METHOD FOR CALCULATION OF NATURAL FREQUENCY OF MULTI-SEGMENT CONTINUOUS BEAM

Abstract

A displacement spring and a rotational spring are arranged on both ends of the multi-segment continuous beam to simulate arbitrary boundary conditions, and a lateral displacement function of the multi-segment continuous beam over a whole segment is constructed. A strain energy, an elastic potential energy of simulated springs at a boundary, a maximum value of a kinetic energy, and a Lagrangian function of the multi-segment continuous beam are calculated. The improved Fourier series of the displacement function is substituted into the Lagrange function. An extreme value of each undetermined coefficient in the improved Fourier series in the Lagrangian function is taken to obtain a system of homogeneous linear equations which is further converted into a matrix. An eigenvalue problem of the standard matrix is solved for to obtain the natural frequency.

Claims

1. A method for calculation of a natural frequency of a multi-segment continuous beam, comprising: (1) arranging a displacement spring and a rotational spring on each of two ends of the multi-segment continuous beam to simulate arbitrary boundary conditions; (2) constructing a lateral displacement function of the multi-segment continuous beam along a full length thereof, and expressing the lateral displacement function in a form of an improved Fourier series, wherein the improved Fourier series is formed by adding four auxiliary functions into the classic Fourier series; (3) calculating a strain energy of the multi-segment continuous beam; (4) calculating an elastic potential energy of the displacement spring and the rotational spring at a boundary of the multi-segment continuous beam; (5) calculating a maximum value of a kinetic energy of the multi-segment continuous beam; (6) calculating a Lagrangian function of the multi-segment continuous beam; (7) substituting the improved Fourier series of the lateral displacement function into the Lagrange function; (8) taking an extreme value of each of undetermined coefficients in the improved Fourier series in the Lagrangian function to let a partial derivative be zero, so as to obtain a system of homogeneous linear equations; (9) converting the system of homogeneous linear equations into a matrix form; and (10) solving for an eigenvalue problem of the matrix to obtain the natural frequency.

2. The method of claim 1, wherein in step (1), a stiffness value of the displacement spring and a stiffness value of the rotational spring at a first boundary are respectively denoted as k.sub.1 and K.sub.1, and a stiffness value of the displacement spring and a stiffness value of the rotational spring at a second boundary are respectively denoted as k.sub.2 and K.sub.2; when the boundary is a clamped boundary, the stiffness value of the displacement spring and the stiffness value of the rotational spring need to be set to infinity at the same time, and the stiffness value of the displacement spring and the stiffness value of the rotational spring are set to 10.sup.13, respectively; when the boundary is a free boundary, the stiffness value of the displacement spring and the stiffness value of the rotational spring are set to zero; when the boundary is a simply supported boundary, the stiffness value of the displacement spring is set to 10.sup.13, and the stiffness value of the rotational spring is 0; and when the stiffness value of the displacement spring and the stiffness value of the rotational spring are finite values, an elastic constraint boundary condition is simulated.

3. The method of claim 1, wherein the lateral displacement function of the multi-segment continuous beam over the full length thereof expressed in the form of the improved Fourier series in step (2) is: $\begin{array}{cc}W\left(x\right)=\underset{n=0}{\overset{9}{.Math.}}{a}_n\cos \left({\lambda }_{n}x\right)+\underset{n=-4}{\overset{-1}{.Math.}}{a}_n\sin \left({\lambda }_{n}x\right);& \left(1\right)\end{array}$ wherein x ∈[0,L]; a.sub.n is an undetermined constant; and λ.sub.n=nπ/L

4. The method of claim 1, wherein the strain energy of the multi-segment continuous beam in step (3) is: $\begin{array}{cc}{V}_P=\frac{1}{2}{E}_1{I}_1{\int }_0^{L_1}{\left(\frac{d^{2}w}{{\mathrm{dx}}^2}\right)}^2\mathrm{dx}+\frac{1}{2}\underset{i=2}{\overset{i=p}{.Math.}}{E}_i{I}_i{\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{\left(\frac{d^{2}w}{d{x}^2}\right)}^2\mathrm{dx};& \left(2\right)\end{array}$ wherein a total length of the multi-segment continuous beam is L; the multi-segment continuous beam is divided into p segments; a length of an i-th segment is L.sub.i; Vp is the strain energy of the multi-segment continuous beam under arbitrary boundary conditions; E.sub.i is an elastic modulus of the i-th segment, and I.sub.i is a moment of inertia of a cross section of the i-th segment.

