Lateral stability of road tankers : Vol. 2 Appendices

S tate n s v ä g - o c h tra fik in s titu t (V T I) • F ack • 5 8 1 0 1 L in k ö p in g N r 1 3 8 A • 1 9 7 8 N a tio n a l Road & Traffic R esearch In s titu te • Fack • S -5 8 1 01 L in k ö p in g • S w e d e n ISSN 0347-6030

138 A

L a te ra l S ta b ility o f R o a d T a n k e rs

V o lu m e II. A p p e n d ic e s

b y L e n n a rt S tra n d b e r g

S ta te n s v ä g - o c h tra fik in s titu t (V T I) • F ack • 5 8 1 0 1 L in k ö p in g N r 1 3 8 A -1 9 7 8 N a tio n a l R oad & Traffic R esearch In s titu te Fack S -5 8 1 01 L in k ö p in g S w e d e n ISSN 0347-6030

138 A

L a te ra l S ta b ility o f R o a d T a n k e rs

V o lu m e II. A p p e n d ic e s

b y L e n n a rt S tr a n d b e r g

Page A N O T A T I O N A1 A . 1 Indices Al A. 2 Sym b o l s A3 B C O N D I T I O N S F O R D Y N AMIC S I M I L A R I T Y B E T W E E N M O D E L A N D FULL SIZE T A N K Bl B.l I m p o r t a n t q u a l i t a t i v e e f f e c t s Bl B. 2 Q u a n t i t a t i v e d y n a m i c d e s c r i p t i o n i n d e p e n d e n t of size B2

B. 3 I n ertial scaling, g e o m e t r i c and k i n e m a t i c r e q u i r e m e n t s B2 B . 4 V i s c o u s sca l i n g B4 B . 5 C a v i t a t i o n s caling B6 B . 6 C h o s e n e x p e r i m e n t a l d e s i g n a n d scale factors B8 C T A N K M O D E L A N D L I Q U I D F O R C E S Cl C.l Br i e f d e s c r i p t i o n of the l a b o r a t o r y e q u i p m e n t Cl C . 2 Forces and m o m e n t s in t he m o d e l C2

C . 3 Forc e s and m o m e n t s in full scale C3

C . 4 E q u i p m e n t da t a C4

D M A T H E M A T I C A L D E S C R I P T I O N OF THE S I M U L A T I O N M O D E L S Dl

D.l G e n eral c o m m e n t s Dl

D. 2 A p p r o x i m a t i o n s and a s s u m p t i o n s for the v e h i c l e m o d e l s D2 D.2.1 L a t e r a l m o t i o n s are p r i m a r y D2 D.2.2 Roll m o t i o n s D2 D.2.3 Y a w m o t i o n s n e g l e c t e d D3 D.2.4 L o n g i t u d i n a l and v e r t i c a l m o t i o n s n e g l e c t e d D3 D. 3 D e f i n i t i o n of the o v e r t u r n i n g risk factor D3 D . 4 N o n r o l l i n g v e h i c l e w i t h r i g i d load D4 D. 5 N o n r o l l i n g v e h i c l e w i t h s l o s h i n g load D5 D . 6 R o l l i n g v e h i c l e w i t h ri g i d load D6 D. 7 R o l ling v e h i c l e w i t h s l o s h i n g load D9

D . 8 V e h i c l e a n d load data DlO

D . 8 .1 V e h i c l e data DIO

Pag e D. 8.2 S l o s h i n g load data D12 D.8.3 Ri g i d load data D12 D. 9 DAVIS, a mo d e l of d r i v e r / m a n o e u v r e D16 D.9.1 G e n e r a l d e s c r i p t i o n D16 D.9.2 I n v e r s e s t e e r i n g p r o c e d u r e D16 D.9.3 M a n o e u v r e s p e c i f i c a t i o n D18 D. 9.4 M a t h e m a t i c a l m a n o e u v r e d e f i n i t i o n D18 E C O M P U T E R P R O G R A M S F O R P H A S E II El E .1 G e n e r a l c o m m e n t s El E. 2 O p e r a t i o n c o n t r o l l i n g c i r c u i t s El E . 3 DAVIS m a n o e u v r e g e n e r a t i n g c i r c u i t E3 E . 4 S l o s h i n g force and m o m e n t c o m p u t i n g c i r c u i t s E4 E. 5 V e h i c l e m o d e l s E5 E . 5 . 1 N o n r o l l i n g v e h i c l e w i t h r i g i d load E5 E . 5.2 N o n r o l l i n g v e h i c l e w i t h s l o s h i n g load E5 E . 5.3 R o l l i n g v e h i c l e w i t h r i g i d load E6 E. 6 C o m p u t e r g e n e r a t i o n of ri g i d liquid forces E7 E. 7 C o m p u t i n g unit a s s i g n m e n t a nd pot sett i n g s E9

