Analysis of Static Instability of an Asymmetric, Rotating Sand-Wich Beam

Abstract

The static stability of an asymmetric, rotating sandwich beam subjected to an axial pulsating load has been investigated for pinned-pinned and fixed-free boundary conditions. The equations of motion and associated boundary conditions have been obtained by using the Hamilton's energy principle. Then, these equations of motion and the associated boundary conditions have been nondimensionalised. A set of Hill's equations are obtained from the nondimensional equations of motion by the application of the general Galerkin method. The static buckling loads have been obtained from Hill's equations. The influences of geometric parameters and rotation parameters on the nondimensional static buckling loads have been investigated.

1. Introduction

The dynamic behavior of rotating beams is of great practical interest in the design of steam and gas turbine blades and helicopter blades. Sandwich constructions with high strength facings and a light weight core have been very popular in aerospace applications. Typical sandwich members used varied from structural panels in aircraft to the helicopter rotor blades. In general, high modulus and light weight characteristics of the sandwich construction normally have great advantages of high movability, power saving, and high strength in robotics applications. Extensive publications have been available concerning the design and analysis of sandwich structures.

Bhat [1] studied the natural frequencies and mode shapes of a rotating uniform cantilever beam with tip mass for different values of rotational speed, hub radius, and setting angle using beam characteristic orthogonal polynomials in the Rayleigh-Ritz method. Liu and Yeh [2] presented the natural frequencies of a nonuniform rotating beam with a restrained base for various values of restraint parameters and rotational speed. Unger and Brull [3] made an analytical investigation to determine the simple and combination resonance regions of shafts mounted in a long and short bearings subjected to pulsating torque applied at the ends and found that the most common and dangerous parametric instabilities arose as a result of combination resonance. Kammer and Schlack [4] obtained the boundaries between the stable and unstable regions of an Euler beam rotating with an angular velocity which has a small periodic component. Sri Namachchivaya [5] derived explicit stability conditions for a rotating shaft under parametric excitation comprising a combination of harmonic terms and stationary stochastic processes using the Routh-Hurwitz criterion and found that the addition of nonwhite noise excitation has a stabilizing effect on harmonically excited shafts. The vibration behavior of a rotating beam, oriented perpendicular to the axis of spin, was investigated by Bauer and Eidel [6]. It was observed that the speed of rotation has a very pronounced influence of the rotating beam and an increase in the speed of rotation may increase or decrease the natural frequencies depending on the boundary conditions. Abbas and Thomas [7, 8] studied the effect of rotational speed and root flexibilities on the first-order simple resonance zones of a rotating Timoshenko’s beam by using the finite element method. Bauchau and Hong [9] utilized the same method to analyze the effect of viscous damping on the response and stability of parametrically excited beams undergoing large deflections and rotations. Dynamic stability of an ordinary rotating beam with various boundary conditions was studied by Kar and Sujata [10]. The same authors [11] also studied the stability of a rotating, pretwisted and preconed cantilever beam. Parametric instability of a rotating pretwisted beam subjected to sinusoidal compressive axial loads was addressed by Tan et al. [12].

As it has been pointed out by many investigators the shear deformation of the core plays an important role in the flexural and dynamic behavior of a sandwich beam; therefore, the flexural rigidity in the core and shear deformation of the facings were neglected in many analysis. The parametric instability of a cantilever beam with magnetic field and periodic axial load has been studied by Pratiher and Dwivedy [13]. The results obtained from perturbation analysis have been verified by solving the temporal equation of motion using fourth-order Runge-Kutta method. Dwivedy et al. [14] have investigated the parametric instability regions for simple supported, clamp-guided, clamp-free riveted, and clamped-free end condition by modified Hsu’s method and found that zones of instability are affected by the shear parameter, core-loss factor and the ratio of core thickness to skin thickness. Zheng et al. [15] investigated the instability of a moving load system with the load approximated as a single-axle mass-spring-damper system. The author showed that instability occurs for a lower mass as compression axial force increases. Metrikine and Verichev [16] introduced two degrees of freedom load and approximated the rail as a Timoshenko beam. They have observed that under these approximations, larger mass of the load lowers the velocity at which instability is observed. The same authors [17] have considered the effect of periodic variations of the foundation stiffness on instability. The parametric instability of viscoelastically supported asymmetric sandwich beam have been studied by Ghosh et al. [18] and found that the viscoelastic constraints improve the stability of the system. The dynamic stability of a rotating sandwich beam using the finite element method has been studied by Lien and Chen [19, 20]. The parametric instability of a rotating asymmetric sandwich beam with viscoelastic core and subjected to a pulsating axial load has been studied by Dash et al. [21].

