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Unformatted text preview: H * impact of a pendulum with a flat rigid surface * L H * packages that have some differential equations to solve and define some utility functions * L Needs @ "DifferentialEquations`NDSolveProblems`" D ; Needs @ "DifferentialEquations`NDSolveUtilities`" D ; Needs @ "DifferentialEquations`InterpolatingFunctionAnatomy`" D ; Needs @ "GUIKit`" D ; ClearAll @ "Global` * " D ; Off @ General:: spell D ; Off @ General:: spell1 D ; H * geometry of the pendulum * L L = 1; H * length of the pendulum @ m D * L g = 9.81; H * gravitational acceleration @ m s^2 D * L ro = 7800; H * density @ kg m^3 D * L R = 0.01; H * radius of the hemisphere end @ m D * L m = Pi R^2 L ro; H * mass @ kg D * L H * angular velocity * L alpha = 8 0, 0, theta'' @ t D< ; H * position of CM: C * L xC = H L 2 L * Cos @ theta @ t DD ; yC = H L 2 L * Sin @ theta @ t DD ; rC = 8 xC, yC, 0 < ; H * position of the tip A * L xA = L * Cos @ theta @ t DD ; yA = L * Sin @ theta @ t DD ; rA = 8 xA, yA, 0 < ; H * velocity of the tip A * L vA = D @ rA, t D ; H * gravitational force at C * L G = 8 0, m g, 0 < ; H * impact angle * L thetai = Pi 6; H * mass moment of inertia * L IC = m * L^2 12; IO = IC + m * H L 2 L ^2; H * equation of motion for free fall * L eqI = Simplify @ IO * alpha- Cross @ rC, G DD@@ 3 DD ; H *------------------* L sol0 = NDSolve @8 eqI 0, theta @ D == 0, theta' @ D == < , theta, 8 t, 0, Infinity < , Method fi 8 EventLocator, "Event" fi H theta @ t D- thetai L<D ; t0 = InterpolatingFunctionDomain @ First @ theta . sol0 DD@@ 1,- 1 DD ; theta0 = H Evaluate @ theta @ t D . sol0 D . t fi t0 L@@ 1 DD ; omega0 = Chop @H Evaluate @ D @ theta @ t D . sol0, t DD . t fi t0 L@@ 1 DDD ; vA0 = vA . 8 theta @ t D-> theta0, theta' @ t D fi omega0 < ; v0x = vA0 @@ 1 DD ; v0y = vA0 @@ 2 DD ; Print @ " " D Print @ "before impact" D H * Print @ "t0 = ",t0," @ s D " D * L Print @ "theta0 = ", theta0, " @ rad D = ", theta0 * 180 Pi, " @ deg D " D Print @ "omega0 = ", omega0, " @ rad s D " D Print @ "v0 = ", vA0, " @ m s D " D ; Print @ " " D H * thetaplot0 = Plot @ Evaluate @ theta @ t D .sol0 D * 180 Pi, 8 t,0,t0 < ,AxesLabel fi 8 "t @ s D ","theta @ deg D " < ,PlotRange fi Automatic D omegaplot0 = Plot @ Evaluate @ D @ theta @ t D .sol0,t DD , 8 t,0,t0 < , AxesLabel fi 8 "t @ s D ","omega @ rad s D " < ,PlotRange fi Automatic D * L AxesLabel fi 8 "t @ s D ","omega @ rad s D " < ,PlotRange fi Automatic D * L before impact theta0 = 0.523599 @ rad D = 30. @ deg D omega0 = 3.83601 @ rad s D v0 = 8- 1.91801, 3.32209, 0 < @ m s D Print @ "impact with a flat surface" D H * elastic compression * L E1 = 200 * 10^9; H * elastic modulus @ Pa D * L Sy = 1.12 * 10^9; H * yield strength @ Pa D * L nu = 0.33; H * Poisson's ratio * L Ep = HH 1- nu^2 L E1 + H 1- nu^2 L E1 L ^- 1; H * equivalent elastic modulus...
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This note was uploaded on 08/29/2011 for the course MECH 7630 taught by Professor Marghitu during the Fall '11 term at University of Florida.
- Fall '11