Keywords

2.1 Introduction

This work adds to a series (see Refs. [1,2,3]) on transverse vibrations of layered beams. The main interest is the vibration analysis, both theoretical and numerical, of segmented damped beams. The media are structures with different materials and varying cross-sections, which are layered in cells and may be uniform or not.

The objective is the determination of frequency response functions (FRFs). Timoshenko beam theory is used for a two-segment configuration under harmonic forcing.

Two approaches are used to address the problem. In the first, analytic solutions are derived for the differential equations for each segment. The constants involved are determined using boundary and interface continuity conditions. The response, at a given location, can then be obtained as a function of forcing frequency (FRF). Note that the procedure can become unwieldy for arbitrary spatial variations. In the second, the discrete cell properties are modeled by continuously varying functions, specifically logistic functions. This provides for a system of differential equations with variable coefficients. The system is then solved numerically utilizing MAPLE®Footnote 1 software.

Similar analytical and numerical approaches were applied in Refs. [1,2,3] (Euler-Bernoulli and Timoshenko models). Overall results showed that the numerical method worked very well when compared to the analytical solutions.

A brief literature review is given next.

For vibrations of layered beams, one may refer to the list given in Refs. [1,2,3]. Solids composed by discrete layers are studied in Refs. [4,5,6,7]. Reference [8] provides a review of articles on this subject.

Reference [9] treats damping effects on Timoshenko beams. The numerical analyses allowed for outlining the relevant influences on the dynamic response associated with any singular damping mechanism and the evaluation of the modal critical damping values. In Ref. [10] a generalized Fourier analysis is applied to damped Timoshenko beam equation to evaluate displacements and bending stresses from random loading. Some of the models were found to be adequate for that purpose. In Ref. [11] the equations of motion for a non-uniform damped Timoshenko beam with distributed axial force are used to extract principal modes of vibration, by numerical means and for specific boundary and orthogonality conditions. Reference [12] deals with Timoshenko beam equations with external damping and internal damping terms plus forcing terms. Different boundary conditions such as pinned ends, hinged-sliding ends, and sliding ends are considered. Unboundedness of solutions of boundary value problems is studied, and it is shown that the magnitude of the displacement of the beam grows up to ∞ as t → ∞ under some assumptions on the forcing term. In Ref. [13], the dynamic behavior of nonlocal viscoelastic damped nanobeams is studied. A Kelvin–Voigt viscoelastic model, velocity-dependent external damping, and Timoshenko beam theory are employed to establish the governing equations and boundary conditions for the vibration of nanotubes. Using transfer function methods (TFMs), the natural frequencies and frequency response functions (FRF) are calculated for different boundary conditions. Reference [14] investigates the effects of axial compressive load and internal viscous damping on the free vibration characteristics of Timoshenko beams using the dynamic stiffness formulation and the differential transformation method. A dynamic stiffness method (DSM) is used where the dynamic stiffness matrix of an axially loaded Timoshenko beam with internal viscous damping is constructed to calculate natural frequencies. The numerical approach compares well to the analytical solutions. Parametric studies on vibrations of twisted Timoshenko damped beams are given in Ref. [15]. Using Timoshenko beam theory and Hamilton’s principle, bending–bending vibrations of axially loaded twisted beams with locally distributed Kelvin–Voigt damping are analyzed. A finite element method is used to reduce the equations of motion into linear second-order ordinary differential equations with constant coefficients. It was asserted that the approach led to a better understanding of the variables affecting the beam vibration characteristics.

2.2 Basic Structure

The segmented beam is shown in Fig. 2.1. It is composed of two cells of different materials. E, ρ, and A may vary in a discontinuous manner. The segments are under transverse loads q 1 and q 2 (force per unit length) and viscous damping forces due to translation and rotation of the cross-section during deflection (per unit length, damping coefficients C T1, C R1, and C T2, C R2).

