procedure. Most of the earlier studies were carried out by considering the assumption of uniform temperature distribution. However, this assumption may not be valid due to the location of the heat source, which may increase the temperature over the structure non-uniformly. Components such as aircraft, space vehicles, steam and gas turbines, and thin printed circuit boards are typical examples where the structure is exposed to non-uniform heating.

Many researchers have investigated the effect of uniform elevated temperature on the static, buckling, and dynamic behavior of structures made up of metal, functionally graded metals, and laminated polymer composites. In most of these studies, numerical and analytical methods have been used to study the effect of elevated temperature effects. However, very few researchers have studied the effect of non-uniform temperature variation on thermal buckling and the dynamic behavior of heated structures.

Experiments were carried out to predict the thermal buckling strength of a clamped metal plate and compared their results with that of a theoretical model.¹ The finite element approach was used to analyze the buckling behavior of uniformly heated laminated composite and sandwich plates² and a similar approach was used to investigate the buckling strength and vibration characteristics of a composite sandwich panel and found that the influence of elevated temperature on free vibration mode shapes is significant.³

The stability and dynamic behavior of a non-uniform heated isotropic plate were investigated and found that the fundamental buckling modes and free vibration modes are significantly altered when exposed to non-uniform heating.⁴ The free vibration behavior of isotropic tapered rectangular plates was analyzed by considering varying thickness in one direction.⁵ The static, buckling, and vibration behavior of carbon nanotube (CNT) reinforced polymer composite beam was investigated under non-uniform heating.⁶

An analytical method is used for analyzing the free vibration response of a plate with an internal crack under thermal load condition.⁷ The effect of temperature variation on buckling and dynamic behavior of anisotropic cylindrical panel was investigated and established a relationship between the critical buckling temperature of non-uniform temperature field and uniform temperature. The experimental investigation is performed to analyze the effect of non-uniform temperature on the buckling temperature of an isotropic beam⁸ and compared the experimental results with finite element-based numerical results. The effect of non-uniform temperature variation on buckling and free vibration behavior of CNT polymer composite plate was analyzed numerically and observed that the thermal buckling strength and free vibration characteristics are more sensitive to the nature of heating compared to the nature of functional grading.⁹ Experimental investigations were performed to study the buckling behavior of anisotropic cylindrical panels with different types of temperature distributions¹⁰ and numerical investigations were made to find the effect of different types of temperature variation on buckling characteristics of the laminated tapered composite plate.¹¹ Similarly, experimental investigations were also made to investigate the buckling behavior of a laminated composite beam under different types of temperature distributions.¹²

Studies on the vibro-acoustic response of heated structures under a steady-state mechanical excitation are limited compared to studies on heated structures’ static, buckling, and dynamic behavior. A comprehensive analysis of the buckling and dynamic (vibration and acoustic) behavior of a uniformly heated isotropic plate and their numerical study revealed that elevated temperature significantly influences the resonant vibration and sound response amplitude.¹³ Later, Jeyaraj et al.¹⁴ extended their numerical approach to analyze the vibro-acoustic response of a uniformly heated laminated composite plate and found that the modal damping increases with elevated temperature. In addition, the resonant amplitude of vibration and acoustic response reduces with an increase in elevated temperature due to the counterbalance effect of reduced structural stiffness and increased modal damping. Finally, an analytical method was used to estimate the effect of thermally induced membrane forces on the vibro-acoustic response of an isotropic plate under uniform heating and observed that the fundamental frequency is quite sensitive to temperature changes.¹⁵

The vibro-acoustic response of a heated sandwich plate under mechanical harmonic excitation were analyzed using an equivalent non-classical theory based analytical approach and found that the resonance peaks float to higher frequency regions with an increase in core layer thickness.¹⁶ Experiments were performed to investigate the vibro-acoustic characteristics of a heated plate and found that the initial deflection influences the natural vibration significantly.¹⁷ Vibro-acoustic characteristics of the heated composite sandwich plate were made using an analytical approach and found that peaks of the responses float to the low frequency with an increase in temperature. Geng and Li¹⁸ presented an analytical solution for the vibration and sound radiation response of an isotropic plate heated uniformly. Buckling and vibro-acoustic response characteristics of the clamped laminated composite plate were also investigated analytically¹⁹ and reported that resonant amplitudes shift to low-frequency regions due to decreased natural frequencies with an increase in elevated temperature. The theoretical model was developed to study the thermoacoustic response behavior of a plate subjected to a temperature variation through the thickness direction.²⁰ The sound transmission loss of heated laminated composite panels was investigated using analytical methods and found that the changes in peaks and tips of the sound transmission loss curve are significant at lower frequency regions with an increase in temperature.²¹ The vibro-acoustic behavior of truncated conical shell using a combined analytical and boundary element method was studied and the results have shown that the anti-resonant peak of the coupled system is shifted to lower frequencies because of the greater mass of water than air.²² The effects of the CNT volume fraction and carbon nanotubes distribution on vibro-acoustic characteristics of laminated FG-CNT composite plates under various temperatures were studied.²³ The effect of temperature profile on dynamic characteristics of aluminum silicon carbide metal matrix composite was examined.²⁴ Non-uniform edge loading effects on dynamic characteristics of FG-graphene composite plate were investigated and found that graphene nanoplatelets distribution and volume fraction significantly altered the resonant amplitudes.²⁵ The effect of variable edge load and porosity on sound radiation characteristics of graphene nanocomposite plate were investigated analytically and given the optimized guidelines for graphene platelets and porosity distribution to achieve less sound power.²⁶

