Find the steady periodic solution for the forcedmass-spring-dashpot system mx\'\

Question

Find the steady periodic solution for the forcedmass-spring-dashpot system mx'' + cx' + kx = F0cost. Find the amplitude C() of steady periodic forcedoscillations with frequency sketch the graph of C()and find the practical resonance frequency () and find thepractical resonance frequency (if any). a.) m = 1, c = 4, k = 5, F0 = 10 PLEASE SHOW STEPS TO SOLVE Find the steady periodic solution for the forcedmass-spring-dashpot system mx'' + cx' + kx = F0cost. Find the amplitude C() of steady periodic forcedoscillations with frequency sketch the graph of C()and find the practical resonance frequency () and find thepractical resonance frequency (if any). a.) m = 1, c = 4, k = 5, F0 = 10 PLEASE SHOW STEPS TO SOLVE

Explanation / Answer

Apparently you're using E&P 4th ed. This is problem 16 onp. 222. You'll find a useful formula (21) on p. 219. But here's the procedure ... First find the characteristic equationand solve it with the quadratic formula. r^2 + 4 r + 5 = 0, r = -2 ± i. The (associated)homogeneous equation has the general solution e^(-2t) (c1 cos(t) + c2 sin(t)). Now we must find the "steadyperiodic function", that is, the particular solution to thenon-homogeneous equation. Assuming not = 1, we can useundetermined coefficients, starting with A cos(t) + Bsin(t). Substitute in your original equation andcollect terms; you'll get the simultaneous equations 4B - A(^2 - 5) = 10 and 4A + B(^2 - 5) =0. Solve them out as functions of . Aand B will have the same denominator, ^4 + 6 ^2 + 25,which is never equal to 0. The numerator of A is 10 (5 -^2) and the numerator of B is 40. Now you cancalculate the amplitude of your steady periodic function,A^2 + B^2. It is 10/(^4 + 6^2 +25). Now take the derivativeof that, and set it to zero to find the extrema. If you lookin the back of the book, you'll see what you'll find. If youfind any extrema, that will be a practical resonancefrequency. If you don't, there is no practical resonancefrequency. If you still have problems, email me atzhandele1@verizon.net, I have a pdf that may help.

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