The free vibration and damping characteristics of rotating shaft with passive constrained layer damping (CLD) are studied. The shaft is made of fiber reinforced composite materials. A composite beam theory taking into account transverse shear deformation is employed to model the composite shaft and constraining layer. The equations of motion of composite rotating shaft with CLD are derived by using Hamilton’s principle. The general Galerkin method is applied to obtain the approximate solution of the rotating CLD composite shaft. Numerical results for the rotating CLD composite shaft with simply supported boundary condition are presented; the effects of thickness of constraining layer and viscoelastic damping layers, lamination angle, and rotating speed on the natural frequencies and modal dampings are discussed.
Rotating shafts are used for power transmission in helicopter drive applications. Weight, vibration, strength, and fatigue have been recognized as serious problem in structural design of this device. Use of fiber reinforced composite materials has received great interest in helicopter drive shaft because composite materials have low weight, high stiffness-to-weight ratio, and vibration damping characteristics compared to metals.
Dynamics of rotating composite material shaft have been an important issue in the field of rotor dynamics. Zinberg and Symonds [
In order to suppress large amplitudes of the shaft of passage through resonance, an effective method is required. CLD treatment has been widely used for damping out structural vibration. Its practical application can be found in aerospace, vehicles, submarines, and so forth. The structures with CLD treatment consist of base structure, viscoelastic material (VEM) layer, and constraining layer (CL). The VEM layer is sandwiched between base structure and CL. The VEM undergoes shear strain as the base structure bends by which the vibration energy of the base structure can be dissipated.
The dynamics of beam, plate, and shell structures with CLD have been studied by many authors. Rao [
Studies on the shaft with CLD, especially the rotating composite shaft with CLD, were relatively limited. Napolitano et al. [
In the present work, an analytical model is developed for free vibration and damping capacity analysis of a rotating CLD composite shaft. The base composite shaft and constraining layer are modelled by using a first-order beam theory in conjunction with the constitutive equations of laminated composite based on three-dimensional continua and the strain-displacement relations. The equations of motion of the rotating CLD composite shaft with extension-twist-bending coupling are derived by employing Hamilton’s principle. The approximate solutions of the shaft system are obtained using the general Galerkin method. Numerical results for the rotating CLD composite shaft with simply supported boundary condition are presented, and the effects of thickness of constraining layer and viscoelastic damping layers, lamination angle, and rotation on the natural frequencies and modal dampings are discussed.
The following assumptions will be adopted for the structural modeling of the constrained layer damping shaft as shown in Figure The base composite shaft and constraining layer satisfy Timoshenko beam assumption. The shaft, constraining layer, and viscoelastic damping layer have the same transverse displacements Only shear strains are considered for the viscoelastic damping layer.
Geometric configuration of the composite shaft with a constrained damping layer.
The constitutive relation for a lamina with respect to the material coordinated system
The constitutive relations for an arbitrary lamina in the base composite shaft and the constraining layer with respect to the cylindrical coordinated system (
In beam theories,
The Timoshenko shear correction factor
The constitutive relations in the viscoelastic layer are
The strain-displacement relations of the viscoelastic layer can be written as
Kinetic energy of the base composite shaft, constraining layer, and viscoelastic layer is
Strain energy of the base composite shaft, constraining layer, and viscoelastic layer is
Taking the variation of (
Define the stress resultants
Using the expressions given in (
Substituting (
Taking the variation of (
The stress resultants of the viscoelastic layer have the following form:
Using (
In order to derive the equations of the motion of the constrained layer damping shaft, the following Hamilton principle can be used:
Substituting (
Substituting (
In present study emphasis is placed on only the problem involving the bending-transverse shear coupling.
The motion equations of bending-transverse shear coupling in terms of displacements can be shown in the form as
The shaft is assumed to be the simply supported boundary conditions for which there are
In order to find the approximate solution of the rotating composite shaft, the quantities
Note that the mode shape functions
From (
In the first example, we investigate the composite shaft considered by Zinberg and Symmonds. The material properties of each ply are
Comparison of critical speed (rpm) for composite shaft.