From the above literature, many researchers have paid significant attention to investigating the effect of temperature on the static, buckling, and dynamic behavior of structures by considering uniform temperature rise. However, studies have rarely been reported about the vibro-acoustic behavior of a non-uniformly heated plate. The literature shows that the developed thermal stress and consequent dynamic behavior of the plate with non-uniform heating are significantly different from the typical uniform temperature rise assumption. Hence, in this work, the effect of non-uniform heating on the vibro-acoustic response of the isotropic plate is considered. The present work aims to perform a numerical investigation on the vibro-acoustic response of a non-uniformly heated isotropic plate.

Methodology

The finite element method is used to obtain thermal buckling and vibration characteristics of different non-uniformly heated plates. Pre-stress developed due to the non-uniform heating is the thermal load in terms of the geometric stiffness matrix. Therefore, the buckling temperature is calculated first, and then pre-stressed modal and harmonic response analyses are performed at different elevated temperatures as a function of the buckling temperature to obtain the free and forced vibration responses, respectively. Finally, the Rayleigh integral has been used to predict the acoustic response characteristics from the forced vibration response. The pictorial representation of the problem analyzed is shown in Figure 1a and the flow chart describing the methodology followed is shown in Figure 1(b).

Commercial finite element software is used to analyze buckling²⁷ and free and forced vibration response characteristics of the non-uniformly heated plate. In present analysis, the assumed temperature profiles are varying along in-plane directions of the plate and thermal stress develops in the plate when exposed to an increase in temperature (ΔT). This problem has been solved with plane stress conditions due to constant temperature through the thickness. The stress–strain constitutive relation for isotropic material plate, considering thermal effects, can be written as follows

{\begin{matrix} σ_{x} \\ σ_{y} \\ σ_{x y} \end{matrix}} = [\begin{matrix} \frac{E}{1 - ν^{2}} & \frac{ν E}{1 - ν^{2}} & 0 \\ \frac{ν E}{1 - ν^{2}} & \frac{E}{1 - ν^{2}} & 0 \\ 0 & 0 & \frac{E}{2 (1 + ν)} \end{matrix}] {\begin{matrix} ϵ_{x} - α Δ T^{'} \\ ϵ_{y} - α Δ T^{'} \\ γ_{x y} \end{matrix}}

(1)

where σ_x, σ_y, and σ_xy are normal and shear stresses along membrane directions; E is Young’s modulus; ν is Poisson’s ratio; α is the coefficient of thermal expansion; and ∈_x, ∈_y, and γ_xy are normal and shear strains along membrane directions. ΔT′ is the elevated temperature, which is the function of in-plane plate direction co-ordinates X and Y, for non-uniform heating (i.e., ΔT′ = ΔT(X, Y)), In general, constitutive relation accounting thermal stress is

{\begin{matrix} σ \end{matrix}} = [D] {\begin{matrix} ϵ - α Δ T^{'} \end{matrix}}

(2)

where [D] is the constitutive matrix. In the following, element type and finite element formulation in this work were adopted directly.²⁷ The element’s structural stiffness matrix can be obtained as follows

[K_{S}] = \iint {[B]}^{T} [D] [B] d x d y

(3)

where [B] is the matrix which relates strain and displacement. The geometric stiffness matrix [K_G] which accounts for thermal stress effect due to non-uniform heating is

[K_{G}] = \iint {[B_{g}]}^{T} [\begin{matrix} σ_{x} & σ_{x y} \\ σ_{x y} & σ_{y} \end{matrix}] [B_{g}] d x d y

(4)

where [B_g] represents the strain displacement matrix which corresponds to the geometric stiffness matrix.

