By J.R. Calvert

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X .!.. (t)dt] First Shifting L[e"'l'{t)] s s = F(s- a) Second Shifting Convolution: L[f"g] L[f(t- a)] =e-"'F(s) {Jf(u)g(t-u)du] = F(s)G(s) Partial Differentiation L[aj~~a)] = aa F(s,a) Time Multiplication L[if(t)] =--- = =--+-(t >a) a dF(s) ds Transform Pairs Function Laplace Transform 1 - H(t-1) =0 =1 tn 3-19 1 s t

0 Parallel axis theorem: X lxx= Iaa + Ayo2 where A is the area of the shape and y 0 is the perpendicular distance between an axis through the centroid, GG, and another parallel axis XX. IGG is the minimum second moment of area for all axes parallel to GG. E. E. E. = Q dt oiJ; oq; oq; oq; I i = 1,2,3 for a 3 degree of freedom system. E. are kinetic, potential and dissipative energy. 3 - - Moments of Inertia Definitions: Moment oflnertia: I xx = Jr dm 2 M r is the perpendicular distance of the element of mass dm from the axis XX; the integral is taken over the whole mass of the body.

Z 2 ,P(z) = - - e 2 J2; z Its cumulative distribution function lP(z) = J~ e -t12 dt -oo is the probability that the random variable is observed to have a value s; z (the shaded area shown in the graph ofN(0,1) on page 4-3). l) "'s ~ , where f(x)dx = 1 = tP -00 4-2 l k (x-p) l u Its cumulative distribution F(x) = e -tr 2 dt = cP( x ~11 ) . Tables of rp(z) and 4\z) may be used for f(x) and F(x) by taking X-11 z=--. 3 Experimental Samples x 1, x 2, ••• Xn denote a set of n observations of a random variable having a normal distribution whose mean 11 is unknown.