p0908

# p0908 -

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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. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ ............................................................. ............ 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.. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. 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 ...
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