Thermal buckling strength and associated buckling mode shape are calculated based on the linear eigenvalue analysis as follows

([K_{S}] + λ_{i} [K_{G}]) {ψ_{i}} = 0

(5)

where λ_i and {ψ_i} are the buckling load parameter and buckling mode shape of the i^th mode, respectively. For the uniform temperature rise case, product of elevated temperature ΔT (above stress-free temperature) and lowest eigenvalue λ_i define the thermal buckling strength (T_cr) which is equal to λ₁ ΔT′. For the non-uniformly heated plate, ΔT′ is the highest temperature of a particular non-uniform temperature profile.

The free vibration frequency and its mode shape of the non-uniformly heated plates are obtained by carrying out a pre-stressed modal analysis as follows

(([K_{S}] + [K_{G}]) - ω_{j}^{2} [M]) {ϕ_{j}} = 0

(6)

where ω_j and {ϕ_j} are the free vibration frequency and associated mode shape, respectively, of j^th mode of the non-uniformly heated plate. [M] is the structural mass matrix as follows

[M] = \iint {[N]}^{T} [ρ] [N] d x d y

(7)

where [ρ] and [N] are inertia and interpolation matrices, respectively.

Forced vibration response of the non-uniformly heated plate under a steady-state mechanical excitation plate is obtained by using the pre-stressed harmonic response analysis given as

[M] {\overset{\cdot\cdot}{U}} + [C] {\overset{\cdot}{U}} + ([K_{S}] + [K_{G}]) {U} = {F^{a}}

(8)

where ${\overset{\cdot\cdot}{U}}, {\overset{\cdot}{U}}, a n d {U}$ are the acceleration, velocity, and displacement of the non-uniformly heated plate, respectively, and {F^a} is the applied mechanical load vector. The structural damping matrix [C] is equal to $(2 ζ / ω) [K_{S}]$ .

Assuming, F^a = F_se^iΩt and U = Ue^iΩt for harmonic motion

(([K_{S}] + [K_{G}]) + i Ω [C] - Ω^{2} [M]) {U} = {F_{s}}

(9)

where F_s is the amplitude of the harmonic load vector.

For the non-uniformly heated vibrating plate, the sound pressure radiated at any point can be obtained using the Rayleigh integral as follows

p (x) = i \frac{ρ c k}{2 π} \int_{S} v_{p} (x_{s}) \frac{- i k R}{R} d S

(10)

where c and ρ are speed of sound in air and its density, S and x represent the surface area of the plate (assumed to lie in XY plane) and the location of observation point, v_p denotes normal velocity at observation point x_s = (x, y), and R gives the distance between the source and observation point which is equal to |x − x_s|. Harmonic time dependence in the form of e^iΩt is assumed for the analysis. When the Rayleigh integral is expressed in discrete form as a sum over elementary source regions, then sound power radiated from the vibrating plate is calculated by using equation (11)

W = \frac{1}{2} \int_{S} Re [p (x) v_{p}^{⋆} (x, y)] d S

(11)

where $v_{p}^{⋆} (x, y)$ indicates the complex conjugate of normal velocity of the plate.

Validation studies

Vibro-acoustic behavior of the uniformly heated aluminum plate with simply supported edges was analyzed.¹⁵ Critical buckling temperature and free vibration characteristics of a plate (0.4 m × 0.3 m × 0.01 m) were obtained using an analytical method based on the classical plate theory. The reader is referred to have the details of material properties.¹⁵ These two cases are used for validation of the proposed methodology. The present method uses commercial finite element software ANSYS to obtain critical buckling temperature and free and forced vibration responses. The element used in the present work is SHELL 181, a four-node quadratic element formulated based on the first-order shear deformation theory. A mesh size of 16 × 16 is used to perform the finite element analysis based on the convergence study. The critical buckling temperature obtained in the present work is 47.35°C and it matches well with the buckling temperature,¹⁵ which is 47.7°C. Furthermore, free vibration frequencies are calculated at different elevated temperatures and compared.¹⁵ As shown in Table 1, the result obtained based on the present method is well matched.¹⁵

Table 1.

Comparison of natural frequencies (Hz) at different temperature¹⁵ results.

Mode	Thermal load
	0°C		15°C		35°C
	Geng and Li¹⁵	Present	Geng and Li¹⁵	Present	Geng and Li¹⁵	Present
(1, 1)	420.2	417.2	348.0	344.5	217.2	211.7
(2, 1)	874.0	870.15	805.3	801.2	703.4	698.68
(1, 2)	1227.0	1224.5	1159.1	1156.5	1061.9	1058.9
(3, 1)	1630.4	1640.1	1563.0	1572.8	1468.3	1478.2

The second example involves the prediction of radiated sound power from the vibrating plate. In order to validate this, the same plate is excited with 1 N harmonic mechanical load at an elevated temperature of 35°C and the sound power is predicted by using the Rayleigh integral. Figure 2 shows the comparison of the predicted sound power¹⁵ and the present method. Again, the agreement is good for the entire frequency range of the analysis.

Results and discussion

To examine the influence of temperature variation on buckling, free vibration, forced vibration, and sound radiation characteristics, an isotropic plate with two different aspect ratios such as square $(a / b = 1)$ and rectangular $(a / b = 2)$ is considered. Four different non-uniform temperature variations (decrease, decrease–increase, increase–decrease and camel-hump) and uniform temperature profiles are considered. An isotropic plate with a = 0.5 m and thickness 0.01 m is considered for the further investigations. Further, different boundary conditions (C-clamped and F-free) are also considered as shown in Figure 3. The analysis begins with buckling studies to obtain the critical buckling temperature and fundamental buckling mode shape. By considering the critical buckling temperature as a function, free and forced vibration analysis are considered next to know the natural frequencies, mode shapes, and spatially average rms velocity. Finally, acoustic response characteristics such as sound power level, overall and $1 / 3$ octave band sound levels are analyzed. The plate is assumed to have the following elastic properties: E = 210 GPa, ν = 0.3, ρ = 7850 kg/m³, and α = 12.6 × 10⁻⁶/°C.

Thermal buckling strength

To show the effect of non-uniform heating on thermal buckling strength, four different types of non-uniform cases are considered as shown in Figure 4. Apart from these non-uniform heating cases, a uniform heating case is also considered for comparison purposes.Temperature variation in all cases considered is one dimensional except for camel-hump temperature variation case, which is two dimensional. The mathematical expression for the camel-hump temperature field is $Δ T^{'} (x, y) = T_{\max} \sin (π x / a) \sin (π y / b) .$ The effect of non-uniform heating on thermal buckling strength of the square and rectangular plates are given in Table 2 and 3. From Table 2, one can observe that the nature of non-uniform heating influence on the thermal buckling strength of the plate is significant. Furthermore, the plate buckling strength of non-uniform heating cases is also found to be higher than the uniform heating case. This may be due to the variation in heat intensity across the plate surface for non-uniform heating cases compared with uniform heating sources, where the intensity is uniform across the plate surface.

Table 2.

Effect of non-uniform heating on thermal buckling strength (°C) of the square plate.

Plate	Uniform	Dec.	Dec.–Inc.	Inc.–Dec.	Camel-hump
CCCC	106.73	210.24	269.79	176.34	193.48
CCFC	113.63	254.06	204.03	194.35	224.84
FCFC	102.44	154.34	190.27	180.27	219.67
CFCF	102.44	204.86	197.41	209.96	219.67

Table 3.

Effect of non-uniform heating on thermal buckling strength (°C) of the rectangular plate.

Plate	Uniform	Dec.	Dec.–Inc.	Inc.-Dec.	Camel-hump
CCCC	79.776	132.15	188.95	113.70	140.41
CCFC	84.736	134.45	156.49	118.32	146.87
FCFC	89.674	129.14	155.70	122.87	153.93
CFCF	26.460	52.917	50.990	54.891	67.419

Thermal buckling strength of the CCCC plate against the decrease–increase heating case is higher than the other cases. For the plates with free edges, the thermal buckling strength is influenced by the intensity of the heat source on the free edges. For example, the CCFC plate exposed to the decrease temperature case has better critical buckling strength as the intensity of heating at the free edge is zero for that case. However, this is not true for the FCFC plate case, for which camel-hump temperature variation yields the highest thermal buckling strength. This is due to the free expansion of membrane displacement in the direction of heating.

Variation of fundamental buckling mode shape with respect to nature of temperature variation is analyzed. For this purpose, the bending amplitude of the centerline (along the X-direction) associated with the buckling mode is extracted for each case and normalized with respect to the highest amplitude associated with the particular case of comparison. In general, the buckling mode shape is essential and has physical significance. The amplitude associated with the buckling mode shape does not have any physical significance. However, the present analysis amplitudes are analyzed to demonstrate the buckling mode shape changes, especially shifting the location of nodal and anti-nodal positions with respect to the nature of temperature variation. Figure 5(a) shows the effect of temperature variation on fundamental thermal buckling mode for the CCCC square plate and Figure 5(b) for the rectangular plate. It is clearly seen that the buckling mode shape of the CCCC square plate is not influenced by non-uniform heating, except a slight shift in peak amplitude for the decreased temperature variation. This may be attributed to the plate’s symmetric boundary condition and geometry along with membrane directions. The fundamental buckling mode of the rectangular CCCC plate is significantly influenced by non-uniform heating, as seen in Figure 5(b). For the rectangular CCCC plate, typical bending deflection variation has been observed except for the decrease heating due to the unsymmetrical temperature variation. However, the bending deflection associated with decrease temperature variation shows a clear shift in the position of maximum amplitude toward the plate edge where the heat is applied for the decrease heating field. Similar behavior in bending amplitude can be seen for the decrease–increase temperature variation and it shows a clear nodal line at the center of the plate.