The present work deals with the static stability of a rotating asymmetric sandwich beam subjected to axial pulsating load, since it has not been studied till now. The equations of motion for transverse vibrations of the beam are obtained using Hamilton’s principle. The general Galerkin method is used to reduce the nondimensional equations of motion to a set of coupled Hill’s equations with complex coefficients. The static buckling loads are obtained from Hill’s equations [22]. The effect of rotation parameters and geometric parameters on the nondimensional static buckling loads is investigated for the pinned-pinned and fixed-free boundary conditions.

2. Formulation of the Problem

2.1. System Configuration

A viscoelastic sandwich beam of length 𝑙 and width 𝐵, set off at a distance 𝑏 from the axis of rotation (𝑧^′-axis) and oriented along the 𝑥-axis, perpendicular to the axis of rotation is shown in Figure 1. The beam rotates about the vertical 𝑧^′-axis at a constant angular velocity Ω₀ and is capable of oscillating in the 𝑥-𝑧 plane. It is asymmetric with respect to the 𝑥-𝑦 plane as layer 1 is not same as layer 3 both geometrically and materially. Note that some authors might call this a “rotating column” configuration. However, it is also customary in the literature to call it a “rotating beam”. This is followed throughout the paper.

Figure 1 

System configuration.

The top layer of the beam is made of an elastic material of thickness 2ℎ₁ and Young’s modulus 𝐸₁, and the bottom layer is made of an elastic material of thickness 2ℎ₃ and Young’s modulus 𝐸₃. The core is made of a linearly viscoelastic material with a shear modulus 𝐺^∗₂=𝐺₂(1+𝑗𝜂), where 𝐺₂ is the in-phase shear modulus, 𝜂 is the core loss factor and 𝑗=√−1.

A pulsating axial load 𝑃(𝑡)=𝑃₀+𝑃₁cos(𝜔𝑡) is applied at the end 𝑥^′=𝑏+𝑙 of the beam. 𝑃₀ and 𝑃₁ are, respectively, the static and dynamic load amplitudes, 𝜔 is the frequency of the dynamic load, and 𝑡 is time.

2.2. Derivation

The following assumptions are made for deriving the equations of motion.(1)The beam deflection 𝑤(𝑥,𝑡), parallel to 𝑧-axis is small and is the same at all points of a given cross-section.(2)The layers are perfectly bonded so that displacements are continuous across the interfaces. The elastic face layers obey the Euler-Bernoulli beam theory.(3)The allowance for rotary inertia is neglected while calculating the kinetic energy of the system. (4)Shear deformation of the facings are neglected.(5)Damping in the viscoelastic core is predominantly due to shear. Bending and extensional effects in the core are neglected.(6)The Kerwin’s assumption [23] is used.