F R E S ULTS F R O M P H A S E II FI

F.l T a b l e s p r e s e n t i n g m a x i m u m v a l u e s Fl F. 2 R e c o r d e r o u t p u t s and p h a s e p o r t r a i t s F2

F . 3 C o m p u t i n g errors F2

F . 4 R e p e a t a b i l i t y F4

A P P E N D I X A

N O T A T I O N

S y m b o l s not found b e l o w a re e x p l a i n e d in c o n n e c t i o n w i t h th e i r a p p e a r a n c e in the text. P a r t i c u l a r l y a p p e n d i ­ ces C, D.9, E a nd F c o n t a i n specific symbols e x p l a i n e d in figure Cl, figure D 5 / t a b l e D 3 , table El, and t able F1/F3 respectively.

Dots above symbols in d i c a t e time d e r i v a t i v e s e.g.

A .1 Indices

c

II) C avitation. Used o n l y in a p p e n d i x B.5

d Dynamic conditions. See page C2 and f o r m u l a (D.6)

D DAVIS manoe u v r e . C o m p a r e index H and a p p e n d i x D.9

E E l l i p t i c - l i k e t a n k cross section. D e f i n e d in fig 3.7

F Full scale c o n t r a r y to model: index M

H H a r m o n i c o s c i l l a t i o n . C o m p a r e index D

£ Load, L i q u i d

L r e f e r r i n g to a point in the roll axis. See f igure Dl

M Model scale c o n t r a r y to full scale: index F or no index

n N a t u r a l frequency, 1st o s c i l l a t i o n m o d e

A2

0

r Ratio b e t w e e n m o d e l and full scale. See a p p e n d i x B . 3 - B . 6

R Rigid load. A liquid is a s s u m e d to be f r o z e n and fixed to the tank in its static p o s i t i o n a n d k e e p i n g its f l u idic v o l u m e

s Stat i c c o n d i tions. See page C2 and f o r m u l a (D.6)

S S u p e r e l l i p t i c - or r e c t a n g u l a r - like t a n k c r o s s section. D e f i n e d in fig 3.7 S S l o s h i n g load. N o r m a l c o n d i t i o n for a l i q u i d l o a d . C o m p a r e index R T , TP r e f e r r i n g to m a s s c e n t r e (c.g.)

y

z tank o r i e n t e d v e r t i c a l c o o r d i n a t e or d i r e c t i o n . See f i g u r e Cl and c o m p a r e figure Dl for a r o l l i n g v e h i c l e

H>

0 n o n r o l l i n g vehicle. I n f i n i t e l y la r g e roll s t i f f ­ ness a s s u m e d

50 50% of tank inner v o l u m e o c c u p i e d by the load

A. 2 S y m b o l s

a I) tank inner rad i u s for c i r c u l a r c y l i n d e r s or spheres. See f i g u r e 2.1

II) tank c o m p a r t m e n t w i d t h for r e c t a n g u l a r cyli- ders. See f o r m u l a (2.4)

III) tank cross s e c t i o n w i d t h for r e c t a n g u l a r cylinders. See f i g u r e 2.5