5. The method of claim 1, wherein the elastic potential energy Vs of the displacement spring and the rotational spring at the boundary of the multi-segment continuous beam in step (4) is: $\begin{array}{cc}{V}_s=\frac{1}{2}\left(k_1{w}^2{|}_{x=0}+{K_1\left(\frac{\partial w}{\partial x}\right)}^2{|}_{x=0}+k_2{w}^2{|}_{x=L}+{K_2\left(\frac{\partial w}{\partial x}\right)}^2{|}_{x=L}\right).& \left(3\right)\end{array}$

6. The method of claim 1, wherein a form of a modal solution of the multi-segment continuous beam is assumed based on a variable separation method in step (2) as:
w(x,t)=W(x)e.sup.iwt   (4); wherein i is an imaginary unit, and ω is the natural frequency of the multi-segment continuous beam.

7. The method of claim 1, wherein the maximum value of the kinetic energy of the multi-segment continuous beam in step (5) is: $\begin{array}{cc}{T}_{\mathrm{ma}x}=\frac{1}{2}\rho \left(x\right){\int }_0^{L}S\left(x\right){\left(\frac{dw}{dt}\right)}^{2}dx=\frac{{\omega }^2}{2}{\int }_0^L\rho \left(x\right)S\left(x\right)w^2\mathrm{dx}.& \left(5\right)\end{array}$

8. The method of claim 1, wherein the Lagrangian function of the multi-segment continuous beam in step (6) is: $\begin{array}{cc}L=V_{\mathrm{ma}x}-T_{m\mathrm{ax}}=V_p+V_s-T_{\mathrm{ma}x}.& \left(6\right)\end{array}$

9. The method of claim 1, wherein in step (8), the partial derivative of the undertermined coefficient an (n=−4, −3, . . . , 9) is calculated item by item in the Lagrangian function, to obtain the system of homogeneous linear equations:
[(M.sub.1+ . . . +M.sub.p)ω.sup.2(Kp.sub.1+ . . . +Kp.sub.p+Ks.sub.1+Ks.sub.2+Ks.sub.3+Ks.sub.4)]A=0   (7); wherein A={a.sub.−4, a.sub.−3, . . . , a.sub.8, a.sub.9}.sup.T, ${\mathrm{Kp}}_1=E_1{I}_1\left[\begin{array}{cccc}{\int }_0^{L_1}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_0^{L_1}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ {\int }_0^{L_1}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_0^{L_1}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_0^{L_1}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_0^{L_1}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\end{array}\right],\mathrm{.Math.}$ ${\mathrm{Kp}}_p=E_p{I}_p\left[\begin{array}{cccc}{\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\end{array}\right]$ ${\mathrm{Ks}}_1=k_1\left[\begin{array}{cccc}{f}_1{f}_1& f_1{f}_2& \mathrm{.Math.}& f_1{f}_m\\ f_1{f}_2& f_2{f}_2& \mathrm{.Math.}& f_2{f}_m\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ f_1{f}_m& f_2{f}_m& \mathrm{.Math.}& f_m{f}_m\end{array}\right]{|}_{x=0},{\mathrm{Ks}}_2=k_2\left[\begin{array}{cccc}{f}_1{f}_1& f_1{f}_2& \mathrm{.Math.}& f_1{f}_m\\ f_1{f}_2& f_2{f}_2& \mathrm{.Math.}& f_2{f}_m\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ f_1{f}_m& f_2{f}_m& \mathrm{.Math.}& f_m{f}_m\end{array}\right]{|}_{x=L},{\mathrm{Ks}}_3=K_1\left[\begin{array}{cccc}\frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_1}{\mathrm{dx}}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_m}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\end{array}\right]{|}_{x=0},{\mathrm{Ks}}_4=K_2=\left[\begin{array}{cccc}\frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_1}{\mathrm{dx}}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_m}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\end{array}\right]{|}_{x=L}{M}_1={\rho }_1{A}_1\left[\begin{array}{cccc}{\int }_0^{L_1}{f}_1{f}_1\mathrm{dx}& {\int }_0^{L_1}{f}_1{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}{f}_1{f}_m\mathrm{dx}\\ {\int }_0^{L_1}{f}_2{f}_1\mathrm{dx}& {\int }_0^{L_1}{f}_2{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}{f}_2{f}_m\mathrm{dx}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_0^{L_1}{f}_m{f}_1\mathrm{dx}& {\int }_0^{L_1}{f}_m{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}{f}_m{f}_m\mathrm{dx}\end{array}\right],\mathrm{.Math.}$ $M_p={\rho }_p{A}_p=\left[\begin{array}{cccc}{\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_1{f}_1\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_1{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_1{f}_m\mathrm{dx}\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_2{f}_1\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_2{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_2{f}_m\mathrm{dx}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_m{f}_1\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_m{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_m{f}_m\mathrm{dx}\end{array}\right].$

