Ch 3-2 HW Solutions - 13 8 Estimate the number of cycles n...

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13 8. Estimate the number of cycles n required for a structural oscillation amplitude to decay to x % of its maximum. (a) What is the expression for the logarithmic decrement in terms of n and x ? (b) Solve for n in terms of and x . (c) Plot n as a function of for (i) x = 70% , (ii) x = 50% , (iii) x = 20% using the exact formula, and then using the approximate formula for = 0 : 1 and = 0 : 5 . Solution: (a) Expression for the logarithmic decrement in terms of n and x : In Example 3.3, the ± = ln x 1 x 2 = n T d = 2 p 1 2 : If instead of the decay over one cycle, we need the decay over n cycles, then following the solution to Example 3.3, we can write ± = 1 n ln 100 x ± = 1 n (ln 100 ln x ) ; where x is in percent. (b) Solve for n in terms of and x : In order to solve for n in terms of and x , we start with ± = 2 p 1 2 = 1 n ln 100 x ± : The number of cycles, n; is then given by n = p 1 2 2 ln 100 x ± : If an approximate value of n is desired, then for small we can approximate p 1 2 by 1 : (c) Plot n as a function of : n as a function of for x = 70% ; 50% ; and 20% : The solid lines are the curves for the exact solution and dotted lines for the approximate solution. The approximate solution is n = 1 2 ln 100 x ± : Note that the approximate solution always overestimates n:
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14 CHAPTER 3 SDOF VIBRATION: WITH DAMPING The number of cycles versus damping ratio for several decays to x % of maximum. clear z=[0:0.01:1]± ; x=[70 50 20]± ; [Z X]=meshgrid(z,x); n=sqrt(1-Z.^2)/2/pi./Z.*log(100./X); na=1./2/pi./Z.*log(100./X); plot(z,n,±k± ,z,na,±k²±) ylabel(±n±) xlabel(± n zeta±)
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22 CHAPTER 3 SDOF VIBRATION: WITH DAMPING 11. A bell-crank mechanism depicted in Figure 3.44 is rotated slightly and released to oscillate in free vibration. Derive the damped frequency of oscillation ! d and the critical damping constant c cr : The mass of the frame can be ignored compared to the mass attached to it at the end. Figure 3.44: Bell-crank mechanism. Solution: Draw a free-body diagram based on positive rotation in the CCW direction. Note that the force exerted by the damper is obtained by F c = c d dt ( b sin ) = cb _ cos Take the moment about O to obtain + X M O = I O ka sin ( a cos ) cb _ cos ( b cos ) mgl sin = ml 2 &;
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23 where the block is treated as a point mass so that I O = ml 2 : Note that if the dimensions of the block are not negligible when compared to length l; we must include it in the expression for the mass moment of inertia of the block. Simplifying, we obtain the equation of motion:
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This note was uploaded on 09/24/2010 for the course MECHANICAL 14:650:443 taught by Professor Benaroya during the Spring '10 term at Rutgers.

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Ch 3-2 HW Solutions - 13 8 Estimate the number of cycles n...

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