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SELF-EXCITED VIBRATION The force acting on a vibrating object is usually external to the system and independent of the motion. However, there are systems in which the exciting force is a function of the motion variables (displacement, velocity or acceleration) and thus varies with the motion it produces (called coupling). Friction-induced vibration (in vehicle clutches and brakes, vehicle-bridge interaction) and flow-induced vibration (circular wood saws, CDs, DVDs, in machining, fluid-conveying pipelines) are examples of self-excited vibration Example 1: a mass supported by a spring is carried by a moving belt through friction. x m v0 The friction coefficient at the mass and belt interface is a function of the relative velocity between the mass and the belt as gradiant – (>0) v The equation of motion of the mass is mx kx mg 0 [1 ( x v0 )]mg 0 (1 v0 )mg 0mgx or mx 0mgx kx 0 (1 v0 )mg negative damping causing (initially) divergent vibration As vibration grows, velocity x and hence relative velocity x v0 . This causes the friction coefficient to decrease (see m – v curve) and then vibration decreases. This cycle of increasing and decreasing vibration repeats itself forever (unless there is structural damping). The moving belt can sustain vibration — selfexcited vibration. Example 2: a solid oscillating in a fluid can interact with the fluid and produce interesting behaviour. The relative airflow against the oscillating solid modifies the velocity vector and thus the lift and drag force acting on the solid. Consider a long cylinder of length l supported by a spring k and a damper c. V k d x The equation of vertical motion of the cylinder in the flow is mx cx kx f (t ) The lift force f is f (t ) 1 V 2dlCL 2 Which term varies with time on the right ? Karman Vortices shedding occurs in Re=[60, 5000] http://www.algor.com/products/analysis_replays/turbulent_flow/d efault.asp?sQuery=vortex%20shedding Due to vibration of the cylinder, the lift coefficient is no longer a constant because of shedding of vortices. It may be expressed as CL CL0 sin t (C L0 1 for cylinder) and 2πksV d Vd 1000, ks 0.21. where ks is the Strouhal number. For Re So when V d 2πks k d n m 2πks the cylinder vibrates violently (in resonance) in the flow. When flow velocity becomes high enough, the flow becomes turbulent, and the lift force becomes random. Flutter is a phenomenon of self-excited vibration. L x V G O V M D A rigid wing attached to a rigid support through a spring Vertical motion: mx kx L cos D sin where L 2 (V 2 x 2 )lcCL 2 (V 2 x 2 )lc CL D 2 (V 2 x 2 )lc CD and x V 0 arctan( ) Assume small displacement so that tan then x V cos 1 V 2 x 2 V 2 dCL dC ( 0 ) cos D ( 0 ) sin ] 2 d d dC dC dCD V 2lc[ L 0 ( L 0 ) ] 2 d d d mx k x finally mx 2 Vlc( V 2 lc[ dCL dCD dC 0 ) x kx V 2lc L 0 d d 2 d V O Torsional vibration (assuming centre of gravity and aerodynamic centre coincide): ( me 2 I ) K Le M where the pitching moment dCL ( 0 ) 2 2 d dC M M V 2 lc 2CM V 2lc 2 ( 0 ) 2 2 d L V 2lc 2CM V 2 lc The above equation becomes dC dC dC dC ( me 2 I ) [ K V 2lc( L e M c)] V 2lc( L e M c)]0 2 d d 2 d d The static divergence speed is Vdiv 2K lc( dC L dC M e c) d d for a symmetric aerofoil Vdiv 2K dC lc L e d The high the Vdiv , the greater speed capacity the aircraft has.