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# Springs Dashpots and the Law of Newton dydy abcy dx dx Suppose we have a spring a mass and a dashpot PDF document - DocSlides

jane-oiler | 2014-12-13 | General

### Presentations text content in Springs Dashpots and the Law of Newton dydy abcy dx dx Suppose we have a spring a mass and a dashpot

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Page 1

Springs, Dashpots, and the Law of Newton dydy abcy dx dx

Page 2

Suppose we have a spring, a mass, and a dashpot.

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According to Hooke’s Law, if we pull the mass units to the right, then there is a restorative force in the the opposite direction equal to –kx where k>0 Fkx

Page 4

The dashpot is a dampening device such as found on a typical building door. It’s effect is proportional to the velocity of the mass over time, and we can express that as below: (0) dx Fkxcc dt !

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Finally, Newton’s Second Law of Motion tells us that force=mass x acceleration. However, acceleration is the second derivative of position, and so we can write the left-hand side of the equation as follows: dxdx mkxc dt dt

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The end result is a second degree homogeneous linear differential equation with constant coefficients. dxdx mckx dt dt

Page 7

What follows are some initial value problems with graphs of the solutions. dxdx mckx dt dt

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560 (0)3,(0)8 dxdx dt dt xx Example 1:

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Example 1: 12 23 12 23 12 560 2,3 () ()23 tt tt rr rr xtCeCe xtCeCe 560 (0)3,(0)8 dxdx dt dt xx

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Example 1: 00 1212 00 1212 (0)3 (0)23238 xCeCeCC xCeCeCC 12 23 12 23 12 560 2,3 () ()23 tt tt rr rr xtCeCe xtCeCe 560 (0)3,(0)8 dxdx dt dt xx

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Example 1: 12 23 1,2 ()2 tt CC xtee 00 1212 00 1212 (0)3 (0)23238 xCeCeCC xCeCeCC 560 (0)3,(0)8 dxdx dt dt xx 12 23 12 23 12 560 2,3 () ()23 tt tt rr rr xtCeCe xtCeCe

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Example 1: 23 ()2 tt xtee 00 1212 00 1212 (0)3 (0)23238 xCeCeCC xCeCeCC 560 (0)3,(0)8 dxdx dt dt xx 12 23 12 23 12 560 2,3 () ()23 tt tt rr rr xtCeCe xtCeCe

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690 (0)3,(0)8 dxdx dt dt xx Example 2: HAVE FUN!

brPage 3br According to Hookes Law if we pull the mass units to the right then there is a restorative force in the the opposite direction equal to kx where k0 Fkx brPage 4br The dashpot is a dampening device such as found on a typical building door ID: 23506

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Page 1

Springs, Dashpots, and the Law of Newton dydy abcy dx dx

Page 2

Suppose we have a spring, a mass, and a dashpot.

Page 3

According to Hooke’s Law, if we pull the mass units to the right, then there is a restorative force in the the opposite direction equal to –kx where k>0 Fkx

Page 4

The dashpot is a dampening device such as found on a typical building door. It’s effect is proportional to the velocity of the mass over time, and we can express that as below: (0) dx Fkxcc dt !

Page 5

Finally, Newton’s Second Law of Motion tells us that force=mass x acceleration. However, acceleration is the second derivative of position, and so we can write the left-hand side of the equation as follows: dxdx mkxc dt dt

Page 6

The end result is a second degree homogeneous linear differential equation with constant coefficients. dxdx mckx dt dt

Page 7

What follows are some initial value problems with graphs of the solutions. dxdx mckx dt dt

Page 8

560 (0)3,(0)8 dxdx dt dt xx Example 1:

Page 9

Example 1: 12 23 12 23 12 560 2,3 () ()23 tt tt rr rr xtCeCe xtCeCe 560 (0)3,(0)8 dxdx dt dt xx

Page 10

Example 1: 00 1212 00 1212 (0)3 (0)23238 xCeCeCC xCeCeCC 12 23 12 23 12 560 2,3 () ()23 tt tt rr rr xtCeCe xtCeCe 560 (0)3,(0)8 dxdx dt dt xx

Page 11

Example 1: 12 23 1,2 ()2 tt CC xtee 00 1212 00 1212 (0)3 (0)23238 xCeCeCC xCeCeCC 560 (0)3,(0)8 dxdx dt dt xx 12 23 12 23 12 560 2,3 () ()23 tt tt rr rr xtCeCe xtCeCe

Page 12

Example 1: 23 ()2 tt xtee 00 1212 00 1212 (0)3 (0)23238 xCeCeCC xCeCeCC 560 (0)3,(0)8 dxdx dt dt xx 12 23 12 23 12 560 2,3 () ()23 tt tt rr rr xtCeCe xtCeCe

Page 13

690 (0)3,(0)8 dxdx dt dt xx Example 2: HAVE FUN!

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