Interpretation of stiffness and damping

Dynamics & Control

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 $\zeta$:

Systemζ [-]
Metals in elastic range0.01
Continuous metal structures0.02 - 0.04
Metal structures with joints0.03 - 0.07
Plastics (hard - soft)0.02 - 0.05
Rubber0.05
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 $F_e$

Interpretation of stiffness and damping - Schematic overview

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: $H_1=\frac{x}{F_e}=\frac{1}{ms^2+ds+c}$

Interpretation of stiffness and damping - Response to the external force

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: $H_1=\frac{x}{F_e}=\frac{ds+c}{ms^2+ds+c}$

Interpretation of stiffness and damping - Response to the input

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 $\omega=0$ : $A_i=\frac{c}{c}\ =1=0\ dB$ 

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