WING INERTIA

The  program  WINGINER.BAS calculates the  spanwise  inertia 
loads,  shears  and moments in balanced  and  accelerated  flight 
along the quarter chord of the wing.  Input data required are the 
inertia factors which are obtained for the selected critical wing 
conditions, the wing panel weight, the ratio of area densities at 
the tip to the root, the wing plan form geometry and the dihedral 
angle  of  the  wing reference plane and  the  waterline  of  its 
intersection  with  the center plane of symmetry at  the  quarter 
chord.

Unsymmetrical Rolling Conditions

The unsymmetrical conditions require special data. For  the 
accelerated  roll, the unbalanced rolling moment is needed.  This 
may be calculated as follows.

According to FAR 23.249(a)(2) the rolling acceleration loads 
may be obtained by modifying the symmetrical condition for condi- 
tion A in the figure of FAR 23.333(d) which is the condition  for 
stalling  speed at limit load factor.  Assume 100 percent of  the 
airload of condition A on one side of the airplane and 70 percent 
on  the  other.  For airplanes of more than  1000  pounds  design 
weight,  the percentage may be increased linearly with weight  up 
to 75 percent at 12500 pounds.

In the sample calculation to follow for accelerated roll, we 
found the bending moment at the plane of symmetry for condition A 
(case 262) is 324,050 inch pounds. We used case 262  because  the 
critical  condition  for accelerated roll was case 280  at  18000 
foot altitude and CG 4 as selected by the program SELECT.BAS. and 
case 262 is condition A for CG 4 at 18000 feet altitude.

Using  the  formula  for percentage on  the  other  side  we 
calculate the other side is 71.43 percent or 231,469 inch pounds. 
Then  the unbalanced moment accelerating the roll is  92581  inch 
pounds.

Input

Sample  calculations for three critical conditions 
using the following data.

Case

22 
280 
138

Cond

PHAA 
ACRL 
TORS

^

3.8

3.24

2.54

NX

-.6065

-.3999 
.1318

Pitch

Accel

0

0

0

are  made

Rolling

Moment

0

92581

0

231

FAR 23 LOADS

"x 22=~2062/340U=~U06^

N., 2go=-825/2063=-.3999

N.. ,-,o=+448/3400=+.1318

Concentrated Weight

The program WINGINER.BAS now accounts for a concentrated 
weight item such as a landing gear or engine.

232

WING INERTIA

3 REM-----------------------------------------------------------------

5 REM          HEADING, COPYRIGHT, PRINTER MARGINS

7 REM-----------------------------------------------------------------

10 CLS:PRINT "WING INERTIA PROGRAM ---WINGINER.BAS, VERSION 2.01"

15 PRINT "COPYRIGHT (C) HAL C. MCMASTER, 1988, 1990"

16 PRINT:INPUT "IS PRINTER READY ";QP$:OP$=LEFT$(QP$,1)

17 IF QP$0"Y" THEN 16

18 LPRINT CHR$(27)CHR$(78)CHR$(6):REM   TO SKIP OVER PERFORATIONS

19 LPRINT CHR$(27)"1"CHR$(2):REM   TO SET LH MARGIN

20 INPUT "HOW MANY POINTS DEFINE THE LEADING EDGE";M$:M=VAL(M$):IF M<2

OR M>100 THEN 20 
30 DIM XLE(M),YLE(M) 
32 REM---------------------------------------------

34 REM           ENTERING AND PRINTING INPUT DATA

36 REM--------------------------------------------------------------

40 FOR 1=1 TO M

50 INPUT "ENTER COORDINATES OF LEADING EDGE STARTING AT INBOARD (XLE,

YLE ---- X IS AFT, Y IS OUTBOARD)";XLE(I),YLE(I)

60 NEXT I

70 INPUT "HOW MANY POINTS DEFINE THE TRAILING EDGE";N$

80 N=VAL(N$):IF N<2 OR N>100 THEN 150

90 DIM XTE(N),YTE(N)

100 FOR 1=1 TO N

110 INPUT "ENTER COORDINATES OF TRAILING EDGE STARTING AT INBOARD (XTE

,YTE ----X IS AFT, Y IS OUTBOARD)";XTE(I),YTE(I)

