Modeling

Vibration Basics

• Free Vibration: $F=0$
• Undamped: $c=0$
• EoM: $m\ddot{y}+ky=0$

Viscous Damping

• $m\ddot{y}=F-ky-c\dot{y}$
• $F_s=ky$
• $F_d=c\dot{y}$
• EoM: $m\ddot{y}+c\dot{y}+ky=F$

Bending loading

• $m\ddot{y}+c\dot{y}+ky=F+mg$
• $k{y}_s=mg$
• EoM: $m\ddot{\hat{y}}+c\dot{\hat{y}}+k\hat{y}=F$

Applied Force with Damping

• $m\ddot{y}=-ky-c\dot{y}+kd+c\dot{d}$
• $m\ddot{y}=-c(\dot{y}-\dot{d})-k(y-d)$
• $m\ddot{y_R}+c\dot{y_R}+k\dot{y_R}=-m\ddot{d}$
• EoM: $F_{\text{applied}}=-m\ddot{d}$

> Initial Conditions at $t=0$

• Can treat as given for all problems (unless otherwise specified)
• $y(0)=y_0$
• $\dot{y}(0)=\dot{y_0}$