p0908 -

Info icon This preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 9.8. CHAPTER 9, PROBLEM 8 355 9.8 Chapter 9, Problem 8 Problem: The motion of a small cart of mass m is governed by two springs, a dashpot and an oscillating attachment, which moves horizontally with a displacement given by xA = a cos ω t. The length a is the magnitude of the oscillation and ω is frequency. Ignore effects of rolling friction on the cart’s motion. (a) Derive the differential equation governing the cart’s motion. HINT: Check that your equation makes sense for the limiting case k1 = k2 and the attachment not moving. (b) Determine the natural frequency of the system. (c) Determine the critical damping coefficient for the system. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. . ............................................................. ............ ............................................................. ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............ ............................................................. ............ ............................................................. ............ ............................................................. 2 ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............................................................. ............ • ............... ... ... ... ... ... ... ... ... ...1... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...•. ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ................................................................ ............ ... ... ... ... . ... .... .... .... .... .... .... .... . .... .... .... .... .... .... .... .... ............................................................. ................................................................................ ....................................................................................................................................................................................................................................................................................................................................... .......................................................................................................................................................................................................................................................................................................................................................................................................................................................... ................ ............ ............ ............ ............ ............ . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ......... . ........ . . . . . . . . . . . . .... ... ... ....... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... .................................................................................................................................................................................................................................................................................................................................................................................................... .................................................................................................................................................................................................................................................................................................................................................................................................... .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. x • • c • • m • k • xA = a cos ω t k f f So ut on: (a) The forces ac ng on he car are as shown n he fo ow ng f gure C ear y for mo on n he pos ve x d rec on bo h spr ng forces (re a ve o a d sp acemen x) and he dashpo force are d rec ed n he nega ve x d rec on The a achmen ’s mo on adds a force n he pos ve x d rec on for pos ve xA cx • • x m • • k2 (xA − xo ) k2 (x − xo ) k1 x The bas c equa on of mo on for h s sys em s mx = −cx − k1 x − k2 (x − xo ) + k2 (xA − xo ) ¨ where xo s he equ br um d sp acemen of Spr ng 2 Th s equa on can be rearranged o read mx + cx + (k1 + k2 )x = k2 xA ¨ F na y us ng he fac ha he d sp acemen of he a achmen s xA = a cos ω t he d fferen a equa on govern ng he car ’s mo on s mx + cx + (k1 + k2 )x = k2 a cos ω t ¨ In m ng case xA = 0 and k1 = k2 = k he equa on of mo on wou d s mp fy o mx + cx + 2kx = 0 ¨ Th s s phys ca y correc s nce we expec o have he spr ngs n ser es for h s m ng case (b) By nspec on he na ura frequency of h s sys em s g ven by 2 ωn = k1 + k2 m =⇒ ωn = k1 + k2 m 356 CHAPTER 9. MECHANICAL VIBRATIONS (c) By definition, the critical damping coefficient is given by cc = 2mωn . Thus, using the value of ωn determined in Part (b), we find cc = 2m k1 + k2 = 2 m(k1 + k2 ) m ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern