15.11 Finite Element Random Response of Plates
561
15.12 The equation of motion of a SDF system is as follows:
¨
x + 2ζ p ˙
x + p
2 x =
F(t)
m
F(t) is a stationary random process with zero mean and a white spectrum
S 0 . Determine E [X
2
] when = ζ0 and 0.1.
15.13 The mass and stiffness matrices of a two degree of freedom system are as
follows:
[M] =
10 0
0 20
and [K ] = 5000
1 −1
−1 2
The free vibration analysis of the system gave the following frequencies and
mode shapes:
{ p} =
12.1
29.2
and [] =
1
1
0.707 −0.707
Assume = ζ0.05 and the excitation force vector as
{F(t)} =
1
0
F e (t)
where F 0 (t) is a stationary random process with zero mean and white spectral
density S 0 .
Find the response spectral density and m the mean square of the displacement
of the first mass.
15.14 Determine the random response at the centre of a uniform flexural beam
simply supported at both ends, when a random concentrated load acts at the
midspan. The load is considered to be random stationary, with zero mean
and with white spectral density S 0 (ω) = S 0 .
References
1. J.G. Bendat and A.G. Piersol, Measurement and Analysis of Random Data, John Wiley and
Sons, New York, 1966.
2. S.H. Crandall, Random Vibration, The Technology Press of MIT, Massachusetts, Cambridge,
1962.
3. F. Dinca and C. Theodosin, Nonlinear and Random Vibrations, Academic Press, 1973.
4. Y.K. Lin, Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, 1967.
5. N.H. Newman and E. Rosenblueth, Fundamentals of Earthquake Engineering, Prentice Hall,
New Jersey, 1973.
6. R.W. Clough and J. Penzien, Dynamics of Structures, 2nd Edition, McGraw-Hill, New York,
1993.
7. A.K. Chopra, Dynamics of Structures, Prentice Hall, New Jersey, 1995.
561
15.12 The equation of motion of a SDF system is as follows:
¨
x + 2ζ p ˙
x + p
2 x =
F(t)
m
F(t) is a stationary random process with zero mean and a white spectrum
S 0 . Determine E [X
2
] when = ζ0 and 0.1.
15.13 The mass and stiffness matrices of a two degree of freedom system are as
follows:
[M] =
10 0
0 20
and [K ] = 5000
1 −1
−1 2
The free vibration analysis of the system gave the following frequencies and
mode shapes:
{ p} =
12.1
29.2
and [] =
1
1
0.707 −0.707
Assume = ζ0.05 and the excitation force vector as
{F(t)} =
1
0
F e (t)
where F 0 (t) is a stationary random process with zero mean and white spectral
density S 0 .
Find the response spectral density and m the mean square of the displacement
of the first mass.
15.14 Determine the random response at the centre of a uniform flexural beam
simply supported at both ends, when a random concentrated load acts at the
midspan. The load is considered to be random stationary, with zero mean
and with white spectral density S 0 (ω) = S 0 .
References
1. J.G. Bendat and A.G. Piersol, Measurement and Analysis of Random Data, John Wiley and
Sons, New York, 1966.
2. S.H. Crandall, Random Vibration, The Technology Press of MIT, Massachusetts, Cambridge,
1962.
3. F. Dinca and C. Theodosin, Nonlinear and Random Vibrations, Academic Press, 1973.
4. Y.K. Lin, Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York, 1967.
5. N.H. Newman and E. Rosenblueth, Fundamentals of Earthquake Engineering, Prentice Hall,
New Jersey, 1973.
6. R.W. Clough and J. Penzien, Dynamics of Structures, 2nd Edition, McGraw-Hill, New York,
1993.
7. A.K. Chopra, Dynamics of Structures, Prentice Hall, New Jersey, 1995.