The problem of predicting the phase angle of two self-synchronized rotors starting from rest is presented. It is shown that with insufficient power the rotors may not reach the final operating speed of the motors and stay locked at one of the lower natural frequencies of the vibrating system, thus producing large amplitude and failure of the equipment

Paz, Mario

English

NASA Technical Reports Server

N88 - 18958 
Electro-Mechanical Vibratory System 
Mario Paz 
The problem of predicting the phase angle of two 
self-synchronized rotors startj.ng from rest, is 
presented in this paper. It is shown that with 
insufficient power the rotors may nct reach the 
final cperating speed of the motors and stay 
locked at one of the lower natural frequencies 
of the vibrating system, thus producinq large 
amplitude and failure of the equipment. 
INTRODUCTION 
Mechanical vibrators are widely i:sed for producing the 
rnoti.cn required in some manufacturing processes such as 
packing cement mixtures, screening, vibration trans- 
portation, vibration pile drivers, etc. The principal 
ccmponent of the mechanical vibratory is an unbalanced mass 
rotated by an electric motor. In some of these 
applications, rectilinear vibratory motion is required. 
This motion may be attained using two synchronized motors 
having eccentric masses rotating in opposite directions. 
SynchrGnization of the rotating masses means coordinating 
the rota-tion tc maintain a defi.nite phase angle between the 
rotation of the two motors. Such synchronization may be 
accomplished by using kinematic coupling such as toothed or 
chair. drivers between the twc rotors. However, some years 
ago, it became apparent the synchronization could be main- 
tained without resorting to any kind of kinematic coupling. 
It was observed that the angular ve1oc:i.ties and phases of 
the motors automatically kept a definite phase rotation 
without the use of coupling. Huygens [l] fj.rst Gbserved a 
case of self-synchronization in the seventeenth century. He 
noticed that two pendulum clocks, regardless Gf their 
initial phase angle, came into synchronization if they were 
mcunted on a common resilient support such as a flexible 
beam. Ee realized that this phenomenon of self- 
synchronization could be explained by the small im- 
perceptible motion in the beam transmitted by the pendulums. 
At the present time, self-synchronization is finding 
wider applications in the design of vibrating machines 121. 
The use of self-synchronization makes forced synchronization 
https://ntrs.nasa.gov/search.jsp?R=19880009574 2020-03-20T08:02:58+00:00Z
unnecessary. The condition of self-synchronization, as well 
as the phase angle maintained by the rotors, has been 
investigated initially by Elekman 131 for one directional 
notion, by Paz [ 4 , 5 ]  for two directional motions and sub- 
sequently by others [6,7,81 . Paz investigated analytically 
and experimentally the problem of predicting t h e  stable 
phase angle and the direction of the resultapt force of any 
two rotors with parallel a x i s  of rotation. He showed that 
the phase angle between two rotors, as we;! as the ccndition 
necessary for a stable s o l u t i c n  could be predicted with the 
application of Hamilton's Principle. 
These developments gave solutions prescribing t h e  
conditions for self-synchronizaticn, hut did not focus in 
the initial transient portion of the motion until it rea-ches 
the final steady-state condition. In this transient state, 
the system may have to transverse some of its natural 
frequencies at which resonance may occnr, locking the motion 
at a speed lower than the operating speea of the motors, 
thus, resulting in large amplitudes of motion and failure of 
the equipment. ?'he electro-mechanical analysis of vibratincj 
equipment during the initial trans1er.t motion, is presented 
in this paper. 
CHARACTERISTICS OF INDUCTION MOTOR 
The torque capacity of an incluction motor is primarily 
a function of its rotating speed. Fig.1 shows a typical 
torque-speed curve for this type of motor. The figure also 
shows an assumed torque-speed curve required to maintain 
rotation as demanded by the load. In this case, the system 
will operate in the transient state until it reaches the 
steady-state condition at which the torque demanded by the 
load equals the torque of the motor, pint P in Fig.1. 
ANGULAR SPEED (rad/ sec) 
Fig. 1 Speed-Torque Curve For Typical Induction Motor 
208 
At any speed 6 (rad/sec), the torque differentialA T between 
that produced by the mGtor and the required by the load is 
