By Howard B. Wilson, Louis H. Turcotte, David Halpern

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This vector has the form % [xmin,xmax,ymin,ymax] when xyz has % only two columns or the form % [xmin,xmax,ymin,ymax,zmin,zmax] % when xyz has three columns. 5 Nonlinear Motion of a Damped Pendulum Motion of a simple pendulum is one of the most familiar dynamics examples studied in physics. The governing equation of motion can be satisfactorily linearized for small oscillations about the vertical equilibrium position, whereas nonlinear effects become important for large deßections. For small deßections, the analysis leads to a constant coefÞcient linear differential equation.

The % transformation is approximated in series form % which converges very slowly near the corners. % % m - number of series terms used % r1,r2,nr - abs(z) varies from r1 to r2 in % nr steps % t1,t2,nt - arg(z) varies from t1 to t2 in % nt steps (t1 and t2 are measured % in degrees) % w - points approximating the square % b - coefficients in the truncated % series expansion which has the % form % % w(z)=sum({j=1:m},b(j)*z*(4*j-3)) % % User m functions called: cubrange %---------------------------------------------- 81: 82: 83: 84: 85: % Generate polar coordinate grid points for the % map.

The translating © 2003 by CRC Press LLC wave solution can be adapted to handle a string of Þnite length l by requiring y(0, t) = y(l, t) = 0. These end conditions, along with initial deßection f (x) ( deÞning F (x) between 0 and l ), are sufÞcient to continue the solution outside the original interval. We write the initial condition for the Þnite length string as y(x, 0) = f (x), 0 < x < l. To satisfy the end conditions, F (x) must be an odd-valued function of period 2l. Introducing a function g(x) such that g(x) = f (x), 0 ≤ x ≤ l and g(x) = −f (2l − x), l < x ≤ 2l leads to F (x) = sign(x)g(rem(abs(x), 2l)) where the desired periodicity is achieved using the MATLAB remainder function, rem.