Fig. 2.1
figure 1

Layered beam

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The equations of motion for a transversely loaded, viscously damped Timoshenko beam are given below. (Details for deriving the equations are discussed in Ref. [2]; see also Refs. [11, 16]). A displacement-velocity and rotation-velocity–dependent viscous damping model is used [13]:

$$ {\displaystyle \begin{array}{l}\rho (x)\;A(x)\;\frac{\partial^2w\left(x,t\right)}{\partial {t}^2}+{C}_{\mathrm{T}}(x)\frac{\partial w\left(x,t\right)}{\partial t}-\frac{\partial V}{\partial x}=q\left(x,t\right)\\ {}\rho (x)\;I(x)\frac{\partial^2\theta \left(x,t\right)}{\partial {t}^2}+{C}_{\mathrm{R}}(x)\frac{\partial \theta \left(x,t\right)}{\partial t}+\frac{\partial M}{\partial x}-V=0\end{array}} $$
(2.1)

Using a linearized curvature model, Eq. (2.1) can be re-written as:

$$ {\displaystyle \begin{array}{l}\rho (x)A(x)\;\frac{\partial^2w\left(x,t\right)}{\partial {t}^2}+{C}_{\mathrm{T}}(x)\frac{\partial w\left(x,t\right)}{\partial t}=q\left(x,t\right)+\frac{\partial }{\partial x}\left[ kA(x)G(x)\left(\frac{\partial w\left(x,t\right)}{\partial t}-\theta \left(x,t\right)\right)\right]\\ {}\rho (x)I(x)\frac{\partial^2\theta \left(x,t\right)}{\partial {t}^2}+{C}_{\mathrm{R}}(x)\frac{\partial \theta \left(x,t\right)}{\partial t}=\frac{\partial }{\partial x}\left[E(x)I(x)\frac{\partial \theta \left(x,t\right)}{\partial x}\right]+ kA(x)G(x)\left(\frac{\partial w\left(x,t\right)}{\partial t}-\theta \left(x,t\right)\right)\end{array}} $$
(2.2)

In Eq. (2.2) the shear force is given by V = kA(x)G(x)γ, where k is the “shear coefficient” (see Ref. [17]), which relates the maximum shear stress to the average value in the cross-section of the beam. Also, \( \frac{\partial w}{\partial x}=\theta +\gamma \) and \( M=-\mathrm{EI}\frac{\partial \theta }{\partial x} \).

Approaches for obtaining the steady-state response, due to harmonic forcing, are sought next.

2.3 Solution Approaches

Non-dimensional versions of Eq. (2.2) can be obtained by taking τ = Ω0 t, \( \nu =\frac{\Omega}{\Omega_0} \), \( \xi =\frac{x}{L} \), \( Y=\frac{w}{L} \), E(x) = E 1 f 1(ξ), I(x) = I 1 f 2(ξ), ρ(x) = ρ 1 f 3(ξ), A(x) = A 1 f 4(ξ), and kG(x) = k 1 G 1 f 5(ξ). This leads to:

$$ {\displaystyle \begin{array}{l}{f}_3\left(\xi \right)\! {f}_4\left(\xi \right)\frac{\partial^2Y\left(\xi, \tau \right)}{\partial {\tau}^2}+{C}_1(x)\frac{\partial Y\left(\xi, \tau \right)}{\partial \tau }=Q\left(\xi, \tau \right)+{\alpha}_1\frac{\partial }{\partial \xi}\left({f}_4\left(\xi \right)\!{f}_5\left(\xi \right)\left(\frac{\partial Y\left(\xi, \tau \right)}{\partial \xi }-\theta \left(\xi, \tau \right)\right)\right)\\ {}{f}_3\left(\xi \right)\!{f}_2\left(\xi \right)\frac{\partial^2\theta \left(\xi, \tau \right)}{\partial {\tau}^2}+{C}_2(x)\frac{\partial \theta \left(\xi, \tau \right)}{\partial \tau }={\gamma}_1\frac{\partial }{\partial \xi}\left({f}_1\left(\xi \right){f}_2\left(\xi \right)\frac{\partial \theta \left(\xi, \tau \right)}{\partial \xi}\right)+\left({\alpha}_1{\gamma}_1{f}_4\left(\xi \right)\!{f}_5\left(\xi \right)\right)\left(\frac{\partial Y\left(\xi, \tau \right)}{\partial \xi }-\theta \left(\xi, \tau \right)\right)\end{array}} $$
(2.3)

