Abstract

This paper presents an analytical method and provides design guidelines for reducing vibrations in fully covered constrained three-layer beams. The analysis is based on the previous well-established studies carried out to investigate the passive vibration control of structures using constrained layer damping treatment. The main objective of this paper is to present guidance for accurate and appropriate practice in the design of a damped cantilever beam using constrained viscoelastic layers subjected to a harmonic force at its free end. The analysis provides the frequency-reacceptance responses, modal natural frequencies, and loss factors. The provided results depict this information for the first three resonance conditions for a wide range of geometries and material properties. The accuracy of the computed results is validated by comparing them with other published results.

Introduction

While designing structural members subjected to dynamic load, it is essential to consider the possibility of vibration and its dissipation. Excessive vibration brings about unpleasant working conditions, such as noise pollution, fatigue failure, and undesirable displacement. It should be noted that the elimination of vibratory forces is not usually possible, and the total balancing of all the moving parts in the machinery is not simple to achieve. Particularly in high-speed rotating machinery, a very small amount of in-balance mass may cause a large vibration to the structure. Although the application of isolators may reduce the transmitted vibration to the supporting structure, nevertheless, it cannot fully eliminate the vibration. Moreover, adding an isolator can cause additional new resonant conditions in the system. It has been proven that an effective practical method for vibration energy dissipation in structural components is utilizing passive damping. This can be done by employing structural members with large internal damping and relatively low strength. These rubber-like materials dissipate energy into heat when subjected to alternating stresses and are often called viscoelastic.

The energy dissipation can be performed either by the free viscoelastic layer or the constrained viscoelastic layer. In the free layer form, one surface of the viscoelastic layer is bonded to a high-strength structural material, and the other surface is free. In this case, when the flexural deformation of the structural member takes place, energy is dissipated by the extension and the compression of the viscoelastic layer. In the case of the constrained layer, the viscoelastic layer is sandwiched between two high-strength materials. In this case, when flexural deformation of the member occurs, energy is dissipated primarily by the deformation of the viscoelastic layer in shear. Theoretical and experimental analyses have proven that the constrained layer type of composite construction is generally more effective than the free layer [1,2]. The application of the constrained layer damping treatment introduces considerable damping to the system. Also, it modifies the stiffness and mass of the system. Consequently, it changes the natural frequencies of the system. Thus, the analysis and predictions of the dynamic response of the constrained layered structures are essential.

Over the years, numerous kinds of research have been conducted on passive vibration control of structures using constrained layer damping treatment. Kerwin [13] was the foremost investigator who studied the constrained layer damping. He considered the propagation of bending waves in a composite bar and their attenuation by employing a complex bending stiffness in his analysis. Ross et al. [4] considered the three-layer sandwich beam, assuming that either or both of the added layers were dissipative, and developed a model to determine the loss factor for the composite beam. Parfitt et al. [5] and Yin et al. [6] conducted experimental investigations on the damping treatment of beams with different configurations. DiTaranto [7] analyzed the flexural vibrations of three-layered sandwich beams for different conditions by assuming no slipping at the interfaces. DiTaranto was the first investigator to formulate static and dynamic bending equations for viscoelastic sandwich beams. He used some basic assumptions set forth by Kerwin to develop a sixth-order, complex, homogeneous differential equation for longitudinal displacements. Using the following assumptions, he was able to determine the natural frequencies and damping loss factors for viscoelastic sandwich beams:

  1. The intersection of the neutral axis with the given cross-sectional area varies with exciting frequency.

  2. There is no slipping between the elastic and viscoelastic layers at their interfaces.

  3. The major part of the damping is due to the shear deformation of the viscoelastic material.

  4. The elastic layers have the same lateral displacements.

  5. The boundary conditions of the three-layered beam are simply supported, or the beam is infinitely long (boundary conditions are ignored).

  6. The acting normal forces in the viscoelastic layer are negligible.

  7. Only the transverse inertia of the sandwich beam is considered, and the rotatory and longitudinal inertia are ignored.

