SolSec 6.7 - 6-Problems and Solutions Section 6.7(6.53...

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Unformatted text preview: 6-Problems and Solutions Section 6.7 (6.53 through 6.63)6.53Calculate the response of Example 6.7.1 for l= 1 m, E= 2.6 ×1010N/m2and ρ= 8.5 ×103kg/m3. Plot the response using the first three modes at x= l/2, l/4, and 3l/4. How many modes are needed to represent accurately the response at the point x= l/2?Solution:w(x,t)=0.02λ2σν2(-129ν+1ε-0.01ϖντχοσϖ2ντσινσνξν=1∞∑Whereσν=(2ν- 129π2λϖν=σνΕρϖδν= 0.9999ϖνForl= 1 mE= 2.6 × 1010Ν/μ2ρ= 8.5 × 103κγ/μ3Response using first three modes at x=λ2,λ4,3λ4plotted below.Three modes accurately represents the response atx=λ2. The error between a three and higher mode approximation is less than 0.2%.526-536-6.54Repeat Example 6.7.1 for a modal damping ratio of ζn= 0.01.Solution:Using ζn= 0.01 and the frequency given in the exampleϖδν=ϖν1-ζν2= 0.995ϖν,ϖν=2ν- 12λΕρ=σνΕρThe time response is then Tn(t)=Ανε-0.1ϖντσιν(ϖδντ+φν29and the total solution is:w(x,t)=Ανε-0.1ϖντσιν(ϖδντ+φν29ν=1∞∑σιν(2ν- 1292λπ ξThe initial conditions are:w(x,0)= 0.01ξλμανδϖτ(ξ,029= 0Therefore:0.01xl=ΑνσινφνσινσνξMultiply by sinσmxand integrate over the length of the bar to get0.01(-129μ+1λσμ2=Αμσινφμλ2μ= 1,2,3,...From the velocity initial conditionwt(x,0)= 0 =Αν-0.1ϖνσινφν+ϖδνχοσφνν=1∞∑σινσνξAgain, multiply by sinσmxand integrate over the length of the bar to getAm(-0.1ϖνσινφν+ϖδνχοσφν29λ2= 0Since Amis not zero this yields:tanφν=σινφνχοσφν=1-ζν30.1= 9.9499 ⇒φν= 1.4706 ραδ= 84.3°Substitution into the equation from the displacement initial condition yields:Am=0.01λ2σμ2(-129μ+11σινφν=0.0201λ2σμ2(-129μ+1The solution is thenw(x,t)=0.01λ2σμ2(-129μ+1ε-0.1ϖντσιν(ϖδντ+φν29ν=1∞∑σιν(2ν- 1292λπ ξ546-6.55Repeat Problem 6.53 for the case of Problem 6.54. Does it take more or fewer modes to accurately represent the response at l/2?Solution: Use the result given in 6.54 andl= 1 μΕ= 2.6 × 1010Ν/μ2ρ= 8.5 × 103κγ/μ3The response is plotted below atx=λ4,λ2,3λ4. An accurate representation of the response is obtained with three modes. The error between a three mode and a higher mode representation is always less than 0.2%.The results here are from Mathcad:556-566-6.56Calculate the form of modal damping ratios for the clamped string of equation (6.151) and the clamped membrane of equation (6.152).(6....
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SolSec 6.7 - 6-Problems and Solutions Section 6.7(6.53...

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