The Governing Equation(s) for a Spring-Mass-System Bernd Schr oder - - PowerPoint PPT Presentation

the governing equation s for a spring mass system
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The Governing Equation(s) for a Spring-Mass-System Bernd Schr oder - - PowerPoint PPT Presentation

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction The Governing Equation(s) for a Spring-Mass-System Bernd Schr oder logo1 Bernd Schr oder Louisiana Tech University, College


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SLIDE 1

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

The Governing Equation(s) for a Spring-Mass-System

Bernd Schr¨

  • der

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 2

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 3

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒ m

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 4

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

  • m

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 5

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

  • equilibrium

m

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 6

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

  • equilibrium

m x ✻ ❄

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 7

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

  • equilibrium

m x ✻ ❄ ❄ x′

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 8

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

−kx equilibrium m x ✻ ❄ ❄ x′

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 9

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

−kx equilibrium m x ✻ ❄ ❄ x′

mx′′ = −kx

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 10

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

−kx equilibrium m x ✻ ❄ ❄ x′

mx′′ = −kx mx′′ +kx = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 11

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

−kx equilibrium m x ✻ ❄ ❄ x′

mx′′ = −kx mx′′ +kx = 0 Free harmonic oscillator

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 12

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

−kx equilibrium m x ✻ ❄ ❄ x′

mx′′ = −kx mx′′ +kx = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 13

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

✻ −kx −cx′ equilibrium m x ✻ ❄ ❄ x′

mx′′ = −kx mx′′ +kx = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 14

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

✻ −kx −cx′ equilibrium m x ✻ ❄ ❄ x′

mx′′ = −kx−cx′ mx′′ +kx = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 15

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

✻ −kx −cx′ equilibrium m x ✻ ❄ ❄ x′

mx′′ = −kx−cx′ mx′′ +cx′ +kx = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 16

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

✻ −kx −cx′ equilibrium m x ✻ ❄ ❄ x′

mx′′ = −kx−cx′ mx′′ +cx′ +kx = 0 Free harmonic oscillator with damping

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 17

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

✻ −kx −cx′ equilibrium m x ✻ ❄ ❄ x′

mx′′ = −kx−cx′ mx′′ +cx′ +kx = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

slide-18
SLIDE 18

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

✻ −kx −cx′ equilibrium m x ✻ ❄ ❄ Fext ❄ x′

mx′′ = −kx−cx′ mx′′ +cx′ +kx = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

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SLIDE 19

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

✻ −kx −cx′ equilibrium m x ✻ ❄ ❄ Fext ❄ x′

mx′′ = −kx−cx′ +Fext mx′′ +cx′ +kx = 0

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

slide-20
SLIDE 20

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

✻ −kx −cx′ equilibrium m x ✻ ❄ ❄ Fext ❄ x′

mx′′ = −kx−cx′ +Fext mx′′ +cx′ +kx = Fext

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

slide-21
SLIDE 21

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

✻ −kx −cx′ equilibrium m x ✻ ❄ ❄ Fext ❄ x′

mx′′ = −kx−cx′ +Fext mx′′ +cx′ +kx = Fext Forced harmonic oscillator with damping

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

slide-22
SLIDE 22

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

✻ −kx −cx′ equilibrium m x ✻ ❄ ❄ Fext ❄ x′

mx′′ = −kx−cx′ +Fext mx′′ +cx′ +kx = Fext

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

slide-23
SLIDE 23

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

−kx equilibrium m x ✻ ❄ ❄ Fext ❄ x′

mx′′ = −kx−cx′ +Fext mx′′ +cx′ +kx = Fext

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

slide-24
SLIDE 24

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

−kx equilibrium m x ✻ ❄ ❄ Fext ❄ x′

mx′′ = −kx+Fext mx′′ +kx = Fext

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

slide-25
SLIDE 25

logo1 The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction

✓ ✒ ☎ ✆ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ✓ ✒ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ☎ ✆ ✓ ✒

−kx equilibrium m x ✻ ❄ ❄ Fext ❄ x′

mx′′ = −kx+Fext mx′′ +kx = Fext Forced harmonic oscillator without damping

Bernd Schr¨

  • der

Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System