  8. Shear deformations of the facing plates are neglected.

DiTaranto and Blasingame [810] were able to compute natural frequencies and modal loss factors for three-layer beams with various beam geometries and material loss factors. Mead and Markus [11] derived a six-order partial differential equation for determining the lateral forced vibrations of the three-layer sandwich beams. They also concluded that the DiTaranto [7] sixth-order differential equation should not be used for free vibration analysis. Mead and Markus [11] also utilized the following three assumptions:

  1. The viscoelastic layer bends in the transverse direction exactly as the base layer.

  2. The viscoelastic layer undergoes pure shear deformation.

  3. The thickness of the viscoelastic layer does not change during deformation.

They concluded that for non-simply supported boundary conditions, the system produces modes of vibrations that may or may not be in phase with one another. They suggested using complex variables to describe the modes of vibration.

Lu and Douglas [12] compared the harmonic response of Mead and Markus [11] with their experimental results and observed good agreement for low-frequency responses but lost accuracy at higher frequencies. Nakra and Grootenhuis [13] investigated the effect of extensional deformations of the s on the viscoelastic layers damping characteristics of unsymmetrical multi-layer beams.

Rao and Nakra [14] considered a sandwich beam model with two outer relatively stiff face layers compared with the constrained layer. Their model used Euler-Bernoulli beam assumptions for the face-plate layers and the Timoshenko beam model for the core layer. They also assumed no displacement slip along the interface. Nakra [15] provided a comprehensive historical background related to constrained layer damping treatment. Based on the previous work of Kerwin [13] and other investigators, Torvik [16] presented the fundamental principles of constrained three-layer beams and listed a broad survey of the previous contributions by other researchers. Moreover, he developed a six-order partial differential equation as a governing equation for analyzing this kind of damping treatment for generalized boundary conditions. Usually, in applications of the three-layered sandwich beams, a soft core is used between the two stiff elastic layers. In addition to the soft foam cores, other types of core layers, such as lattice cores, corrugated cores, or honeycomb cores are also considered by Feng et al. [17] and Ashby [18,19]. Inclusive literature surveys on composite and sandwich beams subjected to bending, buckling, and free vibrations are reported by Ghugal and Shimpi [20] and Sayyad and Ghugal [21]. The flexural vibration of three-layered sandwich beams with a viscoelastic core layer has also been analyzed using the finite element method. A full survey of these approaches is provided by Hamidieh [22] and Schoeftner [23].

The main objective of this work is to implement the Torvik [16] approach and analytically present the vibration response of a three-layered sandwich cantilever beam with a viscoelastic core layer subjected to a harmonic force at its free end. Also, to determine modal information such as resonant frequencies, resonant amplitude, and modal loss factors for the first three modes of vibration. The analysis is conducted for a wide range of material properties and geometries of each layer. The depicted results should provide adequate and appropriate guidelines for the design of constrained layer damping treatment of cantilever beams.

Governing Equations.

The analytical governing equations of motion for the three-layered sandwich beam were developed by Torvik16. In this research work, Torvik's sixth-order differential equation of motion is used in terms of transverse displacement for the harmonic flexural vibrations of a constrained damped beam. The solution to the equation of motion is formulated based on satisfying the boundary conditions of the sandwich beam, the non-dimensional parameters, and ξ is the dimensionless longitudinal distance along the beam).
(1)
where W is the dimensionless transverse displacement, g is the shear parameter, Y is the geometric parameter, and Ω is the dimensionless frequency.

Boundary Conditions

A three-layered continuously sandwich cantilever beam with a core viscoelastic layer shown in Fig. 1 has been considered.

