Using Eqs. (E.4) and (E.6), Eq.(E.5) can be solved to obtain (From Eq. (3.30))
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-114-3.11
Self-Excitation and Stability Analysis
3.11
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-115-3.11 Self-Excitation and Stability Analysis
• The force acting on a vibrating system is usually external to the system and independent of the motion. However, there are systems for which the exciting force is a function of the motion parameters of the system, such as displacement, velocity, or acceleration.Such systems are called self-excited vibrating systems, since the motion itself produces the exciting force.
• The instability of rotating shafts, the flutter (振顫) of turbine blades, the flow-induced vibration of pipes, and the automobile wheel shimmy(汽車的異常振動), and the aerodynamically (空氣動力)-induced motion of bridges are typical examples of self-excited vibrations.
• The motion diverges and the system becomes unstable if energy is fed into the system through self-excitation.
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-116-3.11 Self-Excitation and Stability Analysis
• Dynamic Stability Analysis
Consider the equation of motion of a single degree of freedom system:
If a solution of the form is assumed, this leads to a characteristic equation
The roots of the equation are:
)
The motion will be diverging and aperiodic if the roots s1
and s2are real and positive.
This situation can be avoided if c/m and k/m are positive.
( ) st x t Ce
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-117-3.11 Self-Excitation and Stability Analysis
• Dynamic Stability Analysis Let the roots be expressed as
where p and q are real numbers so that
Hence,
) 110 . 3 (
, 2
1 p iq s p iq
s
) 111 . 3 ( 0 )
( ) )(
( 1 2 2 1 2 12 2 m s k m s c s s s s s s s s s s
) 112 . 3 (
, 2 )
(1 2 s1s2 p2 q2
m p k s m s
c
Eq.(3.112) show that for negative p, c/m must be positive and for positive p2+q2, k/m must be positive. Thus the system will be dynamically stable if c and k are positive (assuming that m is positive).
The motion will also diverge if the roots s1 and s2 are complex conjugates with positive real parts.
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-118-3.11 Self-Excitation and Stability Analysis
Example 3.10Instability of Spring-Supported Mass on Moving Belt
Consider a spring-supported mass on a moving belt as shown in the Figure (a). The kinetic coefficient of friction between the mass and the belt varies with the relative (rubbing) velocity as shown in Figure (b).
As rubbing velocity increases, the coefficient of friction first decreases from its static value linearly and then starts to increase. Assuming that the rubbing velocity, v, is less than the transition value, vQ, the coefficient of friction can be expressed as
0 av 0 ( )
W Wa V x
Where a is a constant and W=mg is the weight of the mass. Determine the nature of the free vibration about the equilibrium position of the mass.
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-119-3.11 Self-Excitation and Stability Analysis
Example 3.10
Instability of Spring-Supported Mass on Moving Belt
Motion of a spring-supported mass due to belt friction
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-120-3.11 Self-Excitation and Stability Analysis
Example 3.10
Instability of Spring-Supported Mass on Moving Belt Solution
Let the equilibrium positionof mass m correspond to an extension of x0of the spring. Then,
The rubbing velocity v is given by:
0
0 0
W kx W
W aV
x k k k
x V v
0
v V x V
x at the equilibrium position
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-121-3.11 Self-Excitation and Stability Analysis
Example 3.10
Instability of Spring-Supported Mass on Moving Belt Solution
The equation of motion for free vibration is
The characteristic roots
0 0 0 the motion given by Eq.(E.1) will be unstable
-122-3.11 Self-Excitation and Stability Analysis
Example 3.10Instability of Spring-Supported Mass on Moving Belt Solution
The solution is given by
where C1and C2are constants Thus, of x increases with time. It increases from . After this, the will have a positive slope, and hence the nature of the motion will be different.
