Question 16.8: Derive the frequency equation for uniform pinned-clamped bea......

The initial parameters and kinematical conditions are shown in Fig.16.16. At the right support the displacement and slope are zero. Therefore we need to use expressions for displacement and slope at any section x

\begin{aligned}& Y(x)=Y_0 S(k x)+\theta_0 \frac{1}{k} T(k x)+\frac{M_0}{k^2 E I} U(k x)+\frac{Q_0}{k^3 E I} V(k x) \\& \theta(x)=Y_0 k V(k x)+\theta_0 S(k x)+\frac{M_0}{k E I} T(k x)+\frac{Q_0}{k^2 E I} U(k x)\end{aligned}      (a)

Taking into account initial conditions \left(Y_0=0, M_0=0\right)  expressions for right fixed support are

\begin{aligned}& Y(l)=\theta_0 \frac{T(k l)}{k}+Q_0 \frac{V(k l)}{k^3 E I}=0 \\& \theta(l)=\theta_0 S(k l)+Q_0 \frac{U(k l)}{k^2 E I}=0\end{aligned}       (b)

Thus, the homogeneous linear algebraic equations with respect to \theta_0  and Q_0  are obtained. This system has a nontrivial solution if and only if the following determinant, which represents the frequency equation, is zero

\left[\begin{array}{cc}\frac{T(k l)}{k} & \frac{V(k l)}{k^3 E I} \\S(k l) & \frac{U(k l)}{k^2 E I}\end{array}\right]=0 \rightarrow T(k l) U(k l)-S(k l) V(k l)=0            (c)

According to Krylov-Duncan relationships (16.26, second row), we can present this equation in the trigonometric form

\begin{aligned} & S T-U V=\frac{1}{2}(\cosh k x \sin k x+\sinh k x \cos k x) \\ & T U-S V=\frac{1}{2}(\cosh k x \sin k x-\sinh k x \cos k x) \\ & S^2-U^2=\cosh k x \cos k x ; \quad T^2-V^2=2\left(S U-V^2\right)=\sinh k x \sin k x \\ & U^2-T V=\frac{1}{2}(1-\cosh k x \cos k x) ; \quad S^2-T V=\frac{1}{2}(1+\cosh k x \cos k x) \\ & T^2-S U=S U-V^2=\frac{1}{2} \sinh k x \sin k x ; \quad 2 S U=T^2+V^2 \end{aligned}         (6.26)

\cosh k l \sin k l-\sinh k l \cos k l=0 \rightarrow \tan k l=\tanh k l

The roots of this transcendental equation are \lambda_1=k_1 l=3.9266, \quad \lambda_2=k_2 l=7.0686, \cdots

The frequencies vibration are \omega_i=k_i^2 \sqrt{\frac{E I}{m}}=\frac{\lambda_i^2}{l^2} \sqrt{\frac{E I}{m}}, \quad i=1,2, \cdots  Corresponding first and second modes (eigenfunctions) are shown in Fig. 16.16. The nodal points N of eigenfunctions for the first mode are located at the supports; for second modeat the supports and additional point N in the span. For third form the nodal points at the span located at x / l=0.308  and 0.616 (Table A.24).