120 NEXT I 
130 INPUT "ENTER THE BL OR WING STATION OF THE INBOARD RIB OF THE WING

PANEL";RSTA$

140 RSTA=VAL(RSTA$):IF RSTA<0 OR RSTA>YLE(M) THEN 130 
150 INPUT "ENTER THE NUMBER OF SPANWISE ELEMENTS THE WING IS TO BE

DIVIDED INTO";H$

160 H=VAL(H$):IF H<2 OR H>100 THEN 150 
170 DIM YE(H)

180 DY=(YLE(M)-YLE(1))/H:YE(1)=YLE(1)+DY/2 
190 FOR L=2 TO H 
200 YE(L)=YE(L-1)+DY 
210 NEXT L

390 DIM XF(H),XA(H),CE(H),C25X(H+1),C50X(H) 
400 FOR J=1 TO H 
402 IF M>2 THEN 410

406 XF(J)=(XLE(2)-XLE(1))*(YE(J)-YLE(1))/(YLE(2)-YLE(1))+XLE(1) 
408 GOTO 510 
410 FOR 1=1 TO (M-1)

420 IF YE(J)<YLE(I) OR YE(J)>YLE(1+1) THEN 460 
440 XF(J)=(XLE(I+1)-XLE(I))*(YE(J)-YLE(I))/(YLE(I+1)-YLE(I))+XLE(I) 
460 NEXT I

510 IF N>2 THEN 520

514 XA(J)=(XTE(2)-XTE(1))*(YE(J)-YTE(1))/(YTE(2)-YTE(1))+XTE(1) 
516 GOTO 555 
520 FOR K=1 TO (N-1)

530 IF YE(J)<YTE(K) OR YE(J)>YTE(K+1) THEN 550 
540 XA(J)=(XTE(K+1)-XTE(K))*(YE(J)-YTE(K))/(YTE(K+1)-YTE(K))+XTE(K) 
550 NEXT K 
555 NEXT J

233

FAR 23 LOADS

570 INPUT "ENTER WL WING REF PLANE AT PLANE OF SYMMETRY OF AIRPLANE";

WL

580 INPUT "ENTER SLOPE OF WRP IN DEGREES":SL

590 DIM DA(H),Z(H+1)

610 FOR L=1 TO H

620 CE(L)=XA(L)-XF(L):DA(L)=CE(L)*DY:C25X(L)=XF(L)+.25*CE(L):C50X(L)=

XF(L)+.5*CE(L)

630 Z(L)=WL+TAN(SL/57.3)*YE(L)

640 A=A+DA(L)

690 NEXT L

691 PRINT: INPUT "IF THERE IS A CONCENTRATED WEIGHT ON THE WING, ENTER 
ITS' WEIGHT ELSE ENTER 0 ";CWT

692 INPUT "ENTER THE COORDINATES OF THE CONCENTRATED WEIGHT X,Y,Z ELSE 
ENTER 0,0,0 ";XCWT,YCWT,ZCWT

700 INPUT "WHAT IS WEIGHT OF WING PANEL (DO NOT INCLUDE CONCENTRATED 
WE IGHT) "'W

701 INPUT "WHAT IS RATIO OF DENSITY OF TIP AREA RELATIVE TO ROOT AREA 
--(.9 IF ROOT IS .2 LBS/SQR IN AND TIP IS .18 LBS/SQR IN)";DR