According to the above assumption, 𝐸₁𝐴₁𝑈_1,𝑥+𝐸₃𝐴₃𝑈_3,𝑥=0

For the system 1𝑇=2𝑚^𝑙₀𝑤²_𝑡1𝑑𝑥,𝑉=2^𝑙₀𝐸₁𝐴₁𝑈²_1,𝑥1𝑑𝑥+2^𝑙₀𝐸₃𝐴₃𝑈²_3,𝑥+1𝑑𝑥2𝐸₁^𝑙₀𝐼₁𝜕²𝑤𝜕𝑥²²1𝑑𝑥+2𝐸₃^𝑙₀𝐼₃𝜕²𝑤𝜕𝑥²²+1𝑑𝑥2𝐺^∗₂^𝑙₀𝐴₂𝜈²₂+1𝑑𝑥2𝑚Ω²₀^𝑙₀(𝑏+𝑥)^𝑥₀𝜕𝑤𝜕𝑥²1𝑑𝑥𝑑𝑥,𝑊=2^𝑙₀𝑃(𝑡)𝑤²_𝑥𝑑𝑥, (1) where𝜈₂=((𝑈₁−𝑈₃−𝑐𝑤_𝑥)/2ℎ₂), which is derived according to Kerwin’s assumption, where 𝑇 is the kinetic energy of the system, 𝑉 is the strain energy of the system and 𝑊 is the work done on the system by the external force.Using Hamilton’s principle, 𝛿^𝑡2_𝑡1(𝑇−𝑉+𝑊)𝑑𝑡=0. (2) The equations of motion and the associated boundary conditions are obtained as follows:𝑤̈+𝐸₁𝐼₁+𝐸₃𝐼₃1𝑚+3Ω²₀2𝐸₁𝐴₁+𝐸₃𝐴₃𝑓−(𝑥+𝑏)²𝑤^′′′′𝐸−3¹𝐼₁+𝐸₃𝐼₃𝐸₁𝐴₁+𝐸₃𝐴₃Ω²₀(𝑥+𝑏)𝑤^′′′+𝐸−3¹𝐼₁+𝐸₃𝐼₃2𝐸₁𝐴₁+𝐸₃𝐴₃Ω²₀−12Ω²₀𝑓−(𝑥+𝑏)²×𝐺^∗₂𝐴₂𝑐²𝑚2ℎ₂²+𝑃(𝑡)𝑚𝑤^′′+12Ω²₀(𝑥+𝑏)𝑤^′+𝐺^∗₂𝐴₂𝑐(1+𝛼)𝑚2ℎ₂²𝑢^′₁𝑢=0_1′′−𝐺^∗₂𝐴₂(1+𝛼)2ℎ₂²𝐸₁𝐴₁+𝛼²𝐸₃𝐴₃(1+𝛼)𝑢₁−𝑐𝑤^′=0. (3)

The boundary conditions at 𝑥=0 and 𝑥=ℓ are 𝑤^′′=0or𝑤^′𝐸=0,₁𝐼₁+𝐸₃𝐼₃𝜕𝜕𝑥1+3𝑚Ω²₀2𝐸₁𝐴₁+𝐸₃𝐴₃𝑓−(𝑥+𝑏)²𝑤^′′−𝑚Ω²₀2𝑓−(𝑥+𝑏)²+𝐺^∗₂𝐴₂𝑐²2ℎ₂²(𝑡+𝐺+𝑃)𝑤′_∗2𝐴₂𝑐(1+𝛼)2ℎ₂²𝑢₁=0, (4) or, 𝑢𝑤=0_′1=0or𝑢₁=0. (5)

