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Unformatted text preview: 6Problems and Solutions Section 6.8 (6.64 through 6.68)6.64Calculate the response of the damped string of Example 6.8.1 to a disturbance force of f(x,t) = (sin x/l) sin10t.Solution:f(x,t)= 10.Assume a solution of the form:wn(x,t)=(29(29whereXn(x)= Substitute into (6.158)&&+&2 = 10Multiply by sinn and integrate over the length of the string:&&+&+22==1101Only the particular solution is of interest since we are looking for the response to the disturbance force. Therefore, dropping the subscripts:&&+&+2= 10&&+&+2=10=Solution isT=(2029646whereA=1222 1002+10022=2222 1002(292+10024= 110222 100= 110 222 1002(,29=(1029 where Aand are given above.6566.65Consider the clampedfree bar of Example 6.3.2. The bar can be used to model a truck bed frame. If the truck hits an object (at the free end) causing an impulsive force of 100 N, calculate the resulting vibration of the frame. Note here that the truck cab is so massive compared to the bed frame that the end with the cab is modeled as clamped. This is illustrated in Figure P6.65.Solution: Assume constant area and constant material properties. Equation of motion:=(,29= 100(29(29Mode shapes (eigenvalues) of a fixedfree bar are (Table 6.1)Xn(x)= (2 129 2Assume a solution of the form:wn(x,t)=(29(29. Substitute into the equation of motion:&&Tn(2 129222(2 129 2= 100(29(29&&+2{}(2 129 2= 100(29(29wherec2==(2 1292. Multiply by sin(2n 129 2and integrate over the length of the rod:&&Tn+2= 2100 (2 129 2(29(29= 200(2 1292(29which has the solution:Tn(t)= 200(2 1292The total solution is:wn(x,t)= 400(2 129(2 1292=1sin(2n 1292(2 129 26666.66A rotating machine sits on the second floor of a building just above a support column as indicated in Figure P6.66. Calculate the response of the column in terms of E, A, and of the column modeled as a bar.of the column modeled as a bar....
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This note was uploaded on 11/28/2010 for the course ME 4440 taught by Professor Hill during the Winter '09 term at Detroit Mercy.
 Winter '09
 Hill

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