Keywords

2.1 Introduction

This work adds to a series (see references [1, 2]) on transverse vibrations of layered beams. Here the main interest is the vibration analysis, both theoretical and numerical, of segmented damped beams. The media of interest 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). Here, Euler-Bernoulli theory is used for a two-segment configuration under harmonic forcing.

To treat the problem two approaches are discussed. 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 the advantage of working with a single differential equation (albeit one with variable coefficients). The differential equation is then solved numerically by means of MAPLE®Footnote 1 software.

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

In the following a brief literature review is given. For vibrations of layered beams one may refer to the list given in references [1] and [2]. Solids composed by discrete layers are studied in references [3], [4], [5] and [6]. Also, a recent article [7] provides a review of references on this subject. Vibrations of damped Euler-Bernoulli beams are treated in reference [8]. There the model consists of a uniform elastically supported beam, which incorporates different dissipation mechanisms. The oscillatory character of solutions is investigated. In reference [9] estimates for the solutions of an abstract second order evolution equation are given. Applications to models of an elastic beam (possibly non-homogeneous), with a frequency-proportional damping, are discussed. Damping mechanisms on beams are studied in reference [10], where different models of dissipation are presented to account for experimentally observed behavior. Damping effect on beam vibrations is also discussed in reference [11], where the dissipation mechanism is due to air. Minimal influence is observed. The response of internally damped cantilever beams to sinusoidal vibration is given in reference [12]. An expression for the magnitude of the force transmitted to the ends of the beams is derived when both a primary and a secondary force are employed to excite the beams at arbitrary positions. It is demonstrated that the transmitted force can be attenuated significantly through broad ranges of frequency when the primary and secondary forces are of equal magnitude and their location is chosen suitably. Reference [13] treats the spectrum of Euler–Bernoulli beam equation with Kelvin–Voigt damping. Under some assumptions on the coefficients, it is shown that the essential spectrum contains continuous spectrum only, and the point spectrum consists of isolated eigenvalues of finite algebraic multiplicity. The asymptotic behavior of eigenvalues is presented. Damping effects on Timoshenko beams can be found in reference [14]. The numerical analyses provided for outlining the relevant influences on the dynamic response associated to any singular damping mechanism and also the evaluation of the modal critical damping values.

2.2 Basic Structure

The equation of motion for a viscously damped Euler-Bernoulli beam is given below. Figure 2.1 exhibits the underlying variables.

Fig. 2.1
figure 1

Beam element

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$$ \frac{\partial^2}{\partial {x}^2}\left(E(x)I(x)\frac{\partial^2w\left(x,t\right)}{\partial {x}^2}\right)+\rho (x)A(x)\frac{\partial^2w\left(x,t\right)}{\partial {t}^2}+C(x)\frac{\partial w\left(x,t\right)}{\partial t}=F\left(x,t\right) $$
(2.1)

The beam is assumed to be under a damping force, with C(x) representing a varying viscous damping coefficient per unit length. Material type can be either homogeneous or a non-homogeneous.

In the following the configuration discussed consists of a beam composed of two cells. Consider the beam shown in Fig. 2.2 which has two cells of different materials. E, ρ, and A may vary in a discontinuous manner. The segments are under transverse loads f 1 and f 2 (force per unit length) and viscous damping forces (per unit length, damping coefficients C 1 and C 2).

Fig. 2.2
figure 2

Layered beam

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

2.3 Solution Approaches

In non-dimensional form, eq. (2.1) can be written as:

$$ \frac{\partial^2}{\partial {\xi}^2}\left({f}_1\left(\xi \right){f}_2\left(\xi \right)\frac{\partial^2u\left(\xi, \tau \right)}{\partial {\xi}^2}\right)+{f}_3\left(\xi \right){f}_4\left(\xi \right)\frac{\partial^2u\left(\xi, \tau \right)}{\partial {\tau}^2}+(CD){f}_5\left(\xi \right)\frac{\partial u\left(\xi, \tau \right)}{\partial \tau }=g\left(\xi, \tau \right) $$
(2.2)

