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Unformatted text preview: ME 563 Mechanical Vibrations Lecture #15 Finite Element Approximations for Rods and Beams Continuous system vibration equations of motion are appropriate for applications where the crosssectional properties of the component are nearly constant. However, if the geometry or material properties change as a function of x (e.g., stepped shaft, cracked component, etc.), then a modeling technique is needed to describe these discontinuities. Finite element models are one means to describe such discontinuities. These models represent continuous systems using discrete (lumped parameter) models. Need for Finite Elements 1 Separation of variables? Rod Elements 2 Consider a rod undergoing longitudinal vibrations. If we wish to model the rod using lumped elements, how do we choose the mass and stiffness values? We use energy methods: KE model =KE rod PE model =PE rod Rod Element Equations 3 The resulting forcemotion relationships for a rod element are given by: Elemental stiffness matrix: Elemental mass matrix: Per unit length! Uniform Rod Example 4 Lets consider a rod with uniform crosssectional properties as an example of how to utilize finite elements. For a rod of total length L , crosssectional area A , modulus E , and density per unit volume , we can calculate the stiffness and mass lumped parameters for each element of the rod on the previous slide. We will use 2 elements to describe the rod: L L/2 L/2 = Node 1 Node 2 Node 3 Per unit vol! Uniform Rod Example...
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 Fall '09

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