available to accelerate the motor. Consequently, applying 
Newton's second law for a rotating mass, this differential 
torque is 
(1) 
dO 
dt 
A T  = ;*- 
in which S is the mass moment of inertia of the rotating 
parts of the rmtor and 0 the angular velocity of the rotor. 
KECHANICAI, VIBRATING SYSTEM 
The equations of motions of a mechanical vihrating 
system my be obtained by direct applicatj.on of Newton's 
Law of motion or alternatively by using an energy method 
such as the application of Lagrange's equation. In any 
case, the equations of motion for a linear system may, in 
general, be expressed in matrix notation as 
I M l C ~ 3  + I C I r ~ 3  + [KIEYI = CF(t)l (2) 
where [MI, [C] and [K] are respectively the mass, damping 
and stiffness matrices; and EY3, E33 {%3 the displacement, 
velocity and acceleration vectors of the generalized 
coordinates. 
In the solution of Eq. ( 2 ) ,  i.t is convenient to use 
modal superposition method to trmsform these equations to a 
system of uncoupled equation of the form 
(i =1,2 ... N) ( 3 )  1 C  2 zi + 2/iwi2i + wizi = Pi(t) 
where Zi = modal displacement 
i = modal velocity 
Zi = nicdal acceleration w = natural frequency 
fi = mod21 damping ratio 
i 
and Pi(t) = modal force 
which is given by 
N 
Pi(t) = 1 F j ( t )  4ji ( 4 )  
j= l  
where 4 ji is the j-component of i-th eigenvector. 
The solution of the uncoupled equation, Eq.(3) during 
the transient motion may be obtained by numerical in- 
tegration in terms of the modal coordinates Zi. 
209 
The motion in function of the generalized coordinates is 
then given by the transformation 
EYI = [ f l C Z l  
where [ 1 is the modal matrix of the system. E 
Fig.2 Mathematical model for the example of a vibrating 
system excited by two self-  synchronizing rotors. 
TORQUE DEMANDED BY THE I,@F,I! 
The torque demanded by the load may be determined from 
the power consumed by the exciting forces of the rotors. 
The forces Fl(t) and F (t), developed by the two rotors 
exciting the system are diven by 
* 2  Fl(t) = m'e O 1  
a 2  
2 F2(t) = m'e 0 
where m' and e, are the eccentric mass and its corresponding 
eccentricity; 0 and 6 the angular velocities of the rotors. 
The power demanded by the load may be calculated as the 
product of the rotor fcrces times the correspcnding linear 
velocities of the points of application of these forces. It 
is convenient to express this power separately for the two 
force components in the plane of the rotating forces as 
1 2 
= Fl(t) sin 0 (t) 1 ( 7 )  
2 10 
and 
F21(t) = F2(t) sin0 2(t) 
F22(t) = F2(t) cos0 ,(t) 
where Fi. is the j-component of the i-th motor; 0 and 0 the angular displacements of the rotors. 1 2 
The power demanded by the load is given by 
PI1 (t) = Fll (t) .ill (t) 
(9) 
and 
, 
P21 (t) = F2+) . X Z l ( t )  
( 1 0 )  
P22 (t) = FZ2(t) .i2*(t) 
where Pi = Power of j force component of the i.-th motor 
9 = LiAar velocity of  the force component j of the i-th 
mAi!or. 
Finally, the torque T, demanded by the load is obtained by 
simply dividing the po'ker components 
the corresponding angular velocity 
sequently, the total torque demanded 
2 
k= 1 
by 
= 1 (t) + Pk2(t) 1 k kl TI* 
in Eqs.(9) and (10) by 
of the rotors. Con- 
by the loads is given 
ANGULAR VEIiOCITY OF THE RO'ICFS 
The angular velocity f o r  t h e  rotors i s  de te rmined  by 
numerical integraticp of Ec.(l) which expressed in finite 
differences form may be written as 
where A b(t) is the incremental angular velocity of the 
rotors attained in tine step A t and A T(t) is the torque 
available a t  accelerate the rotcrs. As illustrated in 
Fig.1, t h i s  tcrque is given by the difference between the 
torque of the motor TM(t) and the torque T @demanded by the 
load: thus for each motor, the differentia$ torque is given 
hY 
AT(t) = TM(t) - TL(t) ( 1 3 )  
211 
The angular velocity for each rotor is then calculated, 
startj ng with initial velocity zero ( 8 ( 0 )  =0) , through the 
recurrence formula 
6 (t+A t) =($ (t) + A 6  (t) (14) 
The angulsr displacement of the rotors, el, and is 
calculated a s  
starting with the initial displacements of the rotors 
O1 (t=O) and O 2  (t=O) . 
The phase angle 4 t )  between the rotors is then given 
by the differeRce between the angul6r displacements of the 
rotors. Hence: 
NUMERICAL EXAMPLE 
Figure 2 shows the mathematical model of a vibrating 
mnveyor, excited by two motors with eccentric ciasses 
rotating in opposite directions. The differential equations 
of motion for the six degrees Gf freedom in this model were 