where: τ = Ω0t, \( {\Omega}_0=\sqrt{\frac{E_1{I}_1}{\rho_1{A}_1{L}^4}} \), \( \xi =\frac{x}{L} \), \( u\left(\xi, \tau \right)=\frac{w\left(x,t\right)}{L} \), \( CD={C}_1\sqrt{\frac{L^4}{\rho_1{A}_1{E}_1{I}_1}} \), \( g\left(\xi, \tau \right)=F\left(x,t\right)\frac{L^3}{E_1{I}_1} \), E(x) = E 1f 1(ξ), I(x) = I 1f 2(ξ), ρ(x) = ρ 1f 3(ξ), A(x) = A 1f 4(ξ), C(x) = C 1f 5(ξ). 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.)

For harmonic forcing with frequency λ:

$$ g\left(\xi, \tau \right)=G\left(\xi \right){e}^{\lambda \tau} $$
(2.3)

One can assume solutions of the following form:

$$ u\left(\xi, \tau \right)=R\left(\xi \right){e}^{\lambda \tau} $$
(2.4)

This leads to:

$$ \frac{d^2}{d{\xi}^2}\left({f}_1\left(\xi \right){f}_2\left(\xi \right)\frac{d^2R\left(\xi \right)}{d{\xi}^2}\right)+{\lambda}^2{f}_3\left(\xi \right){f}_4\left(\xi \right)R\left(\xi \right)+\lambda (CD){f}_5\left(\xi \right)R\left(\xi \right)=G\left(\xi \right) $$
(2.5)

Taking λ = (a + bI) and separating real and imaginary parts, after some manipulation, gives:

$$ \frac{d^2}{d{\xi}^2}\left({f}_1\left(\xi \right){f}_2\left(\xi \right)\frac{d^2R\left(\xi \right)}{d{\xi}^2}\right)-\left({b}^2{f}_3\left(\xi \right){f}_4\left(\xi \right)+\frac{CD^2{\left({f}_5\left(\xi \right)\right)}^2}{4{f}_3\left(\xi \right){f}_4\left(\xi \right)}\right)R\left(\xi \right)=G\left(\xi \right) $$
(2.6)

The result, eq. (2.6), is a non-homogeneous ordinary differential equation with variable coefficients. Analytic and numerical solutions are discussed next.

2.3.1 Analytical Approach

For constant properties in each segment, eq. (2.6) can be written as a system of n-equations, where n is the number of cells. For two cells:

$$ \frac{d^4{R}_i\left(\xi \right)}{d{\xi}^4}-\left({K}_{1,i}{b}^2+{K}_{2,i}\frac{CD^2}{4}\right){R}_i\left(\xi \right)={K}_{3,i}{G}_1\left(\xi \right),i=1,2 $$
(2.7)

where: K 1, 1 = 1, K 2, 1 = 1, K 3, 1 = 1, \( {K}_{1,2}=\frac{a_3{a}_4}{a_1{a}_2} \), \( {K}_{2,2}=\frac{a_5^2}{a_1{a}_2{a}_3{a}_4} \), \( {K}_{3,2}=\frac{a_6}{a_1{a}_2} \), \( {a}_1=\frac{E_2}{E_1} \), \( {a}_2=\frac{I_2}{I_1} \), \( {a}_3=\frac{\rho_2}{\rho_1} \), \( {a}_4=\frac{A_2}{A_1} \), \( {a}_5=\frac{CD_2}{CD_1} \), \( {a}_6=\frac{G_2}{G_1} \).

General solutions to the linear differential eqs. (2.7) involve solutions to the homogeneous equations and “particular integrals”. For arbitrary forcing G i(ξ), finding tractable particular solutions may pose a problem. Consequently, a constant spatial force is discussed here. (Non-constant spatial forcing is treated later numerically.)

