Ulud. Univ. Zir. Fak. Oerg., (1993) 10:1-8 
Modeling of Natural Frequency and Amplitude 
on a Non-Driven Vibrating Potato 
Harvaster Shank 
Rasim OKURSOY * 
SUMMARY 
A non driven vibrating potato harvester has been attempted to design to 
produce exactly same type of vibration function with the forced vibratory soil 
digging shank The analysis examines the rotation angle and position characteristics 
of the shank as the soil force that acts on the b/ade. Because the soil digging 
shank was assumed to be clamped with an elastic material on to a solid potato 
harvester [rame, the shank was assumed in a certain vibration in the soil during 
to soil digging. A system of expressinn was generated to deseribe the general motion 
oft he shank, and equations oft he motion were solved by analytically in canceming 
with necessary assumptions. 
Key words: Natural Frequency, Amplitude, Vıbration, Potato Harvester, 
Modeling. 
ÖZET 
Tahriksiz Patates Hasat Makinalannan Titre§iminde 
Dogal Frekansm ve Genli~in Modellenınesi 
Titreşim/i toprak işleme aletlerinin toprakta yaptığı titreşim hareketine 
benzer bir titreşim yapabilecek bir patates hasat makinasına ait patates sökme 
organı tasar/anmaya çalışılmıştır. Analiz, söldicü bıçağa etki eden toprak 
kuvvetlerinden hareketle sökiicü organın dönme açısı ile durum karakteristiklerini 
* Öğr. Göf. Dr.; U.Ü. Ziraat Fakiiltes~ Tanm Makinalan Bölümü 
- ı -
saptar. Aletin toprak kazan kısmma ait taşıyıcı organın, e/astik bir materyal ile 
birlikte ana şaseye sabitleştiri/miş olması yüzllnden toprak kazıcmm toprak işleme 
süresi boyunca belirli bir titreşim gösterdiği kabul edilmiştir. Sistem kazıcı ayağın 
genel hareket eşitlikleri ile tanımlanmış ve eşitlikler denklemlerin sınır koşullan 
dikkate alınarak analitik yolla çözllmlenmiştir. 
Anahtar sözcükler: Doğal Frekans, Gen/ik, Titreşim, Patates Hasat 
Makinası, Modelleme. 
INTRODUCTION 
A non-driven vibrating soil digger for proposed use on a potato 
barvester generates natural vibration frequency when it moves through the soil 
with a certain velocity. The main discussion point is to determine the natural 
frequency and amplitude of a non-driven vibrating blade oscillation in a specific 
soil conditions. Generally, potat o harvesters are designed by using a mechanical 
driver which forces the vibration of the digger blades and tines (Hammerle, 
1970). Therefore, their construction are complex and are resulted higher design 
cost compared with the non-driven vibrating machines. 
Most non-driven vibrating potato harvesters are usually designed by 
~ing S-type shanks which serve as their own spring (Johnson, 1974). Elastomer 
clamps are also widely used as non-driven vibrating soil digging devices in order 
to generate similar dynamic situations. 
In this work, all analysis and calculations were performed on a 
elastomer clamp type soil digging blade in the sandy loam. The dynamic soil 
resistance force was derived as a function of the soil and tool parameters such 
as soil-tool friction, working depth and velocity. This force causes the rotation 
of the machine axis from x-y to x' -y' (Figure le). When the system rotates at 
certain rotation angle, the elastic material is squeezed by the clamps and it acts 
as a belical spring with a spring constant k. Therefore, deformed elastomer 
stores energy that creates vibration on the system. As a result, the solution of 
the problem takes place for defining dynamic soil forces on the blade and 
deriving equations of motion about the system under these forces. 
MATERIAL AND METHOD 
The soil-tool system is under influence of several factors which are 
related either soil or tool and soil-tool interaction (Sial, 1977). In the model, 
the soil depth, soil cohesion, the soil internal friction angle, the soil bulk 
density and the soil surface forward failure angle are used as soil parameters. 
Similarly, the tool sharpness angle, blade dimension (thickness, width, length) 
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and soil-blade friction angle are also used as blade parameters. The main 
approach is to make some assumptions in order to drive equations of the 
dynamic soil forces acting o n the blade in specific soil condition. These 
equations are solved analytically by concerning soil and tool parametcrs. 
The soil forces on inclined tool was given in Figure la. By the force 
equilibrium concept, the total fo rce acting on the center of the blade in 
horiw ntal directian is, 
(1) 
where R is draft force, ~· is soil metal friction coefl1cient, N0 the normal load 
on the inclined tool, k' is pure cutting resistance of soil per unit width, b is 
tool width and ô is lift angle of the too l (Gill and Vanderberg, 1968). The 
normal load on a inclined soil tillage tool was deseribed as a function of the 
weight of the soil segment (G) and other soil and tool parameters such as 
cohesion of the so il, lift angie  of the b Iade ( ô ), angie  of forward fa ilure surface 
angle (P) ete. 
