PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Vibro-acoustic behavior of plates considering static load effectYue-ming LI; Di WANG; Qian GENG; Xiong-wei YANG; Lei WU
State Key Laboratory for Strength and Vibration of Mechanical Structures,
Shaanxi Key Laboratory of Environment and Control for Flight Vehicle,
School of Aerospace Engineering, Xi’an Jiaotong University, Xi’an, 710049, People’s Republic of China

ABSTRACTIn this paper, we present the general form of governing equations of plates considering thethermal/static load effect, and study the effect of thermal/static load on the vibro-acoustic
behavior of plates. The static state under thermal/static load is obtained firstly, and then thevibration and acoustic radiation are calculated at the reference static position. Numerical andexperimental approaches are also displayed. The results show that both the inner stress andgeometry shape will be changed under thermal/static load. With the temperature increasing, thesoftening effect of thermal stress in the pre-buckling state leads to the decrease of frequencies,
while in the post-buckling state the stiffening effect of thermal buckling deformation leads tothe increase of frequencies. The variation of vibration and acoustic radiation responses are inaccordance with that of the frequencies. Finally, the vibro-acoustic behavior under static load is also
discussed in detail.
Keywords: Thermal load effect, Static load effect, Natural characteristics, Vibration response, Acoustic
radiation

1. INTRODUCTION
With the big progress in hypersonic aerocraft, the research on vibro-acoustic characteristics ofstructures under extreme thermal load, static load, acoustic excitation and mechanical excitation
becomes more and more important. The material properties, geometry shape, inner stress will all bechanged under the thermal load and static load, which makes the vibro -acoustic analysis much moredifficult. Lots of efforts have been made on the related works.
Using the combined FEM/BEM method, Jeyaraj et al detailedly studied the thermal effect on thevibration and acoustic radiation responses of different plates, such as isotropic plate [ 1 ],fiber-reinforced composite plate [2], viscoelastic sandwich plate [3] and so on. In the research, thecritical buckling temperatures and the vibration of the plates are carried out by FEM, while theacoustic radiation is derived by BEM. The method is also utilized in the structures under the water [4].
The temperature dependent material properties are considered in the free vibration analysis offunctionally graded plates in thermal environment [5]. Geng et al [6-8] further investigated the effect ofthermal environments on the vibro-acoustic responses of plates with different boundary conditions
through theoretical, numerical and experimental approaches. The results shown that in the
pre-buckling state, the natural frequencies decrease and the response peaks shift to lower frequency
range with the temperature increasing. Liu and Li [9] studied the vibration and acoustic responses ofsimply supported sandwich plates considering thermal effect using equivalent non -classical theory.
Li and Li [10] conducted the vibration and sound radiation analysis of composite plates in thermal
environment. Li and Yu [ 11 , 12 ] investigated the vibro-acoustic responses of the sandwich andcomposite panels in thermal environment using aa proposed Piecewise shear deformation theory.
Zhao et al [13] presented numerical analysis of the vibro-acoustic responses of laminated composite
plates in thermal environment based on finite element formulation.
The above research mainly focused on the softening effect of thermal stress in pre-buckling state.
However, when the temperature is higher than the critical buckling temperature, the effect ofbuckling configuration should be paid more attention on. In the experiment about the vibro-acoustic

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characteristics of a plate in pre- and post-buckling state, Geng et al [ 14 ] found that with thetemperature increasing, the natural frequencies decrease before the critical temperature and increase
after. The reason is that the effect of buckling configuration plays a more important role than theeffect of thermal stress, which leads to the stiffening of the bending stiffness. Numerical analysis ofthermally-buckled plates also presented the increasing of natural frequencies in the post-bucklingstate [15, 16]. Du et al [17] investigated the vibration and acoustic radiation of laminated plates withtemperature gradient along thickness. It is shown that the deflection under temperature gradient
which results in the increasing of natural frequencies. The same as thermal load, static load will also
result in the change of both inner stress state and geometry shape. The difference is that either
compressive stress or tensile stress may appear under static load depending on the direction of staticload and the initial curvature of the plate. Takabatake studied the effect of dead l oad on the vibrationof beams [18] and plates [19]. Wang et al [20] investigated the effect of static load on vibro-acoustic
behavior of clamped plates with different geometric imperfections.
In this paper, we present a review of the research on vibro-acoustic behavior of plates underthermal/static load carried out by our group.

2. THEORETICAL APPROACH
As we In the theoretical formulation, considering a plate under thermal load or static load, thedisplacements can be expressed as the sum of nonlinear static deflections under thermal/static load
and the linear dynamic displacements[20]. According to the boundary conditions, the displacements
are assumed as the superposition of spatial functions. Thus, by applying Hamilton ’s principle, we
can derive two sets of equations in the following form [17, 20]:

K - K T + K NL1  q s   K NL 2  q s  q s = Q s + QTd0
Mq  t  + K - K T + 2K NL1  q   3K NL 2  q 0  q d  t  = Q d  t 

(1)
(2)

From Eq.(1), we can derive the nonlinear static deflection under thermal/static load. Then substituting thestatic deflection into Eq.(2), we can derived the vibration governing equation considering the initial stressand deformation effects under thermal/static load. Introduce Rayleigh integral[l6, 17]:p  xp , yp , z p , t  

j0 jt v  x, y, t   e
e 

2
R

 jka RdA

(3)

and we can calculate the acoustic radiation pressure at the observation point.

3. NUMERICAL APPROACH
Numerical analysis is carried out by FEM/BEM method [6]. Firstly, the finite element model of theplate is established in Nastran for the modal analysis. Then the mode resul ts are imported into VA
One as shown in Fig. 1. Thus, the vibro-acoustic responses can be calculated by the combined FEM
and BEM solver of VA One.

