Abstract
The dynamic stability of an asymmetric sandwich beam with viscoelastic core resting on a sinusoidal varying Pasternak foundation subjected to parametric vibration is observed. The effects of different parameters such as temperature gradient of each elastic layer, the ratio of modulus of the shear layer of Pasternak foundation to Young’s modulus of the elastic layer, core loss factor, stiffness of Pasternak foundation and elastic foundation parameter on the dynamic stability are investigated. Hamilton’s principle, generalized Galerkin’s method and Hill’s equations are utilized, followed by Saito–Otomi conditions to obtain the results.
Keywords
1 Introduction
The problem of beams on elastic foundations has an important place in modern structural and foundation engineering. Studies were done on asymmetric sandwich beam to reduce the weight of the system by Dash et al. [1], without compromising the stability of the system. Many research works focused on beams’ stability and its vibration when the beams are on springs. The extensive static research work is in Hetenyi’s book [2]. Wang and Stephan [3] studied the natural frequencies for Timoshenko beam and found that shear deformation, rotary inertia and foundation constants for various boundary conditions were affecting it. Kar and Sujata [4] studied the influence of constant temperature gradient on a Timoshenko beam and found that the stability was exaggerated. The temperature effect on Young’s modulus of a nonuniform beam in parametric vibration was illustrated by the same authors [5]. Nayak et al. [6] examined the temperature effect on the sandwich beam, considering two different boundary conditions along with temperature gradient. Pradhan et al. [7] considered the effect of temperature gradient in case of a sandwich beam and observed that the stability was affected by it. It was exposed that the temperature gradient was affecting the stability of a tapered asymmetric sandwich beam by Pradhan et al. [8]. Pradhan et al. [9] studied the effect of the temperature gradient on a sandwich beam, asymmetric in nature, placed on Pasternak foundation and observed that stability was affected due to the stiffness of the foundation and temperature gradient. Ray and Kar [10] also investigated parametric instability of sandwich beams for a number of end conditions. Kerwin [11] observed the damping characteristic of the constrained layer. Then, Rao and Stuhler [12] considered symmetric sandwich beam and studied about loss factor and frequency. Saito and Otomi [13] investigated the parametric response of a beam that is viscoelastically supported. Dash and Nayak [14] determined the profile of an asymmetric rotating sandwich beam which has maximum stability and will be economical.
From the accessible literature, it has been discovered that till now, most of the work has been done for linear and parabolic variation of Pasternak foundation with no work done to study the effect of sinusoidal Pasternak foundation.
So, the present analysis explains the dynamic stability of an asymmetric sandwich beam resting on a sinusoidal varying Pasternak foundation under pinned—pinned and clamped—free conditions at the ends. The objective of using sinusoidal foundation is that any type of deflected shape, whether linear or parabolic, can be expressed in sinusoidal form. The effect of various parameters on the regions of parametric instability is examined by the computational method, and the results are graphically presented.
2 Problem Design
A generalized sandwich beam of length ‘\(l\)’ resting on a sinusoidal Pasternak foundation is shown in Fig. 1. It consists of two elastic layer and middle viscoelastic layer. Here, \(G_{2}\) is the in-phase shear modulus of the viscoelastic core, and \(G_{2}^{*}\) is the complex shear modulus. Hence, \(G_{2}^{*} = G_{2} (1 + j\eta )\) where ‘\(\eta\)’ is the core loss factor of the core and \(j = \sqrt { - 1}\).
Since the system is assumed to be only in the pure bending condition in one plane with a bending plane in the x-z plane, the axial load \(P(t) = P_{0} + P_{1} \cos \omega t\) is applied at the C.G. of the transverse cross section. In the above expression, ‘ω’ is the frequency of the excitation and ‘\(P_{1}\)’ and ‘\(P_{0}\)’ are dynamic and static load amplitudes, and ‘\(t\)’ is time. The system is resting on a foundation, which is made of close and equally placed springs having variable stiffness \(k\left( x \right)\) for the springs, added with a shear layer of modulus of rigidity ‘\(G_{S}\)’.
