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 Fabio Santoro Contents 1 W ave equation 2 2 Stationary solution 3 2.1 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 functions 10 B Recalls of Linear Algebra 10 B.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 figure 1. It is a representation of a beam with lengthL and height h. The space variable is x and 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, where E is 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…