Introducing the nondimensional variables 𝑥=𝑥/𝑙,𝑤=𝑤/𝑙,𝑢₁=𝑢₁/𝑙, and 𝑡=𝑡/𝑡₀, where 𝑡₀=𝑚𝑙⁴/(𝐸₁𝐼₁+𝐸₃𝐼₃) and simplifying, the following nondimensional equations of motion are obtained:̈‌𝜆𝑤+1+²⁰1+𝐸₃₁ℎ³₃₁𝑙²_ℎ11+𝐸₃₁ℎ₃₁𝑓𝑙₂−𝑥+𝑏²𝑤^′′′′−2𝜆²₀1+𝐸₃₁ℎ³₃₁𝑙²_ℎ11+𝐸₃₁𝑙₃₁𝑥+𝑏𝑤^′′′+𝜆²₀1+𝐸₃₁ℎ³₃₁𝑙²_ℎ11+𝐸₃₁𝑙₃₁−𝜆²₀𝑓𝑙²−𝑥+𝑏²−3𝑔^∗ℎ1+¹²+ℎ₃₂2²+𝑃𝑡𝑤^′′+𝜆²₀𝑥+𝑏𝑤^′+32𝑔^∗𝑙_ℎ1ℎ₁₂ℎ1+¹²+ℎ₃₂2×(1+𝛼)𝑢^′₁=0,𝑢₁^′′−𝑔^∗4ℎ²₁₂1+𝐸₃₁ℎ³₃₁1+𝛼²𝐸₃₁ℎ₃₁()×)1+𝛼(1+𝛼𝑢₁−21+ℎ₁₂+ℎ₃₂/2𝑙_ℎ1ℎ₁₂𝑤^′=0. (6)

The associated nondimensional boundary conditions at 𝑥=0 and 𝑥=1 are 𝑤^′′=0or𝑤^′𝜕=0,𝜕𝑥1+𝜆₁²𝑓𝑙²−𝑥+𝑏²𝑤^′′−𝜆²₀𝑓𝑙²−𝑥+𝑏²𝑤^′+3𝑔^∗ℎ1+¹²+ℎ₃₂2²+𝑃𝑡𝑤^′+32𝑔^∗𝑙_ℎ1ℎ₁₂()ℎ1+𝛼1+¹²+ℎ₃₂2𝑢₁=0, (7) or𝑤=0,𝑢^′₁=0or𝑢₁=0 (8)

The various parameters. are defined as𝐸₃₁=𝐸₃𝐸₁,ℎ₃₁=ℎ₃ℎ₁,𝑙_ℎ1=𝑙ℎ₁,ℎ₁₂=ℎ₁ℎ₂,𝑔^∗=𝐺=𝑔1+𝑗𝜂_∗2ℎ₂₁𝑙²_ℎ1𝐸₁1+𝐸₃₁ℎ³₃₁, (9) where g being the shear parameter𝜆₀=𝑚Ω²₀𝑙⁴2𝐸₁𝐼₁+𝐸₃𝐼₃,𝜆₁=3𝑚Ω²₀𝑙²2𝐸₁𝐴₁+𝐸₃𝐴₃. (10)

These are rotation parameters𝑃𝑡=𝑃₀+𝑃₁cos𝜔𝑡=𝑃(𝑡)𝑙²𝐸₁𝐼₁+𝐸₃𝐼₃. (11) This is the nondimensional load.

2.3. Approximate Solution

Approximate solution to the nondimensional equations of motion are assumed as 𝑤𝑥,𝑡=^𝑖=𝑁_𝑖=1𝑓_𝑖𝑡𝑤_𝑖(𝑥),𝑢₁𝑥,𝑡=^𝑘=2𝑁_𝑘=𝑁+1𝑓_𝑘𝑡𝑢_1𝑘(𝑥), (12) where 𝑓_𝑟 (𝑟=1,2,…,2𝑁) are the generalized coordinates and 𝑤_𝑖 and 𝑢_1𝑘 are the coordinate functions satisfying as many boundary conditions as possible [24]. For the pinned-pinned case, the shape functions chosen are𝑤_𝑖(𝑥)=sin(𝑖𝜋𝑥),𝑢_1𝑘(𝑥)=cos𝑘𝜋𝑥,𝑘=𝑘−𝑁. (13) For𝑖=1,2,…,𝑁,and𝑘=𝑁+1,𝑁+2,…,2𝑁.