where \( {\Omega}_0=\sqrt{\frac{E_1{I}_1}{\rho_1{A}_1{L}^4}} \) (reference frequency), \( {C}_1={C}_T\sqrt{\frac{L^4}{\rho_1{A}_1{E}_1{I}_1}} \), \( {C}_2={C}_R\sqrt{\frac{A_1{L}^4}{\rho_1{E}_1{I_1}^3}} \), \( Q\left(\xi, \tau \right)=q\left(x,t\right)\frac{L^3}{E_1{I}_1} \), \( {\alpha}_1=\frac{k_1{G}_1{A}_1{L}^2}{E_1{I}_1} \), and \( {\gamma}_1=\frac{A_1{L}^2}{I_1} \). f i are functions representing the transitions from one cell to another. (For the continuous variation approach, logistic functions will be utilized; details are given below.)

Assuming harmonic forcing with frequency \( \hat{\lambda} \), \( Q\left(\xi, \tau \right)={q}_1\left(\xi \right)\mathit{\exp}\left(\hat{\lambda}\tau \right) \), one can take \( Y\left(\xi, \tau \right)=Z\left(\xi \right)\mathit{\exp}\left(\hat{\lambda}\tau \right) \) and \( \theta \left(\xi, \tau \right)=\phi \left(\xi \right)\mathit{\exp}\left(\hat{\lambda}\tau \right) \). This leads to:

$$ {\displaystyle \begin{array}{l}{f}_3\left(\xi \right)\!{f}_4\left(\xi \right)\;{\hat{\lambda}}^2Z\left(\xi \right)+{C}_1\hat{\lambda}Z\left(\xi \right)-{\alpha}_1\frac{d}{d\xi}\left({f}_4\left(\xi \right)\!{f}_5\left(\xi \right)\left(\frac{d Z\left(\xi \right)}{d\xi}-\phi \left(\xi \right)\right)\right)-{q}_1\left(\xi \right)=0\\ {}{f}_3\left(\xi \right)\!{f}_2\left(\xi \right)\;{\hat{\lambda}}^2\phi \left(\xi \right)+{C}_2\hat{\lambda}\phi \left(\xi \right)-{\gamma}_1\frac{d}{d\xi}\left({f}_1\left(\xi \right)\!{f}_2\left(\xi \right)\frac{d\phi \left(\xi \right)}{d\xi}\right)-\left({\alpha}_1{\gamma}_1{f}_4\left(\xi \right)\!{f}_5\left(\xi \right)\right)\left(\frac{d Z\left(\xi \right)}{d\xi}-\phi \left(\xi \right)\right)=0\end{array}} $$
(2.4)

Taking \( \hat{\lambda}=\left(a+ bI\right) \) and separating real and imaginary parts, after some manipulation, gives:

$$ {\displaystyle \begin{array}{l}\left({f}_3\left(\xi \right)\!{f}_4\left(\xi \right){b}^2+\frac{{C_1}^2}{4{f}_3\left(\xi \right)\!{f}_4\left(\xi \right)}\right)Z\left(\xi \right)+{\alpha}_1\frac{d}{d\xi}\left({f}_4\left(\xi \right)\!{f}_5\left(\xi \right)\left(\frac{d Z\left(\xi \right)}{d\xi}-\phi \left(\xi \right)\right)\right)+{q}_1\left(\xi \right)=0\\ {}\left({f}_3\left(\xi \right)\!{f}_2\left(\xi \right)\!{b}^2+\frac{{C_2}^2}{4{f}_3\left(\xi \right)\!{f}_2\left(\xi \right)}\right)\phi \left(\xi \right)+{\gamma}_1\frac{d}{d\xi}\left({f}_1\left(\xi \right)\!{f}_2\left(\xi \right)\frac{d\phi \left(\xi \right)}{d\xi}\right)+\left({\alpha}_1{\gamma}_1{f}_4\left(\xi \right)\!{f}_5\left(\xi \right)\right)\left(\frac{d Z\left(\xi \right)}{d\xi}-\phi \left(\xi \right)\right)=0\end{array}} $$
(2.5)

The result, Eq. (2.5), is a set of non-homogeneous ordinary differential equations with variable coefficients. Analytic and numerical solutions are discussed next.