10. The method of claim 1, wherein a condition for the system of the homogeneous linear equations to have a nontrivial solution in the step (8) is: a value of coefficient determinant of the system of the homogeneous linear equations is zero to obtain a frequency equation.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0030] The present disclosure will be further described in detail in conjunction with the accompanying drawings.

[0031] The figure is a schematic diagram of a multi-segment continuous beam model under arbitrary boundary conditions according to an embodiment of the present disclosure.

DETAILED DESCRIPTION OF EMBODIMENTS

[0032] In order to make the technical means, inventive features, objectives and effects of the present disclosure easy to understand, the present disclosure will be further illustrated below in conjunction with specific embodiments.

[0033] As shown in the figure, the embodiment provides a method for calculation of a natural frequency of a multi-segment continuous beam, including the following steps.

[0034] (1) A displacement spring and a rotational spring are arranged on each of a left end and a right end of the multi-segment continuous beam to simulate arbitrary boundary conditions.

[0035] (2) A lateral displacement function of the multi-segment continuous beam over a whole segment is constructed and consists of an undetermined mode shape function and an exponential function of an undetermined vibration frequency. The undetermined mode shape function is expressed in a form of an improved Fourier series, where the improved Fourier series is formed by adding four auxiliary functions into a classic Fourier series.

[0036] (3) A strain energy of the multi-segment continuous beam is calculated. The multi-segment continuous beam is a straight rod, such as Bernoulli-Euler beams.

[0037] (4) An elastic potential energy of simulated springs at a boundary of the multi-segment continuous beam is calculated.

[0038] (5) A maximum value of a kinetic energy of the multi-segment continuous beam is calculated.

[0039] (6) A Lagrangian function of the multi-segment continuous beam is calculated.

[0040] (7) The improved Fourier series of the lateral displacement function is substituted into the Lagrange function.

[0041] (8) An extreme value of each undetermined coefficient in the improved Fourier series in the Lagrangian function is taken to let a partial derivative be zero, so as to obtain a system of homogeneous linear equations.

[0042] (9) The system of homogeneous linear equations is converted into a matrix form.

[0043] (10) An eigenvalue problem of the standard matrix is solved for through Mathematica, to obtain a natural angular frequency of each order of the multi-segment continuous beam.

[0044] In step (1), a stiffness value of the displacement spring and a stiffness value of the rotational spring at a left boundary are respectively denoted as k1 and K.sub.1, and a stiffness value of the displacement spring and a stiffness value of the rotational spring at a right boundary are respectively denoted as k.sub.2 and K.sub.2. When the boundary is a clamped boundary, the stiffness value of the displacement spring and the stiffness value of the rotational spring need to be set to infinity at the same time, and the stiffness value of the displacement spring and the stiffness value of the rotational spring are set to 10.sup.13, respectively. When the boundary is a free boundary, the stiffness value of the displacement spring and the stiffness value of the rotational spring can be set to zero. When the boundary is a simply supported boundary, the stiffness value of the displacement spring is set to 10.sup.13, and the stiffness value of the rotational spring is set to 0. When the stiffness value of the displacement spring and the stiffness value of the rotational spring are finite values, an elastic constraint boundary condition can be simulated.

[0045] A form of a modal solution of the multi-segment continuous beam is assumed based on a variable separation method as:

w(x,t)=W(x)e.sup.iwt   (4);

[0046] where i is an imaginary unit; W(x) is the vibrational model function; and co is the natural frequency of the multi-segment continuous beam.

[0047] The vibrational model function W(x) is expressed in a form as follows:

[00007]$\begin{array}{cc}W\left(x\right)=\underset{n=0}{\overset{9}{.Math.}}{a}_n\cos \left({\lambda }_{n}x\right)+\underset{n=-4}{\overset{-1}{.Math.}}{a}_n\sin \left({\lambda }_{n}x\right);& \left(1\right)\end{array}$

[0048] where x ∈[0,L]; a.sub.n(n=−4, −3, . . . , 9) is an undetermined constant; and λ.sub.n=nπ/L.

[0049] The strain energy of the multi-segment continuous beam consists of strain energies of segments of the multi-segment continuous beam and is expressed as:

[00008]$\begin{array}{cc}{V}_P=\frac{1}{2}{E}_1{I}_1{\int }_0^{L_1}{\left(\frac{d^{2}w}{{\mathrm{dx}}^2}\right)}^2\mathrm{dx}+\frac{1}{2}\underset{i=2}{\overset{i=p}{.Math.}}{E}_i{I}_i{\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{\left(\frac{d^{2}w}{d{x}^2}\right)}^2\mathrm{dx};& \left(2\right)\end{array}$

[0050] the strain energy of each segment of the multi-segment continuous beam is expressed as:

[00009]$V_{P1}=\frac{1}{2}{E}_1{I}_1{\int }_0^{L_1}{\left(\frac{d^{2}w}{{\mathrm{dx}}^2}\right)}^{2}dx{V}_{P2}=\frac{1}{2}{E}_2{I}_2{\int }_{L_1}^{L_1+L_2}{\left(\frac{d^{2}w}{d{x}^2}\right)}^{2}dx;$$\mathrm{.Math.}$

[0051] where a total length of the multi-segment continuous beam is L; the multi-segment continuous beam is divided into p segments; and a length of the i-th segment is Li; Vp is the strain energy of the multi-segment continuous beam under arbitrary boundary conditions; Ei is an elastic modulus of the i-th segment, and is a moment of inertia of of a cross section of the i-th segment.

[0052] The elastic potential energy Vs of the simulated spring at the boundary of the multi-segment continuous beam is:

[00010]$\begin{array}{cc}{V}_s=\frac{1}{2}\left(k_1{w}^2{|}_{x=0}+{K_1\left(\frac{\partial w}{\partial x}\right)}^2{|}_{x=0}+k_2{w}^2{|}_{x=L}+{K_2\left(\frac{\partial w}{\partial x}\right)}^2{|}_{x=L}\right).& \left(3\right)\end{array}$

[0053] The maximum kinetic energy of the multi-segment continuous beam is:

[00011]$\begin{array}{cc}{T}_{\max }=\frac{1}{2}\rho \left(x\right){\int }_0^{L}S\left(x\right){\left(\frac{dw}{dt}\right)}^{2}dx=\frac{{\omega }^2}{2}{\int }_0^L\rho \left(x\right)S\left(x\right)w^{2}dx.& \left(5\right)\end{array}$

[0054] The Lagrangian function of the multi-segment continuous beam is:

L=V.sub.max−T.sub.max=V.sub.p.sub.+V.sub.s.sub.−T.sub.max   (6).

[0055] The partial derivative of the undetermined coefficient an (n=−4, −3, . . . , 9) is calculated item by item in the Lagrangian function to obtain the system of homogeneous linear equations:

[(M.sub.1+ . . . +M.sub.p)ω.sup.2(Kp.sub.1+ . . . +Kp.sub.p+Ks.sub.1+Ks.sub.2+Ks.sub.3+Ks.sub.4)]A=0   (7);

