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Generalized differential quadrature (GDQ) method is used to analyze the vibration of sandwich beams with different boundary conditions. The equations of motion of the sandwich beam are derived using higher-order sandwich panel theory (HSAPT). Seven partial differential equations of motions are obtained through the use of Hamilton’s principle. The GDQ method is utilized to solve the equations of motion. Experiments are conducted to validate the proposed theory. The results from the analytical model are also compared to those from the literature and finite element method (FEM). Parametric studies are conducted to investigate the effects of different parameters on the natural frequency and response of the sandwich beam under various boundary conditions.

Many engineering applications in aerospace, aeronautics, and robotics use multibody systems that consist of flexible appendages containing sandwich panels. These appendages experience large deflections due to vibration excitations. Sandwich panel structures with soft cores made of foam or low strength honeycomb, like aramid or nomex, are used in various industrial applications. They consist of two composite or metallic layers that are separated by a thick and lightweight core. The types of cores in sandwich beam can vary from honeycomb, web core, and balsa wood to foamed polymer. Such configurations in sandwich panels result in high stiffness, light weight characteristics, and high energy absorption capability. These kinds of properties make sandwich structures very useful in aerospace application. Herrmann et al. gave a brief overview on sandwich beam, explaining the importance of sandwich panels in the aerospace industry [

The behavior of sandwich panels can be described using various theories based on the type of core. In the analysis of sandwich panels with very rigid cores, it is a common practice to neglect the transverse deformation of the core. Several authors have investigated these types of sandwich panels. An early theory of sandwich structures, known as the first-order shear deformation theory (FSDT), employs the plate theory by taking the shear rigidity of the core into account but assumes that the longitudinal deformation is linear in the thickness coordinate and that the core is infinitely rigid in the transverse direction. Although this model is simple, the assumptions made are validated by the fact that the sandwich core is statically loaded and stiff in the thickness coordinate.

In reality, the cores of modern sandwich panels are flexible in all directions. Hohe et al. acknowledged this phenomenon by investigating the effect of the transverse compressibility of the core of sandwich beams on the transient dynamic response of structural sandwich panels under rapid loading conditions [

The second model is HSAPT (displacement), which assumes cubic longitudinal displacement and quadratic vertical displacement through the thickness of the sandwich beam. The formulation of this model considers shear strain instead of shear stress. Regardless of axial stress in the core and according to the static equilibrium equation, the second model will consider a constant shear distribution within the core thickness. This type of approximation for sandwich beam construction with a soft core is appropriate for static problems. Experimental investigations have shown that HSAPT accurately predicts displacements and axial strain at the surface, points of support, and regions of concentrated loading. In certain regions of the core, using HSAPT, results are acceptable approximations for shear stress and axial strain values through thickness distributions that are away from points of support and areas of concentrated loading. In contrast, obtaining the same values in regions of the core adjacent to concentrated loading and points of support using HSAPT yields inaccurate approximations of the same values mentioned previously. Since axial stress through the length of the core in HSAPT is neglected, this theory can be used for the study of composite beams and composite plates with soft cores possessing only lateral stress. Frostig et al. studied the free vibration of a unidirectional sandwich panel, which consists of compressible and incompressible core by using various computational models [

Different methods can be used to solve equations of motion of HSAPT models. The most common methods are dynamic stiffness method (DSM) and GDQ method. Damanpack and Khalili investigated the higher-order free vibration of sandwich beams with flexible cores using DSM [

The GDQ method is adopted in this paper. The objective is to

Figure

Geometry of sandwich beam.

The general assumptions for the derivation of the governing equation of motion are as follows [

All deformations and strains are very small.

The face sheets and the core of the beam are made of isotropic and homogeneous materials.

The sandwich beam is assumed to be symmetric.

Transverse normal strains are negligible in the face sheets.

There is no slippage between layers.

The face sheets are modeled by Euler-Bernoulli beam theory and the core is modeled using 2D elasticity theory.