For the clamped-free case, the approximating functions are𝑤_𝑖(𝑥)=(𝑖+2)(𝑖+3)𝑥^𝑖+1−2𝑖(𝑖+3)𝑥^𝑖+2+𝑖(𝑖+1)𝑥^𝑖+2,𝑢_1𝑘(𝑥)=𝑥^𝑘−𝑘𝑘+1𝑥^𝑘+1. (14)

Substitution of the series solutions in the nondimensional equations of motion and subsequent application of the general Galerkin method [24] leads to the following matrix equations of motion:[𝑚]𝑓_𝑗̈+𝑘₁₁𝑓_𝑗+𝑘₁₂𝑓_𝑙={0}𝑘₂₂,𝑓_𝑙+𝑘₂₁𝑓_𝑗={0}, (15)(16) where 𝑗=1,2,…,𝑁 and 𝑙=𝑁+1,…,2𝑁. The various matrix elements are given by𝑚_𝑖𝑗=¹₀𝑤_𝑖𝑤_𝑗𝑑𝑘𝑥,_11𝑖𝑗=¹₀1+𝜆₁𝑓𝑙²−𝑥+𝑏²𝑤_𝑖^′′𝑤_𝑗^′′𝑑𝑥+𝜆²₀¹₀𝑓𝑙²−𝑥+𝑏²𝑤^′_𝑖𝑤^′_𝑗𝑑𝑥+3𝑔^∗ℎ1+¹²+ℎ₃₂2²−𝑃𝑡¹₀𝑤^′_𝑖𝑤^′_𝑗𝑑𝑥,𝑘_12𝑗𝑙3=−2𝑔^∗𝑙_ℎ1ℎ₁₂()ℎ1+𝛼1+¹²+ℎ₃₂2×¹₀𝑢_1𝑙𝑤^′_𝑗𝑑𝑥,𝑘_21𝑙𝑖3=−2𝑔^∗𝑙_ℎ1ℎ₁₂()ℎ1+𝛼1+¹²+ℎ₃₂2×¹₀𝑢_1𝑙𝑤^′_𝑖𝑑𝑥,𝑘_22𝑘𝑙=3𝑙²_ℎ11+𝛼²𝐸₃₁ℎ₃₁1+𝐸₃₁ℎ³₃₁¹₀𝑢^′_1𝑘𝑢^′_1𝑙𝑑𝑥+34𝑔^∗𝑙²_ℎ1ℎ²₁₂(1+𝛼)²¹₀𝑢_1𝑘𝑢_1𝑙𝑑𝑥. (17) From (16) 𝑓_𝑙=−𝑘₂₂⁻¹𝑘₂₁𝑓_𝑗, (18) where𝑘₂₁=𝑘₁₂^𝑇. (19) Substitution of above in (15) and subsequent simplification leads to[𝑚]𝑓_𝑗̈+[𝑘]−𝑃₀[𝐻]𝑓_𝑗−𝑃₁cos𝜛𝑡[𝐻]𝑓_𝑗={0}, (20) where[𝑘]=𝑘−𝑘₁₂𝑘₂₂⁻¹𝑘₁₂^𝑇,𝑘_𝑖𝑗=¹₀1+𝜆₁𝑓𝑙²−𝑥+𝑏²𝑤_𝑖^′′𝑤_𝑗^′′𝑑𝑥+𝜆²₀¹₀𝑓𝑙²−𝑥+𝑏²×𝑤^′_𝑖𝑤^′_𝑗𝑑𝑥+3𝑔^∗ℎ1+¹²+ℎ₃₂2²¹₀𝑤^′_𝑖𝑤^′_𝑗𝑑𝐻𝑥,_𝑖𝑗=¹₀𝑤^′_𝑖𝑤^′_𝑗𝑑𝑥. (21)

2.4. Static Buckling Loads

Substitution of 𝑃₁ =0 and {𝑓_𝑗̈}={0} in (20) leads to the eigen value problem [𝑘]⁻¹[𝐻]{𝑓_𝑗}=1/𝑃₀{𝑓_𝑗}. The static buckling loads (𝑃₀)_crit for the first few modes are obtained as the real parts of the reciprocals of the eigen values of [𝑘]⁻¹[𝐻].