2.3.1 Analytical Approach

For constant properties in each segment, Eq. (2.5) can be written as:

$$ {\displaystyle \begin{array}{l}\left({f}_3{f}_4{b}^2+\frac{{C_1}^2}{4{f}_3\!{f}_4}\right)Z\left(\xi \right)+{\alpha}_1{f}_4{f}_5\frac{d}{d\xi}\left(\left(\frac{d Z\left(\xi \right)}{d\xi}-\phi \left(\xi \right)\right)\right)+{q}_1\left(\xi \right)=0\\ {}\left({f}_3{f}_2{b}^2+\frac{{C_2}^2}{4{f}_3\!{f}_2}\right)\phi \left(\xi \right)+{\gamma}_1{f}_1{f}_2\frac{d^2\phi \left(\xi \right)}{d{\xi}^2}+\left({\alpha}_1{\gamma}_1{f}_4{f}_5\right)\left(\frac{d Z\left(\xi \right)}{d\xi}-\phi \left(\xi \right)\right)=0\end{array}} $$
(2.6)

Note that, for the beam segments, the functions f i are constants and can be calculated from:

$$ {\displaystyle \begin{array}{l}{f}_{k,i}=1,\kern0.36em k=1\cdots 5,\kern0.72em i=1\\ {}{f}_{1,i}=\frac{E_i}{E_{i-1}},{f}_{2,i}=\frac{I_i}{I_{i-1}},{f}_{3,i}=\frac{\rho_i}{\rho_{i-1}},{f}_{4,i}=\frac{A_i}{A_{i-1}},\ {f}_{5,i}=\frac{K_i{G}_i}{K_{i-1}{G}_{i-1}},\ i=2\cdots n\end{array}} $$
(2.7)

Equations (2.6) and (2.7) are valid for cases with multiple segments (n).

Solutions to the system of ODEs (2.6), with constant coefficients, involve solutions to the homogeneous equations and particular integrals. Depending on the forcing function q 1(ξ), tracking particular solutions may pose a problem. In this section attention is directed to obtaining solutions for the case of constant spatial forcing. (Non-constant spatial forcing is treated later numerically.)

Here this is done using MAPLE®. The solutions consist of exponential functions containing unknown coefficients, which must be determined based on the set of boundary conditions for the problem. In addition, solutions must match at the interface between the elements. Interface continuity requires matching of displacement, slope, moment, and shear force.

These conditions provide a set of algebraic equations for the unknown coefficients. The FRFs can then be obtained by monitoring solutions, for different values of the forcing frequency b, at a specific point of the beam (here this is taken at the center).

Solutions require that the boundary conditions be defined. Two sets are discussed below.

2.3.2 Numerical Approach

For the numerical approach, a continuous variation model is used. With this model, transitions from one cell to another are modeled via logistic functions. Here these functions, f i, in non-dimensional form are taken to be:

$$ {f}_i\left(\xi \right)=1+\left(\frac{\delta_2-{\delta}_1}{\delta_1}\right)\left(\frac{1}{2}+\frac{1}{2}\tanh \left(K\left(\xi -\frac{1}{2}\right)\right)\right),i=1,2,3,4,5 $$
(2.8)

δ j represents a material property, geometric property, or damping (E, I, ρ, kG, A, or C). K controls the sharpness of the transition from one cell to another in the function. A larger value corresponds to a sharper transition at \( \xi=\frac{1}{2} \).

Substituting Eqs. (2.8) into (2.5) leads to a system of differential equations, with variable coefficients, which may not have analytic solutions. Given the material layout and cross-section variation, i.e., the corresponding logistic functions, a MAPLE® routine can be used to obtain numerical approximations to the FRF of the system. This is done by monitoring the response for different values of the frequency b. Resonances can also be obtained via a forced-motion approach (see Ref. [18]). It consists of using MAPLE®’s two-point boundary value solver to solve a forced-motion problem. A constant value for the forcing function G is assumed and the frequency b is varied. By observing the mid-span deflection of the beam, resonant frequencies can be found on noting where changes in sign occur.

The approaches are illustrated in the following numerical examples.