Figure 6 shows fundamental buckling mode shapes associated with the CCFC plate exposed to different temperature fields. Unlike the CCCC plate, the thermal buckling mode shapes of the CCFC plates are highly sensitive to non-uniform heating, irrespective of their aspect ratio. This happens due to the free edge associated with the CCFC plates and the nature of the temperature variation. One can observe from Figure 6 that the maximum amplitude of thermal buckling mode shapes do not occur at the free edge of the CCFC plate except for decrease–increase (for both square and rectangular plates) and uniform temperature (for the square plate) variation in which the heating source is located at the free edge. One can observe in Figure 6(b) that the free edge of the rectangular CCFC plate has no bending displacement due to the higher aspect ratio and non-uniform heating of the plate. However, this effect is not significant for the CFCF plates, as seen in Figure 7. This is due to the restriction of free expansion of the plates in the direction of temperature variation. The effect of non-uniform temperature on buckling mode shapes of the CFCF plates is shown in Figure 7. This can be attributed to the free expansion of the CFCF plate along the direction of temperature variation. From Figure 8, it is clear that the fundamental buckling mode shapes reflect the shape of temperature variation for the FCFC plates.

From the studies carried out on the thermal buckling mode shapes, it is concluded that the variation of the buckling mode shapes is dependent on the nature of temperature variation, structural boundary condition (free edge direction), and aspect ratio of the plates.

Free vibration studies

To examine the effect of temperature variation on free vibration characteristics, critical buckling temperature is considered as a parameter. The pre-stressed modal analysis was performed to include pre-stress effects due to the temperature variation. First, five-mode shapes associated with CCCC and CCFC plates exposed to uniform and decrease heating conditions are considered to demonstrate the influence of an increase in temperature on natural frequency and results are given in Table 4. Modal indices mentioned in Table 4 were obtained when the plate was not exposed to thermal load. One can observe a reduction in natural frequencies due to the reduction in structural stiffness with elevated temperature, as reported by several researchers. The present study reveals the same trend also for the non-uniformly heated plate. The same behavior of natural frequency with temperature variation has also been observed for the CFCF and FCFC plates.

Table 4.

Free vibration frequency (Hz) variation of CCCC and CCFC plates with uniform and decrease heating.

Plate	Mode	0T_cr	Uniform		Decrease
Plate	Mode	0T_cr	0.5T_cr	0.99T_cr	0.5T_cr	0.99T_cr
CCCC square	(1, 1)	358	255	37	256	37
	(1, 2)	733	623	490	623	484
	(2, 1)	733	623	490	623	495
	(2, 2)	1077	963	835	965	841
	(2, 2)*	1234	1211	1087	1211	1082
CCCC rectangle	(1, 1)	246	176	33	184	14
	(2, 1)	318	236	103	252	164
	(3, 1)	448	360	246	376	290
	(4, 1)	636	547	442	562	479
	(1, 2)	648	571	482	578	483
CCFC square	(1, 1)	239	173	25	194	32
	(2, 1)	397	338	266	331	267
	(1, 2)	634	550	447	571	446
	(3, 1)	765	711	631	693	614
	(2, 2)	803	721	652	721	653
CCFC rectangle	(1, 1)	227	168	25	184	28
	(2, 1)	260	197	97	222	169
	(3, 1)	337	269	180	282	232
	(4, 1)	466	397	314	406	339
	(1, 2)	628	558	472	576	483

Variation in free vibration mode shapes with respect to different heating conditions and elevated temperature levels is analyzed similar to the fundamental buckling mode analysis. In general, free vibration mode shapes are not sensitive when the plate is square and the boundary conditions are symmetric, which does not have any free edge along the direction of temperature variation. Due to this reason, the free vibration mode shapes associated with square CCCC and CFCF plates are not affected by either the magnitude of thermal load or the nature of in-plane temperature variation. Even though the structural stiffness of these two plates reduces with an increase in temperature, it is not reflected in the mode shapes. However, free vibration mode shapes of the square CCFC and FCFC plates, exposed to decrease heating, are affected by the magnitude of thermal load, as seen in Figure 9 and Figure 10, respectively. Displacement in the thickness direction along the central line of the plate is obtained at different temperature levels of a particular mode shape exposed to a particular temperature variation. In the case of modes, in which the central line of the plate becomes the vibration nodal line, for example, mode (1, 2) of the CCFC plate, displacement in the normal direction is obtained along the line located at ${3 / 4}^{t h}$ of the plate width.

