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 of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✒ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✒ m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✒ m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ equilibrium ✒ ✆ ✓ ☎ ✒ ✆ ✒ m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ equilibrium ✒ ✆ ✓ ☎ ✒ ✆ ✻ ✒ x ❄ m logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ equilibrium ✒ ✆ ✓ ☎ ✒ ✆ ✻ ✒ x ❄ m x ′ ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ − kx equilibrium ✒ ✆ ✓ ☎ ✻ ✒ ✆ ✻ ✒ x ❄ m x ′ ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ = − kx ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ − kx equilibrium ✒ ✆ ✓ ☎ ✻ ✒ ✆ ✻ ✒ x ❄ m x ′ ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ = − kx ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ + kx = 0 ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ − kx equilibrium ✒ ✆ ✓ ☎ ✻ ✒ ✆ ✻ ✒ x ❄ m x ′ ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ = − kx ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ + kx = 0 ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ − kx Free harmonic oscillator equilibrium ✒ ✆ ✓ ☎ ✻ ✒ ✆ ✻ ✒ x ❄ m x ′ ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ = − kx ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ + kx = 0 ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ − kx equilibrium ✒ ✆ ✓ ☎ ✻ ✒ ✆ ✻ ✒ x ❄ m x ′ ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ = − kx ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ + kx = 0 ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ − kx equilibrium ✒ ✆ ✓ ☎ ✻ ✒ ✆ ✻ − cx ′ ✒ x ✻ ❄ m x ′ ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ = − kx − cx ′ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ + kx = 0 ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ − kx equilibrium ✒ ✆ ✓ ☎ ✻ ✒ ✆ ✻ − cx ′ ✒ x ✻ ❄ m x ′ ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ = − kx − cx ′ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ + cx ′ + kx = 0 ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ − kx equilibrium ✒ ✆ ✓ ☎ ✻ ✒ ✆ ✻ − cx ′ ✒ x ✻ ❄ m x ′ ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System

The Parts Intrinsic Forces, No Friction Friction Friction and Outside Forces Outside Forces and No Friction � � � � � � � � � � � � � � � � � � � � ✓ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ = − kx − cx ′ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ mx ′′ + cx ′ + kx = 0 ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ ✒ ✆ ✓ ☎ − kx Free harmonic oscillator equilibrium ✒ ✆ ✓ ☎ ✻ ✒ ✆ ✻ − cx ′ ✒ with damping x ✻ ❄ m x ′ ❄ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Governing Equation(s) for a Spring-Mass-System