702 LPRINT TAB(22)"WING INERTIA":LPRINT

703 LPRINT "INPUT DATA":LPRINT

711 LPRINT "WING PANEL WEIGHT ON ONE SIDE = ";W;" LBS"

712 LPRINT "RATIO OF DENSITY PER SQ FT OF TIP AREA TO ROOT AREA = ";DR

713 REM---------------------------------------------------------------

714 REM         CALCULATE ROOT DENSITY

715 REM-------------------------------------------------------------

716 1=0 
720 1=1+1

725 IF YE(I)<RSTA THEN 720 
730 YINBD=YE(I):II=I 
735 DENSR=.02

737 DIM W(H)

738 TW=0

739 PRINT "CALCULATING DENSITIES, PLEASE WAIT"

740 FOR 1=11 TO H

750 W(I)=DA(I)*DENSR*(1-(YE(I)-RSTA)*(1-DR)/(YLE(M)-RSTA))

760 TW=TW+W(I)

770 NEXT I

772 D=DY/2-(YE(II)-RSTA)

774 DW=D*CE(II)*DENSR

776 TW=TW-DW

778 W(II)=W(II)-DW

780 IF TW>.99*W AND TW<1.01*W THEN 820

790 IF TW=>1.01*W THEN DENSR=DENSR-.00001

800 IF TW=<.99*W THEN DENSR=DENSR+.00001

810 GOTO 738

820 DENST=144*DR*DENSR:DENST=INT(DENST*1000)/1000

830 DENSR=144*DENSR:DENSR=INT(DENSR*1000)/1000

840 LPRINT "DENSITY AT ROOT = ";DENSR;" LB/SQ FT":LPRINT "DENSITY

AT TIP = ";DENST;" LB/SQ FT":LPRINT:LPRINT:GOTO 890

881 REM-------------------- -------------------------------------------

882 REM         CALCULATE 1 G VERTICAL INERTIA LOADS

883 REM---------------------------------------------------------------

890 PRINT "CALCULATING 1G VERTICAL INERTIA LOADS" 
898 DIM SZ(H+1),MXX(H),TYY(H),DTYY(H)

234

WW INERTIA

900 SZ=0:SZ(H+1)=0:MXX=0:TYY=0

910 FOR I=H TO 1 STEP -1

920 SZ=SZ+W(I)

930 SZ(I)=SZ

940 MXX=MXX+SZ(I+1)*DY

950 MXX(I)=MXX

955 DTYY(I)=-W(I)*(C50X(I)-C25X(I))

960 TYY=TYY-SZ(I+1)*(C25X(I+1)-C25X(I))+DTYY(I)

9 70 TYY(I)sTYY

971 NEXT I

972FOR I=H TO 1 STEP -1

973 IF YE(I)<YCWT THEN SZ(I)=SZ(I)+CWT

974 IF YE(I)<YCWT THEN MXX(I)=MXX(I)+CWT*(YCWT-YE(I))

975 IF YE(I)<YCWT THEN TYY(I)=TYY(I)+CWT*(C25X(I)-XCWT)

980 NEXT I

981 REM--------------- ----------------------,-------------

982 REM          CALCULATE 1 G DRAG LOADS

983 REM-- --------------------------------------------------

990 PRINT "CALCULATING 1G DRAG INERTIA LOADS"

997 DIM SX(H+1),MZZ(H),TVYY(H)

1000 SX=0:C25X(H+1)=C25X(H):MZZ=0:SX(H+1)=0:Z(H+1)=Z(K):TVYY=0

1010 FOR I=H TO 1 STEP -1

1020 SX=SX+W(I)

1030 SX(I)=SX

1040 MZZ=MZZ+SX(I+1)*DY

1050 MZZ(I)=MZZ

1055 REM DVYY(I)=0

1060 TVYY=TVYY+SX(I+l)*(ZlI+l)-Z(I))

1070 TVYY(I)=TVYY

1071 NEXT I

1072 FOR I=H TO L STEP -1

1073 IF YE(I)<YCWT THEN SX(I)=SX(I)+CWT

1074 IF YE(I)<YCWT THEN MZZ(I)=MZZ(I)+CWT*(YCWT-YE(I))

1075 IF YE(I)<YCWT THEN TVYY(I)=TVYY(I)+CWT*(ZCWT-Z(I))

1080 NEXT I

1081 REM-----------------------------------------------

1082 REM         CALCULATE UNBALANCED aOLLING INERTIA

1083 REM---------------------------------------------

1090 PRINT "CALCULATING 100000 IN-LB UiNBALANCED ROLL INERTIA" 
1097 DIM FZ(H),SUZ(H+1),MUXX(H),TUYY(H),DVYY(H) 
1100 FOR 1=1 TO H 
1110 IXX=IXX +W(I)*YE(I)"2

1120 NEXT I

1121 IF CWTOO THEN IXX=IXX+CWT*YCWT~2 
1130 IWXX=2*IXX 
1150 FOR 1=1 TO H

1160 FZ(I)=W(I)*YE(I)*lUOOOOi/IWXX 
1170 NEXT I

" '71 FZCWT=CWT*YCWT*100000!/IWXX

, "5 REM   1180 TO 1270 SAME AS 900-970 EXCEPT FZ(I) REPLACES W(I) 
J 1' 0 SUZ=0:MUXX=0:SUZ(H+1)=-0:TUYY=0 
1200 FOR I=H TO 1 STEP -1 
1210 SUZ=SUZ+FZ(I) 
1220 SUZ(I)=SUZ