A tank inner cr o s s s e c t i o n Are a

c half the e f f e c t i v e tr a c k width. D i s t a n c e from the road c o n t a c t point of the o u t e r w h e e l s (when large sidef o r c e s are acting) to the v e r t i c a l and l o n g i t u d i n a l p l a n e of

s y m metry of the vehicle. See figure Dl

C roll d a m p i n g c o n s t a n t [moment per roll angle v e l o c i t y unit]

c . g . C e n t r e of Gravity, m a s s c e n t r e

d D i a m e t r e for c i r c u l a r tank c y l i n d e r

DAVIS D r i v e r A p p r o x i m a t i o n for V e h i c l e I n v e s t i g a t i o n by Simulation. A b b r e v i a t i o n for stee r i n g

p r o c e d u r e e x p l a i n e d in ap p e n d i x D.9

°D7 D e n o m i n a t o r in formula (D.7)

e roll axis d i s t a n c e ab o v e road s urface

f o s c i l l a t i o n F requency, cycles per s econd

fD DAVIS F r e q u e n c y d e f i n e d by f o r m u l a (D.34)

F 'Fy ' F z For c e s from liquid a c t i n g on t he t a n k walls. See page 37

f a 'f b,f c For c e s a c t i n g u p o n the sus p e n d e d u n i t s in

figure Cl

g

h I) liqu i d surface H e i g h t from tank bottom. See figure 2.1, 2.2

ii) d i s t a n c e b e l o w s u r face for a local po i n t in the liquid. Used o n l y in a p p e n d i x B.5

J L roll m o m e n t of i n e rtia of sprung m a s s and (rigid) load about the roll axis

J L roll m o m e n t of i n e rtia of sprung m a s s w i t h o u t load about the roll axis

A4

VTI REPORT NO. 1 3 8 A

J0 1 ' J L1 roll m o m e n t of inertia of r i g i d load (1) ab o u t tank c e n t r e (0) or roll axis (L)

k, kvf roll s t i f f n e s s constant. A n t i - r o l l m o m e n t from s p r ings a n d wheel s u s p e n s i o n s per roll an g l e unit

K D4 c o n s t a n t d e f i n e d by f o r mula (D.4a)

k d i o c o n s t a n t d e f i n e d in formula (D.lOa)

K 1 ' K 2 ratio b e t w e e n full scale and m o d e l l i q u i d forces (1) a nd m o m e n t s (2). See page C3 for f u r ther d e f i n i t i o n

K F A C r o l l i n g F A C t o r d e f i n e d in f o r mula (3.5), page 40

£ I) II)

t a n k c y l i n d e r inner Length. See fig u r e 2.1 c h a r a c t e r i s t i c len g t h - m o s t l y r e f e r r i n g to the tank c r o s s section pl a n e in a p p e n d i c e s B3-B.6 and D.8

L tank c y l i n d e r or c o m p a r t m e n t inner Length. See p a g e B8 and formula (D.16)

m total Mass (s p r u n g + unsprung) of the v e h i c l e w i t h o u t load

m" sprung Mass of the v e h i c l e w i t h o u t load

m " u n s p r u n g v e h i c l e Mass

m i load Mass

M o l i q u i d force Mome n t ab o u t tank c e n t r e

N D7 N o m i n a t o r in f o r mula (D.7)

*0 _a **

< 0 P r e s s u r e s y m b o l s d e f i n e d in a p p e n d i x B.5

p d D y n amic v e r t i c a l c o n t a c t force from all w h e e l s on the left side of the vehicle. See figure Dl

p

s Static v e r t i c a l c o n t a c t force from all w h e e l s on e i t h e r side of the vehicle. See f o r m u l a

(D.l)

r, r tank a c c e l e r a t i o n and v e l o c i t y v e c t o r s r e s p e c t i v e l y

R o v e r t u r n i n g (Risk)factor d e f i n e d in fo r m u l a (3.1) page 35. See also indices R, S, 0, on page 38

S h o r i z o n t a l S i d e f o r c e from the r o a d sur f a c e a c t i n g u p o n all w h e e l s of the v e h icle. See f i g u r e Dl and f o r mula (D.12)

SA l a t eral a c c e l e r a t i o n of a p o i n t fixed to the m o d e l tank or a full scale v e h i c l e

sal l a t eral a c c e l e r a t i o n in roll axis. See

a p p e n d i x D.6

sal im o v e r t u r n i n g limit, i.e. the m a x i m u m

a p p l i c a b l e lateral a c c e l e r a t i o n w i t h o u t i r r e v e r s i b l e o v e r t u r n i n g . D e f i n i t i o n s in f o r m u l a (3.2) or figure 3.8, page 36

