Chapter 14: Vibration Generations Mechanisms: Self Excited Vibration

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1 hapter 14: ibration Generations echanisms: Self Ecite ibration Introuction: Self-ecite systems begin to vibrate of their own accor spontaneously, the amplitue increasing until some nonlinear effect limits any further increase. The energy supplying these vibrations is obtaine from a uniform source of power associate with the system which, ue to some mechanism inherent in the system, gives rise to oscillating forces. The force acting on a vibrating object is usually eternal to the system an inepenent of the motion. However, there are systems in which the eciting force is a function of the motion variables isplacement, velocity or acceleration an thus varies with the motion it prouces calle coupling. Friction-inuce vibration in vehicle clutches an brakes, vehicle-brige interaction an flow-inuce vibration circular woo saws, s, s, in machining, flui-conveying pipelines are eamples of self-ecite vibration. Selfecite vibration represents an important phenomenon in physical an mechanical systems. There eist ifferent sources of self-ecitation, which nee ifferent mathematical moels escribing important properties of the self-ecitation. In most cases the self- ecite vibration represents a anger for the save run of ifferent systems an evices. Therefore, it is necessary to use means for vibration suppressing or, at least, for reucing the vibration intensity. The occurrence of self-ecite vibration in a physical system is intimately associate with the stability of equilibrium positions of the system. If the system is isturbe from a position of equilibrium, forces generally appear which cause the system to move either towar the equilibrium position or away from it. There are systems where the ecitation comes from within, ue to its own isplacement. When a system is isturbe, the free vibration uner certain conitions can cause an ecitation that makes the system vibrate further. An increase in vibration thus can cause a further increase in the ecitation an therefore the vibratory amplitue runs out of control. When the amplitues become larger an larger the restoring force goes on increasing further. The increase restoring force fights the increasing amplitues until a balance is reache amongst the ecitation generate by the amplitue an the restoring

2 force. Then the vibration remains sustaine at this value, calle a limit cycle vibration. The ecitation gets remove when the amplitue of vibration becomes zero for some reason an then the system comes to rest. Once a vibration is initiate, an ecitation comes into effect an the system runs off with increasing amplitues until a limit cycle is reache. Such a vibratory motion is calle Self Ecite ibration. The function of many machines an evices, such as power picks, rilling sets, compacting equipment, hammers, etc. is base on self-ecite vibrations prouce by the action of compresse air. Such machines can harly be epecte to operate efficiently without a thorough unerstaning of the problems of eistence an stability of perioic motions, gaine by research stuies. These problems arise as a result of the non-linear epenence of pneumatic forces on the motion of the mechanical system. Another strong non-linearity is ae to the motion whenever impacts occur in the system. achining an measuring operations are invariably accompanie by vibration. To achieve higher accuracy an prouctivity vibration in machine tool must be controlle. For analysis of ynamic behavior of machine tool rigiity an stability are two important characteristics. achine tool vibrations may be ivie into 3 basic types as Free or transient vibration, Force vibration an Self ecite vibration achine tool chatter. hatter is a self-ecite vibration which is inuce an maintaine by forces generate by the cutting process. It effects surface finish, tool life, prouction rate an also prouces noise. hatter resistance of a machine tool is usually characterize by a maimum stable i.e., not causing chatter vibration epth of cut b lim. achine-tool chatter is essentially a problem of ynamic stability. A machine tool uner vibration-free cutting conitions may be regare as a ynamical system in steay-state motion. Systems of this kin may become ynamically unstable an break into oscillation aroun the steay motion. In self-ecite vibration the alternating force that sustains the motion is create or controlle by the motion itself; when the motion stops, the alternating force isappears. In a force vibration the sustaining alternating force eists inepenent of the motion an persists when the vibratory motion is stoppe.

3 The vibration behaviour of a machine tool can be improve by a reuction of the intensity of the sources of vibration by enhancement of the effective static stiffness an amping. By appropriate choice of cutting regimes, tool esign, an work-piece can be esigne properly. Abatement of the sources is important mainly for force vibrations. Stiffness an amping are important for both force an self ecite chatter vibrations. Both parameters, especially stiffness, are critical for accuracy of machine tools, stiffness by reucing structural eformations from the cutting forces, an amping by accelerating the ecay of transient vibrations. Self-ecite vibrations are characterize by the presence of a mechanism whereby a system will vibrate at its own natural or critical frequency, essentially inepenent of the frequency of any eternal stimulus. In mathematical terms, the motion is escribe by the unstable homogeneous solution to the homogeneous equations of motion. In contraistinction, in the case of force, or resonant, vibrations, the frequency of the oscillation is epenent on equal to, or a whole number ratio of the frequency of a forcing function eternal to the vibrating system e.g., shaft rotational spee in the case of rotating shafts. In mathematical terms, the force vibration is the particular solution to the non-homogeneous equations of motion. Self-ecite vibrations pervae all areas of esign an operations of physical systems where motion or time-variant parameters are involve aeromechanical systems flutter, aircraft flight ynamics, aeroynamics separation, stall, mu sical win instruments, iffuser an inlet chugging, aerothermoynamics flame instability, combustor screech, mechanical systems machine -tool chatter, an feeback networks hyraulic, an electromechanical servomechanisms. Ehrich, 1999 pneumatic, The mechanisms of self-ecitation in rotating machinery, which have been ientifie, can be categorize as follows: Whirling or Whipping Hysteretic whirl Flui trappe in the rotor ry friction whip Flui bearing whip Seal an blae-tip-clearance effect in turbomachinery Propeller an turbomachinery whirl

