1. Kjell Simonsson 1 Vibrations in linear 1-dof systems; III. damped systems (last updated 2011-08-28)

2. Kjell Simonsson 2 Aim The aim of this presentation is to give a short review of basic vibration analysis of damped linear 1 degree-of-freedom (1-dof) systems. For a more comprehensive treatment of the subject, see any book on vibration analysis.

3. Kjell Simonsson 3 A simple example Let us study the effect of linear damping by considering the following simple example. where F d is a damping force, having any kind of physical origin. By making a free body diagram of the (point) mass, we get where the coordinate x describes the position of the mass (equal to the deflection of the rod), and where the force S represents the interaction between the rod and the point mass. Elementary solid mechanics now gives

4. Kjell Simonsson 4 Thus, the lateral motion of the mass as a function of time is governed by a linear 2'nd order ordinary differential equation with constant coefficients! The solution is given by the sum of the homogeneous solution and the particular solution,, where the former corresponds to the self vibration of the mass caused by the initial conditions (placement and velocity), while the latter is caused by the applied force. A simple example; cont. By assuming a linear damping we get or

5. Kjell Simonsson 5 A simple example; cont. Homogeneous solution For the homogeneous solution we have Depending on the size of ζ we may now get two different imaginary roots (for ζ <1); case of weak damping two equal real roots (for ζ =1); case of critical damping two different real roots (for ζ >1); case of strong damping The reason for the labels weak, critical and strong will be motivated below.

6. Kjell Simonsson 6 A simple example; cont. Homogeneous solution; weak damping (ζ <1) As can be seen, the homogeneous solution decays as a periodic function with decreasing amplitude.

7. Kjell Simonsson 7 A simple example; cont. Homogeneous solution; critical damping (ζ =1) As can be seen, the homogeneous solution decays as a non-periodic/aperiodic function. For this case we may define the critical damping parameter c crit

8. Kjell Simonsson 8 A simple example; cont. Homogeneous solution; strong damping (ζ >1) As can be seen, the homogeneous solution again decays as an aperiodic function. A final comment is that the label "critical damping" by no way means that it is critical in some way– it only indicates that it is the lowest value of ζ (and thus for c) for which an aperiodic motion can exist.

9. Kjell Simonsson 9 A simple example; cont. Particular solution For the particular solution, valid at stationary conditions, we have

10. Kjell Simonsson 10 A simple example; cont. Particular solution; cont.

11. Kjell Simonsson 11 A simple example; cont. Particular solution; cont.

12. Kjell Simonsson 12 A simple example; cont. Particular solution; cont.

13. Kjell Simonsson 13 A simple example; cont. Particular solution; cont. For the damped case we note that there is no resonance frequency in a strict sense, since the damping will make the stationary vibration amplitude finite for all applied frequencies the dynamic impact factor and thus the vibration amplitude may however become very large if the damping is small