The damped dynamic vibration absorber Fig. 5.20 shows the primary system with a viscous damped absor
Fig. 5.20 shows the primary system with a viscous damped absorber added. The equations of motion are
Fig. 5.20. System with damped vibration absorber.
MX¨=F sin vt−KX−k(X−x)−c(X¨−x¨)
and
mx¨=k(X−x)+c(X˙−x˙).
Substituting X = Xo sin vt and x = xo sin (vt - Ø) gives, after some manipulation,
X0=F√[(k−mv2)2+(cv)2]√{[(k−mv2)(K+k−Mv2)−k2]2+[(K−Mv2−mv2)cv2]2}
It can be seen that when c = 0 this expression reduces to that given above for the undamped vibration absorber. Also when c is very large
X0=FK−(M+m)v2
For intermediate values of c the primary system response has damped resonance peaks, although the amplitude of vibration does not fall to zero at the original resonance frequency. This is shown in Fig. 5.21.
Fig. 5.21. Effect of absorber damping on system response.
The response of the primary system can be minimized over a wide range of exciting frequencies by carefully choosing the value of c, and also arranging the system parameters so that the points P1 and P2 are at about the same amplitude. However, one of the main advantages of the undamped absorber, that of reducing the vibration amplitude of the primary system to zero at the troublesome resonance, is lost.
A design criterion that has to be carefully considered is the possible fatigue and failure of the absorber spring: this could have severe consequences. In view of this, some dampedabsorber systems dispense with the absorber spring and sacrifice some of the absorber effectiveness. This has particularly wide application in torsional systems, where the device is known as a Lanchester Damper.
It can be seen that if k = 0,
X0=F√(m2v4+c2v2)√{[(K−Mv2)mv2]2+[(K−Mv2−mv2)cv2]}.
When c = 0
X0=FK−Mv2(no absorber)
and when c is very large,
X0=FK−(M+m)v2
These responses are shown in Fig. 5.22 together with that for the optimum value of c.
Fig. 5.22. Effect of Lanchester damper on system response.
The springless vibration absorber is much less effective than the sprung absorber, but has to be used when spring failure is likely, or would prove disastrous.
Vibration absorbers are widely used to control structural resonances. Applications include:
Machine tools, where large absorber bodies can be attached to the headstock or frame for control of a troublesome resonance.
Overhead power transmission lines, where vibration absorbers known as Stockbridge dampers are used for controlling line resonance excited by cross winds.
Engine crankshaft torsional vibration, where Lanchester dampers can be attached to the pulley for the control of engine harmonics.
Footbridge structures, where pedestrian-excited vibration has been reduced by an order of magnitude by fitting vibration absorbers.
Engines, pumps and diesel generator sets where vibration absorbers are fitted so that the vibration transmitted to the supporting structure is reduced or eliminated.
Not all damped absorbers rely on viscous damping; dry friction damping is often used, and the replacement of the spring and damper elements by a single rubber block possessing both properties is fairly common.
A structure or mechanism that has loosely fitting parts is often found to rattle when vibration takes place. Rattling consists of a succession of impacts, these dissipate vibrational energy and therefore rattling increases the structural damping. It is not desirable to have loosely fitting parts in a structure, but an impact damper can be fitted.
An impact damper is a hollow container with a loosely fitting body or slug; vibration causes the slug to impact on the container ends, thereby dissipating vibrational energy. The principle of the impact damper is that when two bodies collide some of their energy is converted into heat and sound so that the vibrational energy is reduced. Sometimes the slug is supported by a spring so that advantage can be taken of resonance effects. Careful tuning is required, particularly with regard to slug mass, material and clearance, if the optimum effect is to be achieved. Although cheap and easy to manufacture and install, impact dampers have often been neglected because they are difficult to analyse anddesign, and their performance can be unpredictable. They are also rather noisy in operation, although the use of PVC impact surfaces can go some way towards reducing this. Some success has been achieved by fitting vibration absorbers with impact dampers. The significant advantage of the impact vibration absorber over the conventional dynamic absorber is the reduction in the amplitude of the primary system both at resonance and at higher frequencies.