#### Introduction

This sheet gives some insight about stiffness and damping and their effect on the dynamics of mechanical systems.

#### Influence of stiffness

Stiffness increases the tracking behavior the displacement x of the end-effector (mass m) in relation to the input $x_{in}$ (a stiff actuator). Moreover, it decreases the influence of the external force $F_e$, which is often a disturbance to the system.

#### Influence of damping

Damping is difficult! Damping can be regarded as loss of energy. However, the positive effect of damping is that it damps oscillations and resonances.

#### Damping prediction

The damping of mechanical systems is hard to predict. Rule of thumb: damping decreases with increasing frequency. Joints and other system impurities increase damping.

$d=2\zeta\sqrt{cm}$ with viscous damping ratio $\zeta$:

System | ζ [-] |
---|---|

Metals in elastic range | 0.01 |

Continuous metal structures | 0.02 - 0.04 |

Metal structures with joints | 0.03 - 0.07 |

Plastics (hard - soft) | 0.02 - 0.05 |

Rubber | 0.05 |

Sintered material (piezos) | 0.05 |

Airpots (vibration isolation tables) | 0.07 |

#### Schematic overview

1 mass $m$, 1 spring $c$, 1 damper $d$, input $x_{in}$ , external force $F_e$

#### Differential equation

$m\ddot{x}+d({\dot{x}-\dot{x}}_{in})+c(x-x_{in})=F_e$

#### Tracking – design rule

When designing a system that has to track the input $x_{in}$ and that needs to be insensitive to disturbance force $F_e$, then design ‘light and stiff’.

#### Vibration isolation – design rule

When designing a system that needs to be insensitive to vibrations $x_{in}$ (such as ground vibrations), then design ‘heavy and soft’.

#### Eigen frequency

At this point the spring energy is converted into kinetic energy: $cx=m\ddot{x}$ hence: $c\hat{x}=m{\hat{x}\omega}^2$ and thus: $\omega=\sqrt{c/m}$

#### Response to the external force: $H_1=\frac{x}{F_e}=\frac{1}{ms^2+ds+c}$

Properties:

- Eigen frequency: $\omega_{ei}=\sqrt{\frac{c}{m}}\sim\sqrt c\sim\frac{1}{\sqrt m}$
- Gain at $\omega=0 : A_i=\frac{1}{c}\sim\frac{1}{c}$

#### Response to the input: $H_1=\frac{x}{F_e}=\frac{ds+c}{ms^2+ds+c}$

Properties:

Eigen frequency: $\omega_{ei}=\sqrt{\frac{c}{m}}\sim\sqrt c\sim\frac{1}{\sqrt m}$

2nd cross-over frequency: $\omega_{ei}=\frac{c}{d}\sim c\sim\frac{1}{d}$

(from $cx=d\dot{x}$ hence: $c\hat{x}=d\hat{x}\omega$ and thus: $\omega=c/d$ )Gain at $\omega=0$ : $A_i=\frac{c}{c}\ =1=0\ dB$