The following forcing is considered: G 1(ξ) = G 1,0, G 2(ξ) = G 2,0; G 1,0, G 2,0 constants.

Then the general solutions can be written:

$$ {R}_1\left(\xi \right)={B}_1\cos \left(\alpha \xi \right)+{B}_2{e}^{\alpha \xi}+{B}_3\sin \left(\alpha \xi \right)+{B}_4{e}^{-\alpha \xi}-\frac{G_{1,0}}{\alpha^4} $$
(2.8)
$$ {R}_2\left(\xi \right)={B}_5\cos \left(\beta \xi \right)+{B}_6{e}^{\beta \xi}+{B}_7\sin \left(\beta \xi \right)+{B}_8{e}^{-\beta \xi}-{K}_{3,2}\frac{G_{1,0}}{\beta^4} $$
(2.9)

Where B i are constants to be determined and \( {\alpha}^4=\left({K}_{1,1}{b}^2+{K}_{2,1}\frac{CD^2}{4}\right) \), \( {\beta}^4=\left({K}_{1,2}{b}^2+{K}_{2,2}\frac{CD^2}{4}\right) \).

The overall analytic solution requires that the boundary conditions be defined. Two sets are considered below.

2.3.1.1 Free-Fixed Boundary Conditions

For these conditions the moment and shear free end at ξ = 0 gives: \( {\left.\frac{d^2{R}_1\left(\xi \right)}{d{\xi}^2}\right|}_{\xi =0}=0 \) and \( {\left.\frac{d^3{R}_1\left(\xi \right)}{d{\xi}^3}\right|}_{\xi =0}=0 \). The fixed boundary condition at ξ = 1 gives: R 2(1) = 0 and \( {\left.\frac{{d R}_2\left(\xi \right)}{d\xi}\right|}_{\xi =1}=0 \). Interface continuity conditions (assuming the cells have the same length) provide: R 1(ξ) = R 2(ξ), ξ = 0.5 (displacement continuity), \( \frac{{d R}_1\left(\xi \right)}{d\xi}=\frac{{d R}_2\left(\xi \right)}{d\xi},\xi =0.5 \) (slope continuity), \( \frac{d^2{R}_1\left(\xi \right)}{d{\xi}^2}={a}_1{a}_2\frac{d^2{R}_2\left(\xi \right)}{d{\xi}^2},\xi =0.5 \) (moment continuity) and \( \frac{d^3{R}_1\left(\xi \right)}{d{\xi}^3}={a}_1{a}_2\frac{d^3{R}_2\left(\xi \right)}{d{\xi}^3},\xi =0.5 \) (shear continuity). These conditions lead to a system of algebraic equations, which in matrix form can be written as:

$$ \left[\mathrm{A}\right]\left\{\mathrm{B}\right\}=\left\{G\right\} $$
(2.10)

Where:

$$ \left[A\right]=\left[\begin{array}{cccccccc}-{\upalpha}^2& {\upalpha}^2& 0& {\upalpha}^2& 0& 0& 0& 0\\ {}0& {\upalpha}^3& -{\upalpha}^3& -{\upalpha}^3& 0& 0& 0& 0\\ {}0& 0& 0& 0& \cos \left(\upbeta \right)& {\mathrm{e}}^{\upbeta}& \sin \left(\upbeta \right)& {\mathrm{e}}^{-\upbeta}\\ {}0& 0& 0& 0& -\upbeta\;\sin \left(\upbeta \right)& \upbeta\;{\mathrm{e}}^{\upbeta}& \upbeta\;\cos \left(\upbeta \right)& -\upbeta\;{\mathrm{e}}^{-\upbeta}\\ {}\cos\;\left(0.5\;\upalpha \right)& {\mathrm{e}}^{0.5\kern0.24em \upalpha}& \sin \left(0.5\;\upalpha \right)& {\mathrm{e}}^{-0.5\;\upalpha}& -\cos \left(0.5\;\upbeta \right)& -{\mathrm{e}}^{0.5\;\upbeta}& -\sin \left(0.5\;\upbeta \right)& -{\mathrm{e}}^{-0.5\;\upbeta}\\ {}-\upalpha\;\sin \left(0.5\;\upalpha \right)& \upalpha\;{\mathrm{e}}^{0.5\;\upalpha}& \upalpha\;\cos \left(0.5\;\upalpha \right)& -\upalpha\;{\mathrm{e}}^{-0.5\;\upalpha}& \upbeta\;\sin \left(0.5\;\upbeta \right)& -\upbeta\;{\mathrm{e}}^{0.5\;\upbeta}& -\upbeta\;\cos \left(0.5\;\upbeta \right)& \upbeta\;{\mathrm{e}}^{-0.5\;\upbeta}\\ {}-{\upalpha}^2\;\cos \left(0.5\;\upalpha \right)& {\upalpha}^2\;{\mathrm{e}}^{0.5\;\upalpha}& -{\upalpha}^2\;\sin \left(0.5\;\upalpha \right)& {\upalpha}^2\;{\mathrm{e}}^{-0.5\;\upalpha}& al\;a 2\;{\upbeta}^2\;\cos \left(0.5\;\upbeta \right)& - al\;a 2\;{\upbeta}^2\;{\mathrm{e}}^{0.5\;\upbeta}& al\;a 2\;{\upbeta}^2\;\sin \left(0.5\;\upbeta \right)& - al\;a 2\;{\upbeta}^2\;{\mathrm{e}}^{-0.5\;\upbeta}\\ {}{\upalpha}^3\;\sin \left(0.5\;\upalpha \right)& {\upalpha}^3\;{\mathrm{e}}^{0.5\;\upalpha}& -{\upalpha}^3\;\cos \left(0.5\kern0.24em \upalpha \right)& -{\upalpha}^3\;{\mathrm{e}}^{-0.5\;\upalpha}& - al\;a 2\;{\upbeta}^3\;\sin \left(0.5\;\upbeta \right)& - al\;a 2\;{\upbeta}^3\;{\mathrm{e}}^{0.5\;\upbeta}& al\;a 2\;{\upbeta}^3\;\cos \left(0.5\;\upbeta \right)& al\;a 2\;{\upbeta}^3\;{\mathrm{e}}^{-0.5\;\upbeta}\end{array}\right]\vspace*{-12pt} $$
$$ \left\{B\right\}=\left[\begin{array}{c}{B}_1\\ {}{B}_2\\ {}{B}_3\\ {}{B}_4\\ {}{B}_5\\ {}{B}_6\\ {}{B}_7\\ {}{B}_8\end{array}\right]\left\{G\right\}=\left[\begin{array}{c}0\\ {}0\\ {}\frac{K_{3,2}{G}_{1,0}}{\upbeta^4}\\ {}0\\ {}-\frac{\upalpha^4\;{G}_{1,0}\;{K}_{3,2}-{\upbeta}^4\;{G}_{1,0}}{\upalpha^4\;{\upbeta}^4}\\ {}0\\ {}0\\ {}0\end{array}\right] $$

Note that natural frequencies can found on setting the determinant of [A] to zero.

2.3.1.2 Fixed-Fixed Boundary Conditions

For fixed-fixed conditions, R 1(ξ) = 0, ξ = 0 and \( {\left.\frac{{d R}_1\left(\xi \right)}{d\xi}\right|}_{\xi =0}=0 \). In this case, the matrices in eq. (2.10) are:

$$ \left[A\right]=\left[\begin{array}{cccccccc}1& 1& 0& 1& 0& 0& 0& 0\\ {}0& \upalpha & \upalpha & -\upalpha & 0& 0& 0& 0\\ {}0& 0& 0& 0& \cos \left(\upbeta \right)& {\mathrm{e}}^{\upbeta}& \sin \left(\upbeta \right)& {\mathrm{e}}^{-\upbeta}\\ {}0& 0& 0& 0& -\upbeta\;\sin \left(\upbeta \right)& \upbeta\;{\mathrm{e}}^{\upbeta}& \upbeta\;\cos \left(\upbeta \right)& -\upbeta\;{\mathrm{e}}^{-\upbeta}\\ {}\cos\;\left(0.5\;\upalpha \right)& {\mathrm{e}}^{0.5\kern0.24em \upalpha}& \sin \left(0.5\;\upalpha \right)& {\mathrm{e}}^{-0.5\;\upalpha}& -\cos \left(0.5\;\upbeta \right)& -{\mathrm{e}}^{0.5\;\upbeta}& -\sin \left(0.5\;\upbeta \right)& -{\mathrm{e}}^{-0.5\;\upbeta}\\ {}-\upalpha\;\sin \left(0.5\;\upalpha \right)& \upalpha\;{\mathrm{e}}^{0.5\;\upalpha}& \upalpha\;\cos \left(0.5\;\upalpha \right)& -\upalpha\;{\mathrm{e}}^{-0.5\;\upalpha}& \upbeta\;\sin \left(0.5\;\upbeta \right)& -\upbeta\;{\mathrm{e}}^{0.5\;\upbeta}& -\upbeta\;\cos \left(0.5\;\upbeta \right)& \upbeta\;{\mathrm{e}}^{-0.5\;\upbeta}\\ {}-{\upalpha}^2\;\cos \left(0.5\;\upalpha \right)& {\upalpha}^2\;{\mathrm{e}}^{0.5\;\upalpha}& -{\upalpha}^2\;\sin \left(0.5\;\upalpha \right)& {\upalpha}^2\;{\mathrm{e}}^{-0.5\;\upalpha}& al\;a 2\;{\upbeta}^2\;\cos \left(0.5\;\upbeta \right)& - al\;a 2\;{\upbeta}^2\;{\mathrm{e}}^{0.5\;\upbeta}& al\;a 2\;{\upbeta}^2\;\sin \left(0.5\;\upbeta \right)& - al\;a 2{\upbeta}^2\;{\mathrm{e}}^{-0.5\;\upbeta}\\ {}{\upalpha}^3\;\sin \left(0.5\;\upalpha \right)& {\upalpha}^3\;{\mathrm{e}}^{0.5\;\upalpha}& -{\upalpha}^3\;\cos \left(0.5\kern0.24em \upalpha \right)& -{\upalpha}^3\;{\mathrm{e}}^{-0.5\;\upalpha}& - al\;a 2\;{\upbeta}^3\;\sin \left(0.5\;\upbeta \right)& - al\;a 2\;{\upbeta}^3\;{\mathrm{e}}^{0.5\;\upbeta}& al\;a 2\;{\upbeta}^3\;\cos \left(0.5\;\upbeta \right)& al\;a 2\;{\upbeta}^3\;{\mathrm{e}}^{-0.5\;\upbeta}\end{array}\right] $$
$$ \left\{B\right\}=\left[\begin{array}{c}{B}_1\\ {}{B}_2\\ {}{B}_3\\ {}{B}_4\\ {}{B}_5\\ {}{B}_6\\ {}{B}_7\\ {}{B}_8\end{array}\right]\left\{G\right\}=\left[\begin{array}{c}\frac{G_{1,0}}{\alpha^4}\\ {}0\\ {}\frac{K_{3,2}{G}_{1,0}}{\upbeta^4}\\ {}0\\ {}-\frac{\upalpha^4\;{G}_{1,0}\;{K}_{3,2}-{\upbeta}^4\;{G}_{1,0}}{\upalpha^4\;{\upbeta}^4}\\ {}0\\ {}0\\ {}0\end{array}\right] $$

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:

$$ {f}_i\left(\xi \right)=1+\left(\frac{\delta_2-{\delta}_1}{\delta_1}\right)\left(\frac{1}{2}+\frac{1}{2}\tanh \left(K\left(\xi -\frac{1}{2}\right)\right)\right),i=1,2,3,4,5 $$
(2.11)

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

Substituting eqs. (2.11) into (6) leads to a differential equation, 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 reference [15]). 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. 2.2 and assume the following materials: Aluminum (E 1 = 71 GPa, ρ 1 = 2710 Kg / m 3) and Silicon Carbide (E 2 = 210 GPa, ρ 2 = 3100 Kg / m 3). These values are taken from a paper in the field [16].

