Technion - Israel Institute of Technology Faculty of Mechanical Engineering

Dynamics of rotating Machinery, Dr. I. Bucher Spring 2003 Dynamics Laboratory

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Dynamics of rotating machinery The discipline of dynamics of rotating machinery (rotordynamics) refers to systems where at least one part rotates with a significant angular momentum. This discipline is applied to analyze the behavior of freely rotating objects as well as constrained (by bearings) ones ranging from projectiles, satellites in orbit to pumps, turbines.

Grade composition

Homework %35 Quiz 25% Final project 40 %

Technion - Israel Institute of Technology Faculty of Mechanical Engineering

Dynamics of rotating Machinery, Dr. I. Bucher Spring 2003 Dynamics Laboratory

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Course topics

• Intoduction • Review of vibration theory 1-DOF, N-DOF • free vibration, forced vibration, mode shapes, modal decomposition / superposition, orthogonality, Lagrange equations • Jeffcot rotor • 1 DOF rotating shaft + disk, complex notation, stationary and rotating co-

ordinates • free response - eigenvalues, forward/backward whirl, unbalance - forced

response, asymmetric supports, Campbell diagram, internal damping vs. external damping - stability, acceleration through critical speeds,

• Continuous systems • Torsional vibration • Continuous Models, limitations • Discrete Models - simple FE model • coupled systems, connected shafts • interference diagram - engine orders • Shaft bending vibration • Euler/Timoshenko beam • Kinetic and potential energy of a rotating shaft • shaft with rigid disks and flexible supports • gyroscopic effects • Continuous systems • flexible disks • forward/backward whirl • Campbell diagram, measurements interpretation • static loads, synchronous loads • Experimental verification • Critical speeds • Natural frequencies and modeshapes • Unbalance response & balancing • Measurements and signal processing • Experimental identification, bearings, shaft, damping

Technion - Israel Institute of Technology Faculty of Mechanical Engineering

Dynamics of rotating Machinery, Dr. I. Bucher Spring 2003 Dynamics Laboratory

Page 0

Importance of rotordynamics for performance and design- examples Example no. 1: Stored kinetic energy in a typical rotating disc (dimensions/speed are typical of pump/compressor) Consider a ring made of steel rotating at 10,000 Rev/min. The ring D=0.6 m, d=0.4 m , w=0.04, m=p*(0.3^2-0.2^2)*0.04*8000= 50.3 Kg Jp= ½*m*(0.3^2-0.2^2)=1.26 Kg x m^2 Ω=10000 Rev/min 1/60 min/sec 2π Rad/Rev = ~1000 Rad/s T= ½*Jp*Ω^2= 6.89*105 Joule What would happen if, for some reason the angular momentum (Kinetic energy) would be converted into linear momentum? How high/far would the ring fly? V= mgh = T h(max) = 6.89*10^5/(9.81*50)

h =~ 700 m (for upwards fly) at 30 ° (neglect aerodynamic drag) a=-g mh''=-mg h'=-g*t+c1, h=-1/2*g*t^2+c1*t th=hmax=sqrt(2*hmax/g)=~12 Sec; c1=g*sqrt(2*hmax/g) = sqrt(2*hmax*g); vx=~200 m/s, -> xmax=2400 m =2.4 Km !!

The amount of energy storage in a state-of-the-art flywheel is about equal to the amount of energy stored in a good battery.

The large energy is a safety hazard and is much larger than the deformation that any internal mode can absorb (failure)

Ω

h

L

Technion - Israel Institute of Technology Faculty of Mechanical Engineering

Dynamics of rotating Machinery, Dr. I. Bucher Spring 2003 Dynamics Laboratory

Page 0

Example no.2 Stress in a rotating ring Stress in a rotating hoop 222 RV Ωρ=ρ=σ For our example, R=0.3 m , Ω=1000 Rad/s Steel hoop MPa720=σ (Would yield most materials) A full disk would have the maximal (bi-axial) stress at its middle,

and 22

8)3( RΩρ

υ+=σ

** Kinetic energy per unit volume

2

21 VT ρ= ,

energy stored due to elastic deformation (per unit volume) E

U2

2σ=

σ

==E

UT

PEKE ,

Speed of sound in materialρ

=Ec , Since 2Vρ=σ , 2

2

Vc

UT

=

If the rotor is sped up to a point near yield, the ration of kinetic to elastic energy, would be

y

EUT

σ= , As for most materials this ratio would be of order 100- 1000, thus if a

mechanism arises that transfers rotational energy into vibrational energy, only a tiny fraction of the available energy would be required to initiate failure in the vibratory mode.