Question 16.12: The simply supported uniform massless beam (m=0) of length L......

The origin is placed at the right end of the beam, axis x is directed to the left. The numbering of the portions is from right to left, i.e., AC and CB. Vector of initial parameters (right support) and vector of a state at left support are

\textbf Y _0=\left[\begin{array}{c} y_0 \\ \varphi_0 \\ M_0 \\ Q_0 \end{array}\right]=\left[\begin{array}{c} 0 \\ \varphi_0 \\ 0 \\ Q_0 \end{array}\right] ; \quad \textbf Y _{x=2 l}=\left[\begin{array}{c} y_B \\ \varphi_B \\ M_B \\ Q_B \end{array}\right]=\left[\begin{array}{c} 0 \\ \varphi_B \\ 0 \\ Q_B \end{array}\right]

Relationships between initial parameters and parameters at the left end (final point) can be constructed as multiplication of corresponding matrices (initial parameters and transfer matrices) considering them successively from right to left. The concept of the transfer matrix is presented by the following relationship between the initial (point A) and final (point B) states

\textbf Y _{x=2 l}= \textbf A _{C B} \textbf A _M \textbf A _{A C} \textbf Y _0

Here \textbf A _{C B} and \textbf A _{A C} are matrices of massless portions CB and AC, while \textbf A _M is a matrix of lumped mass  \textbf M. In expanded form, this matrix ratio is written as

\underbrace{\left[\begin{array}{c} 0 \\ \varphi_B \\ 0 \\ Q_B \end{array}\right]}_{ \textbf Y _L}=\underbrace{\left[\begin{array}{cccc} 1 & l & l^2 / 2 E I & l^3 / 6 E I \\ 0 & 1 & l / E I & l^2 / 2 E I \\ 0 & 0 & 1 & l \\ 0 & 0 & 0 & 1 \end{array}\right]}_{ \textbf A _{C B}} \cdot \underbrace{\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ M \omega^2 & 0 & 0 & 1 \end{array}\right]}_{ \textbf A _M} \cdot \underbrace{\left[\begin{array}{cccc} 1 & l & l^2 / 2 E I & l^3 / 6 E I \\ 0 & 1 & l / E I & l^2 / 2 E I \\ 0 & 0 & 1 & l \\ 0 & 0 & 0 & 1 \end{array}\right]}_{ \textbf A _{A C}} \cdot \underbrace{\left[\begin{array}{c} 0 \\ \varphi_0 \\ 0 \\ Q_0 \end{array}\right]}_{ \textbf Y _0}

The result of the intermediate operation of multiplication is

  \underbrace{\left[\begin{array}{c} 0 \\ \varphi_B \\ 0 \\ Q_B \end{array}\right]}_{ \textbf Y _L}=\underbrace{\left[\begin{array}{cccc} 1+M \omega^2 l^3 / 6 E I & l & l^2 / 2 E I & l^3 / 6 E I \\ M \omega^2 l^2 / 2 E I & 1 & l / E I & l^2 / 2 E I \\ M \omega^2 l & 0 & 1 & l \\ M \omega^2 & 0 & 0 & 1 \end{array}\right]}_{ \textbf A _{C B} \textbf A _M} \cdot \underbrace{\left[\begin{array}{c} \varphi_0 l+Q_0 l^3 / 6 E I \\ \varphi_0+Q_0 l^2 / 2 E I \\ Q_0 l \\ Q_0 \end{array}\right]}_{ \textbf A _{A C} \textbf Y _0}

Since at the left support B are y_B=0 and M_B=0, then the first and third equations lead to set of two homogeneous equations with respect to unknown initial parameters \varphi_0and Q_0. After isolation of these unknown initial parameters, we obtain the following system of homogeneous equations

\begin{aligned} & \left(2 l+\frac{M \omega^2 l^4}{6 E I}\right) \varphi_0+\left[\frac{l^3}{6 E I}\left(1+\frac{M \omega^2 l^3}{6 E I}\right)+\frac{7}{6} \frac{l^3}{E I}\right] Q_0=0 \\ & \left(M \omega^2 l^2\right) \varphi_0+\left(2 l+\frac{M \omega^2 l^4}{6 E I}\right) Q_0=0 \end{aligned}

Nontrivial solution occurs if determinant of coefficients at unknowns is equal to zero, so the frequency equation becomes

\left(2 l+\frac{M \omega^2 l^4}{6 E I}\right)^2-M \omega^2 l^2\left[\frac{8}{6} \frac{l^3}{E I}+M \omega^2\left(\frac{l^3}{6 E I}\right)^2\right]=0 .

This equation may be presented as follows

(2 l+k)^2-k(8 l+k)=0, \quad \frac{M \omega^2 l^4}{6 E I}=k

Solution of this equation is k=l, so the frequency of free vibration becomes \omega=\sqrt{\frac{6 E I}{M l^3}}. In terms of the total span L=2 l we get well-known result \omega=\sqrt{\frac{48 E I}{M L^3}}.

Obviously, this result can be obtained in other simpler ways. However, the transfer matrix method is out of competition in the numerical analysis of complex systems.

Example of some complex system may be a rotor. Its design diagram is a continuous beam, which contains portions of varying length, different stiffness, lumped and distributed masses, as well as some additional features, for example, dynamical absorber, etc. When analyzing such systems using the transfer matrix method, you should select a specific portion with basic parameters (l,EI,m) and similar parameters for the remaining portions should be expressed in terms of basic parameters. All matrix procedures are performed using modern software.

Detailed catalog of the transfer matrices is presented in book (Ivovich 1981).