A homogeneous thin body of mass M, shown in Fig. 11.8, is suspended in the vertical plane by a thin wire of mass m

Question

A homogeneous thin body of mass M, shown in Fig. 11.8, is suspended in the vertical plane by a thin wire of mass [latex]m \ll M[/latex], length a, and with its ends hinged in smooth bearings. (i) Derive the equations for small vibrations of the body about its vertical equilibrium position [latex]\phi= heta=0[/latex]. Identify the inertia and stiffness matrices. (ii) Sketch the general formulation of the solution. Then let [latex]a=l,[/latex], and suppose that the body is a thin circular disk of radius [latex]R=\sqrt{2} l[/latex]. Determine the normal mode frequencies [latex]P_m[/latex], the ratios of the amplitudese [latex]C_{l m},[/latex] and thus derive the solution of the coupled equations of motion . Identify the normal equations of motion and their normal mode solutions as functions of the generalized variables.

Accepted Answer

Solution of (i). The system is scleronomic with two degrees of freedom [latex] heta[/latex] and [latex]\phi[/latex] defined in Fig. 11.8. Since the wire has negligible mass compared with M, its contribution to the total kinetic and potential energies is negligible. The total velocity of the center of mass G referred to the body frame [latex]\psi=\left\{G ; \mathbf{e}_r, \mathbf{e}_\phi ight\}[/latex] is given by [latex]\mathbf{v}^*=\mathbf{v}_H + \omega imes \mathbf{l}=a \dot{ heta} \mathbf{t} + l \dot{\phi} \mathbf{e}_\phi,[/latex] where [latex]t = \sin(\phi - heta)e_r + \cos(\phi - heta)e_q,[/latex] The total kinetic energy of the system, by (10.101), is

[latex]K(\mathscr{B}, t)=\frac{1}{2} m \mathbf{v}^* \cdot \mathbf{v}^* + \frac{1}{2} \boldsymbol{\omega} \cdot \mathbf{I}_C \boldsymbol{\omega}[/latex] (10.101)

[latex]T=\frac{1}{2} M\left[a^2 \dot{ heta}^2 \sin ^2(\phi - heta) + (a \dot{ heta} \cos (\phi - heta) + l \dot{\phi})^2 ight] + \frac{1}{2} I_G \dot{\phi}^2[/latex], (11.92a)

and the total gravitational potential energy is given by

[latex]V=M g[a(1 - \cos heta) + l(1 - \cos \phi)][/latex]. (11.92b)

We note that no coordinates are absent. For small vibrations, only terms of second order in all of the small quantities [latex]\phi, \dot{\phi}, heta, \dot{ heta}[/latex] are retained in (11.92a) and (11.92b). Therefore , the first term in (11.92a) is of higher order and may be neglected ; and with [latex]\cos (\phi - heta) \cong 1 - (\phi - heta)^2 / 2[/latex], to terms of second order the total kinetic and potential energies of the system for small vibrations about the vertical equilibrium configuration are thus given by

[latex]\begin{aligned}T &=\frac{1}{2}\left(I_G + M l^2 ight) \dot{\phi}^2 + \frac{1}{2} M\left(2 l a \dot{\phi} \dot{ heta} + a^2 \dot{ heta}^2 ight), \V &=\frac{1}{2} M g\left(l \phi^2 + a heta^2 ight)\end{aligned}.[/latex] (11.92c)

The potential energy is a positive definite, homogeneous quadratic function of the generalized coordinates [latex] heta[/latex] and [latex]\phi[/latex]. Hence, clearly, the equilibrium configuration [latex]\phi= heta=0[/latex] is infinitesimally stable. Similarly, the total kinetic energy is a homogeneous quadratic function of the generalized velocities [latex]\dot{ heta}[/latex] and [latex]\dot{\phi}[/latex] With [latex]I_H=I_G + M l^2[/latex] in accordance with the parallel axis theorem in (11.92c), the symmetric inertia and stiffness matrices in (11.84) and (11.83) are thus identified as

[latex]T\left(\dot{q}_r ight)=\frac{1}{2} M_{k l} \dot{q}_k \dot{q}_l[/latex] (11.84)

[latex]V\left(q_r ight)=\frac{1}{2} K_{k l} q_k q_l[/latex] (11.83)

[latex]\left[M_{k l} ight]=\left[\begin{array}{ll}I_H & M l a \M l a & M a^2\end{array} ight], \quad\left[K_{k l} ight]=\left[\begin{array}{ll}M g l & 0 \0 & M g a\end{array} ight] .[/latex] (11.92d)

We note in passing that the stiffness matrix is diagonal and all of its nontrivial components are positive; so, it is quite evident that [latex]K u \cdot u>0[/latex] holds for all [latex]u eq 0[/latex].

The hinge constraints are workless, so the system is conservative with a Lagrangian function

[latex]L=\frac{1}{2} I_H \dot{\phi}^2 + \frac{1}{2} M\left(2 l a \dot{\phi} \dot{ heta} + a^2 \dot{ heta}^2 ight) - \frac{1}{2} M g\left(l \phi^2 + a heta^2 ight),[/latex] (11.92e)

and Lagrange's equations (11.66) now yield the equations of small vibration,

[latex]\frac{d}{d t}\left(\frac{\partial L}{\partial \dot{q}_r} ight) - \frac{\partial L}{\partial q_r}=0, \quad r=1,2, \ldots, n[/latex] (11.66)

[latex]\begin{gathered}I_H \ddot{\phi} + M a l \ddot{ heta} + M g l \phi=0, \M l a \ddot{\phi} + M a^2 \ddot{ heta} + M g a heta=0\end{gathered}.[/latex] (11.92f)

Solution of (ii). The natural frequencies for the small vibrations are determined by (11.89). With (11.92d), we have

[latex]\operatorname{det}\left|K_{k l} - p^2 M_{k l} ight|=0[/latex] (11.89)

[latex]\operatorname{det}\left[\begin{array}{ll}M g l - p^2 I_H & -p^2 M l a \-p^2 M l a & M g a - p^2 M a^2\end{array} ight]=0[/latex].