where A={a.sub.−4, a.sub.−3, . . . , a.sub.8, a.sub.9}.sup.T,

[00012]${\mathrm{Kp}}_1=E_1{I}_1\left[\begin{array}{cccc}{\int }_0^{L_1}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_0^{L_1}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ {\int }_0^{L_1}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_0^{L_1}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_0^{L_1}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_0^{L_1}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\end{array}\right],\mathrm{.Math.}$$K{p}_p=E_p{I}_p\left[\begin{array}{cccc}{\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\end{array}\right]$${\mathrm{Ks}}_1={k_1\left[\begin{array}{cccc}{f}_1{f}_1& f_1{f}_2& \mathrm{.Math.}& f_1{f}_m\\ f_1{f}_2& f_2{f}_2& \mathrm{.Math.}& f_2{f}_m\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ f_1{f}_m& f_2{f}_m& \mathrm{.Math.}& f_m{f}_n\end{array}\right]}_{x=0},{\mathrm{Ks}}_2={k_2\left[\begin{array}{cccc}{f}_1{f}_1& f_1{f}_2& \mathrm{.Math.}& f_1{f}_m\\ f_1{f}_2& f_2{f}_2& \mathrm{.Math.}& f_2{f}_m\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ f_1{f}_m& f_2{f}_m& \mathrm{.Math.}& f_m{f}_n\end{array}\right]}_{x=L},{\mathrm{Ks}}_3={K_1\left[\begin{array}{cccc}\frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_1}{\mathrm{dx}}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_m}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\end{array}\right]}_{x=0},{\mathrm{Ks}}_4={K_2\left[\begin{array}{cccc}\frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_1}{\mathrm{dx}}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_m}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\end{array}\right]}_{x=L}{M}_1={\rho }_1{A}_1\left[\begin{array}{cccc}{\int }_0^{L_1}{f}_1{f}_1\mathrm{dx}& {\int }_0^{L_1}{f}_1{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}{f}_1{f}_m\mathrm{dx}\\ {\int }_0^{L_1}{f}_2{f}_1\mathrm{dx}& {\int }_0^{L_1}{f}_2{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}{f}_2{f}_m\mathrm{dx}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_0^{L_1}{f}_m{f}_1\mathrm{dx}& {\int }_0^{L_1}{f}_m{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}{f}_m{f}_m\mathrm{dx}\end{array}\right],\mathrm{.Math.}{M}_p={\rho }_p{A}_p\left[\begin{array}{cccc}{\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_1{f}_{1}dx& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_1{f}_{2}dx& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_1{f}_{m}dx\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_2{f}_1\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_2{f}_{2}dx& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_2{f}_{m}dx\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_m{f}_{1}dx& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_m{f}_{2}dx& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_m{f}_{m}dx\end{array}\right].$

[0056] A condition for the system of the homogeneous linear equation to have a nontrivial solution is: a value of the coefficient determinant of the system of the homogeneous linear equation is zero, to obtain a frequency equation.

[0057] In the embodiment, through the above-mentioned calculation steps, a stiffness matrix and a mass matrix of the multi-segment stepped beam with the following common boundary conditions can be derived, and the natural frequency values of the beam in different orders can be obtained by solving for the eigenvalue problem. The boundary conditions include:

[0058] (1) one end of the beam is simply supported and hinged, and the other end of the beam is clamped;

[0059] (2) one end of the beam is a simply supported and hinged, and the other end of the beam is free;

[0060] (3) one end of the beam is clamped, and the other end is free;

[0061] (4) both ends of the beam are clamped;

[0062] (5) one end of the beam is simply supported and hinged, and the other end of the beam is constrained by a wire spring and a torsion spring;

[0063] (6) one end of the beam is clamped, and the other end of the beam is constrained by a wire spring and a torsion spring; and

[0064] (7) both ends of the beam are constrained by a wire spring and a torsion spring.

[0065] For a multi-segment continuous beam with any one of the above-mentioned boundary conditions, after obtaining the expressions of its mass matrix and stiffness matrix, its cross-sectional shape, cross-sectional dimensions, total length and lengths of segments of the beam, and material parameters of the segments of the beam can be changed arbitrarily, and the circular frequency values of each order of the multi-segment continuous beam under corresponding changes can be quickly obtained by using Mathematica, thereby solving the problem that there is no calculation formula or method to calculate the natural circular frequencies of different orders of the current multi-segment continuous beams with different lengths, sizes and materials under given elastic boundary conditions.

Embodiment 1

[0066] Taking the cantilever multi-segmental continuous beam shown in the figure as an example, after a mass matrix and a stiffness matrix of its bending vibration are given, the natural circular frequency can be calculated through the matrix eigenvalue problem. This method is suitable for cantilever multi-segment beams with different segment lengths, different cross-sectional shapes and different cross-sectional dimensions.

