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Having trouble with this impedance problem

  1. Nov 15, 2005 #1
    The problem follows:
    Suppose you have a massless dashpot having two moving parts 1 and 2 that can move relative to one another along the x direction, which is transverse to the string direction z. Friction is provided by a fluid that retards the relative motion of the two moving parts. The friction is such that the force needed to maintain relative velocity [tex] x_1 - x_2 [/tex] between the two moving parts is [tex] Z_d*(x_1 - x_2) [/tex], where [tex] Z_d [/tex] is the impedance of the dashpot. The input (part 1) is connected to the end of a string of impedance Z_1 stretching from -infinity to 0. The output (part 2) is connected to a string of impedance [tex] Z_2 [/tex] that extends to z = infinity. Show that a wave incident from the left experiences an impedance at z = 0 which is as if the impedances [tex] Z_d [/tex] and [tex] Z_2 [/tex] where connected in parallel.
    I'm thinking:
    The wave incident from -infinity will hit the dashpot and experience a force in the opposite direction of [tex] x_1 * Z_L [/tex], where Z_L is the "load" impedance we are trying to show is equal to Z_d*Z_2 / (Z_d + Z_2). This means that a force [tex] x_1 * Z_L [/tex] is exerted on "part 1" of the dashpot. The second part of the dashpot experiences a force [tex] Z_2 * {x_2} [/tex] on it. Therefore, the dashpot has a "tension" of [tex] x_1 * Z_L + Z_2 * x_2 [/tex] and the two parts of the dashpot will be moving with relative velocity [tex] x_1 - x_2 [/tex]. So I have the equation [tex] x_1 * Z_L + Z_2 * x_2 [/tex] = [tex] Z_d (x_1 - x_2) [/tex]
    Problem is that this doesn't get me to my desired answer of Z_L = Z_d*Z_2/(Z_d+Z_2). I was wondering what other information I need to solve the problem.
    edit: I can't seem to get \\dot{x} to work. Please pretend all the x's have dots over them
    Last edited: Nov 15, 2005
  2. jcsd
  3. Nov 17, 2006 #2
    I'm not sure, but if you say that the tension of the dashpot is [tex] Z_d*(x_1 - x_2) [/tex], then this is the magnitude of the force the dashpot exerts on the strings. Using newtons third law, the strings will exert an opposite and equal force on the dashpot. Thus, we would have [tex] Z_d*(x_1 - x_2) [/tex] = [tex] Z_L*(x_1) [/tex] and [tex] Z_d*(x_1 - x_2) [/tex] = [tex] Z_2*(x_2) [/tex]. Using the 2nd eq., we can then express x2 in terms of x1 for the first equation, then divide by x1, and acquire the intended result. Somebody please correct me if I'm wrong. (Note, there should be dots on all the x's)
    Last edited: Nov 17, 2006
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