2.4 Numerical Examples

Consider the beam shown in Fig. 2.1. Here the segment cross-sections are assumed to have the same geometry and to be joined at the center of the beam. The cross-section is taken to be a S 24 X 121 (ASTM A6 – American Standard Beam),Footnote 2 and the following materials are used: aluminum (E 1 = 71 GPa, ρ 1 = 2710 Kg/m3, υ 1 = 0.33, G 1 = 26.69 GPa, k 1 = 0.89) and silicon carbide (E 2 = 210 GPa, ρ 2 = 3100 Kg/m3, υ 2 = 0.16, G 2 = 90.52 GPa, k 2 = 0.87). These values are taken from Refs. [17, 19].

2.4.1 Free/Fixed Boundary Conditions

Consider a free/fixed set. The boundary conditions are as follows (i subscript refers to the segment number).

The moment and shear free end at ξ = 0 gives \( {\left.\frac{d{\phi}_1\left(\xi \right)}{d\xi}\right|}_{x=0}=0 \) and \( {\left.\frac{d{Z}_1\left(\xi \right)}{d\xi}\right|}_{\xi =0}={\phi}_1\left(\xi \right) \). The conditions at the fixed end give: Z 2(ξ) = 0 and ϕ 2(ξ) = 0. Interface continuity conditions are Z 1(ξ) = Z 2(ξ), ξ = 0.5 (displacement continuity), ϕ 1(ξ) = ϕ 2(ξ), ξ = 0.5 (slope continuity), \( \frac{d{\phi}_1\left(\xi \right)}{d\xi}={f}_1{f}_2\frac{d{\phi}_2\left(\xi \right)}{d\xi},\xi =0.5 \) (moment continuity), and \( \left(\frac{d{Z}_1\left(\xi \right)}{d\xi}-{\phi}_1\left(\xi \right)\right)={f}_4{f}_5\left(\frac{d{Z}_2\left(\xi \right)}{d\xi}-{\phi}_2\left(\xi \right)\right),\xi =0.5 \) (shear continuity). q 1(ξ) is set to 1.

Utilizing the analytical approach described above allows for the calculation of the FRF for the system. Setting ξ = 0.50 (beam mid-span), amplitudes can be calculated for different values of the non-dimensional frequency b.

The frequency response function, spanning the first two natural frequencies, for the mid-point of the beam is shown in Fig. 2.2. (The following parameters apply: f 1 = 2.9577, f 2 = 1.0000, f 3 = 1.1439, f 4 = 1.0000, f 5 = 3.3368, C 1 = 10.0000, and C 2 = 10.0000.)

Fig. 2.2
figure 2

FRF for non-homogeneous Timoshenko beam at mid-point: free/fixed

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A comparison between this damped and an undamped version of the same beam is given in Fig. 2.3. Note the shift on the frequencies and changes of amplitude.

Fig. 2.3
figure 3

FRFs for damped and undamped non-homogeneous Timoshenko beam: free/fixed

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For the continuous variation model and using the numerical values given above, the continuously varying functions are shown in Fig. 2.4 (note: K = 500).

Fig. 2.4
figure 4

Relative property variation for two-cell Timoshenko beam

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Assuming a value of 1 for the external forcing q 1(ξ) and using the forced-motion approach [18], the resultant deflections are plotted bellow for two distinct values of the frequency b.

The resonance frequency is taken to occur at b = 1.55, as seen in Fig. 2.5.

Fig. 2.5
figure 5

Beam deflections for distinct values of b: free/fixed – first resonance

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Fig. 2.6
figure 6

Results comparison: numerical and analytical approaches – free/fixed

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Amplitudes for the response at the center of the beam can be monitored from Eq. (2.5). The approach leads to the numerical FRF shown in Fig. 2.6. The figure shows the results from the numerical simulation and an overlap of those with the analytical results. Excellent agreement is seen; the first two resonances and amplitude values correspond very well.

From the numerical FRF, the damping ratio of the system, corresponding to the assumed non-dimensional values of damping (C 1 = 10.0000, C 2 = 10.0000), can be estimated. The method used here is the half-power bandwidth [20], applied to the first mode, which, although only applicable to lightly damped single degree of freedom systems, is frequently applied to well-separated modes of multi-degree of freedom systems. It leads to a ratio of approximately 2%.

