### Free transverse vibration multi-layer segmental Timoshenko and Euler–Bernoulli beams

In this section, the analytic approach to study the free vibration of the lumbar spine as a muti-layer segmental beam is developed. As previously indicated, the beam has almost uniform cross section but is composed of the components with different geometry and mechanical properties. Figure 3 gives the reader an idea about the model of the multi-layer segmental beam to derive free vibrational equations in this work.

#### Free transverse vibration analysis of Timoshenko beam

In order to solve the governing equations of a Timoshenko beam, we consider both the slope and deflection profiles, *ϕ*(*x*,*t*) and *w*(*x*,*t*) as the following form:

$$ \left\{\begin{array}{l}w\left(x,t\right)=W(x){e}^{i\lambda t}\\ {}\phi \left(x,t\right)=\Phi (x){e}^{i\lambda t}\end{array}\right. $$

(15)

where *λ* is the frequency of oscillation. By solving Eqs. (8) and (9), we have:

$$ {\displaystyle \begin{array}{l}{W}_j(x)={c}_{1j}\sin {\xi}_1x+{c}_{2j}\cos {\xi}_1x+{c}_{3j}\sinh {\xi}_2x+{c}_{4j}\cosh {\xi}_2x\\ {}j=1,2,3,\dots, n\end{array}} $$

(16-a)

$$ {\displaystyle \begin{array}{l}{\Phi}_j(x)={k}_a{c}_{2j}\sin {\xi}_1x-{k}_a{c}_{1j}\cos {\xi}_1x+{k}_b{c}_{3j}\cosh {\xi}_2x+{k}_b{c}_{4j}\sinh {\xi}_2x\\ {}j=1,2,3,\dots, n\end{array}} $$

(16-b)

In the above equation, *j* relates to the free vibration solution of the *n*th segment of the beam. Also, the following parameters are defined as:

$$ {\displaystyle \begin{array}{l}{\xi}_1={\left(\frac{\eta_1+\sqrt{\eta_1^2-4{\eta}_2}}{2}\right)}^{\frac{1}{2}}\ \\ {}{\xi}_2={\left(\frac{-{\eta}_1+\sqrt{\eta_1^2-4{\eta}_2}}{2}\right)}^{\frac{1}{2}}\kern0.5em \end{array}} $$

(17)

$$ {\eta}_1=\frac{\Lambda_1}{\Lambda_2}\kern0.75em ;\kern0.75em {\eta}_2=\frac{\Lambda_3}{\Lambda_2} $$

where

$$ \left\{\begin{array}{l}{\Lambda}_1={k}_1{k}_3+{k}_2{k}_4-{k}_3^2\\ {}{\Lambda}_2={k}_2{k}_3\\ {}{\Lambda}_3={k}_1{k}_4\end{array}\right. $$

(18)

$$ {\displaystyle \begin{array}{l}{k}_1=-\left({\rho}_{\mathrm{in}}{I}_{\mathrm{in}}+{\rho}_{\mathrm{out}}{I}_{\mathrm{out}}\right){\lambda}^2+k\left({G}_{\mathrm{in}}{A}_{\mathrm{in}}+{G}_{\mathrm{out}}{A}_{\mathrm{out}}\right);{k}_2=-\left({E}_{\mathrm{in}}{I}_{\mathrm{in}}+{E}_{\mathrm{out}}{I}_{\mathrm{out}}\right);\\ {}{k}_3=k\left({G}_{\mathrm{in}}{A}_{\mathrm{in}}+{G}_{\mathrm{out}}{A}_{\mathrm{out}}\right);{\mathrm{k}}_4=\left({\rho}_{\mathrm{in}}{A}_{\mathrm{in}}+{\rho}_{\mathrm{out}}{A}_{\mathrm{out}}\right){\lambda}^2;\end{array}} $$

Also *k*_{a} and *k*_{b} in Eq. (16-b) are:

$$ \left\{\begin{array}{l}{k}_a=\frac{k_3{\xi}_1}{k_1-{k}_2{\xi}_1^2}\\ {}{k}_b=\frac{-{k}_3{\xi}_2}{k_1+{k}_2{\xi}_2^2}\end{array}\right. $$

(19)

