Laboratory notes Report 1

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MECHANICAL SYSTEM DYNAMICS Assignment 1 Part 1A –Model of a cantilever beam I. La Paglia Computethenaturalfrequenciesandthemodeshapesofacantileverbeam,bymakingreferencetothestandingwavesolutionofaslenderbeaminbendingvibration. Studytheforcedresponseofthesystem,bycomputingitsFrequencyResponseFunction(forassignedinputandoutputpositions). Contents: ▪Data of the reference structure (geometry and material properties) ▪Vibration modes of the cantilever beam ▪Frequency Response Function Target 2 I. La PagliaMechanical Systems Dynamics Data of the reference structure 3 L b h Aluminum beam with rectangular cross-section I. La PagliaMechanical Systems Dynamics 4 I. La PagliaMechanical Systems Dynamics Vibration modes of the cantilever beam 1.Standing wave solution 𝑤𝑥,𝑡=[𝐴cos𝛾𝑥+𝐵sin𝛾𝑥+𝐶Cosh𝛾𝑥+𝐷Sinh(𝛾𝑥)]cos(𝜔𝑡+𝜑) 2.Boundary conditions 3.Matrix formulation 𝐻(𝜔)𝑧=0(𝑧vector of unknown coefficients) 4.Solution of the characteristic equation det𝐻(𝜔)=0→𝜔𝑖 (numerical solution in Matlab) 5.Mode shapes computation𝜔𝑖→𝐻(𝜔𝑖)𝑧(𝑖)=0→Φ𝑖𝑥 6.Plot the mode shapes with the associated natural frequencies L b h 5 I. La PagliaMechanical Systems Dynamics A normalization is recommended for the visualization of the mode shapes. Vibration modes of the cantilever beam Mode shapes 6 I. La PagliaMechanical Systems Dynamics Frequency Response Function w (xi , t) 𝑭𝒌𝒕=𝑭𝒌𝟎𝒆𝒋𝜴𝒕 1.Frequency Response Function 𝐺(𝑗Ω)=෍𝑖=1 𝑛Φ𝑖𝑥𝑖Φ𝑖(𝑥𝑘)/𝑚𝑖 −Ω2+𝑗2𝜉𝑖𝜔𝑖Ω+𝜔𝑖2 2.Choose input and output positions (𝑥𝑘and 𝑥𝑖respectively) 3.Define proper damping values (in the order of 1%) 𝜉𝑖 4.Compute the modal mass (hint: trapz.mMatlabfunction)𝑚𝑖=׬0 𝐿𝑚Φ𝑖2𝑥𝑑𝑥 5.Plot the FRF 7 I. La PagliaMechanical Systems Dynamics Frequency Response Function 8 I. La PagliaMechanical Systems Dynamics Assignment 1 –Part 1A…