SAO f t a n k c e n t r e o r i e n t e d lateral a c c e l e r a t i o n d u r i n g roll m o t i o n i n c l u d i n g gravity. See f o r m u l a (D.9)

sc

SFAC S l o s h i n g FACtor d e f i n e d in f o r m u l a (3.4), p a g e 38

t , t p p r o b l e m T i m e (real Time) o f t e n d i f f e r i n g f r o m c o m p u t e r time x a nd m o d e l time t^

u" spr u n g m a s s c.g. (without load) d i s t a n c e a b o v e roll axis

u " u n s p r u n g m a s s c.g. d i s t a n c e a b o v e roll axis

U 1 L o a d (assumed to be rigid) c.g. d i s t a n c e a b o v e roll axis

u

o t a n k c e n t r e d i s t a n c e above roll axis

V load (characteristic) V e l o c i t y r e l a t i v e to the tank for a liquid point used in a p p e n d i x B .4 and B .5

V t a n k inner V o l u m e

x o r e c t i l i n e a r h a r monic o s c i l l a t i o n a m p l i t u d e

for a po i n t fixed to a tank c o n t a i n e r

A6

• • ••

y ,y ,y,y l a t eral d e v i a t i o n from initial c o u r s e a n d its t i m e d e r i v a t i v e s . See f i g u r e D5 y d ' V V y d p e a k v a l u e s of Y , Y , Y , Y in DAVI S ' d o u b l e lane c h a n g e m a n o e u v r e d e f i n e d in c h a p t e r D . 9.4 y h ' V y h p e a k v a l u e s of Y , Y , Y in H a r m o n i c o s c i l l a ­ tion m a n o e u v r e ZTP1 r i g i d load c.g. d i s t a n c e a b o v e t a n k c e n t r e 3 r a t i o b e t w e e n c o m p u t e r time and p r o b l e m (real) time VI d y n a m i c visco s i t y . See p a g e B4 V k i n e m a t i c v i s c o s i t y . See p a g e B4

7T the c o n s t a n t 3 . 1 4159 or symbol for d i m e n ­ si o n l e s s te r m s used in a p p e n d i x B p liquid d e n s i t y T c o m p u t e r time w h e n unind e x e d . C o m p a r e p r o b l e m time: t t

g

f

CO a n g u l a r velocity, a n g u l a r f r e q u e n c y in

general. In a p p e n d i x D.9: o s c i l l a t o r f r e ­ q u e n c y [rad/s] for DAVIS' d o u b l e lane c h a n g e

A P P E N D I X B

C O N D I T I O N S F O R D Y N A M I C S I M I L A R I T Y B E T W E E N M O D E L A N D F U L L SIZE TANK

B . 1 I m p o r t a n t q u a l i t a t i v e e f f e c t s

M e a s u r i n g the liqu i d forces on a p h y s i c a l tank m o d e l in w e l l - k n o w n m o t i o n r e l e a s e s us from f i n d i n g a l g e b r a i c e x p r e s s i o n s of th e s e forces. However, we m u s t m a k e an e f f o r t to d e s i g n the e x p e r i m e n t s for d y n a m i c s i m i l a r i t y w i t h c o r r e s p o n d i n g full scale c onditions. And, of

course, we have to find the a p p r o p r i a t e e x p r e s s i o n s for t r a n s f o r m a t i o n of m e a s u r e d v a r i a b l e s (Model - i n dex M) to full scale v a r i a b l e s (index F ) .