It should be noted that the core is more soft through thickness in comparison with the face sheets and the normal stress of the core is negligible. The axial and transverse displacements of a point on the middle of the top face sheet are

Here X and Y are displacement fields in x and y directions. By using the compatibility conditions given as

Here

Here,

The kinetic energy is given as

where

The GDQ method is applied in order to discrete the equation of motion and boundary condition along the length of the beam. Applying GDQ method to the equation of motion (see (

The solution of the system is achieved by employing a local version of the well-known GDQ method. With respect to the global form, this local approach considers localized interpolating basis function. The numerical technique in GDQ is able to evaluate the

The coefficient of

The higher-order derivatives are obtained as

There are also different arrangements of grid points which are used in GDQ method. It has been shown in the literature that using equal spacing sample of grid points gives inaccurate results. Therefore, the Chebyshev-Gauss-Lobatto distribution is employed to discretize the spatial domain as follows [

Other grid distributions such as expanded Chebyshev [

The properties of the sandwich beam used for conducting the numerical simulation are listed in Table

Parameters of sandwich beam.

References | | | | | | | | |
---|---|---|---|---|---|---|---|---|

[ | | 22.1 | | 60 | 1.9 | 34.8 | 59.9 | 260 |

| ||||||||

[ | | 20.0 | | 52.1 | 0.5 | 20.0 | 20.0 | 300 |

| ||||||||

[ | | 12.8 | | 58.5 | 5 | 19 | 20 | 152 |

| ||||||||

Experiment | | 22.1 | | 60 | 1.9 | 15 and 25 | 59.9 | 260 |

To examine the validity of the present formulation, Table

The five low natural frequencies for a simply supported sandwich beam (rad/s).

Mode | Present method (GDQ) | Exact [ | Model (C) [ |
---|---|---|---|

1 | 2048.46 | 2048.41 | 2048.19 |

2 | 5189.73 | 5189.67 | 5183.37 |

3 | 8250.24 | 8250.19 | 8224.06 |

4 | 11225.32 | 11225.27 | 11159.63 |

5 | 14139.22 | 14139.19 | 14009.12 |

The five low natural frequencies for different boundary conditions (Hz).

BCs | Mode | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|---|

C-F | ABAQUS [ | 327.40 | 1079.1 | 2138.0 | 2175.3 | 2216.4 |

GDQ | 329.18 | 1083.22 | 2152.54 | 2201.58 | 2258.15 | |

QEM [ | 329.66 | 1088.43 | 2164.59 | 2183.78 | 2225.72 | |

HQEM (full [M]) [ | 327.82 | 1082.4 | 2153.9 | 2175.9 | 2217.9 | |

| ||||||

C-C | ABAQUS [ | 839.92 | 1956.7 | 2255.6 | 2702.6 | 3477.8 |

GDQ | 844.17 | 1976.58 | 2274.56 | 2709.82 | 3541.27 | |

QEM [ | 848.71 | 1988.13 | 2262.66 | 2725.13 | 3563.96 | |

HQEM (full [M]) [ | 844.03 | 1978.1 | 2257.1 | 2717.5 | 3547.5 | |

| ||||||

S-S | ABAQUS [ | 618.56 | 1349.1 | 1503.3 | 2170.4 | 2358.4 |

GDQ | 619.05 | 1348.25 | 1508.81 | 2212.64 | 2413.09 | |

QEM [ | 622.22 | 1352.86 | 1515.45 | 2176.86 | 2369.77 | |

HQEM (full [M]) [ | 618.91 | 1349.3 | 1508.2 | 2171.2 | 2362.3 | |

| ||||||

F-F | ABAQUS [ | 1290.2 | 1366.9 | 2153.7 | 2169.2 | 2214.6 |

GDQ | 1292.10 | 1366.87 | 2197.87 | 2203.86 | 2245.86 | |

QEM [ | 1299.19 | 1372.39 | 2162.80 | 2176.40 | 2225.19 | |

HQEM (full [M] [ | 1291.8 | 1367.6 | 2154.2 | 2170.1 | 2215.2 | |

| ||||||

S-C | ABAQUS [ | 715.45 | 1714.4 | 2196.4 | 2514.4 | 3117.1 |

GDQ | 717.25 | 1725.64 | 2233.9 | 2541.3 | 3157.28 | |

HQEM (full [M]) [ | 717.18 | 1725.8 | 2197.5 | 2521.2 | 3160.7 |

The first two mode shapes of sandwich beam with cantilever ((a) and (b)), simply-simply ((c) and (d)), and clamped-clamped ((e) and (f)) supported edges.

