Question 7.4: Two mass points (equal mass m) lie on a frictionless horizon...
Two mass points (equal mass m) lie on a frictionless horizontal plane and are fixed to each other and to two fixed points A and B by means of springs (spring tension T , length l).
(a) Establish the equation of motion.
(b) Find the normal vibrations and frequencies and describe the motions.
(a) For the vibrating chain with n mass points, which are equally spaced by the distance l, the equations of motion
\frac{d^{2}y_{N}}{dt^{2}}= \frac{T}{ml}(y_{N−1} −2y_{N} +y_{N+1}) (N= 1, . . . , n)were established. For the first and second mass point, we have
\ddot{y}_{1} = k(y_{0} − 2y_{1} +y_{2}) = k(y_{2} − 2y_{1}),\ddot{y}_{2} = k(y_{1} − 2y_{2} +y_{3}) = k(y_{1} − 2y_{2}) (7.40)
with k = T/ml; the chain is clamped at the points A and B, i.e., y_{0} = y_{3} = 0.
(b) Solution ansatz: y_{1} = A_{1} cos ωt, y_{2} = A_{2} cos ωt (ω = eigenfrequency). Insertion into (7.40) yields
(2k −ω^{2})A_{1} −kA_{2} = 0,(2k −ω^{2})A_{2} −kA_{1} = 0. (7.41)
To get the nontrivial solution, the determinant of coefficients must vanish, i.e.,
D=\begin{vmatrix}2k −ω^{2} & −k \\ −k & 2k −ω^{2}\end{vmatrix}=0,i.e., ω^{4} +3k^{2} − 4kω^{2} = 0, from which it follows that ω^{2}_{1}= 3k, ω^{2}_{2}= k.
Insertion in (7.41) yields A_{1} = A_{2} for ω_{2} and A_{1} = −A_{2} for ω_{1}. This is an opposite-phase and an equal-phase vibration, respectively. We note that the vibration with the higher frequency has opposite phases and a “node,” while the vibration with lower frequency has equal phases and a “vibration antinode.”