Response of st ructur es to ra ndom excitation The classical method of dete r mining the r esponse of a

structure to excitation requires the solution of the

differential equation (Newland [39]) relating the excitation x(t) and the response y(t) in

r, n„ ,nv dnx0

Y

_{•• i}

_{• j}

TT

This relationship however cannot be used in random vibration work because

(i) inadequate data is available to determine the

coefficients A,B & C so that the complete differential equation is not available.

(ii) even if the equation is known, a complete time history of x(t) is not obtainable because of its

random nature.

Analysis of the response of any structure to random . excitation is therefore best carried out using the form of its response power spectral density, described in the previous chapter. This requires the determination of the

structural frequency response curve commonly called the

receptance (Bishop & Johnson [4]) of the structure.

The receptance of the structure is defined as the response d of the structure to a unit applied sinusoidal force p of frequency f. It should be noted that the receptance is different at different points of the structure and relates only two specific points on the structure, A the point at which the force is applied and B the point at which the

response is measured i.e.

dfi = oCAB(if)PA C3.ll)

Determination of oZ-(if) will therefore give the response of the structure to any given force p. The receptance and hence the response is complex, with a real and an imaginary part, the physical significance, being that there is a time lag between the application of the force p and the occurance of

the response d. For a single degree of freedom structure with a hysteretic damping factor the equation of motion may be written as

mx + k(l + i^)x = p sin cot (3.12)

from which the receptance is found to be 1

■C*(if) = --- r---2--- (3.13) k - 4 - i r m f + i ^ k

Robson [2l shows that the receptance of a multi-degree of freedom structure can be expressed in terms of its natural frequencies and mode shapes (equation 1.2). For a structure with light damping and hence pronounced resonance peaks the

receptance becomes

lor-(if) | 2 = £ Ewr-(xA)]2 [wr (xB)2]___________ (3.14)

r lV2[ l ^4{(fr2 " f2)2+rirfr4 }]

where wr (x) is the r ,th mode shape

fr is the r th natural frequency

and Mp is the generalised mass at mode r

r L. 2,. •

given by Jowr (x)mdx

Here structural damping is taken to be hysteretic only (applicable to light structures) and any coupling between the structure and a medium such as air is assumed to be negligible. ^ the hysteretic damping factor for the

structure is calculated from its response curves and is

discussed in more detail later.

The natural frequencies and mode shapes of a simple struc­ ture can be determined using the exact method of solving differential equations or by approximations using various ener ethods. Ftbr more complicated structures however the

finite element method which is basically an enery method,

.described in chapter 2,is increasingly used because of difficulties encountered with other methods. In this work therefore the natural frequencies and mode shapes of the box structure are predicted using a finite element computer program package developed by the author and given in Appendix 3. Once the receptance of the structure is known a prediction

of the response spectral density is possible using the

relationship given previously in equation (1.1) provided the input spectral density is also known.

Sd(f) =|«dp(if)i2 Sp(f)

(1.1)

Therefore once the receptance of the structure is known, the

response spectral density can be obtained for any given excitation spectral density, from which other parameters describing the response such as the mean square value and probability distribution are obtained using equations (3.7) and (3.8).