Figure 1 – FEM/BEM model[6]25764. EXPERIMENT SETUPAs shown in Fig. 2, a series of experiments about the vibro-acoustic characteristics underthermal/static load have been conducted based on the experimental setup.
In the modal test, the single-input single-output hammer (PCB hammer 086C04) based impact
method is used. Test points are uniformly distributed on the plate. An accelerometer (PCB 333B32) islocated on a proper test point to receive the transient signals when all the test points are impacted by
the hammer in sequence. The Impact Testing module of the software package LMS TEST.LAB is used
for signal acquisitions and data analysis[8].
In the vibration response measurement, an accelerometer (PCB 333B32) is located at theobservation point on the plate to receive the vibration signals under mechanical excitation (MB Exciter
Modal 2) or acoustic excitation (Alpine SPS-170A). In addition, a microphone (B&K type 4958) is
located above the plate to receive the acoustic radiation pressure from the plate under mechanicalexcitation.

(a) vibration under thermal load[8](b) vibration under static load[20]Figure 2 – Experimental setup

5. RESULTS DISCUSSION
Based on the above theoretical, numerical and experimental approaches, we have studied thevibro-acoustic characteristics of different plates, revealing the influence mechanism of thermal andstatic load.
5.1 Effect of uniform thermal environment
A clamped plate is shown in Fig. 3. Detailed information about the material properties, geometry
dimensions and load cases can be seen in Ref.[6-8, 14].
Fig. 4 gives the first five theoretical natural frequencies of the plate in different thermal
environments before critical buckling temperature. It is shown that all the frequencies decrease withthe temperature increasing. The vibration and acoustic radiation results in Fig. 5 show the same
variation trend with that of the frequencies. The response peaks shift to lower frequency range.
The reason is that in the pre-buckling state, only thermal stress appears in the plate and there’s notransverse deformation. Thermal stress results in the softening of bending stiffness and the natural
frequencies decease hereof.
From Fig. 6 we can see that, with the temperature getting higher than critical temperature, theplate is buckled and the buckling deflection gets larger and larger. The stiffening effect of buckling
deformation becomes stronger than the softening effect of thermal stress. T he frequencies gradually
begin to increase. At the same time, the response peaks shift to higher frequency range as shown in
Fig. 7. Moreover, the third natural frequency line crosses the fourth natural frequency line.

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Figure 3 – Scratch of the plate

Figure 4 – Natural frequencies before critical buckling temperature [6] (theoretical results)(a) vibration responses[8](b) acoustic radiation[20]Figure 5 – Vibration and acoustic radiation responses of the plate subjected to point excitation before
critical buckling temperature (theoretical results)(a) before critical buckling temperature(b) after critical buckling temperature
Figure 6 – Center buckling deflection and natural frequencies [14] (experimental results)

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(a) before critical buckling temperature(b) after critical buckling temperature
Figure 7 – Vibration responses of the plate subjected to point excitation before critical buckling
temperature [14] (experimental results)
5.2 Effect of thermal gradient
The vibro-acoustic characteristics analysis of laminated plates under temperature gradients in Fig.
8(a) [17] shows that the deflection gets larger with the temperature gradient increasing. Fig. 8(b) showsthat the resonant frequency increases as the temperature gradient increases. This is because initialdeflections caused by temperature gradient intensify the stiffness of the structure. Accordingly,
resonant frequency is enhanced.

(a) center deflection(b) resonant frequency
Figure 8 – Center deflection and resonant frequency under different temperature gradients [17](theoretical results)
5.3 Effect of static load
Different from the thermal load which leads to compress stress in the plate, the static load may leadto both compress/tensile stress depending on the static load direction and the initial imperfection. In
Ref.[20], the vibro-acoustic responses of three plates with different imperfections are studied. The
plates are shown in Fig. 9. The deflections and natural frequencies shown in Fig. 10 indicate that when
the plate is flat or the directions of the imperfection and the static load are the same, the plate isstiffened and the natural frequencies increase with the static load gets larger. However, when the
directions are the opposite as shown in Fig. 9(c), compress stress appears and the curvature becomes
small at the beginning. The plate is softened and the natural frequencies decrease. Then the compress
stress gradually becomes tensile stress and the curvature turns to increase. The plate is stiffened andthe natural frequencies increase.

Figure 9 – (a) flat plate, (b) plate with geometric imperfection  w
geometric imperfection  w

00

in -z direction, (c) plate within z direction

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(a) center deflections(b) first order(c) second order(b) third orderFigure 10 – Center deflection and first three natural frequencies of the three plates versus static load
(theoretical results)

6. CONCLUSIONSIn this paper, we review the recent works of our group about the vibro-acoustic responses ofplates under thermal/static load. Usually, both the inner stress state and the geometry shape may bechanged under thermal/static load. Under thermal load, compress stress is the dominat ing effect in the
pre-buckling state, resulting in the softening of the plate and the decreasing of the natural frequencies.
While, the effect of buckling deformation is more significant in the post-buckling state and it results instiffening of the plate and increasing of the natural frequencies. The deflection and natural frequencies
increase with the temperature gradient increasing. Different from that under thermal load, bothcompress and tensile stress will appear under static load. It depends on the static load direction and theinitial imperfection. The works reviewed in the paper lay the foundation for the vibro -acoustic analysis
of hypersonic aerocrafts under extreme thermal load and static load.

ACKNOWLEDGEMENTS
This work was supported by the National Natural Science Foundation of China [Grant Numbers
91016008, 91216107, 11772251].

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