Reference [ |
Reference [ |
Reference [ |
Reference [ |
Reference [ |
Reference [ |
Reference [ |
Present |
---|---|---|---|---|---|---|---|
5780 | 5500 | 4942 | 5762 | 5620 | 5555 | 5435 | 5619.6 |
Next, a thin tube made from a single layer which is considered by Singh and Gupta [
Comparison of natural frequencies (Hz) for composite thin-walled tube.
|
1st | 2nd | 1st, EMBT [ |
2nd, EMBT [ |
1st, LBT [ |
2nd, LBT [ |
---|---|---|---|---|---|---|
0 | 426.81 | 1224.92 | 426.92 | 1225 | 426.92 | 1224.9 |
15 | 429.84 | 1404.43 | 429.95 | 1404.5 | 429.95 | 1404.5 |
30 | 324.99 | 1203.21 | 325.11 | 1203.3 | 325.12 | 1203.3 |
45 | 216.52 | 832.52 | 216.62 | 832.61 | 216.62 | 832.61 |
60 | 164.95 | 638.56 | 165.17 | 638.65 | 165.18 | 638.66 |
75 | 145.69 | 558.88 | 145.79 | 558.95 | 145.8 | 558.95 |
90 | 139.94 | 526.95 | 140.13 | 527.04 | 140.13 | 527.04 |
In the following example, the geometry and material properties of the composite shaft are as follows:
Figures
The first three natural frequencies versus fiber angle for various thicknesses of constraining layer (
First mode
Second mode
Third mode
Figures
The first three dampings versus fiber angle for various thicknesses of constraining layer (
First mode
Second mode
Third mode
Figures
The first three natural frequencies versus fiber angle for various thicknesses of viscoelastic damping layer (
First mode
Second mode
Third mode
Figures
The first three dampings versus fiber angle for various thicknesses of viscoelastic damping layer (
First mode
Second mode
Third mode
Figures
The first three natural frequencies versus rotating speed for various thicknesses of constraining layer (
First mode
Second mode
Third mode
From these figures it can be seen that as
Figures
The first three dampings versus rotating speed for various thicknesses of constraining layer (
First mode
Second mode
Third mode
As shown in these figures, the thicknesses of constraining layer have a clear influence on the modal dampings. The variation trends of the modal dampings with thicknesses of constraining layer are similar to that presented in Figures
The effects of the thickness of viscoelastic damping layer on the curves of natural frequencies and modal dampings are presented in Figures
The first three natural frequencies versus rotating speed for various thicknesses of viscoelastic damping layer (
First mode
Second mode
Third mode
The first three modal dampings versus rotating speed for various thicknesses of viscoelastic damping layer (
First mode
Second mode
Third mode
Figures
The critical speed versus fiber angle for various thicknesses of constraining layer (
The critical speed versus fiber angle for various thicknesses of viscoelastic damping layer (
Based on first-order shear deformation theory of composite beam in conjunction with Hamilton’s principle and general Galerkin’s method, the free vibration equations of the composite rotating shaft with CLD treatment are derived and used to predict the free vibration and damping capacity of the rotating CLD composite shafts. The present numerical result is compatible with those available in the literature. The frequencies and modal dampings are obtained using the present model. From the present analysis and the numerical results, the following main conclusions were drawn: The natural frequencies and modal dampings of the rotating CLD composite shafts increase with the thickness of constraining layer. As the thickness of viscoelastic damping layer increases, the natural frequencies decrease and the modal dampings increase. The modal dampings are more likely to be affected by constraining layer in a certain variation range of The modal dampings have same branching phenomenon as in the natural frequencies due to rotation effect. The effect of rotation on the modal dampings is insignificant for thin viscoelastic damping layer or thin constraining layer.
The mass, gyroscopic, and stiffness matrices in (
Matrix coefficients
The authors declare that they have no competing interests.
The research is funded by the National Natural Science Foundation of China (Grant no. 11272190) and Shandong Provincial Natural Science Foundation of China (Grant no. ZR2011EEM031).