One can observe from Figure 9 that, with increase in temperature, vibration anti-nodal position of modes ((2, 1) and (1, 2)) and vibration nodal position of mode (2, 1) move toward the clamped edge. Square FCFC plate vibration modes under decrease heating also exhibit the same behavior as seen in Figure 10. But this is not the case for the square CCFC and FCFC plates which are exposed to other types of temperature variations. To demonstrate this, the effect of decrease–increase heating on mode shapes of square CCFC and FCFC plates is given in Figure 11 and Figure 12. This can be attributed to the unsymmetric variation of temperature associated with the decrease type temperature variation. As the plate is square and temperature variation is also symmetric, for other cases of temperature fields, the effect is not significant on the mode shapes of square CCFC and FCFC plates.

The effect of thermal load and non-uniform heating on vibration mode shapes of rectangular plates is severe when they have any free edge along the direction of temperature variation and where the nature of temperature variation is not symmetric. To demonstrate this, some mode shapes of the rectangular CCCC plate exposed to uniform and decrease temperature variations are shown in Figure 13 and Figure 14, respectively. One can observe from Figure 13(a), there is no significant change in free vibration modes associated with the rectangular CCCC plate exposed to uniform heating. This is due to the symmetry associated with both uniform temperature rise and structural boundary conditions. There is a significant variation in terms of shifting of nodal and anti-nodal positions of the free vibration when the rectangular CCCC plate is exposed to decrease heating. This can be clearly seen for modes (1, 1) and (2, 1) in Figure 14.

Significant variation in vibration mode shapes of the rectangular CCFC plate is observed for both uniform and decrease temperature fields as shown in Figure 15 and Figure 16, respectively, due to the synergy of both the free edge and temperature variation. Shifting of nodal and anti-nodal positions of the free vibration modes is common for the CCFC plate under both the temperature variations. However, the effect is significant for the decrease temperature field compared to uniform temperature rise. From the investigation on the mode shapes of non-uniformly heated plates, the following observations are made: (1) influence of elevated temperature is significant for the rectangular plates compared to the square plates, (2) vibration nodal position of a mode gets shifted toward the clamped edge at higher temperatures, and (3) modes of the rectangular plate with boundary conditions which allow in-plane free expansion is not very sensitive to both the thermal load and non-uniform heating.

Forced vibration response

Next, the same square and rectangular plates are used to investigate the effects of elevated temperature, boundary conditions, and temperature fields on forced vibration response. These effects are included in the analysis with the help of pre-stressed harmonic response analysis. The frequency range of interest for the analysis is from 0 to 1200 Hz. Harmonic load with a magnitude of 1 N is applied at (0.375 m, 0.375 m) and (0.75 m, 0.375 m) from the corner for the square and rectangular plates. Finally, to show the influence of non-uniform temperature variation on forced vibration, the overall global vibration response (spatially averaged root mean square (rms) velocity) is computed from the average sum of the square of normal velocities of the structure. Spatially averaged root mean square (rms) velocity can be calculated as a function of excitation frequency (Ω) and elevated temperature (ΔT) is given by

V_{r m s}^{2} (Ω, Δ T) = \frac{1}{n} [V_{1}^{2} + V_{2}^{2} + \dots + V_{n}^{2}]

(12)

where n is the number of finite element nodes and V_i is the normal velocity of i^th node.

Irrespective of temperature fields, the resonant amplitude of the particular mode is increased with the variation in temperature for the square CCCC and CFCF plates. This may be due to the reduction in structural stiffness associated with that particular mode. Further, the free vibration studies have shown that the free vibration modes of these plates are not influenced much by both the thermal load and non-uniform heating. So the resonant amplitude is varied due to reduction in structural stiffness only. However, this is not true for the CCFC and FCFC plates having a free edge(s) along the direction of the temperature variation. Their mode shapes are highly sensitive to elevated temperatures. For example, consider the resonant amplitude behavior of the (2, 1) mode associated with the square CCFC plate exposed to decrease and decrease–increase temperature fields. Figure 17 shows that the spatially averaged rms velocity response of the square CCFC plate increases significantly with temperature for the decrease heating while it is not significant for decrease–increase heating. By increasing the temperature for the (2, 1) mode under decrease temperature field, one can see the significant changes in the resonant amplitude. However, the structural stiffness reduction is not that significant. Conversely, the associated variation in the resonant amplitude is not significant for the (2, 1) mode under decrease–increase temperature field as shown in Figure 17(b). Similar behavior is observed for the (3, 1) mode of the square FCFC plate under decrease and decrease–increase temperature fields as shown in Figure 18.

