hw_1_solu

hw_1_solu - ME163 Mechanical Vibrations(Winter 2009 HW-1...

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Unformatted text preview: ME163 Mechanical Vibrations (Winter 2009) HW-1 Due 1.15.08 in class-25% if late within 24 hours-50% if later than that 1. This problem is a review of some necessary background from mathematics. Consider the following differential equation ¨ x ( t ) + 2 ˙ x ( t ) + 2 x ( t ) = ˙ y ( t ) + 2 y ( t ) Such equation are often solved by taking the Laplace transform of the input y ( t ) and output x ( t ) which is defiend as follow X ( s ) = integraldisplay ∞ e- st x ( t ) dt Y ( s ) = integraldisplay ∞ e- st y ( t ) dt where s is a complex number. Take the Laplace transform of both sides of the equation with x (0) = ˙ x (0) = 0 ,y (0) = 0 to obtain X ( s )( s 2 + 2 s + 2) = Y ( s )( s + 2) Divide X ( s ) by Y ( s ) to obtain a quantity called the Transfer Function H ( s ). H ( s ) = X ( s ) Y ( s ) = s + 2 s 2 + 2 s + 2 Later we will see that it determines how the output x ( t ) depends on the input y ( t ).We shall see more about H ( s ) later in the course, for now we would like to play with complex number • We call the roots of the polynomial in the numerator of the transfer functin as zeors and that of the denominator as poles . Find the zeros and poles of the transfer function. Zero is at s =- 2 Poles is at s =- 1 ± ı • Represent the complex poles obtained above in the Euler form (Hint: re ıθ ) and compute the sum, the product and the ratio of the two. In Polar coordinates Z 1 = √ 2 e ı 3 π 4 Z 2 = √ 2 e ı- 3 π 4 Z 1 + Z 2 = √ 2 e ı 3 π 4 + √ 2 e ı- 3 π 4 =- 2 Z 1 · Z 2 = √ 2 e ı 3 π 4 · √ 2 e ı- 3 π 4 = 2 Z 1 Z 2 = e ı π 2 =- ı ME163 Mechanical Vibrations 1 HW1 • Express the sum, product and ratio in Cartesian coordinates. In Cartesian coordinates Z 1 =- 1 + ı 1 Z 2 =- 1- ı 1 Z 1 + Z 2 =- 2 Z 1 · Z 2 = 2 Z 1 Z 2 =- ı the ratio can be the different sign depending on which you choose as the numerator 2. The purpose of this problem is to remind you on how to work with the2....
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hw_1_solu - ME163 Mechanical Vibrations(Winter 2009 HW-1...

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