It seems r e a s o n a b l e to a s s u m e that the t a n k _ w a l l _ d e - Î2ï ™ § t i o n s _ a n d _ s u r f a ç e _ t e n s i o n _ e f f e ç t s c a n be n e g l e c t e d - at least for c o n v e n t i o n a l l i q u i d load c o n t a i n e r s on road v e h i c l e s a nd t h e i r m o d e l s - s c a l e d a p p r o x i m a t e l y 1:10. Thus f o l l o w i n g d i s c u s s i o n will be c o n c e n t r a t e d o n t h e d y n amic b e h a v i o u r of the fluids in the c o n t a i ­ ner. Further, o ur s t u dies w e r e l i m i t e d to tank c o n f i ­ g u r a t i o n s and fluid g r o u p s w h e r e we c o u l d n e g l e c t the d i f f e r e n c e s b e t w e e n m o d e l a nd full scale c o m p r e s s i b i ­ lity of gas and liquid.

P r o v i d e d that the g e o m e t r i c and k i n e m a t i c v a r i a b l e s w e r e p r o p e r l y c o n t r o l l e d by the e x p e r i m e n t e r the p r i o ­ r i t y o r d e r of the r e m a i n i n g e f f ects w as r e g a r d e d to be:

I) Inertial

II) V i s c o u s III) C a v i t a t i o n

D a l zell (1966a) p r e s e n t s a c o m p r e h e n s i v e d i s c u s s i o n on e x p e r i m e n t a l d e s i g n w h e r e these e f f ects a re c o n s i d e r e d s e p a r a t e l y and s i m u l t a n e o u s l y . In the same d o c u m e n t

B2

Dalzell and S i l v e r m a n (1966) have c o m p i l e d t a b l e s on p h y s i c a l p r o p e r t i e s for fluids s u itable in m o d e l e x p e ­ riments. Some of these d a t a are found in t a b l e B l .

B .2 Q u a n t i t a t i v e d y n a m i c d e s c r i p t i o n i n d e p e n d e n t of size

D a l z e l l (1966a) d e s c r i b e s t h o r o u g h l y h o w to f o r m d i m e n ­ s i o n l e s s terms (so c a l l e d iT-terms) from the q u a n t i t i e s that are n e c e s s a r y to c h a r a c t e r i z e the d y n a m i c s of the e x p e r iment. By k e e p i n g e v e r y such ir-term at a c o n s t a n t n u m e r i c v a l u e - the same for the m o d e l as in full s c a ­ le - d y n a m i c s i m i l a r i t y will be achieved.

W h e n the c h a r a c t e r i z i n g q u a n t i t i e s (ChQ) h a v e b e e n l i s ­ ted a g r o u p of so c a l l e d r e g e a t i n2_ v a r i a b l e s (ReV) sho u l d be selected. The r e p e a t i n g v a r i a b l e s m u s t i n c l u ­ de all the f u n d a m e n t a l d i m e n s i o n s found in t h e C h Q - list. A n o t h e r r e s t r i c t i o n is, that it shou l d n ot be p o s s i b l e to f o r m any d i m e n s i o n l e s s e x p r e s s i o n from the ReVs alone.

T he f u n d a m e n t a l d i m e n s i o n s mass, length, and time are s u f f i c i e n t to form all the geome t r i c , k inematic, i n e r ­ tial, viscous, a nd c a v i t a t i o n v a r i a b l e s that m i g h t be i n c l u d e d in our ChQ-list. T h e s e d i m e n s i o n s are r e p r e ­ sen t e d in table B2 as well as the ReVs, w h i c h w e r e r e ­ g a r d e d as m o s t c o n v e n i e n t for this kind of s l o s h i n g e x p e r i m e n t s .

B .3 I n e rtial scaling, g e o m e t r i c and k i n e m a t i c r e q u i r e m e n t s

N o w any g e o m e t r i c v a r i a b l e can be t r a n s f o r m e d to c o r r e ­ sp o n d i n g iT-term by d i v i s i o n w i t h a p p r o p r i a t e p o w e r of the c h a r a c t e r i s t i c leng t h l. In this case no t r o u b l e should o c c u r if the model a nd its m o t i o n a m p l i t u d e s have strict g e o m e t r i c s i m i l a r i t y a nd r e p r e s e n t the same ratio of full scale in all dire c t i o n s .

