Question
Asked Oct 31, 2019
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Take mx′′+cx′+kx=F0cos(ωt).mx″+cx′+kx=F0cos⁡(ωt). Fix c>0,c>0, k>0,k>0, and F0>0.F0>0. Consider the function C(ω).C(ω). For what values of mm (solve in terms of c,c, k,k, and F0F0) will there be no practical resonance (that is, for what values of mm is there no maximum of C(ω)C(ω) for ω>0ω>0)?

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Expert Answer

Step 1

First find the particular solution for the differential equation-

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mx" +cx + loc F cos ().

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Step 2

Auxiliary solution is

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x)Acos()Bsin (ex) differentiate it once and twice x'(t)-Aosin(r)+ Bøcos (x) x"(t)A cos()-Bo sin()

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Step 3

Substitute the obtained solution...

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m(-A cos(at)-Ba sin(at))+c(-Awsin (ax)+Bcos (at)) k(Acos(at)+Bsin (ox)) = F cos (ar) mA cos (ax)-mBø° sin(@x)+ -cAwsin (at)+ cBwcos() kA cos (at)kB sin (ct)= F cos(at) (mACBkA cos(a)+(-mBa -cA+kB)sin (ax)= F cos(ax)

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Advanced Math