The assumptions considered to obtain the governing equations of motion are as [7].
According to Kerwin’s [11] assumption, \(E_{1} A_{1} \left( x \right)U_{1,x} + E_{3} A_{3} \left( x \right)U_{3,x} = 0\)
The kinetic energy \(\left( T \right)\), potential energy \(\left( V \right)\) and work done \(\left( {W_{P} } \right)\) expressions are given by
and
The shear strain in the middle layer, \(\gamma_{2} = \frac{{u_{1} - u_{3} }}{{2h_{2} }} - \frac{{cw,_{x} }}{{2h_{2} }}\).
By Kerwin’s assumption [11], \(u_{3}\) is eliminated. The expression for nondimensional equations of motion is obtained by applying the extended Hamilton’s energy principle.
where \(\bar{w},\;_{{\bar{x}\bar{x}\bar{x}\bar{x}}} = \frac{{\partial^{4} \bar{w}}}{{\partial \bar{x}^{4} }},\bar{w},\;_{{\bar{x}\bar{x}}} = \frac{{\partial^{2} \bar{w}}}{{\partial \bar{x}^{2} }},\gamma_2,\;{{\bar{x}\bar{x}\bar{x}}} = \frac{{\partial^{3} \gamma_{2} }}{{\partial \bar{x}^{3} }},\gamma_2,\;{{\bar{x}\bar{x}}} = \frac{{\partial^{2} \gamma_{2} }}{{\partial \bar{x}^{2} }} .\)
The geometric parameter, \(Y = \frac{{E_{1} \left( x \right)A_{1} c^{2} }}{D(1 + \alpha )}\) where \(\alpha = {{(E_{1} A_{1} )} \mathord{\left/ {\vphantom {{(E_{1} A_{1} )} {(E_{3} A_{3} )}}} \right. \kern-0pt} {(E_{3} A_{3} )}}\), \(D = E_{1} \left( x \right)I_{1} + E_{3} \left( x \right)I_{3}\) and \(\overline{m} = 1 + \left( {\frac{{\rho_{2} }}{{\rho_{1} }}} \right)\left( {\frac{{h_{2} }}{{h_{1} }}} \right) + \left( {\frac{{\rho_{3} }}{{\rho_{1} }}} \right)\left( {\frac{{h_{3} }}{{h_{1} }}} \right)\)
Equation (5) can be simplified as
The nondimensional end conditions at \(\bar{x} = 0\) and \(\bar{x} = 1\) are given by
or
or
or
In the above, \(\bar{x} = {x \mathord{\left/ {\vphantom {x l}} \right. \kern-0pt} l}\), \(h_{31} = h_{30} /h_{10}\), \(h_{21} = h_{20} /h_{10}\), \(\bar{P}\left( {\bar{t}} \right) = \bar{P}_{0} + \bar{P}_{1} \cos \left( {\bar{\omega }\bar{t}} \right)\), \(\bar{P}_{0} = {{P_{0} l^{2} } \mathord{\left/ {\vphantom {{P_{0} l^{2} } {\left( {E_{1} \left( x \right)I_{1} + E_{3} \left( x \right)I_{3} } \right)}}} \right. \kern-0pt} {\left( {E_{1} \left( x \right)I_{1} + E_{3} \left( x \right)I_{3} } \right)}}\), \(\bar{P}_{1} = {{P_{1} l^{2} } \mathord{\left/ {\vphantom {{P_{1} l^{2} } {\left( {E_{1} \left( x \right)I_{1} + E_{3} \left( x \right)I_{3} } \right)}}} \right. \kern-0pt} {\left( {E_{1} \left( x \right)I_{1} + E_{3} \left( x \right)I_{3} } \right)}}\), \(\bar{w}_{{,\;\bar{x}}} = \frac{{\partial \bar{w}}}{{\partial \bar{x}}}\) and \(\bar{w}_{{,\;\bar{t}}} = \frac{{\partial \bar{w}}}{{\partial \bar{t}}}\), etc.