2.4.2 Fixed/Fixed Boundary Conditions

For fixed/fixed conditions, the left-side (ξ = 0) constraints change to Z 1(ξ) = 0 and ϕ 1(ξ) = 0, whereas all the other remain the same. The FRF for this case is shown in Fig. 2.7.

Fig. 2.7
figure 7

FRF for non-homogeneous Timoshenko beam at mid-point: fixed/fixed

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Fig. 2.8
figure 8

FRFs for damped and undamped non-homogeneous Timoshenko beam: fixed/fixed

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A comparison between the damped and undamped results for this beam is given in Fig. 2.8. Note the shift on the frequencies and changes of amplitude.

Assuming the same external forcing q 1(ξ) as above and using the forced-motion approach, the resultant deflections are shown in Fig. 2.9, for two distinct values of the frequency b.

Fig. 2.9
figure 9

Beam deflections for distinct values of b: fixed/fixed – first resonance

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The resonance frequency is taken to occur at b = 1.55.

Monitoring amplitudes at the center of the beam leads to the numerical FRF shown in Fig. 2.10. Results show the numerical simulation and an overlap of these with the analytical results. As in the previous case, excellent agreement is observed.

Fig. 2.10
figure 10

Results comparison: numerical and analytical approaches – fixed/fixed

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Consider next a case in which the spatial force is non-constant. For example, a variable force given by the exponential function: \( G\left(\xi \right)={e}^{-{\xi}^2} \) is assumed. The results can be found using the continuous variation model. The FRFs for this case are seen in Fig. 2.11.

Fig. 2.11
figure 11

FRFs for exponential force: free/fixed (a), fixed/fixed (b)

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2.5 Assumed Modes Method

In this section, an approach similar to the one described in Ref. [2] is pursued. The solution to Eq. (2.1) is assumed to have the form of a Rayleigh-Ritz expansion:

$$ w\left(x,t\right)=\sum \limits_{i=1}^n{\hat{U}}_i(t){\eta}_i(x),\kern0.6em \theta \left(x,t\right)=\sum \limits_{i=1}^n{\hat{V}}_i(t){\lambda}_i(x) $$
(2.9)

where the generalized coordinates \( {\hat{U}}_i \) and \( {\hat{V}}_i \), in the linear combination of shape functions η i and λ i, are functions of time.

The shape functions must form a linearly independent set, possessing derivatives up to the order appearing in the strain energy expression for the problem. They also must satisfy the prescribed boundary conditions.

For the undamped case, one can use the expressions for the kinetic energy, strain energy, and the external work done by the transverse loads, for each segment [2], in Lagrange’s equations. This leads to a set of n differential equations for the generalized coordinates [21, 22].

Then the discrete non-dimensional mass and stiffness matrices [23] can be obtained from:

$$ {\displaystyle \begin{array}{l}{M^{\mathrm{I}}}_{i,j}=\int\limits_0^{\frac{1}{2}}{\eta}_i{\eta}_j\ dx+{A}_r{\rho}_r\int\limits_{\frac{1}{2}}^1{\eta}_i{\eta}_j\kern0.5em dx,\kern0.75em {M^{\mathrm{I}\mathrm{I}}}_{i,j}=\int\limits_0^{\frac{1}{2}}{\lambda}_i{\lambda}_j\ dx+{I}_r{\rho}_r\int\limits_{\frac{1}{2}}^1{\lambda}_i{\lambda}_j\kern0.5em dx,\\ {}\ {K^{\mathrm{I}}}_{i,j}=\int\limits_0^{\frac{1}{2}}{\eta}_i^{\prime }{\eta}_j^{\prime }\ dx+{k}_r{G}_r{A}_r\int\limits_{\frac{1}{2}}^1{\eta}_i^{\prime }{\eta}_j^{\prime }\ dx,\kern0.5em {K^{\mathrm{I}\mathrm{I}}}_{i,j}=-\int\limits_0^{\frac{1}{2}}{\eta}_i^{\prime }{\lambda}_j\ dx-{k}_r{G}_r{A}_r\int\limits_{\frac{1}{2}}^1{\eta}_i^{\prime }{\lambda}_j\ dx,\\ {}{K^{\mathrm{I}\mathrm{I}\mathrm{I}}}_{i,j}=\int\limits_0^{\frac{1}{2}}{\lambda}_i^{\prime }{\lambda}_j^{\prime }\ dx+{E}_r{I}_r\int\limits_{\frac{1}{2}}^1{\lambda}_i^{\prime }{\lambda}_j^{\prime }\ dx+\int\limits_0^{\frac{1}{2}}{\lambda}_i{\lambda}_j\ dx+{k}_r{G}_r{A}_r\int\limits_{\frac{1}{2}}^1{\lambda}_i{\lambda}_j\ dx,\kern0.5em {K^{\mathrm{I}\mathrm{V}}}_{i,j}=-\int\limits_0^{\frac{1}{2}}{\eta}_j^{\prime }{\lambda}_i\ dx-{k}_r{G}_r{A}_r\int\limits_{\frac{1}{2}}^1{\eta}_j^{\prime }{\lambda}_i\ dx\end{array}} $$
(2.10)