V x 0 to V x vQ
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-123-• Dynamic Instability Caused by Fluid Flow
• The vibration caused by a fluid flowing around a body is known as flow-induced vibration, in which the vibration of the system continuously extracts energy from the source, leading to larger and larger amplitude of vibration.
• For example, tall chimneys (煙囪), submarine periscopes (潛望鏡), electric transmission lines, and nuclear fuel rods are found to vibrate violently under certain conditions of fluid flow around them.
Similarly, water and oil pipelines and tubes in air compressors under severe vibration under certain conditions of fluid flow through them.
3.11 Self-Excitation and Stability Analysis
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-124-•Dynamic Instability Caused by Fluid Flow
(旋渦脫落、旋渦分離 、渦流溢放 )
機翼 疾馳
顫動
(Induce the instability)
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-125-3.11 Self-Excitation and Stability Analysis
• Dynamic Instability Caused by Fluid Flow
The figure illustrates the phenomenon of galloping of wires:
Fig. 3.28 Galloping of a wire
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-126-3.11 Self-Excitation and Stability Analysis
• Dynamic Instability Caused by Fluid Flow
The figure illustrates the phenomenon ofsinging of wires:
Experimental data show that regular vortex sheddingoccurs strongly in the range of Reynolds number (Re) from about 60 to 5000. In this case,
) 113 . 3 (
Re
Vd
Smooth cylinder
Fluid flow past a cylinder
where d is the diameter of the cylinder, is the density, V is the velocity, and is the absolute viscosity of the fluid
Under certain conditions, alternating vortices (旋渦) in a regular pattern are formed downstream, which are called Karman vortices. They are alternately clockwise and counterclockwise and thus cause harmonically varying lift forces on the cylinder perpendicular to the velocity of the fluid.
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-127-• Vortex shedding (旋渦脫落、旋渦分離 、渦流溢放 )(影片欣賞)
In fluid dynamics,vortex shedding is an unsteady oscillating flowthat takes place when a fluid such as air or water flows past a blunt cylindrical body at certain velocities, depending to the size and shape of the body. In this flow, vortices are created at the back of the body and detach periodically from either side of the body.The fluid flow past the object creates alternating low-pressure vortices on the downstream side of the object.The object will tend to move toward the low-pressure zone.If the cylindrical structure is not mounted rigidly and the frequency of vortex shedding matches the resonance frequency of the structure, the structure can begin to resonate, vibrating with harmonic oscillations driven by the energy of the flow.This vibration is the cause of the "singing" of overhead power line wires in a wind, and the fluttering of automobile whip radio antennas at some speeds. Tall chimneys constructed of thin-walled steel tube can be sufficiently flexible that, in air flow with a speed in the critical range, vortex shedding can drive the chimney into violent oscillations that can damage or destroy the chimney.
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-128-3.11 Self-Excitation and Stability Analysis
• Dynamic Instability Caused by Fluid Flow
For Re > 1000, the dimensionless frequency of vortex shedding, expressed as a Strouhal number (St), is approximately equal to 0.21.
The harmonically varying lift force (F)is given by
where
) 114 . 3 ( 21 . 0 St
V fd
) 115 . 3 ( 2 sin
) 1
(t c V2A t
F
where f is the frequency of vortex shedding
c = constant (c 1 for a cylinder)
A = projected area of the cylinder perpendicular to the direction of V ω = circular frequency and t is the time.
台灣師範大學機電科技學系 C. R. Yang, NTNU MT
-129-3.11 Self-Excitation and Stability Analysis
From a design point of view, we have to ensure the following:1.The magnitude of the force exerted on the cylinder, given by Eq.
(3.115), is less than the static-failure load.
2.Even if the magnitude of force F is small, the frequency of oscillation (f) should not cause fatigueduring the expected lifetime of the structure (or cylinder).
3.The frequency of vortex shedding (f) does not coincide with the natural frequency of the structure or cylinder to avoid
resonance.
台灣師範大學機電科技學系 C. R. Yang, NTNU MT