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Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk mode 1 mode 2 mode 3 mode 4 EULER BENDING VIBRATION

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Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk z y Equation of motion: For vibration, assume y(x,t)=Y(x)cos( t), so This has general solution Boundary condition for a fee end at z=0: mass per unit length m flexural rigidity EI, length L Free vibration of a beam

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Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk so i.e. C=A and D=B Boundary condition for a free end at z=L: so and or, in matrix form,

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Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk For a non-trivial solution, the determinant must be zero, so 0 1 Exact solutions for L:

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Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk From aL the frequencies of free vibration are found using a j = 22.37, 61.67, , ,... or a j The corresponding mode shapes are obtained by substituting j into the matrix equation to find the ratio between A and B so that The location of nodal points is then found by looking for where Y(z)=0

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Hugh Hunt, Trinity College, Cambridgewww.hughhunt.co.uk The location of the nodal points needs to be computed numerically, and the values are: Position of nodal points for a beam of L=1000mm (measured in mm from one end) mode 1: mode 2: mode 3: mode 4: mode 5: mode 6: mode 7: mode 8:

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