( a) ( b) (c ) 
Figure 1 
(a) Soil Forces A cting to inclined Tine, (b) A Non-Driven Vibrating 
Potato Harvester Shank, (c) Simulation of the Shank 
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The second approach for finding natural frequency and the amplitude 
of the system is to simulate elastomer clamp as a set of helical spring which is 
assumed to be fixed in the clamp. From the Figure lb and Figure le, the total 
moment at the point O is, 
2 
lo d6(t) = RS3 Cos(e) -2[kS1 Sin(e)] [Sı Cos(e)]-9.8mgS2 Sin(6)] 
dt 2 
(2) 
where J0 is the mass moment of inertia of the system, k is spring constant, m 
is system mass, e is rotation angle on the x-y axis. For smail oscillation that 
means the smail rotation angle, we can assume Sin6 = O and Cos6 = 1 then 
the equation becomes, 
2 
2kS + 9.8mgS2 RS1 3 (3) 
6=-
1o 1o 
If the system has a damping, the analysis should be done by considering 
the damping coefficient of the system. Therefore, dynamic equations of the 
system con be rewritten by including the damping coefficient which is a 
function of several factors such as soil blade friction, soil moisture content, 
plasticity and adhesion. In general, if D and A. represent the damping coefficient 
and the damping ratio, the equation can be reduced in smail fraction. 
Therefore, 
2 
1 de(ı) 2 - [ .!!._ ı da(t) = _RS3  Cos(e) _2_kS_ı  Sin(6) Cos(6) 
0 
dıı 'o dt lo lo 
(4) 
9.8nıgS2 
- Sin(e) 
'o 
where 2l=D/J0. Assuroing Sine = O and Cose = ı for the smail osdilation 
(Zill, 1982), the final equation of the motion is turn out to second order linear 
differential equation with force function. That is also, 
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(5) 
where the 2 c.> is given by the following formula; 
2 
2kS + 9.8mgs2 
cl=1 ---- (6) 
The differential equation of the system solution was performed with 
computer program that is written in FORT RAN. The flowchart of the program 
is given in Figure 2. In the Appendix: ı, the program is also given. 
Figure 2 
Flowchart of the Computer Program 
DISCUSSION AND RESULT 
The solution of the linear second order differential equations with force 
function consist on the particular and the homogenous solution with necessary 
initial and boundary condition. For calculations, dimension and the total weight 
of the sbank and clamp have been arbitrarily chosen. The elastic material in the 
clamp was assumed as a set of belical spring which has a spring constant as ıo5 
N/m. The spring mass and the soil adhesion force were neglected for the 
calculations. The initial conditions were assumed as, 8(0) = O and d8(0)/dt = 
o. For the calculation, the blade cutting angle, soil cohesion coefficient, the soil 
internal friction angle and soil-metal friction angle were chosen as 16°, 700 
N/m, 25° and 20°, respectively. The soil force cakulation were performed for 
the sandy loam, which has 1660 N/m density, and the blade velocity was 5 km 
per hour in that soil. 
The biggest difficulty was to determine the soil damping coefficient, 
because it is a function of the soil plasticity, the soil moisture content. 
However, the equations of motion about the system were solved using several 
damping ratio which are shown in Figure 3. The frequency model has maximum 
amplitude around 0.07 radian if the damping coefficient is 0.25. The system can 
reach steady state in 2 second. The lower value in damping ratio produced high 
amplitude and frequency as it was expected. The critical value for the damping 
ration is 1, and there is no osdilation in that case. The system is also able to 
produce a sinus cycles which has no steady state equilibrium if the damping of 
the system is not taking account. 
O... . 
1\-- ı 1 ,_ 1 .. o.ıs 
1
~ '-"(• 0.5 
0.05 1 
~ - / ...... /V 7 
o.. .. '! (\ ---
V 
..,...... V; '-- ..,~ 0.75 AA~ r/ 7 :r1.0 
'/ 
0.02 
1 
0.01~ 
oo  0.4 OB 1.2 1.6 2.0 2.4 
Time (see) 
Figure 3 
System Vibration for Critica/ and Underdamping Situation 
- 6 - . 
REFERENCES 
GILL, W.R. and G.E. V ANDERBERG, 1968. Soil Dynamics in Tillage and 
Traction. Agriculture Handbook, No: 316. Agricultural Research 
Service, USDA. Washington, DC. 
HAMMERLE, J. 1970. The Design of Sweet Potato Machinery. Transaction of 
the ASAE, p. 281-270. 
JOHNSON, L.F. 1974. Vibrating Blade for Potat o Harvester. Transaction of the 
ASAE, p. 867-872. 
SIAL, S.F. 1977. Longitudinal - Vertical Behavior of Soil Engaging Implements. 
North Carolina State Univers ity, ASAE Annual Meetings, June 26-29, 
Raleigh, NC. 