2.4.1 Free-Fixed Boundary Conditions

For the free-fixed case, the determinant of [A] in eq. (2.10) leads to the following values for the first two non-dimensional natural frequencies: b = 5.7228 and b = 27.3930. (The following parameters apply: a 1 = 2.9577, a 2 = a 4 = a 5 = a 6 = 1.0000, a 3 = 1.1439, G 1, 0 = 1.0000, CD = 1.0000).

Using eq. (2.10) to determine the values of the constants B i, and eq. (2.8), 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. 2.3.

Fig. 2.3
figure 3

FRF for non-homogeneous beam at mid-point – 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 properties variation for two-cell beam

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Assuming a value of 1 for the external forcing G(ξ) and using the forced-motion approach ([15]), the resultant deflections are plotted bellow for two distinct values of the frequency b.

The resonance frequencies are taken to occur at b = 5.9 and b = 27.5, as seen in Fig. 2.5.

Fig. 2.5
figure 5

Beam deflections for distinct values of b – Free-Fixed: first and second resonance

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Amplitudes for the response at the center of the beam can be monitored from eq. (2.6). The approach leads to the numerical FRF shown in Fig. 2.6 (dots on the plot).

Fig. 2.6
figure 6

Results comparison – numerical and analytical approaches – Free/Fixed

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The figure shows an overlap of the numerical results and the results from the analytical approach (eq. (2.8)). It is seen that very good agreement is obtained, the first two resonances and amplitude values correspond well.

From the numerical FRF, the damping ratio of the system, corresponding to the non-dimensional value of damping CD = 1.0000, can be estimated. The method used here is the half-power bandwidth [17], 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 the fixed-fixed case, the determinant of [A] in eq. (2.10) gives the natural frequencies: b = 27.40 and b = 79.49. Eqs. (2.10) and (2.8) lead to the FRF for the system, which is shown in Fig. 2.7.

Fig. 2.7
figure 7

FRF for non-homogeneous beam at mid-point – Fixed/Fixed

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The numerical FRF for this case is seen in Fig. 2.8. The overlap of the numerical results and the results from the analytical approach show that good agreement is obtained for the first frequency and amplitude values, with the second one not quite given the amplitudes seen in the analytical approach (similar results were obtained for the non-damped case, see reference [1]).

Fig. 2.8
figure 8

Results comparison – numerical and analytical approaches – Fixed/Fixed

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An example is now given in which the spatial force is non-constant. Consider a variable force given by the exponential function: \( G\left(\xi \right)={e}^{-{\xi}^2} \). The results can be found using the continuous variation model. The FRFs for this case are seen in Fig. 2.9.

Fig. 2.9
figure 9

FRFs for exponential force – Free/Fixed (a), Fixed/Fixed (b)

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2.5 PDE Direct Numerical Approach

Equation (2.2) could be tackled directly via a PDE solver. To this end, here an attempt is made using MAPLE® software.

On running the numerical routine, it was noticed that convergence of the PDE solver is better achieved with less abrupt changes on the logistic functions. Therefore, here, these functions will be used with a value of K = 100. In addition, the beam is subjected to harmonic forcing: g(ξ, τ) =  sin (ντ) and CD = 0.1000 (approximately a damping ratio of 0.2%).

Figure 2.10 shows overlaps for the FRFs obtained using the ODE and PDE solutions. Both cases are illustrated. Note that good agreement is seen for the resonances with some variation on the amplitudes.