Taking the external transverse loads to be sinusoidal with frequency ν, the generalized external forces can be calculated from:

$$ {P_i}^{\mathrm{I}}=\left[\int\limits_0^{\frac{1}{2}}{p}_1{\eta}_i dx+\int\limits_{\frac{1}{2}}^1{p}_2{\eta}_i\ dx\right]\sin \left(\nu \tau \right),{P_i}^{\mathrm{I}\mathrm{I}}=\left[\int\limits_0^{\frac{1}{2}}{p}_1{\lambda}_i dx+\int\limits_{\frac{1}{2}}^1{p}_2{\lambda}_i\ dx\right]\sin \left(\nu \tau \right) $$
(2.11)

where p i is the amplitude of the force acting on the i-segment.

Damping is introduced by assuming proportional damping:

$$ \left[\begin{array}{cc}\left[{D}^{\mathrm{I}}\right]& \left[{D}^{\mathrm{I}\mathrm{I}}\right]\\ {}\left[{D}^{\mathrm{I}\mathrm{V}}\right]& \left[{D}^{\mathrm{I}\mathrm{I}\mathrm{I}}\right]\end{array}\right]=\alpha \left[\begin{array}{cc}\left[{M}^{\mathrm{I}}\right]& \left[0\right]\\ {}\left[0\right]& \left[{M}^{\mathrm{I}\mathrm{I}}\right]\end{array}\right]+\beta \left[\begin{array}{cc}\left[{K}^{\mathrm{I}}\right]& \left[{K}^{\mathrm{I}\mathrm{I}}\right]\\ {}\left[{K}^{\mathrm{I}\mathrm{V}}\right]& \left[{K}^{\mathrm{I}\mathrm{I}\mathrm{I}}\right]\end{array}\right] $$
(2.12)

where α and β are constants of mass and stiffness proportionality, respectively.

The resulting system of equations can be written as:

$$ \left[\begin{array}{cc}\left[{M}^{\mathrm{I}}\right]& \left[0\right]\\ {}\left[0\right]& \left[{M}^{\mathrm{I}\mathrm{I}}\right]\end{array}\right]\left\{\begin{array}{c}\left\{{\ddot{U}}_i\right\}\\ {}\left\{{\ddot{V}}_i\right\}\end{array}\right\}+\left[\begin{array}{cc}\left[{D}^{\mathrm{I}}\right]& \left[{D}^{\mathrm{I}\mathrm{I}}\right]\\ {}\left[{D}^{\mathrm{I}\mathrm{V}}\right]& \left[{D}^{\mathrm{I}\mathrm{I}\mathrm{I}}\right]\end{array}\right]\left\{\begin{array}{c}\left\{{\dot{U}}_i\right\}\\ {}\left\{{\dot{V}}_i\right\}\end{array}\right\}+\left[\begin{array}{cc}\left[{K}^{\mathrm{I}}\right]& \left[{K}^{\mathrm{I}\mathrm{I}}\right]\\ {}\left[{K}^{\mathrm{I}\mathrm{V}}\right]& \left[{K}^{\mathrm{I}\mathrm{I}\mathrm{I}}\right]\end{array}\right]\left\{\begin{array}{c}\left\{{U}_i\right\}\\ {}\left\{{V}_i\right\}\end{array}\right\}=\left\{\begin{array}{c}\left\{{P^{\mathrm{I}}}_i\right\}\\ {}\left\{{P^{\mathrm{I}\mathrm{I}}}_i\right\}\end{array}\right\} $$
(2.13)

The undamped natural frequencies can be evaluated via an eigenvalue problem for the undamped case. The overall system response to external forcing can be estimated through modal analysis.