4 Parametric Instability Asymmetric shafting Pulsating torque Pulsating longituinal loaing Stick-Slip Rubs an hatter Self-ecite oscillations are oscillations that are ecite by the motion of the system. Self-ecite oscillations are inuce by nonlinear forms of amping where the amping term is negative over a certain range of motion. echanical system that ehibits negative amping, where the free oscillations amplitue grows, is shown in Fig. Fig. 4.4 System with negative amping The instability of rotating shafts, the flutter of turbine blaes, the flow inuce vibration of pipes, an the automobile wheel shimmy an aeroynamically inuce motion of briges are typical eamples of self-ecite vibrations. A system is ynamically stable, if the motion or isplacement converges or remains steay with time. On the other han, if the amplitue of isplacement increases continuously iverges with time it is sai to be ynamically unstable. The motion iverges an the system becomes unstable if energy is fe into the system through self-ecitation. For a ampe free vibration system as shown in Fig. 4.4 the characteristics equations becomes; m + c + k =

5 k m Xt m mg m mg Substituting, t = a e λ in equation, get, amλ e λ + cλe λ + ke λ = here a an e λ hence, mλ + cλ + k = λ + c m λ + k m = The solution of equation 1.8 yiels as follows λ, = c m ± 1 c m 4 k m Since, the solution is assume to be t = e t the motion will be iverging an aperioic if the roots s 1 an s are real an positive. This situation can be avoie if c/m an k/m are positive. The motion will also iverge if the roots λ 1 an λ are comple conjugates with positive real parts. Thus, the funamental criterion of stability in linear systems is that the roots of the characteristic equation have negative real parts, thereby proucing ecaying amplitues. The whirling spee at onset of instability is the shaft s natural or critical frequency, irrespective of the shaft s rotational spee rpm.the irection of whirl may be in the same rotational irection as the shaft rotation forwar

6 whirl or opposite to the irection of shaft rotation backwar whirl, epening on the irection of the estabilizing force. Fig. 4.5 A Illustration showing a ecaying vibration stable corresponing to negative real parts of the comple roots B Increasing vibration corresponing to positive real parts of the comple roots unstable Eample 4.1: A mass supporte by a spring is carrie by a moving belt through friction. m v The friction coefficient at the mass an belt interface is a function of the relative velocity between the mass an the belt as

7 Graient > v The equation of motion of the mass is m k mg v mg v mg mg [ 1 ] 1 or m mg k 1 v mg negative amping causing initially ivergent vibration As vibration grows, velocity an hence relative velocity v. This causes the friction coefficient to ecrease see m v curve an then vibration ecreases. This cycle of increasing an ecreasing vibration repeats itself forever unless ther e is structural amping. The moving belt can sustain vibration self-ecite vibration. Eample 4.: a soli oscillating in a flui can interact with the flui an prouce interesting behaviour. The relative airflow against the oscillating soli moifies the velocity vector an thus the lift an rag force acting on the soli.

8 onsier a long cyliner of length l supporte by a spring k an a amper c. k The equation of vertical motion of the cyliner in the flow is m c k f t The lift force f is f 1 t l Karman ortices sheing occurs in Re=[6, 5] ue to vibration of the cyliner, the lift coefficient is no longer a constant because of sheing of vortices. It may be epresse as an sin t 1 for cyliner πk s

9 where k s is the Strouhal number. For Re 1, k. s 1. So when k n πk m πk s s the cyliner vibrates violently in resonance in the flow. When flow velocity becomes high enough, the flow becomes turbulent, an the lift force becomes ranom. Flutter is a phenomenon of self-ecite vibration. G O Fig. 4.5 A rigi wing attache to a rigi support through a spring ertical motion: m k cos sin where an arctan Assume small isplacement so that

10 1 cos tan then ] [ ] sin cos [ k m finally k m Torsional vibration assuming centre of gravity an aeroynamic centre coincie: e K I me where the pitching moment The above equation becomes ] ] [ c e c e K I me The static ivergence spee is O

11 iv K e c for a symmetric aerofoil iv K e The high the iv, the greater spee capacity the aircraft has.