NORMAL MODES
9.8 (It is recommended that the reader does not attempt this question until exercise 9.6
has been studied.)
Find a real linear transformation that simultaneously reduces the quadratic
forms
3 x^2 +5y^2 +5z^2 +2yz+6zx− 2 xy ,
5 x^2 +12y^2 +8yz+4zx
to diagonal form.
9.9 Three particles of massmare attached to a light horizontal string having fixed
ends, the string being thus divided into four equal portions each of lengthaand
under a tensionT. Show that for small transverse vibrations the amplitudesxi
of the normal modes satisfyBx=(maω^2 /T)x,whereBis the matrix
2 − 10
− 12 − 1
0 − 12
.
Estimate the lowest and highest eigenfrequencies using trial vectors( 343 )T
and( 3 − 43 )T. Use also the exact vectors(
1
√
21
)T
and(
1 −
√
21
)T
and compare the results.
9.10 Use the Rayleigh–Ritz method to estimate the lowest oscillation frequency of a
heavy chain ofNlinks, each of lengtha(=L/N), which hangs freely from one
end. (Try simple calculable configurations such as all links but one vertical, or
all links collinear, etc.)
9.5 Hints and answers9.1 See figure 9.6.
9.3 (b)x 1 =(cosωt+cos
√
2 ωt),x 2 =−cos√
2 ωt, x 3 =(−cosωt+cos√
2 ωt).
At various times the three displacements will reach 2, , 2 respectively. For exam-
ple,x 1 canbewrittenas2cos[(√
2 −1)ωt/2] cos[(√
2+1)ωt/2], i.e. an oscillation
of angular frequency (√
2+1)ω/2 and modulated amplitude 2cos[(√
2 −1)ωt/2];
the amplitude will reach 2after a time≈ 4 π/[ω(√
2 −1)].
9.5 As the circuit loops contain no voltage sources, the equations are homogeneous,
and so for a non-trivial solution the determinant of coefficients must vanish.
(a)I 1 =0,I 2 =−I 3 ; no current inPQ; equivalent to two separate circuits of
capacitanceCand inductanceL.
(b)I 1 =− 2 I 2 =− 2 I 3 ; no current inTU; capacitance 3C/2 and inductance 2L.
9.7 ω=(2. 634 g/l)^1 /^2 or (0. 3661 g/l)^1 /^2 ;θ 1 =ξ+η,θ 2 =1. 431 ξ− 2. 097 η.
9.9 Estimated, 10/ 17 <Maω^2 /T < 58 /17; exact, 2−
√
2 ≤Maω^2 /T≤2+