HW7_10_solns - ME 563 HOMEWORK # 7 SOLUTIONS Fall 2010...

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ME 563 HOMEWORK # 7 SOLUTIONS Fall 2010 PROBLEM 1: Given the mass matrix and two undamped natural frequencies for a general two degree-of-freedom system with a symmetric stiffness matrix, find the stiffness matrix, modal mass and stiffness for each mode of vibration, and the modal vectors (normalize the modal vectors so that the largest coefficient in each vector is of unity magnitude). First, note that the system has a rigid body mode, which we assume is described by the modal vector [1 1] T (i.e., semi-definite system). Also, recall that modal orthogonality can be expressed in the following way for undamped systems like this one with distinct modal frequencies: The equation K 1 /M 1 =0 implies that the modal stiffness for mode 1 is zero, so with the known form of the modal vector, we can find the stiffness relationship: (1) But we also know that for semi-definite systems, the stiffness matrix is singular: (2) With three unknowns and two equations (1) and (2), a third equation is needed that relates the stiffness parameters. This equation is found by moving to the second mode orthogonality expression: (3) However, this equation contains two more unknowns, the second modal vector coefficients, that must now be associated with two additional equations to solve the set. These two equations are also found using mass orthogonality relationships:
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2 (4) and (5) These equations can now be solved simultaneously to obtain the five desired unknowns: From Eq. (4), . Substituting this result into Eq. (3) yields , which when added to Eq. (1) yields or . Then this result can be back- substituted into Eq. (1) to determine that . Thus, the stiffness matrix is of the form . When these results are all substituted into Eq. (2), the following conclusion is drawn: , which is an identity and provides no further useful information. From Eq. (5), . The stiffness matrix is therefore given by . It was already determined in the previous development that the modal stiffness for mode 1 is zero and that the modal vectors are defined according to the ratios, . Remember that modal vectors are only defined to within a scale factor (they are under- determined). With these particular modal vector scaling selections, the full set of modal mass and modal stiffness values can be summarized as follows:
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4 PROBLEM 2: Calculate, plot (in MATLAB), and describe the response of the single story building in Figure 1 to the two different blast excitations, f(t) , shown. Assume M=100,000 kg , K=100 kN/m , C=1000 N-s/m , t o =4 sec , T o =12 sec , and F o =10 kN . What happens to the responses as t o approaches zero? The first thing we have to do is describe the excitations above analytically; only then can we use the frequency response function to calculate the response. We use a different approach for each input. On the left, the excitation is a transient excitation – it starts at a certain time and ends a later time. On the right, the excitation is a periodic excitation – it starts at -
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HW7_10_solns - ME 563 HOMEWORK # 7 SOLUTIONS Fall 2010...

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