WING  INERTIA

1493 LPRINT "WING CL TO TIP IS DIVIDED INTO ";H;" ELEMENTS"

1494 LPRINT "WL REF PLANE AT PLANE OF SYMMETRY IS ";WL

1495 LPRINT "SLOPE OF WRP IS ";SL;" DEGREES":LPRINT "RATIO OF

DENSITY OF TIP TO ROOT IS ";DR

1496 LPRINT "WING PANEL (TIP TO TIP) INERTIA IXX = ";IWXX;" LB-IN 
-2"

1497 LPRINT "WEIGHT OF WING PANEL IS ";W:LPRINT "CONCENTRATED 
WEIGHT IS ";CWT;" LBS":LPRINT "AT COORDINATES X, Y, Z    ";-XCWT;

";YCWT;",  ";ZCWT

1498 LPRINT:LPRINT K;" CASES ARE CALCULATED"

1499 LPRINT "CASE NO","NZ","NX","UNBAL MOM"

1500 FOR 1=1 TO K

1501 LPRINT CASE(I),NZ(I),NX(I),UNB(I )

1502 NEXT I

1503 LPRINT:LPRINT:LPRINT "OUTPUT DATA":LPRINT

1504 PRINT "PRINTING OUT LOAD, SHEAR AND TORSION INERTIA DISTRIBUTIONS 
n

1505 P$="+###.ft# +###.ft# +^#.#ft +ft### +#ft## +##ftft +MM +ft##ftft 
+##^## +W#^f +##^#"

1506 PCWT$="+###.## +^#.^ +###.ft# +ftfft# +#ft^" 
1530 FOR J=1 TO K

1533 THETADOT(J)=UNB(J)*386/IWXX

1534 THETADOT(J)=INT(THETADOT(J)* 1000)/1000

1535 LPRINT.-LPRINT "CASE = ";CASE(J)"  NZ = ";NZ(J);"  NX = ";

NX(J);" THETADOT = ";THETADOT(J);" UNBAL MOM = ";UNB(J) 
1537 LPRINT "   X       Y       Z       FX    FZ  DMYY     SX     SZ

MXX     MYY   MZZ" 
1540 FOR I=H TO 1 STEP -1 
1550 LPRINT USING P$;C25X(I);YE(I);Z(I);T2FX(J,I);TFZ(J,I);DMYY(J,I) ;

TXS(J,I);TSZ(J,I);TMXX(J,I);TTYY(J,I);TZMZ(J,I)

1560 NEXT I

1561 LPRINT "CONCENTRATED WEIGHT"

1562 LPRINT USING PCWT$;XCWT;YCWT;ZCWT;FXCWT(J);FZCWT(J) 
1570 NEXT J 
1580 END

237

FAR 23 LOADS

Example input for Wing Inertia:

WING INEBTIA

INPUT DATA

WING PANEL WEIGHT ON ONE SIDE = 165 LBS

RATIO OF DENSITY PER SQ FT OF TIP AREA TO ROOT AREA = .95 
DENSITY AT ROOT = 2.213 LB/SQ FT 
DENSITY AT TIP = 2.102 LB/SQ FT

3  POINTS DEFINE LEADING EDGE OF WING

ITEM          X Y

1             45 0

2             64.31301 46.5

3             72 201

2  POINTS DEFINE TRAILING EDGE OF WING 
ITEM          X             Y

1             146           0

2             116           201

BL OF INBOARD RIB IS  23

WING CL TO TIP IS DIVIDED INTO  20  ELEMENTS

WL REF PLANE AT PLANE OF SYMMETRY IS  78.5

SLOPE OF WRP IS  6  DEGREES

RATIO OF DENSITY OF TIP TO ROOT IS  .95

WING PANEL (TIP TO TIP) INERTIA IXX =  4330074  LB-IN'2

WEIGHT OF WING PANEL IS  165

CONCENTRATED WEIGHT IS  0  LBS

AT COORDINATES X, Y, Z     0 ,   0 ,   0

6 CASES ARE "CALCULATED

CASE NO 
22 
280 
138

1001

1002

1003

NZ

-3.8

-3.24

-2.54

NX

.6065 
.3999 
-.1318 
0

UNBAL MOH 
0

-92581 
0 
0 
0

-100000

238