obtained using Lagrange's equation. The numerical values 
for the components of this system are indicated in Table 1. 
Discrete values of the torque-speed function for the motors, 
provided by the manufacture, are given in Table 2 .  
A time step A t =0.01 sec. was selected and damping in the 
system was assumed equlal to 5 %  of the critjcal damping in 
each mode. 
TAELE 1 NUMERICAL VALUES VIBRATING SYSTEM OF FIG.2 
BASE : 
2 Mass: M1= 31.863(lb-sec /in) 
Mass moment of inertia: 
11= 42,772(lb-sec -in) 2 
EXCITER: 
Gass: M 2 =  15.236 (lb-sec2/in) 
Mass ncment of inertia: 
I,= .- 4 , 5 7 8  (3b-sec2-in) 
212 
Springs costant: 
-----.- -  ____I 
Speed 0 47.12 162.3 184.3 188.5 (rad/sec) 
Torque 428.4 321.3 616.75 (lb-in) 
142.8 0 
r 
K1 (axial) = 3k 
K, (axial) = 9k 
(k = 9,40O(lb/in) 
K; (transverse) = 30k 
K2 (transverse)= 90k 
Distance: 
TIME (sec) 
Fig.3 Motor speed and phase angle of the rotors - Motors 
having full torque. 
The analysis of the system modeled as shown in Fig. 2 
provided the following three non-zero values for the naturnl 
frequencies: 
213 
w1 =114.7 rad/sec, w 2  =195.7 rad/sec and w3 = 7 8 3 . 6  rad/sec. 
The solution output ir.cludes the motion ot thcl system and of 
t,he rotors, as a function of time, during the initial 15 
seconds of the transient state. The solution also includes, 
the torques of the motors and t.h,e torques demanded by the 
load as well a s  the phase angle between the rotors in the 
transient state. 
For the motors having the speed-torque characteristic 
depicted in Fig.1 (Numerical values in Table 2 ) ,  Fig.3 
shows the plots of the phase ar.gle between rotors and  the 
angular speed of the motors during the injtial 40 seconds 
motion, while Fig.4 shcws these plots assuming that the 
motors can provide only one-half of the torque indicated in 
Table 2. The jnitial phase angle w a s  6 @  degrees due to the 
physjcal arrangement of the rotors under the effect- of 
gravity. 
. 
: .~ .. .... 'f . , . 
. . . a  . . . .  
TIME (sec) 
Fig.4 Motor speed and phase angle of the rotors - Motors 
limited to 50% of the full torque. 
DISCUSSION 
It may be observed in Fig.3 that for the motors having 
the speed-torque characteristics of Fig.1, the motors 
ra.pidly, in about five seconds reach their operating speed 
iri the neighborhood of b =I65 rad/sec and that rotors attain 
self-synchronization with a phase angle o(=O degrees. On the 
other hand, Fig.4 shows that the motors do not have 
2 14 
su.fficient pGwer to reach the operating speed and become 
locked in the neighborhood of resonant speed w = 114.7 
(rad/sec) attaining out of phase self-synchronizaticn of 
the rotors (4=i8Cu). In the first case, t h e  self- 
synchronized machine will perform as intencleci in the design, 
while in the second case t h e  system will vibrate erratically 
at the resonant condition with large amplitudes resulting in 
failure of the equipment. 
r 
REFERENCES 
1. 
2. 
3 .  
4 .  
5. 
6. 
7. 
8. 
Panovko, Y. G. and Gubanova, I. I., "Stability a n d  
Oscillations of Elastjc Systems, Consultants Bureau, 
New York, 1965, pp. 234.  
Paz, Mario a ~ d  PGrris, John, "The Use of Vibration for 
Katerial Handling, li Joclrnal of Engineering for Irdustry 
ASME, PaFer No. 73-MH-8, pp. 735-740, August, 1974. 
Bleckmar, , I. I., "Rotation of an Unhalanced Rotor 
Produced by Iiarmonic Oscil lati on of the Axis. 'I 
Bulletin of Academy of Sciences, USSR, Division of 
Technical Sciences No.8, 1954. 
Paz, Nario, "Sel f-Synchronization of Two Eccentric 
Rotors on Plane Motion." Shcck and Vibration, Bulletin 
4 1  , Part 6 , Naval Research Laboratory, Washington, 
D.C., Dec. 1970, pp. 159-162. 
Paz, Eiario, "Synchronization and Phase Angle of Two 
Unbalanced Retors." Schcck and Vibration, Bulletin 44, 
Fart 5, Naval Research Labcratory, Washington, D.C., 
August 1974, pp. 83-88. 
Blackman, R . L . ,  Schrader, P.H., and Paz, Mario, 
"Predicticn cf Self-Synchronization of Unbalanced 
Rotors," Wcrld Congress on the T h e o r y  of Machines and 
Mechanisms, Vo1.2, Part 2, C u b l i n ,  Ireland, Sept. 1974, 
pp. 375-384. 
Inque, J., Araki, Y., and Wantanabe, Y., "On the 
Multiple S e l  f-Synchronization of Mechanical Vibrators" 
Proc. of the World 4th Congress on the Theory of 
Machines and Kechanisms, University of Newcastle, 
September 1575, pp. 451-457. 
Bangchun, Wen , Xiangyang, Lin, "Vibratory 
Synchronization Transmissi.on," Chinese Journal of 
Mechanical Engineering, Vol. 20, No. 3 ,  September 1584, 
pp. 26-42 .  
215 


Electro-mechanical vibratory system

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server