6.22 Lagrange’s Equation
255
6.11 Determine the natural frequencies and modes of the system shown in the
figure for the two-storey frame idealised as a shear building. Normalise the
modes.
6.12 The properties of a three-storey shear building are given in the figure. Derive
the Rayleigh damping matrix such that the damping ratio is 5% for the first
and third modes. Compute the damping ratio for the second mode.
w = 100 k = 168
{ p}
T
= {12.01 25.47 38.90} rad/s
[] =
⎡
⎣
0.6375 0.95527 1.5778
1.2750 0.9829 − 1.1270
1.9125 − 1.9642 0.4508
⎤
⎦
6.13 The mass and stiffness matrix of a MDF system is given by
[M] = m
⎡
⎣
1 0 0
0 1 0
0 0 2
⎤
⎦ , [K ] = k
⎡
⎣
2 − 1 0
− 1 3 − 2
0 − 2 2
⎤
⎦
The modal matrix has worked out to be equal to
255
6.11 Determine the natural frequencies and modes of the system shown in the
figure for the two-storey frame idealised as a shear building. Normalise the
modes.
6.12 The properties of a three-storey shear building are given in the figure. Derive
the Rayleigh damping matrix such that the damping ratio is 5% for the first
and third modes. Compute the damping ratio for the second mode.
w = 100 k = 168
{ p}
T
= {12.01 25.47 38.90} rad/s
[] =
⎡
⎣
0.6375 0.95527 1.5778
1.2750 0.9829 − 1.1270
1.9125 − 1.9642 0.4508
⎤
⎦
6.13 The mass and stiffness matrix of a MDF system is given by
[M] = m
⎡
⎣
1 0 0
0 1 0
0 0 2
⎤
⎦ , [K ] = k
⎡
⎣
2 − 1 0
− 1 3 − 2
0 − 2 2
⎤
⎦
The modal matrix has worked out to be equal to