This paper puts forward the concept of double semi-active vibration isolation systemutilizing the 'on-off' active damper and the 'on-off' active spring. Applying KBM Asympto-tic method, the vibration of the system in case of weak semi-active control is studied. ln pcactice, there are two types of vibration isolation. One is to isolate the transmitting of the cyclicunbalanced force generated by the revolving machine to the surroundings. The other is to isolatethe transmitting of the vibration displacement of the surroundings to precise equipment. Deduc-ing the influence of the roadway unflatness on vehicles also belongs to this type. For the firsttype of isolation system, the damper always dissipates energy, and the total work done by the spring in a vibration cycle is zero. For the second type of isolation system, the work done by the damper sometimes is positive, and sometimes negative. The damper dissipates less energy. The work done by the spring in a vibration cycle isn't zero, and it is usually positive. ln thispaper, the vibration isolation is refered to the second type. .
The form invariance of Routh equations in holonomic systems is studied. The definition and criterion for the form invariance under the infinitesimal transformations are given. The relation of the form invariance with the Noether symmetry and the Lie symmetry is discussed.