\(g^{*} = \frac{{G_{2}^{*} l_{h1}^{2} \left( {1 + E_{31} h_{31} } \right)}}{{4E_{3} \left( x \right)h_{21} h_{31} }}\) is the complex shear parameter and also \(g^{*} = g(1 + j\eta )\)
2.1 Approximate Series of Solution
For Eqs. (4) and (6), the series of approximate solutions are assumed in the form
Here, the shape functions are \(w_{i}\) and \(\gamma_{k}\) . \(f_{i}\) and \(f_{k}\) are generalized coordinates. \(w_{i}\) and \(\gamma_{k}\) are chosen so that they satisfy the equations for motion and maximum number of possible end conditions [12]. The shape functions given in Ray and Kar [10] used for the following end conditions.
-
1.
Pinned-Pinned (P–P): \(w_{i} \left( {\bar{x}} \right) = \sin \left( {i\pi \bar{x}} \right)\), \(\gamma_{k} \left( {\overline{x} } \right) = \cos \left( {k\pi \overline{x} } \right)\)
-
2.
Clamped-Free (C–F): \(w_{i} \left( {\bar{x}} \right) = (i + 2)(i + 3)\bar{x}^{i + 1} - 2i(i + 3)\bar{x}^{i + 2} + i(i + 1)\bar{x}^{i + 3}\) \(\gamma_{k} \left( {\bar{x}} \right) = \bar{x}^{{\bar{k}}} - [\bar{k}/(\bar{k} + 1)]\bar{x}^{{\bar{k} + 1}}\)
where \(i = 1,2,3, \ldots p, \, \overline{k} = p + 1,p + 2, \ldots 2p .\)
Substituting the above-mentioned shape functions in Eqs. (4) and (6), the following equations of motion in matrix form are obtained by means of Galerkin’s method.
where
The various matrix elements are
In the above, \(u_{k} = \frac{{2h_{2} }}{c}\gamma_{k}\), \(u_{l} = \frac{{2h_{2} }}{c}\gamma_{l}\) and \(w_{i} ' = \frac{{\partial w_{i} }}{\partial x}\), \(\lambda_{s} = \left( {\frac{kl}{{E_{1} }}} \right)\), \(\phi = \frac{{3\lambda_{s} l_{{h_{1} }}^{3} }}{{2\left( {1 + E_{31} h_{31}^{3} } \right)}}\),
Equations (15) and (16) further simplified to
where
and
2.2 Instability Regions
Let for \(\left[ m \right]^{ - 1} \left\{ k \right\}\),\(\left[ L \right]\) be a modal matrix. Introducing the linear transformation \(\left\{ {Q_{1} } \right\} = \left[ L \right]\left\{ u \right\}\), where a set of new generalized coordinates is \(\left\{ u \right\}\) and a set of Hill’s equation with complex coefficient is obtained from (24).
where the distinct eigenvalues for \(\left[ m \right]^{ - 1} \left\{ k \right\}\) are \(\omega_{N}^{*2}\) and are given by \(\varepsilon = \frac{{\bar{P}_{1} }}{2} < 1\)
Equation (28) can be rewritten as
where \(N\) varies from \(1,2 \ldots p\); \(b_{NM}\) are the elements of \([B]\); the complex quantities \(\omega_{N}^{*}\) and \(b_{NM}\) are given by
The main and combination resonances are obtained by conditions of Saito and Otomi [13].
3 Discussion and Results
In this case, the stiffness of the springs is of sinusoidal type and is given as \(K\left( {\overline{x} } \right) = K_{0} \left( {1 - \gamma_{e} \sin \left( {\pi \xi } \right)} \right)\) where \(\gamma_{e}\) denote the foundation parameter and \(K_{0}\) is the constant spring stiffness. Unless stated, the following parameter values have been considered for sandwich beam.