This section discusses the time and frequency responses of the sandwich beam with various boundary conditions. All the numerical simulations in this section are based on a harmonic applied load

Tip deflection of free edge of cantilever sandwich beam (

The validation of the forced vibration analysis is demonstrated in Figure

The effect of geometric parameters on the time response at the tip of the cantilever sandwich beam.

The effect of core by GDQ method

The effect of core by SolidWorks

The effect of face sheets by GDQ method

The effect of face sheets by SolidWorks

The effect of width by GDQ method

The effect of width by SolidWorks

It is also observed in Figure

Figures

Vibration amplitude at the tip of the cantilever beam with respect to applied frequency.

The effect of core thickness (

The effect of face sheets thickness (

The effect of width

Vibration amplitude at the middle of the simply-simply supported edges sandwich beam with respect to applied frequency.

The effect of core thickness (

The effect of face sheets thickness (

The effect of width

Vibration amplitude at the middle of the clamped-clamped edges sandwich beam with respect to applied frequency.

The effect of core thickness (

The effect of face sheets thickness (

The effect of width

An experiment is conducted to determine the frequency response curve of two cantilever sandwich beams with different core thicknesses. The material properties of the tested sandwich beams are listed in Table

Schematic of experimental setup.

To reduce experimental error, the fixture is designed in SolidWorks such that its fundamental frequency is well above the maximum excitation frequency. A signal generated from the controller (Laser USB, 1.4V) is fed to the power amplifier (LSD SPA 16K) and then to the vibration shaker. The signal is fed back to the controller to ensure precise measurement. Two accelerometers (B & K 8325) are employed for input and output measurements. One is placed at the top of the fixture to measure the input acceleration and the other is placed at the tip of the beam to measure the output response. The test is carried out with a constant velocity of 2 mm/s and a sine sweep is performed for a frequency range of 10 to 2000 Hz. Measurements are sent to the data acquisition system through a signal analyzer. The frequency response plots are obtained and the frequencies corresponding to the peaks of these plots are the natural frequencies of the tested sandwich beams. The experimental results are shown in Figure

The natural frequencies (Hz) for cantilever beam with two different thicknesses.

Modes | | | ||
---|---|---|---|---|

GDQ method | Experiment | GDQ method | Experiment | |

1 | 118.00 | 111.19 | 143.92 | 143.02 |

2 | 381.00 | 364.60 | 452.27 | 475.21 |

3 | 719.84 | 680.80 | 821.86 | 824.33 |

4 | 1141.94 | 1063.50 | 1261.56 | 1323.81 |

5 | 1675.51 | 1652.07 | 1800.17 | 1772.60 |

Vibration amplitude of cantilever sandwich beam for different core thicknesses by experiment.

This paper presents the vibration analysis of sandwich beams with various boundary conditions. The governing equations of motion are derived using Hamilton’s principle. The GDQ method is utilized for solving the problem. SolidWorks simulation and experimental analyses are conducted to assess the validity of the GDQ method and the results show very good agreement. Parametric studies are conducted to examine the role of geometric properties on the time and frequency response curves. The results indicate that vibration amplitude decreases with increasing both core and face sheet thicknesses, whereas natural frequency increases with increasing core sheet thickness but can increase or decrease with varying face sheet thickness. The results also show that vibration amplitude decreases with increasing beam width, whereas no effect is observed on the natural frequency for varying the width of the beam. Numerical examples also demonstrate that all studied boundary conditions exhibit similar effect with varying the aforementioned geometric properties. The findings in this paper are anticipated to be appealing to research communities because of the experimental and FEM validation of the GDQ method for the free and forced vibration analyses of sandwich beams under various boundary conditions.

The rest of the six equations of motion are given as follows:

The axial forces

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

The authors declare that they have no conflicts of interest.