By Dana Bell

Air strength shades Vol 1: 1926-42

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14) one can connect the whole bridge between the force and displacement matrices. e. 15) Now, we want to describe the stiffness matrix in a local coordinate system defined in such a way that x1 = y1 = y2 = 0: y 3 1 [K ]e ⎡ y 3 λ 1 x 23 2 + ⎢ x2 y3 ⎢ x2 ⎢ λ 2 x32 ⎢ − x2 ⎢ ⎢ y 3 λ 1 x3 x 23 + ⎢− Et x2 x2 y3 = ⎢ 2(1 − ν 2 ) ⎢ νx3 λ 1 x32 ⎢ x + x 2 ⎢ 2 λ x ⎢ − 1 23 ⎢ y3 ⎢ ⎢ −ν ⎢⎣ © 2005 by T. H. Kwon x 2 λ y x 23 + 1 3 x2 y3 x2 νx32 λ 1 x3 + x2 x2 x3 x 23 λ 1 x3 − x2 y3 x2 2 − λ1 − x 23 y3 29 symm.

2 ⎠ ∂φ x ∂x ∂φ x ∂φ ∂x ∂φ x ∂x ∂φ x ∂φ y ∂x * Gauss Theorem (Green’s Theorem in plane) r r ∇ ⋅ V d Ω = V ∫ ∫ ⋅ nˆdS Ω ∂Ω ∂φ ∂φ ∫ ∂x dxdy = ∫ φn ds, ∫ ∂y dxdy = ∫ φn ds x A © 2005 by T. H. Kwon ∂A y ∂A A 45 Example 3 : Hamilton’s Principle and Lagrange’s equation in dynamics Dynamic motion of a rigid body system can be described in several ways. Newton’s 2 law of motion is one of them. Other methods are based on energy concept . 22) (or kinetic potential) where qi : generalized coordinate T : kinetic energy V : potential energy One can easily derive Lagrange’s equation via calculus of variation from Hamilton’s principle that is represented by the following (case B in the summary).

F. n For equilibrium Fi = f i A + f i B + f i C + f i D ⎛ NDOF ⎞ ⎛ NDOF ⎞ ⎛ NDOF ⎞ = ⎜ ∑ K inA x nA ⎟ + ⎜ ∑ K inB x nB ⎟ + L + ⎜ ∑ K inD x nD ⎟ ⎝ n ⎠ ⎝ n ⎠ ⎝ n ⎠ Using the nodal compatibility xiA = xiB = xiC = xiD = xi one can rearrange the equilibrium equation as ⎛ NDOF ⎞ ⎛ NDOF ⎞ ⎛ NDOF ⎞ Fi = ⎜ ∑ K inA x nA ⎟ + ⎜ ∑ K inB x nB ⎟ + L + ⎜ ∑ K inD x nD ⎟ ⎝ n ⎠ ⎝ n ⎠ ⎝ n ⎠ A B C D A B C = K ii + K ii + K ii + K ii xi + K ii′ + K ii′ + K ii′ + K iDi′ xi′ ( + (K + K + (K + 0 +( 0 + 0 +( 0 +K A ij B ij A ik B im + 0 +K C ik +K C il + 0 ) ( ) + 0 )x + (K + K + 0 + 0 )x + 0 )x + (K + 0 + K + 0 )x + K )x + ( 0 + 0 + K + K )x + K )x + ( 0 + K + 0 + K )x D il D im j A ij ′ k A ik ′ B ij ′ C il ′ l m j′ C ik ′ B im ′ k′ D il ′ D im ′ l′ m′ which becomes one of the equations in the global matrix equation [K ]{x} = {F } .