The temperature referring to reference temperature, anywhere \(\xi\), is \(\psi = \psi_{0} \left( {1 - \xi } \right)\). Choosing \(\psi = \psi_{0}\) as the reference temperature, the variation of Young’s modulus of the beam, \(E\left( \xi \right) = E\left[ {1 - \lambda \psi_{1} \left( {1 - \xi } \right)} \right] = E_{1} T\left( \xi \right)\), \(0 \le \lambda \psi_{1}\) < 1
where \(\lambda\) is the coefficient of thermal expansion of the beam material, \(\delta = \lambda \psi_{1}\) and \(T\left( \xi \right) = \left[ {1 - \delta \left( {1 - \xi } \right)} \right]\). Here,
where \(\delta_{1}\) and \(\delta_{2}\) are thermal gradient in the top and bottom elastic layer, respectively (Figs. 2, 3 and 4).
The increase of \(\delta_{2}\) and decrease of \(\delta_{1}\) shift the instability zones away from the origin as in Figs. 2 and 3. Increase in \(\delta_{2}\) and decrease in \(\delta_{1}\) improve the rigidity of the system. Increase of \(\eta\) is due to the increase of intermolecular friction. This increase results in loss of energy possessed by the system which ultimately makes the system more stable. This nature has been established in Fig. 4 . The increase of \(K_{0}\)(constant spring stiffness) and decrease of \(\gamma_{e}\) are both responsible for the decrease in maximum deflection of the system. Because of this reason, the effect of the increase in \(K_{0}\) and decrease in \(\gamma_{e}\) is making the system to have better stability as in Figs. 5 and 6. All the results are developed with pinned-pinned boundary conditions. To limit the number of figures, the zones of instability for clamped-free conditions are developed for \(\delta_{1}\) only. The nature of graphs for clamped-free boundary condition is the same as that of pinned-pinned boundary condition. However, from all the above figures, it can be inferred that sandwich beams with pinned-pinned boundary conditions will have greater stability as compared to clamped-free boundary conditions (Fig. 7).
4 Conclusion
Two types of boundary conditions have been studied, and these are pinned-pinned and clamped-free. It is found that increase of \(\delta_{2}\), \(K_{0}\), \(\eta\) and decrease of \(\gamma_{e}\), \(\delta_{1}\) is responsible for improvising dynamic stability for the above two boundary conditions. In all the cases, the systems with the pinned-pinned condition in comparison with the clamped-free condition are found to have better stability. This is obvious because the earlier one is more rigid than the later one.
Abbreviations
- \(A_{i} (i = 1,2,3)\) :
-
Cross-sectional Area of ith layer
- \(B\) :
-
Beam width
- \(E_{i} (i = 1,3)\) :
-
Young’s modulus of ith elastic layer
- \(g\) :
-
Shear parameter
- \(G_{s}\) :
-
Foundation’s shear layer modulus
- \(G_{2}^{*}\) :
-
Complex shear modulus of core
- \(h_{i} (i = 1,2,3)\) :
-
Ith layer’s thickness at ‘x’
- \(I_{i} (i = 1,2,3)\) :
-
Second moment of inertia about relevant axis
- \(l\) :
-
Length of beam
- \(l_{h1}\) :
-
\(l/h_{10}\)
- \(d\) :
-
Shear layer thickness of foundation
- \(m\) :
-
Mass per unit length of beam
- \(\rho_{i}\) :
-
Ith layer’s density
- \(\overline{\omega }\) :
-
Nondimensional forcing frequency
- \(\delta_{i} (i = 1,3)\) :
-
Constant temperature gradient of ith layer
- \(t\) :
-
Time
- \(\overline{t}\) :
-
Nondimensional time
- \(w(x,t)\) :
-
Lateral deflection of beam at ‘x’
- \(p\) :
-
Number of functions
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Nayak, D.K., Pradhan, M., Jena, P.K., Dash, P. (2020). Dynamic Stability Analysis of an Asymmetric Sandwich Beam on a Sinusoidal Pasternak Foundation. In: Deepak, B., Parhi, D., Jena, P. (eds) Innovative Product Design and Intelligent Manufacturing Systems. Lecture Notes in Mechanical Engineering. Springer, Singapore. https://doi.org/10.1007/978-981-15-2696-1_10
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