Mechanical Engineering - Mechanical Systems Dynamics

M01-Transversal vibrations of a beam

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Mechanical System Dynamics - Lecture Notes MSc. Mechanical Engineering A.A. 2022-2023 Vibration analysis of one-dimensional continuous systems Tranverse Vibrations of a beam Teacher: Prof. Stefano Melzi Trainer: Eng. Binbin Liu FabioSantoroContents 1 Wave equation2 2 Stationary solution32.1 Case: Pinned-Pinned Beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5 2.2 Case: Cantilever. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6 2.3 Case: multiple span beam. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8 3 Application of ICs (not required for the exam)9 A Hyperbolic sine and cosine functions10 B Recalls of Linear Algebra10B.1 Matrix inverse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10 B.2 Array as combination of bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 B.3 Orthogonal bases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11 1 1.Wave equation To study the transversal vibrations of a beam, let’s consider the model represented in figure1. It is a representation of a beam with lengthLand heighth. The space variable isxand the goal is to identify the expression of the vertical displacementw(x, t)Figure 1:Pinned-pinned beam model It is necessary to assess some hypothesis:1.small displacements 2.no damping (no dissipation) 3.no concentrated loads (constraints) along the span (just on the boundaries) 4.linear elastic behaviour:σ=E·ε, whereEis the Young’s modulus; isotropic relationship between stress and strain in the beam material (typical behaviour for metallic materials) 5.homogeneous material:•constant transversal areaA •constant mass per unit lengthm •constant Young’s modulusE 6.pure planar bending (no torsion) 7.slender beam (h/L≪1(