The Differential Equation for a Vibrating String Bernd Schr oder - PowerPoint PPT Presentation

Model Forces The Equation The Differential Equation for a Vibrating String Bernd Schr¨ oder logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Modeling Assumptions logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Modeling Assumptions 1. The string is made up of individual particles that move vertically. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Modeling Assumptions 1. The string is made up of individual particles that move vertically. 2. u ( x , t ) is the vertical displacement from equilibrium of the particle at horizontal position x and at time t . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Modeling Assumptions 1. The string is made up of individual particles that move vertically. 2. u ( x , t ) is the vertical displacement from equilibrium of the particle at horizontal position x and at time t . u = 0 � u > 0 � � � � � � � � � ✠ � � � � � � � � � � � � u < 0 � ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✰ � F t ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✛ ✰ � F t ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✛ � F v ✰ ❄ � F t ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✛ α � F v ✰ ❄ � F t ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✛ α � F v ✰ ❄ � F t ✲ x x + ∆ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✿ � F t ✛ α � F v ✰ ❄ � F t ✲ x x + ∆ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✿ ✻ � � F v F t ✛ α � F v ✰ ❄ � F t ✲ x x + ∆ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✲ ✿ ✻ � � F v F t ✛ α � F v ✰ ❄ � F t ✲ x x + ∆ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✲ ✿ ✻ α ˜ � � F v F t ✛ α � F v ✰ ❄ � F t ✲ x x + ∆ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation Decomposing the Tensile Force ✲ ✿ ✻ α ˜ � � F v F t ✛ α � F v ✰ ❄ � F t ✲ x x + ∆ x F ( x ) ≈ F v ( x + ∆ x ) − F v ( x ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ = F t sin ( ˜ α ) − F t sin ( α ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ = F t sin ( ˜ α ) − F t sin ( α ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ = F t sin ( ˜ α ) − F t sin ( α ) 0 . 25 ≈ 14 . 3 ◦ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ � � = F t sin ( ˜ α ) − F t sin ( α ) sin ( θ ) ≈ tan ( θ ) , θ small logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ � � = F t sin ( ˜ α ) − F t sin ( α ) sin ( θ ) ≈ tan ( θ ) , θ small F t tan ( ˜ α ) − F t tan ( α ) ≈ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ � � = F t sin ( ˜ α ) − F t sin ( α ) sin ( θ ) ≈ tan ( θ ) , θ small F t tan ( ˜ α ) − F t tan ( α ) ≈ ✻ ✲ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ � � = F t sin ( ˜ α ) − F t sin ( α ) sin ( θ ) ≈ tan ( θ ) , θ small F t tan ( ˜ α ) − F t tan ( α ) ≈ ✻ ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ � � = F t sin ( ˜ α ) − F t sin ( α ) sin ( θ ) ≈ tan ( θ ) , θ small F t tan ( ˜ α ) − F t tan ( α ) ≈ ✻ ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ � � = F t sin ( ˜ α ) − F t sin ( α ) sin ( θ ) ≈ tan ( θ ) , θ small F t tan ( ˜ α ) − F t tan ( α ) ≈ ✻ ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ � � = F t sin ( ˜ α ) − F t sin ( α ) sin ( θ ) ≈ tan ( θ ) , θ small F t tan ( ˜ α ) − F t tan ( α ) ≈ ✻ 1 ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String

Model Forces The Equation The Vertical Force at a Point F ( x ) F v ( x + ∆ x ) − F v ( x ) ≈ � � = F t sin ( ˜ α ) − F t sin ( α ) sin ( θ ) ≈ tan ( θ ) , θ small F t tan ( ˜ α ) − F t tan ( α ) ≈ ✻ 1 ✲ x logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science The Differential Equation for a Vibrating String