Fig. 1
Three-layer damped sandwich Cantilever beam
Fig. 1
Three-layer damped sandwich Cantilever beam
Close modal

The physical boundary conditions for a fixed-free sandwich beam are described by the following equations:

At the fixed end (ξ=0):

Transverse Displacement = 0
(2)
Rotation (Slope) = 0
(3)
The longitudinal displacement of the face-plate mid-surfaces, u1, and u3, at the fixed end, are restrained due to rigid rivet, which can be expressed by the following equation:
(4)
At the free end (ξ=1): the bending moment = 0
(5)
Considering the shear force of F at the free end, then the non-dimensional representation of the shear force in terms of the other parameters can be expressed by the following equation:
(6)
Considering that the longitudinal displacement of the face-plate mid-surfaces, at the free end, is zero, then this boundary condition can be presented by:
(7)

Analytical Solution.

Equation (1) has the following general solution:
(8)
Substituting the above solution in Eq. (1) represents this sixth-order polynomial equation:
(9)
where S is the root of this characteristic equation. Considering six distinct solutions for Eq. (9), the general solution for Eq. (1) could be expressed by
(10)
To find the sets of An, n = 1–6, the above equation should satisfy all the six boundary conditions mentioned earlier.
(11)
(12)
(13)
(14)
(15)
(16)
Equations (11)(16) can be generalized by the following form:
(17)
The above set of equations is obtained after considering the fixed-free boundary conditions. This set of equations could be solved for any particular mode. Here,
(18)
(19)
(20)
(21)
(22)
(23)
The unknown coefficients are the vector An and the known boundary conditions are represented in a vector Bn with values of [0,0,0,0,1,0]T. The system of equations can be solved to determine the values of An.
(24)

The coefficients An are calculated for different values of frequencies. As a result, the displacement frequency response is obtained from the solution of the system of equations. The resonant frequency and its corresponding receptance were computed for three modes of vibration. Then, modal parameters for the three-layer sandwich beam are calculated using the receptance frequency response data. When the frequency response of the beam for each mode is plotted in Nyquist format, it produces a circle approximately through which the modal parameters are obtainable. Modal parameters are extracted from the resulting circle that can be fitted through the points around the resonance frequency using the circle fitting method. Figure 2 illustrates exemplary frequency response and Nyquist plot for a specific case characterized by the parameters: g = 50, Y = 20, and β=0.1.

Fig. 2
(a) Resonant frequencies and (b) Nyquist plot illustrating the imaginary part versus the real part of the response for three modes
Fig. 2
(a) Resonant frequencies and (b) Nyquist plot illustrating the imaginary part versus the real part of the response for three modes
Close modal

Results

The dimensionless resonant frequencies and their corresponding dimensionless displacement and modal loss factors are computed for the range of shear parameter g (5–100), geometric parameters Y (5–100), and material loss factors β (0.05–0.2). The ranges over which these non-dimensional parameters vary are sufficient to cover most cases that may be encountered in practice for the first three modes of vibration of a three-layered sandwich beam.

Figures 3 and 4 show the range of variation of (a) the dimensionless frequency, (b) modal loss factor, and (c) receptance for the first, second, and third modes of vibration versus shear parameter, g, and versus geometric parameters, Y, considering β = 0.05 and 0.2, respectively. Also, Figs. 58 show the same results for the second and third modes of vibration.

Fig. 3
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the first mode of vibration versus shear parameter g and geometric parameters Y, β = 0.05
Fig. 3
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the first mode of vibration versus shear parameter g and geometric parameters Y, β = 0.05
Close modal
Fig. 4
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the first mode of vibration versus shear parameter g and geometric parameters Y, β = 0.2
Fig. 4
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the first mode of vibration versus shear parameter g and geometric parameters Y, β = 0.2
Close modal
Fig. 5
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the second mode of vibration versus shear parameter g and geometric parameters Y, β = 0.05
Fig. 5
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the second mode of vibration versus shear parameter g and geometric parameters Y, β = 0.05
Close modal
Fig. 6
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the second mode of vibration versus shear parameter g and geometric parameters Y, β = 0.2
Fig. 6
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the second mode of vibration versus shear parameter g and geometric parameters Y, β = 0.2
Close modal
Fig. 7
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the third mode of vibration versus shear parameter g and geometric parameters Y, β = 0.05
Fig. 7
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the third mode of vibration versus shear parameter g and geometric parameters Y, β = 0.05
Close modal
Fig. 8
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the third mode of vibration versus shear parameter g and geometric parameters Y, β = 0.2
Fig. 8
(a) Dimensionless frequency, (b) modal loss factor, and (c) receptance for the third mode of vibration versus shear parameter g and geometric parameters Y, β = 0.2
Close modal

The present analysis is verified by comparing it with the works carried out by Soni and Bogner [24], Mace [25], and an experimental report by Drake and Terborg [26]. For comparison, the constrained sandwich beam with a length of 0.1778 m, a width of 1.27 cm, a viscoelastic layer thickness of 0.127 mm, and a main and constraining layer thickness of 1.52 mm were considered. The main and contrasting layers are made of aluminum (modulus of elasticity E = 6.9 × 10 N/m2, density ρ = 2800 Kg/m3, and Poisson's ratio υ = 0.3). The damping layer is 3M-ISD468, which has a mass density of 970 Kg/m3, and Poisson's ratio υ = 0.499 at 17.8 °C. The material is an adhesive viscoelastic, which has temperature and frequency-dependent material properties. Shear modulus and core loss factor are obtained from Mace [25] (ISD468). Table 1 shows the results from those studies and the current study. The presented results indicate satisfactory agreement between the results of the current study and those of other established results.

Table 1

Comparison of frequency and modal loss factor for cantilever sandwich beam for three modes of vibration

ModeFrequency (Hz)Modal Loss Factor
Soni and Bogner [24]Macé [25]Drake and Terborg [26]Current StudySoni and Bogner [24]Macé [25]Drake and Terborg [26]Current study
18277810.00490.00390.0049
25054755105040.01010.00680.0060.0061
313941295140213880.01230.00810.0090.0114
ModeFrequency (Hz)Modal Loss Factor
Soni and Bogner [24]Macé [25]Drake and Terborg [26]Current StudySoni and Bogner [24]Macé [25]Drake and Terborg [26]Current study
18277810.00490.00390.0049
25054755105040.01010.00680.0060.0061
313941295140213880.01230.00810.0090.0114

Conclusion

This work was intended to determine the dynamic response of any passively damped three-layer fully sandwiched cantilever beam excited by harmonic force at its free end. The damping layer was assumed to be viscoelastic with a complex shear modulus. Analysis was conducted by adopting the established governing equation and by satisfying the required boundary conditions. As a result, the frequency response for a wide range of frequencies was achieved, and consequently, the resonant frequencies, as well as the related receptances and modal loss factors, were determined. The most significant achievement of this research work is presenting resonance frequency, related receptance, and the modal loss factor for the first three modes. These results are depicted in terms of two parameters, which represent wide ranges of material properties and the geometries for the three-layer beams that can be used as design guidelines. Moreover, the presented results are well compared with some of the results that are already published.

Conflict of Interest

There are no conflicts of interest. This article does not include research in which human participants were involved. Informed consent not applicable. This article does not include any research in which animal participants were involved.

Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.

Nomenclature

d =

distance between centroids of the constraining layers

g=Gch2(1E1h1+1E3h3)L2 =

shear parameter

x =

longitudinal coordinate

L =

length of sandwich beam

W =

transverse displacement/length of the beam

Y=dDt2(E1h1E3h3E1h1+E3h3) =

geometric parameter:

h1 =

thickness of the main elastic layer

h2 =

thickness of damping material

h3 =

thickness of the constraining layer

Dt=E1h1312+E3h3312 =

uncoupled flexural stiffness

E1 =

elastic modulus of the main layer

E3 =

elastic modulus of constraining segments

GC =

shear modulus of viscoelastic material

β =

loss factor of the viscoelastic material

ηn =

nth modal loss factor

ξ =

non-dimensional longitudinal distance along the beam (x/l)

Ω=L2ωm/Dt =

non-dimensional frequency parameter

ω =

exciting frequency rad/s

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