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consider the mapping w zn where n is an integer a generalization of the mapping w z2 use the polar representation of
find the images of the lines y x and y -x under the mapping w z2 also find the image of the general line through the
find the most general bilinear mapping that maps the unit circle z 1 in the z plane onto the unit circle w 1 in the w
for z x jy find the image region in the w plane corresponding to the semi-infinite strip x gt 0 0 y 2 in the z plane
the mapping w alphaz beta a beta both constant complex numbers maps the point z 1 j to the point w j and the point
show that the mapping w 1z maps the circle z - a a with a being a positive real constant onto a straight line in the
determine the image in the w plane of the circleunder the inversion mapping w
show that the bilinear mappingwhere theta0 is a real constant 0 le theta0 2pi z0 a fixed complex number and z0 its
given the complex mappingwhere w u jv and z x j y determine the image curve in the w plane corresponding to the
the two complex variables w and z are related through the inverse mappinga find the images of the points z 1 1 - j and
a cantilever beam of negligible weight and of length l is clamped at the end x 0 determine the deflection of the beam
a uniform cantilever beam of length l is subjected to a concentrated load w at a point distance b from the fixed end
the response xt of a system to a forcing function ut is determined by the differential equation modela determine the
using the convolution theorem determine the following inverse laplace transforms check your results by first expressing
use the initial- and final-value theorems to find the jump at t 0 and the limiting value as t rarr infin for the
discuss the applicability of the final-value theorem for obtaining the value of i2t as t rarr infin for the circuit of
using the final-value theorem check the value obtained for i2t as t rarr infin for the circuit of example 528example
the response of a given system to a unit step ut 1ht is given bywhat is the transfer function of the
determine the impulse responses of the linear systems whose response xt to an input ut is determined by the following
a feedback control system has characteristic equationwhere k is a constant gain factor determine the range of positive
the behaviour of a system having a gain controller is characterized by the characteristic equationwhere m c k and r are
repeat exercise 34 for a system whose response xt to an input ut is determined by the differential equationrepeat
consider the mass-spring-damper system of figure 518 which may be subject to two input forces u1t and u2t show that the
when testing the landing-gear unit of a space vehicle drop tests are carried out figure 517 is a schematic model of the
given that x 1 and dxdt 0 obtain the solution of the differential equationwhere gt is the piecewise-continuous