This reduces to a quadratic equation in p²:

[latex]\left(M g l - p^2 I_H ight)\left(M g a - p^2 M a^2 ight) - p^4 M^2 l^2 a^2=0[/latex], (11.92g)

which yield s two eigenfrequencies [latex]p_m .[/latex] Use of these in (11.90) provides two sets of equations for the ratios of the coefficients [latex]C_{l m}, m=1,2[/latex]; namely,

[latex]\left(K_{k l}-p_m^2 M_{k l} ight) C_{l m}=0[/latex] (11.90)

[latex]\left[\begin{array}{ll}M g l - p_m^2 I_H & -p_m^2 M l a \-p_m^2 M l a & M g a - p_m^2 M a^2\end{array} ight]\left[\begin{array}{l}C_{1 m} \C_{2 m}\end{array} ight]=0[/latex]. (11.92h)

Finally, use of these results in (11.91) yield s the solutions for [latex]q_1=\phi ext { and } q_2= heta[/latex]. We omit these general detail s and tum to a special case for illustration.

[latex]q_k=\sum_{m=1}^n C_{k m} \sin \left(p_m t + \phi_m ight), \quad k=1,2, \ldots, n[/latex] (11.91)

Consider a circular disk of radius [latex]R=\sqrt{2} l[/latex] and let [latex]a=l ext {. }[/latex] Then [latex]I_G=M R^2 / 2=M l^2[/latex], and [latex]I_H=2 M l^2[/latex]. The characteristic equation (11.92g) thus simplifies to [latex]p^4 - 3 p_0^2 p^2 + p_0^4=0[/latex] with positive roots

[latex]p_1=p_0 \sqrt{\frac{1}{2}(3 + \sqrt{5})}, \quad p_2=p_0 \sqrt{\frac{1}{2}(3 - \sqrt{5})} ext {, }[/latex] (11.92i)

where [latex]p_0 \equiv \sqrt{g / l}[/latex], and ( 11.92h) becomes

[latex]\left[\begin{array}{ll}p_0^2 - 2 p_m^2 & -p_m^2 \-p_m^2 & p_0^2 - p_m^2\end{array} ight]\left[\begin{array}{l}C_{1 m} \C_{2 m}\end{array} ight]=0 .[/latex] (11.92j)

For m = 1, 2 (no sum), we thus obtain the amplitude ratios

[latex]\frac{C_{21}}{C_{11}}=\frac{\left(p_0^2 - 2 p_1^2 ight)}{p_1^2}, \quad \frac{C_{12}}{C_{22}}=\frac{p_2^2}{\left(p_0^2 - 2 p_2^2 ight)}[/latex]. (11.92k)

Substitution here of the characteristic roots (11.92i) yields

[latex]C_{21}=-\frac{1}{2}(1 + \sqrt{5}) C_{11}, \quad C_{12}=\frac{1}{2}(1 + \sqrt{5}) C_{22} .[/latex] (11.92l)

Introducing these in (11.91) in which [latex]q_1=\phi[/latex] and [latex]q_2= heta[/latex], we obtain the general solution

[latex]\begin{aligned}\phi &=C_{11} \sin \left(p_1 t + \phi_1 ight) + \frac{1}{2}(1 + \sqrt{5}) C_{22} \sin \left(p_2 t + \phi_2 ight), \ heta &=-\frac{1}{2}(1 + \sqrt{5}) C_{11} \sin \left(p_1 t + \phi_1 ight) + C_{22} \sin \left(p_2 t + \phi_2 ight)\end{aligned}.[/latex] (11.92m)

The remaining constants [latex]C_{11}, C_{22}, {\phi}_1[/latex], and [latex]\phi_2[/latex] are determined upon specification of the initial data.

The normal mode motions, readily identified from (11.92m), are defined by

[latex]\xi_1=C_{11} \sin \left(p_1 t + \phi_1 ight), \quad \xi_2=C_{22} \sin \left(p_2 t + \phi_2 ight)[/latex], (11.92n)

and upon solving (11.92m) for the [latex]\xi_k \mathrm{~s}[/latex], we obtain the normal coordinates in terms of the original generalized coordinates:

[latex]\begin{aligned}&\xi_1=-\frac{1}{10} \sqrt{5}(2 heta + \phi(1 - \sqrt{5})), \&\xi_2=-\frac{1}{20}(\sqrt{5} - 5)(2 heta + \phi(1 + \sqrt{5}))\end{aligned}.[/latex] (11.92o)

These normal coordinates uncouple the original equations of motion (11.92f) applied to the circular disk so that the normal mode equations of motion are

[latex]\ddot{\xi}_k + p_k^2 \xi_k=0, \quad k=1,2( ext { no sum })[/latex], (11.92p)

in which the normal mode frequencies [latex]p_k[/latex] are given in (11.92i).

Question 11.12: A homogeneous thin body of mass M, shown in Fig. 11.8, is su...

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