1 / 6

mode 1

EULER BENDING VIBRATION. mode 4. mode 3. mode 2. mode 1. Free vibration of a beam. mass per unit length m flexural rigidity EI , length L. y. z. Equation of motion: For vibration, assume y ( x,t )= Y ( x )cos( w t ), so This has general solution

arawn
Download Presentation

mode 1

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

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

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

  4. For a non-trivial solution, the determinant must be zero, so 1 0 Exact solutions for aL: 4.730 7.853 10.996 14.137 Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

  5. From aL the frequencies of free vibration are found using aj= 22.37, 61.67, 120.90, 199.86, ... or aj The corresponding mode shapes are obtained by substituting aj 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 Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

  6. 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: 224 776 mode 2: 132 500 868 mode 3: 94 356 644 906 mode 4: 73 277 500 723 927 mode 5: 60 226 409 591 774 940 mode 6: 51 192 346 500 654 808 949 mode 7: 44 166 300 433 567 700 834 956 mode 8: 39 147 265 382 500 618 735 853 961 Hugh Hunt, Trinity College, Cambridge www.hughhunt.co.uk

More Related