Spatially averaged rms velocity response of the rectangular CCCC plate exposed to uniform and decrease temperature fields is shown in Figure 19(a) and (b), respectively. Resonant amplitudes of the modes, except the fundamental mode (1, 1) of the rectangular CCCC plates, increase with elevated temperature for both the temperature fields as observed in Figure 19. One can expect an increase in the resonant amplitude of velocity response with increased elevated temperature, as the structural stiffness reduces with increased temperature. However, this does not happen for the fundamental mode (1, 1) of the rectangular CCCC plate exposed to decrease temperature field. The resonant amplitude of the fundamental mode of the rectangular CCCC plate under decrease heating experiences a significant reduction in resonant amplitude near the critical buckling temperature. This can be attributed to the significant movement of anti-nodal position associated with the mode (1, 1), away from the chosen excitation location of the harmonic force with an increase in temperature as seen in Figure 14(a). Another reason for this distinct behavior could be the very low natural frequency of the fundamental mode near the critical temperature under decrease heating. This shifting of nodal and anti-nodal positions, away from chosen excitation location, happening for the other free vibration modes such as (2, 1) and (3, 1) with an increase in elevated temperature does not result in similar distinct variations. The nodal and anti-nodal shifts are not so significant for these modes relative to the fundamental mode (1, 1). Hence, the resonant amplitudes of these modes do not exhibit a distinct reduction in resonant amplitude near the critical temperature. Fundamental mode (1, 1) of the rectangular CCFC plate also exhibits the same distinct behavior, as seen in Figure 20(b), as the anti-nodal position shifts away from the excitation location.

The resonant amplitude of the fundamental mode (1, 1) of the rectangular plate exposed to uniform temperature field also increases with the elevated temperature. It reduces when the elevated temperature is near the critical buckling temperature, but the reduction in resonant amplitude is not as significant as observed for the rectangular CCCC under decrease temperature field. This can be attributed to the relative far away shifting of the (1, 1) mode under decrease temperature field than under uniform temperature rise. Similarly, increase in bending amplitude of the free vibration mode (2, 1) (refer Figure 15(b) and Figure 16(b)), of the rectangular CCFC plate, under both the temperature fields results in resonant amplitude of the (2, 1) mode with increase in the elevated temperature. The resonant amplitude of the mode (3, 1) of the rectangular CCFC plate exposed to decrease temperature field increases with an increase in elevated temperature. The resonant amplitude of vibration response of the same mode (3, 1) under a uniform temperature field also increases with elevated temperature, but one cannot find a peak corresponding to the natural frequency of the mode (3, 1) near the critical buckling. This indicates that mode (3, 1) of uniformly heated rectangular CCFC plate does not get excited near the critical temperature. This can be attributed to shifting one of the nodal lines of the mode (3, 1) to the location of harmonic excitation when the plate is heated near the critical temperature.

Acoustic response

Finally, the radiated sound power is calculated from the vibration response using the Rayleigh integral. Figure 21(a) and (b) shows the radiated sound power for the square CCFC plate under decrease and decrease–increase temperature fields. One can observe a distinct sound power level response behavior associated with the fundamental mode from Figure 21. Also, the resonant amplitude increases with temperature and significantly reduces near-critical temperature in both cases. However, the resonant amplitude associated with the second mode of the square CCFC plate under decrease heating increases with a temperature significantly due to the shifting of the nodal line away from the excitation location. This is not the case when the second mode of the square CCFC plate is under decrease–increase heating, as the nodal line of the vibration mode is not sensitive to decrease–increase heating. Similar variations have been observed also for the other modes of the square CCFC plates. Radiated sound power from the square FCFC plate under uniform and decrease temperature fields is shown in Figure 22(a) and (b), respectively. Near to the buckling temperature, one can see the distinct behavior of the radiated sound power from the vibrating FCFC and CFCF plates (see Figure 22). However, the shift in second (2, 1) and third (3, 1) modes of the square FCFC plate under decrease heating leads to an increase in radiated sound power for the second mode and no variation for the third mode, near-critical buckling temperature, as seen in Figure 22.

The influence of uniform and decrease temperature fields on sound power response of different rectangular plates such as CCCC, CCFC, CFCF, and FCFC is shown in Figures 23 –26, respectively. It is observed that the sound power resonant amplitude of the fundamental mode reduces significantly near the critical temperature when the rectangular plate is under decrease heating. This behavior is insensitive to the nature of structural boundary conditions. This can be attributed to the significant movement of anti-nodal lines of the fundamental mode toward the clamped edge or away from the free edge subjected to maximum temperature. Resonant amplitudes of the sound power level of the second and third modes associated with the rectangular CCCC plate reduces with an increase in the magnitude of heating, as seen in Figure 23.