[0067] As shown in the figure, a total length of the beam is L and the beam is divided into 2 segments, where a length of a left segment is L.sub.1; a mass per unit volume of the left segment is ρ.sub.1; an area of a cross section of the left segment is A.sub.1; a moment of inertia of the cross section of the left segment is I.sub.1; and an elastic modulus of the left segment is E.sub.1. A length of a right segment is L.sub.2; a mass per unit volume of the right segment is ρ.sub.2; an area of a cross section of the right segment is A.sub.2; a moment of inertia of a cross section of the right segment is I.sub.2; and an elastic modulus of the right segment is E.sub.2.

[0068] It is assumed that w(x, t) is the lateral displacement of the cross section of the multi-segment continuous beam from the coordinate origin x at the t moment.

[0069] Based on the variable separation method, the modal solution is set as follows:

w(x,t)=W(x)e.sup.iwt   (4).

[0070] The function is set as follows:

[00013]${\mathrm{Kp}}_1=E_1{I}_1\left[\begin{array}{cccc}{\int }_0^{L_1}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_0^{L_1}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ {\int }_0^{L_1}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_0^{L_1}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_0^{L_1}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_0^{L_1}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\end{array}\right],\mathrm{.Math.}$$K{p}_p=E_p{I}_p\left[\begin{array}{cccc}{\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_1}{{\mathrm{dx}}^2}\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_2}{{\mathrm{dx}}^2}\mathrm{dx}& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\frac{d^2{f}_m}{{\mathrm{dx}}^2}\mathrm{dx}\end{array}\right]$${\mathrm{Ks}}_1={k_1\left[\begin{array}{cccc}{f}_1{f}_1& f_1{f}_2& \mathrm{.Math.}& f_1{f}_m\\ f_1{f}_2& f_2{f}_2& \mathrm{.Math.}& f_2{f}_m\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ f_1{f}_m& f_2{f}_m& \mathrm{.Math.}& f_m{f}_n\end{array}\right]}_{x=0},{\mathrm{Ks}}_2={k_2\left[\begin{array}{cccc}{f}_1{f}_1& f_1{f}_2& \mathrm{.Math.}& f_1{f}_m\\ f_1{f}_2& f_2{f}_2& \mathrm{.Math.}& f_2{f}_m\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ f_1{f}_m& f_2{f}_m& \mathrm{.Math.}& f_m{f}_n\end{array}\right]}_{x=L},{\mathrm{Ks}}_3={K_1\left[\begin{array}{cccc}\frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_1}{\mathrm{dx}}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_m}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\end{array}\right]}_{x=0},{\mathrm{Ks}}_4={K_2\left[\begin{array}{cccc}\frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_1}{\mathrm{dx}}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_2}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ \frac{{\mathrm{df}}_1}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \frac{{\mathrm{df}}_2}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}& \mathrm{.Math.}& \frac{{\mathrm{df}}_m}{\mathrm{dx}}\frac{{\mathrm{df}}_m}{\mathrm{dx}}\end{array}\right]}_{x=L}$$M_1={\rho }_1{A}_1\left[\begin{array}{cccc}{\int }_0^{L_1}{f}_1{f}_1\mathrm{dx}& {\int }_0^{L_1}{f}_1{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}{f}_1{f}_m\mathrm{dx}\\ {\int }_0^{L_1}{f}_2{f}_1\mathrm{dx}& {\int }_0^{L_1}{f}_2{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}{f}_2{f}_m\mathrm{dx}\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_0^{L_1}{f}_m{f}_1\mathrm{dx}& {\int }_0^{L_1}{f}_m{f}_2\mathrm{dx}& \mathrm{.Math.}& {\int }_0^{L_1}{f}_m{f}_m\mathrm{dx}\end{array}\right],\mathrm{.Math.}$$M_p={\rho }_p{A}_p\left[\begin{array}{cccc}{\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_1{f}_{1}dx& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_1{f}_{2}dx& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_1{f}_{m}dx\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_2{f}_1\mathrm{dx}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_2{f}_{2}dx& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_2{f}_{m}dx\\ \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}& \mathrm{.Math.}\\ {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_m{f}_{1}dx& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_m{f}_{2}dx& \mathrm{.Math.}& {\int }_{L_1+L_2\mathrm{.Math.}{L}_{i-1}}^{L_1+L_2\mathrm{.Math.}{L}_i}{f}_m{f}_{m}dx\end{array}\right]$