To calculate the natural frequencies of Timoshenko beam, it is essential to satisfy the following boundary conditions:

$$ {\displaystyle \begin{array}{l}\mathrm{Clamped}-\mathrm{Clamped}:{W}_1(0)={\Phi}_1(0)=0;{W}_n(L)={\Phi}_n(L)=0\\ {}\mathrm{Simply}-\mathrm{Simply}:{W}_1(0)={M}_1^T(0)=0;{W}_n(L)={M}_n^T(L)=0\end{array}} $$

(20)

Furthermore, following compatibility conditions should be established in the contact boundary between the different adjacent segments:

$$ {\displaystyle \begin{array}{l}{M}_j^T\left({L}_j^{\ast}\right)={M}_{j+1}^T\left({L}_j^{\ast}\right);{Q}_j^T\left({L}_j^{\ast}\right)={Q}_{j+1}^T\left({L}_j^{\ast}\right);j=1,2,\dots, n-1.\\ {}{W}_j\left({L}_j^{\ast}\right)={W}_{j+1}\left({L}_j^{\ast}\right);{\Phi}_j\left({L}_j^{\ast}\right)={\Phi}_{j+1}\left({L}_j^{\ast}\right);j=1,2,\dots, n-1.\end{array}} $$

(21)

where \( {L}_j^{\ast } \) is the length of the contact place of *j*th segment to (*j* + 1)th one. As it can be observed from above equations, there are 4*n* equations and 4*n* unknown coefficients that must be solved to have a nontrivial set of solution.

#### Free transverse vibration analysis of Euler–Bernoulli beam

In Euler–Bernoulli beam, to obtain the natural frequencies, only the deflection profile is considered. In other words, it is assumed that the deflection profile, *w*(*x*,*t*) has the following form:

$$ w\left(x,t\right)=W(x){e}^{i\lambda t} $$

(22)

By considering Eq. (22) and solving Eq. (12), we have:

$$ {\displaystyle \begin{array}{l}{W}_j(x)={c}_{1j}\sin \xi x+{c}_{2j}\cos \xi x+{c}_{3j}\sinh \xi x+{c}_{4j}\cosh \xi x\\ {}j=1,2,3,\dots, n\end{array}} $$

(23)

where *j* relates to the free vibration solution of the *n*th segment of the beam and *ξ*_{1} and *ξ*_{2} are also given by:

$$ \xi ={\left(\lambda \sqrt{\frac{\left({\rho}_{\mathrm{in}}{A}_{\mathrm{in}}+{\rho}_{\mathrm{out}}{A}_{\mathrm{out}}\right)}{E_{\mathrm{in}}{I}_{\mathrm{in}}+{E}_{\mathrm{out}}{I}_{\mathrm{out}}}}\right)}^{\frac{1}{2}}\kern0.5em ; $$

(24)

The following boundary conditions should be also satisfied to determine the natural frequencies of the Euler–Bernoulli beam:

$$ {\displaystyle \begin{array}{l}\mathrm{Clamped}-\mathrm{Clamped}:{W}_1(0)=\frac{\partial {W}_1(0)}{\partial x}=0;{W}_n(L)=\frac{\partial {W}_n}{\partial x}(L)=0\\ {}\mathrm{Simply}-\mathrm{Simply}:{W}_1(0)={M}_1^E(0)=0;{W}_n(L)={M}_n^E(L)=0\end{array}} $$

(25)

In addition, following compatibility conditions should be considered in the contact boundaries between the adjacent segments:

$$ {\displaystyle \begin{array}{l}{M}_j^E\left({L}_j^{\ast}\right)={M}_{j+1}^E\left({L}_j^{\ast}\right);{Q}_j^E\left({L}_j^{\ast}\right)={Q}_{j+1}^E\left({L}_j^{\ast}\right);j=1,2,\dots, n-1.\\ {}{W}_j\left({L}_j^{\ast}\right)={W}_{j+1}\left({L}_j^{\ast}\right);\frac{\partial {W}_j\left({L}_j^{\ast}\right)}{\partial x}=\frac{\partial {W}_{j+1}\left({L}_j^{\ast}\right)}{\partial x};j=1,2,\dots, n-1.\end{array}} $$

(26)

Similar to what was proceeded for the Timoshenko beam, 4*n* equations and 4*n* unknown coefficients, it was found that in order to attain the nontrivial solution, the determinant of the coefficient matrix must be zero.