3. Numerical Results and Discussion

Numerical results were obtained for various values of the non-dimensional geometric parameters ℎ₃₁,ℎ₁₂,𝑙_ℎ1,𝑏, the modulus ratio 𝐺₂/𝐸₁ and the rotation parameter 𝜆₁. The following values of parameters are used unless otherwise stated: 𝜂=0.05, ℎ₃₁=ℎ₁₂=1.0, 𝑙_ℎ1=40, 𝑏=0.05, 𝐺₂/𝐸₁=0.001, 𝑔=0.1, 𝜆₀=0.05, and 𝜆₁=0.1.

It should be noted that Figures 2(a), 3(a), 4(a), 5(a), 6(a), and 7(a) are for the pinned-pinned case and Figures 2(b), 3(b), 4(b), 5(b), 6(b), and 7(b) are for the clamped-free configuration. Figures from 2(a), 2(b)–7(a), and 7(b) show the variation of the first three nondimensional static buckling loads, (𝑃₀)_crit with ℎ₃₁,ℎ₁₂,𝑙_ℎ1,𝑏,𝐺₂/𝐸₁, and 𝜆₁, respectively. The variation of these loads with 𝑔 and 𝜆₀ are similar to those for 𝐺₂/𝐸₁ and 𝑙_ℎ1, respectively, and are not shown.

(a)

(b)

Figure 2 

(a) Variation static buckling loads with ℎ31 (pinned-pinned). (b) Variation static buckling loads with ℎ31 (clamped-free).

(a)

(b)

Figure 3 

(a) Variation static buckling loads with ℎ12 (pinned-pinned). (b) Variation static buckling loads with ℎ12 (clamped-free).

(a)

(b)

Figure 4 

(a) Variation static buckling loads with 𝑙ℎ1 (pinned-pinned). (b) Variation static buckling loads with 𝑙ℎ1 (clamped-free).

(a)

(b)

Figure 5 

(a) Variation static buckling loads with 𝑏 (pinned-pinned). (b) Variation static buckling loads with 𝑏 (clamped-free).

(a)

(b)

Figure 6 

(a) Variation static buckling loads with 𝐺2/𝐸1 (pinned-pinned). (b) Variation static buckling loads with 𝐺2/𝐸1 (clamped-free).

(a)

(b)

Figure 7 

(a) Variation static buckling loads with 𝜆1 (pinned-pinned). (b) Variation static buckling loads with 𝜆1 (clamped-free).

With an increase in ℎ₃₁ (Figure 2(a)), the buckling loads are seen to decrease monotonically for the pinned-pinned case. For the clamped-free case (Figure 2(b)), these first increase and then decrease. The rate of decrease is maximum for the highest mode for moderate ℎ₃₁ values.

As ℎ₁₂ (Figures 3(a) and 3(b)) is increased,(𝑃₀)_crit for the three modes shows a decreasing trend for low values of ℎ₁₂. It remains fairly constant for moderate values of ℎ₁₂ and increases for higher values.

It is observed that (𝑃₀)_crit increases nonlinearly with 𝑙_ℎ1 for three modes (Figures 4(a) and 4(b)).

The critical loads are almost constant over the entire range of b (Figures 5(a) and 5(b)).

For increasing the 𝐺₂/𝐸₁, Figures 6(a) and 6(b), the (𝑃₀)_crit values are initially constant and rise appreciably for higher values of 𝐺₂/𝐸₁, the rate of increase being higher for the higher modes.

It is seen that (𝑃₀)_crit is almost constant over a wide range of 𝜆₁ (Figures 7(a) and 7(b)) and decrease slightly for the values of 𝜆₁ greater than 0.1.