Fig. 2.10
figure 10

FRFs comparison: ODE versus PDE solutions – Free/Fixed (a), Fixed/Fixed (b)

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2.6 A More Complete Model

For cases involving lighter, stronger and more flexible beams, subjected to vibration problems, energy losses due to damping effects become very important. When dealing with such cases, the viscous damping approach discussed above may not be adequate to fully model the behavior of such structures. The model leads to uniform damping rates, which are generally not observed, since damping rates in beams tend to increase with frequency (see, for instance, [18]).

In order to address this issue, a more complete model for the damped beam is presented in the following. The basic viscously damped Euler-Bernoulli equation is modified to include extra energy dissipation mechanisms. The first one includes a Kelvin-Voigt (internal) mechanism and the second one a structural damping. These mechanisms are discussed in details in reference [19]. Note that in the study it was observed, via experiments, that a predominantly linear relationship between the damping rate and the frequency exists; a behavior which was described as “structural damping”.

The modified equation, in non-dimensional form, then becomes:

$$ \begin{aligned}&\frac{\partial^2}{\partial {\xi}^2}\left({f}_1\left(\xi \right){f}_2\left(\xi \right)\frac{\partial^2u\left(\xi, \tau \right)}{\partial {\xi}^2}\right) +{f}_3\left(\xi \right){f}_4\left(\xi \right)\frac{\partial^2u\left(\xi, \tau \right)}{\partial {\tau}^2}+(CD){f}_5\left(\xi \right)\frac{\partial u\left(\xi, \tau \right)}{\partial \tau }-\gamma {f}_6\left(\xi \right)\frac{\partial^3u\left(\xi, \tau \right)}{\partial \tau\ \partial {\xi}^2}\nonumber\\ &\quad+\eta \frac{\partial^2}{\partial {\xi}^2}\left({f}_2\left(\xi \right){f}_7\left(\xi \right)\frac{\partial^3u\left(\xi, \tau \right)}{\partial \tau\ \partial {\xi}^2}\right)=g\left(\xi, \tau \right)\end{aligned} $$
(2.12)

Where η is a non-dimensional viscosity coefficient, γ is a non-dimensional structural damping coefficient and f 6(ξ) and f 7(ξ) are non-dimensional logistic functions describing variations of these damping mechanisms from one cell to another.

Inclusion of Kelvin-Voigt damping requires that the boundary conditions for the problem be revisited [10]. Here the displacement solutions are assumed to be sufficiently smooth, so that the conditions utilized for the previous model still apply (see [20]).

As discussed above, for the case of stacked beams, analytic solutions for eq. (2.12) may not be feasible.

In this section, the continuous variation approach is employed to obtain direct numerical solutions to the equation. MAPLE® PDE solver is employed.

Consider the same 2-cell beam given above and the following parameters: g(ξ, τ) =  sin (ντ), K = 100,CD = 0.1000 (damping ratio of 0.2%), γ = 0.1000, η = 0.1000. (For simplicity, damping coefficients are taken to have the same values in both cells.) The logistic functions assumed are: f 6(ξ) = f 7(ξ) = 1.0000 (A) and (B).

Figure 2.11 shows the FRFs for these cases with free-fixed boundary conditions. Only the first mode is shown, since the procedure did not capture large amplitudes at the expected second frequency. It is conjectured that the damping caused the amplitudes to be small and not significantly distinct to stand out in the numerical routine. (Decreasing damping values showed increased amplitudes around expected resonance.)

Fig. 2.11
figure 11

FRFs for expanded model – Free/Fixed

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2.7 Conclusions

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 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 a specific case.

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, with some variation on the amplitudes for the FRFs.

Comparing numerical solutions from two approaches, using ODE and PDE solvers and continuously varying functions, also showed good agreement.

Lastly, a model involving internal and structural damping, in addition to viscous damping was introduced. Numerical results were shown to be feasible for specific cases, using the numerical PDE solver provided by MAPLE® software.