Variation of sound power radiated at the first three resonant frequencies of the rectangular CCFC plate (uniformly heated and decrease heating cases) is influenced by changes in the free vibration modes with elevated temperature. From Figure 16(a), one can identify that the fundamental free vibration mode of the rectangular CCFC plate under decrease heating, near the critical buckling temperature, experiences a nodal line near the excitation location. Due to this reason, the distinct behavior has been observed in the resonant amplitude of the fundamental mode of the rectangular plate when the elevated temperature is around the buckling temperature. Similarly, the third mode of the uniformly heated rectangular CCFC plate is not galvanized near the critical temperature. However, this is not the case when the third mode is exposed to decrease temperature field under which the resonant amplitude increases significantly with elevated temperature. This can be attributed to the corresponding shift of nodal lines of the third mode toward/away from the excitation location when the rectangular CCFC plate is under uniform/decrease heating. Similar variation in resonant amplitudes of the sound power level with respect to changes in corresponding free vibration modes has also been observed for the rectangular CFCF and FCFC plates.

The influence of thermal load on the sound power level of the square CCCC and CCFC plates with respect to 1/3 octave band is shown in Figures 27 and 28, respectively. Sound power level increases with a temperature significantly, in the lower frequency bands 0–42 Hz and 45–81 Hz for both the square CCCC and CCFC plates, as seen in Figures 27 and 28, respectively. One can observe a distinct increase in sound power level around 40 dB near-critical buckling in the lowest frequency band. This can be attributed to both the increase in sound power and the reduction in natural frequency of fundamental mode toward zero frequency, with increased temperature. It is also observed from Figures 27 and 28 that sound power of both the square CCCC and CCFC plates increases and then decreases near the critical temperature in the mid-frequency bands 83–162 Hz and 165–315 Hz. This is due to the movement of modes toward the lower frequencies with increased temperature. Similarly, sound power variation with an increase in temperature for the square CCCC and CCFC plates is not significant in frequency bands 318–630 Hz and 630–1200 Hz as seen in Figures 27 and 28. Even though the reduction in natural frequency happens in modes in the higher frequency bands, the participation of more sound radiation modes in the higher frequency bands leads to the same amount of sound power level in the higher frequency bands.

The influence of thermal load on sound power level of the rectangular CCCC and CCFC plates with respect to $1 / 3$ octave band is shown in Figures 29 and 30, respectively. The variation of sound power with frequency bands of the rectangular CCCC and CCFC plates are similar to that of square CCCC and CCFC plates in most of the cases, except the plates exposed to decrease heating as observed in Figures 29 and 30. Due to the anti-nodal movement toward excitation with increase in temperature, results in distinct increase in sound power near the critical temperature for the rectangular CCCC and CCFC plates exposed to decrease heating.

Variation of the overall sound power level of the square and rectangular CCFC plates subjected to different temperature fields with an increase in temperature is shown in Figure 31. It is observed from Figure 31 that overall sound power is not sensitive to different kinds of temperature fields, as the variation is only about 2 dB. This happens due to the influence of a very few dominant sound radiating modes in the chosen frequency range for all the cases. The same kind of variation has been observed for other plates such as CCCC, CFCF, and FCFC, for both square and rectangular cases.

Conclusion

A comprehensive numerical investigation has been carried out on buckling, vibration, and acoustic characteristics of a non-uniformly heated metal plate. The influence of temperature variation and structural boundary conditions on thermal buckling strength, buckling mode shape, free vibration modes, and vibro-acoustic response are studied in detail. It is found that buckling temperature and mode shapes are sensitive to the plate’s non-uniform heating and free edges. Similarly, shifting vibration nodes and anti-nodes due to non-uniform heating has been observed for the free vibration modes of plates with higher aspect ratios and free edges. The resonant amplitude of vibration and sound power level of a particular mode at a particular temperature is dictated by the free vibration mode shape at that temperature. The overall sound power level is not influenced by non-uniform heating as very few modes radiate more sound power in the chosen frequency range. The investigation based on the $1 / 3$ octave band indicates that the sound power level increases with temperature in the lower frequency bands, while there is not much variation in the higher frequency bands.

Supplemental Material

sj-cls-1-nvw-10.1177_09574565221093235 – Supplemental material for Dynamic and sound radiation characteristics of a non-uniformly heated isotropic plate

Supplemental material, sj-cls-1-nvw-10.1177_09574565221093235 for Dynamic and sound radiation characteristics of a non-uniformly heated isotropic plate by Arul Raj Vaiduriyam, Lenin Babu Mailan Chinnapandi and Jeyaraj Pichaimani in Noise & Vibration Worldwide

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 iD

Lenin Babu

Supplemental material

Supplemental material for this article is available online.

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