ZILL, D., A First Course in Differential Equations with Applications. 2nd 
Edition, PWS Publisher, Bostan 1982. 
Appencllx 1. FORTRAN Program 
C*********************************************************** 
C CALCULATION OF NATURAL FREQUENCY AND AMPLITUDE 
C A NON-VIBRATING POTATO HARVESTER SHANK EITHER 
C DAMPED OR UNDAMPED CONDITION 
c 
C Dr. Rasiın OKURSOY 
C ULUDAG VNIVERSITESI ZIRAAT FAKULTESI 
C TARIM MAKINALAR! BOLUMU 
c 
C NOTE: ALL UNITS ARE IN METRIC SYSTEM 
C*********************************************************** 
REAL JO,L,LAM,M,KS,K,MU1,MU2,N1 
DATA VC,PI,WD,ZERO,BW,BL/S.,J.1416,0.25 , 0.,0.55,0.11/ 
DATA GRAV,GAMA,C,STEP/9.81,1660.,700.,0.025/ 
DATA M,S1,S2,S3,J0/1.1151,0.03,0.035,0.03,3.68/ 
OPEN(2,FILE='VIBRA,OUT',STATUS='NEW') 
WRITE(*,*) 'ENTER SPRING CONSTANT AND DAMPING RATIO' 
READ(*,*)KS,DR 
WRITE(*,*)'----------------------------------------' 
V=(100./360)*VC 
K=PI/180. 
RH0=16.*K 
RH01=20.*K 
PHI=25. *K 
BETA~(PI/4.)-(PHI/2.) 
MU1=TAN(RH01) 
MU2=TAN(PHI) 
C*********** PRINT INPUT DATA ****************************** 
- 7 . 
WRITE(2,*) ' BETA,V',BETA,V 
WRITE(2,*) '---- ------ ---- - --- --- ---------- ----- ---- - -' 
WRITE(2,*) 'DR,STEP',DR,STEP 
Bl=(WD*SIN(RHO+BETA))/(SIN(BETA)) 
B2=COS(RHO+BETA)+(TAN(RHO)*SIN(RHO+BETA) 
G=GRAV*GAMA*BW*Bl*)BL+(WD/(2.*SIN( BETA)) )*B2) 
Cl=(GAMA/GRAV)*BW*Bl*( BL+(WD/ (2. *SIN(BETA)))*B2 ) 
C2=SIN(RHO)/SIN(RHO+BETA) 
FA=GRAW*(Cl*C2) 
AF=(WD*BW)/SIN(BETA) 
Al=(COS(RHO)-MU1*SIN(RHO))*(SIN(BETA)+MU2*COS (BETA)) 
A2=(-(SIN(RHO)+MU1*COS(RHO)))*(COS(BETA) -MU2*SIN(BETA)) 
DELTA=Al-A2 
A3=(SIN(BETA)+MU2*COS(BETA))* (G+(C*AF+FA)*SIN(BETA)) 
A4=(COS (BETA) - MU2*SIN(BETA))*(- (C*AF+FA)*COS(BETA)) 
DELTA1=A3-A4 
Nl=DELTAl/DELTA 
R=(Nl*( S IN(RHO)+MUl*COS(RH0))) / 2 
C********** SOIL FORCE CALCULATION ************************ 
ETA=ABS(DR) 
OM=SQRT((2 . *KS*S1**2+M*GRAV*S2*GRAV) / JO) 
LAM=ETA*OM 
VAL=(R*S3/(JO*OM* *2 .) ) 
DO 100 J=l , l OO 
T=J*STEP 
EF=EXP(-LAM*T) 
IF(ETA.EQ.ZERO) GO TO 75 
IF(ETA.LT.l) GO TO 76 
IF(ETA.EQ . l) GO TO 7 7 
IF(ETA.GT. l ) GO TO 78 
76 RT2=SQRT(OM**2. -LAM**2.) 
TETA= (-VAL) *EF*(COS(RT2*T)+(LAM/ RT2)*SIN(RT2*T))+VAL 
GO TO 80 
77 TETA= (-VAL)*EF*( l. + LAM*T )+VAL 
GO TO 80 
78 RT1=SQRT(LAM**2. - 0M**2. ) 
VALl=LAM+RTl 
VAL2=LAM- RT1 
VAL3=( - VALl*EXP(RTl*T)+VAL2*EXP(-RTl*T)) 
TETA=(VAL/(2. *RT1 ))*EF*(VAL3)+VAL 
GO TO 80 
75 TETA=VAL*(l.-COS(OM*T)) 
C********** PRI NT OUTPUT DATA ***************************** 
80 WRITE(2,35) J,T,TETA 
35 FORMAT(1X, I 3,2(1X, F15 3)) 
100 CONTINUE ' 
CLOSE(2,STATUS= 'KEEP') 
STOP 
END 
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