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ECS221L29a

# ECS221L29a - • k A B y C x y D x u = u u = u aω u = u...

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STATICALLY INDETERMINATE PROBLEMS II Summary: Rigid body equilibrium Kinematics of infinitesimal rigid displacements ( u = u 0 + ϖ × r ) Lumped support models e.g. linear and torsional springs x y P F = 0, F = 0, M = 0 x 0x y 0y u = u -ωy, u = u + ωx plane Cartesian compon ents sp sp linear spring relation F = ku torsional spring relation M =βω

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Three Dimensional Problems Rigid body equilibrium Kinematics of infinitesimal rigid displacements u = u 0 + ϖ × r Lumped support models e.g. linear and torsional springs P F = 0 , M = 0 sp sp linear spring relation F = ku torsional spring relation M =βω
The Table Top Four unknown reactions: R A , R B , R C , R D Three equilibrium equations: One Force Constraint Equation x y P z A B C D a b z x y F = 0, M = 0, M = 0

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The Table Top Equilibrium: x y P z A B C D a b A B C D B C P C D P R + R + R + R = 0 -aR - aR + x P = 0 bR + bR - y P = 0
The Table Top Force Constraint: u = u 0 + ϖ × r x y P z A B C D a b ( 29 ( 29 0 z 0z x y u = u + ω×r u = u +ω i + ω j× xi + y j

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Unformatted text preview: • k A B y C x y D x u = u , u = u - aω , u = u + bω - aω , u = u + bω ⇒ C A B D A C B D A C B D R R R R u + u - u - u = 0 +--= 0 k k k k Force Constraint Equation The Table Top x y P z A B C D a b C A B D A C B D R R R R +--= 0 k k k k Force Constraint A B C D B C P C D P R + R + R + R = 0-aR - aR + x P = 0 bR + bR - y P = 0 Equilibrium 4 equations for 4 unknowns! The Table Top (equal stiffnesses) Non-dimensional variables Solution x y P z A B C D a b , ε P P A A B B C C D D ξ = x a, η = y a, = a b, r = R P, r = R P , r = R P r = R P ( 29 ( 29 ( 29 ( 29 A B C D 3 1 1 1 r =-ξ + εη , r = + ξ - εη , 4 2 4 2 1 1 1 1 r = -+ξ + εη , r =-ξ - εη 4 2 4 2 Are there points where the load can be applied where 1 or more reactions vanish? YES!!!...
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ECS221L29a - • k A B y C x y D x u = u u = u aω u = u...

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