In order to generate the damping matrix, the constants of mass and stiffness proportionality must be determined. If the modal damping ratios are known, then the constants can be evaluated from:

$$ {\zeta}_i=\frac{\alpha {m}_{ii}+\beta {k}_{ii}}{2\sqrt{k_{ii}{m}_{ii}}} $$
(2.14)

where m ii and k ii are coefficients from the modal mass and stiffness matrices, respectively. (See also the approach described in reference [24] for the same purpose, which, for the example considered here, led to similar numerical values.)

Here the modal ratios are not known, but they can be estimated from the previous analytical results. Two modes are needed to calculate α and β. For the free/fixed case, shown in Fig. 2.2, two modes are well defined; then the ratios can be evaluated. The approach used is the half-power bandwidth. For the fixed/fixed case, Fig. 2.7, parameters for the second mode are difficult to evaluate, so, for the numerical example, only the former will be considered.

The procedure requires a choice of shape functions, which are taken to be beam characteristic orthogonal polynomials. They are generated by the Gram-Schmidt process [25] as demonstrated by Bhat [26].

For the free/fixed case, the procedure is described in reference [2]. The first polynomial follows the static deflection of a homogeneous beam, under constant distributed load and with these boundary conditions. In non-dimensional form:

$$ {\eta}_1(x)={s}_1\left[\frac{qL}{2 kGA}\left(1-{\xi}^2\right)+\frac{qL^3}{6\mathrm{EI}}\left(\frac{1}{4}{\xi}^4-\xi +\frac{3}{4}\right)\right],{\lambda}_1(x)={r}_1\left[\frac{qL^3}{6\mathrm{EI}}\left({\xi}^3-1\right)\right] $$
(2.15)

The constants s 1 and r 1 are chosen such that:

$$ \int\limits_0^1{\left({\eta}_k\right)}^2 dx=1,\int\limits_0^1{\left({\lambda}_k\right)}^2 dx=1 $$
(2.16)

The remainder polynomials are generated by the Gram-Schmidt approach. In addition, the set is also normalized. They are divided by normalization parameters that are taken to be the inverse of the magnitude of their maximum values in the interval ξ = 0...1.

Assuming non-dimensional values of damping C 1 = 0.1 and C 2 = 0.1 (ζ 1 ≈ 0.0153, ζ 2 ≈ 0.0122), the procedure is tackled using MAPLE®. For 18 polynomials and a non-dimensional time τ = 100, monitoring the amplitudes at ξ = 0.5 leads to the results shown in Fig. 2.12. The figure shows a comparison (overlap) between the analytical results and the assumed modes approach. The eigensolution gives the following first two frequencies: ν 1 = 2.75 and ν 2 = 5.45, whereas the analytical results are ν 1 = 2.50 and ν 2 = 5.20. The results differ by 10% and 5%, respectively.

Fig. 2.12
figure 12

Comparison of FRFs: assumed modes and analytical results – free/fixed

Full size image

2.6 Conclusion

Modeling discrete property variations via continuously varying functions, in conjunction with numerical solutions, has been shown to lead to good results for resonant frequencies and FRFs of viscously damped Timoshenko layered beams subject to harmonic excitation.

A numerical approach was conducted using MAPLE® software, which shows to lead to accurate solutions based on a comparison to analytical results for specific cases.

Two sets of boundary conditions were studied, namely, free/fixed and fixed/fixed for a uniform two-cell beam made of aluminum and silicon carbide.

Very good agreement was observed for both cases.

The continuous variation approach was used to produce solutions for a case with external spatially varying force, which could be intractable analytically.

Finally, an assumed modes approach was used to estimate the FRF for one of the cases. Good agreement was found with maximum error, when compared to the analytical solution, of 10%.