4. Conclusions

In this paper, the static stability of an asymmetric three-layered rotating sandwich beam with viscoelastic core has been considered. The beam is subjected to an axial pulsating load. Both pinned-pinned and clamped-free configurations have been considered. An increase in ℎ₃₁ is seen to have a detrimental effect on the nondimensional static buckling loads. Hence, a symmetric beam is seen to have better resistance against static buckling. Over a wide range of values, ℎ₁₂ is seen to have little effect upon the buckling loads. However, for small values, it has a detrimental effect, and for large values, it improves the buckling loads. 𝑙_ℎ1,𝐺₂/𝐸₁,𝜆₀, and 𝑔 improve static stability, especially for large values of the parameters. The buckling loads are seen to be almost independent of 𝑏 and 𝜆₁.

Nomenclature

𝐴𝑖(𝑖=1,2,3):	Areas of cross section of a three layered beam, 𝑖=1,2, and 3 for top, middle and bottom layer, respectively,
𝐵:	Width of beam
𝑏:	Distance of nearer end of the beam from the axis of rotation
𝑏:	𝑏/𝑙
𝑐:	ℎ1+2ℎ2+ℎ3
𝐸𝑖(𝑖=1,2,3):	Young’s modulli, 𝑖=1,2,3 for top, middle and bottom layer, respectively,
𝑓𝑗̈:	𝜕2𝑓𝑗/𝜕𝑡2
𝐺2:	In-phase shear modulus of the viscoelastic core
𝐺∗2:	𝐺2(1+𝑗𝜂), complex shear modulus of core
𝑔∗:	𝑔(1+𝑗𝜂), complex shear parameter
𝑔:	Shear parameter
2ℎ𝑖(𝑖=1,2,3):	Thickness of the 𝑖th layer, 𝑖=1,2 and 3 for top, middle and bottom layer, respectively
ℎ12:	ℎ1/ℎ2
ℎ31:	ℎ3/ℎ1
𝐼𝑖(𝑖=1,2,3):	Second moments of area of cross section about a relevant axis, 𝑖=1,2 and 3 for top, middle, and bottom layer
𝑗:	√−1
𝑙:	Beam length
𝑙ℎ1:	𝑙/ℎ1
𝑚:	Mass/unit length of beam
𝑃1:	Nondimensional amplitude for the dynamic loading
(𝑃0)crit:	Nondimensional static critical buckling load
𝑡:	Time
𝑡:	Nondimensional time
𝑢1(𝑥,𝑡),𝑈1(𝑥,𝑡):	Axial displacement at the middle of the top layer of beam
𝑢3(𝑥,𝑡),𝑈3(𝑥,𝑡):	Axial displacement at the middle of the bottom layer of beam
𝑤(𝑥,𝑡):	Transverse deflection of beam
𝑤′:	𝜕𝑤/𝜕𝑥
𝑤′′:	𝜕2𝑤/𝜕𝑥2
𝑤:	𝑤/𝑙
𝑤̈‌:	𝜕2𝑤/𝜕𝑡2
𝑤′′:	𝜕2𝑤/𝜕𝑥2
𝑡0:	𝑚𝑙4/(𝐸1𝐼1+𝐸3𝐼3)
𝑢′1:	𝜕𝑢1/𝜕𝑥
𝑢1′′:	𝜕2𝑢1/𝜕𝑥2
𝑓:	(𝑙+𝑏)2, for beam free at 𝑥=𝑙,=(𝑙3/3)+𝑏2+𝑏𝑙 for all other cases
𝜂:	Core-loss factor
𝜆0:	Rotation parameter
𝜆1:	Rotation parameter
(Φ):	A null matrix
𝜔:	𝜔𝑡0
𝜔:	Frequency of forcing function
𝜔:	Nondimensional forcing frequency
Ω0:	Speeds of rotation
Ω0:	Nondimensional rotation parameters
𝛼:	𝐸1𝐴1/𝐸3𝐴3.