Question 15.15: Four mass points of mass m move on a circle of radius R. Eac...

The kinetic energy of the system is given by

$T = \frac{1}{2} m \sum\limits_{ν=1}^{4}{\dot{s}^{2}_{ν}} .$                           (15.44)

For small displacements from the equilibrium position, the potential reads

$V = \frac{1}{2} k \sum\limits_{ν=1}^{4}{(s_{ν+1} −s_{ν} )^{2}}, \ \ \ \ \ s_{4+1} = s_{1}.$                            (15.45)

We set $s_{ν} = R\varphi_{ν}$ , and take the angles $\varphi_{ν}$ as generalized coordinates. Then the Lagrangian is

$L = T −V = \frac{1}{2} mR^{2} \sum\limits_{ν=1}^{4}{\dot{\varphi}^{2}_{ν}}− \frac{1}{2} kR^{2} \sum\limits_{ν=1}^{4} {(\varphi_{ν+1} −\varphi_{ν} )^{2}}.$                                            (15.46)

From the Lagrange equations

$\frac{d}{dt} \frac{∂L}{∂ \dot{\varphi}_{ν}} = \frac{∂L}{∂\varphi_{ν}},$                               (15.47)

we find the equations of motion:

$= \frac{∂L}{∂\varphi_{ν}}.$                                   (15.48)

For the case of four mass points, we then obtain

With the ansatz $\varphi_{ν} = A_{ν} cos ωt, \ddot{\varphi}_{ν} =−A_{ν}ω^{2} cos ωt$, we are led to the following linear system of equations:

$\begin{pmatrix} 2\frac{k}{m}− ω^{2} & -\frac{k}{m} & 0 & -\frac{k}{m} \\ -\frac{k}{m} & 2\frac{k}{m}− ω^{2} & -\frac{k}{m} &0 \\ 0 &-\frac{k}{m} &2\frac{k}{m}− ω^{2} & -\frac{k}{m} \\ -\frac{k}{m}& 0 & -\frac{k}{m} & 2\frac{k}{m}− ω^{2}\end{pmatrix} \begin{pmatrix} A_{1}\\ A_{2} \\ A_{3} \\ A_{4} \end{pmatrix}=0$                               (15.50)

For the nontrivial solutions, the determinant of the coefficient matrix must vanish. This condition leads to the determining equation for the eigenfrequencies:

$\left(2 \frac{k}{m} − ω^{2}\right)^{2} \left(4 \frac{k}{m} −ω^{2}\right) (−ω^{2}) = 0.$                                           (15.51)

The frequencies are

$ω^{2}_{1} = 0, \ \ \ \ \ ω^{2}_{2}= 4 \frac{k}{m}, \ \ \ \ \ ω^{2}_{3}= ω^{2}_{4}= 2 \frac{k}{m}.$                                      (15.52)

To calculate the related eigenvibrations, we insert these frequencies into the system of equations (15.50).

(1) $ω^{2}_{1}= 0: A_{1} = A_{2} = A_{3} = A_{4}$. The system does not vibrate but performs a uniform rotation (Fig. 15.23(a)).

(2) $ω^{2}_{2} = 4k/m: A_{1} = A_{3} =−A_{2} =−A_{4}$. Two neighboring mass points perform an out-of-phase vibration (Fig. 15.23(b)).

(c) $ω^{2}_{3}= ω^{2}_{4}= 2k/m: A_{1} = A_{2} = −A_{3} = −A_{4}    or      A_{1} = A_{4} = −A_{2} = −A_{3}$. Two neighboring mass points vibrate in phase (Fig. 15.24(a,b)).