The t a n k m o d e l r e s u l t a n t a c c e l e r a t i o n r^ c o r r e s p o n d s to the TT-term r-_/gw . B e c a u s e g w = g 1-,f the d e m a n d on

M M M

c o n s t a n t 7T-terms d e f i n e s the a c c e l e r a t i o n ratio (r ) r

r

r 1.0 (B.l)

As the d i m e n s i o n l e s s (or 77-term for) time is f o r m e d by £ and g, we k n o w that the time scale is d e f i n e d as soon as the g e o m e t r i c scale ratio £^ : £^ has b e e n s e ­ lected. C o n s t a n t TT-terms:

cau s e s the time ratio to be the square root of the l e n g t h ratio.

A l s o the scale ratios for f r e q u e n c y (oj) a nd v e l o c i t y (r) are d e f i n e d by the len g t h ratio alone.

If the t h ird r e p e a t i n g v a r i a b l e (p) is i n t r o d u c e d the force and m o m e n t ratios can be d e r i v e d as f u n c t i o n s of the g e o m e t r i c and l i q u i d d e n s i t y scale ratios.

B4

B .4 V i s c o u s s c a l i n g

T h e k i n e m a t i c r e q u i r e m e n t s are m e t and the iner t i a l s c a l i n g is p r o p e r if the e x p e r i m e n t s a re c o n t r o l l e d a n d e v a l u a t e d a c c o r d i n g to c h a p t e r B.3.

If the v i s c o u s e f f e c t s a nd c o r r e s p o n d i n g in t e r n a l f o r ­ ces are c o n s i d e r e d to be larger than - or of the same m a g n i t u d e as - the inertial e f f e c t s and forces, v i s ­ c o u s s c a l i n g is necessary.

As above, this s c a l i n g is simply p e r f o r m e d by k e e p i n g c o n s t a n t the ir-term for the d y n a m i c v i s c o s i t y (y) of the liquid. In a s i m u l a t i o n w i t h the same l e n g t h scale in all t h r e e d i m e n s i o n s this ir-term c an be written:

IT =

y _p 2 _{g £}„3 ( B . 5)

If ir is h e l d c o n s t a n t and if we s u b s t i t u t e the d y

na-y

m i c v i s c o s i t y (y) w i t h the k i n e m a t i c v i s c o s i t y ( v = y / p ) , we will get the 2£ o p e r _ s c a l e _ r a t i o _ f o r _ k i n e m a t i c _ v i s c o -

sity:

Thus the model liquid (and if possible the geometric scale £^) should be selected to satisfy (B.6) for the full scale liquid in question.

Example:

A s s u m e that £ ^ = 0.1 a nd V p = 3 0 c St X (which c o r r e s p o n d s to fuel oil c l a s s i f i e d as EO 3 in S w e d e n ) . Hence, a

m o d e l l i q u i d w i t h « 1 c St. is required. T h i s c o n d i ­ t i o n is a p p r o x i m a t e l y s a tisfied by water. M o r e v i s c o u s

x

1 c St = 10 ^ m^ / s

m o d e l fluids c an e a s i l y be p r o d u c e d by m i x i n g w a t e r and gl y c e r o l (CH2OH CHOH C H 2OH) : 1 - 1200 c St for 0 - 100% g l y c e r o l .

The e x a m p l e ab o v e i n d i c a t e s that an imp o r t a n t g r o u p of l i q u i d s - c o m m o n on the r o a d s - can be i n v e s t i g a t e d w i t h safe, low c o s t and a c c e s s i b l e m o d e l fluids (water and g l y c e r o l ) . However, it is hard to find flu i d s w i t h a v i s c o s i t y t h a t is m u c h b e l o w 1 c St. For i n s t a n c e the k i n e m a t i c v i s c o s i t y of m e r c u r y (Hg) is a p p r o x i m a ­ t e l y 0.1 c St at a t e m p e r a t u r e of +20°C.

Q222ii-i22§_l2E_2§9i§2 ting _ v i s c o u s _ forces

F o r t u n a t e l y v i s c o u s s c a l i n g is not n e c e s s a r y if the ratio b e t w e e n inertial a n d v i s c o u s forces is s u f f i c i e n t ­ ly small. This ratio is c o m m o n l y e x p r e s s e d in R e y n o l d s n u m b e r :

w h e r e v is the c h a r a c t e r i s t i c v e l o c i t y for the fluid r e l a t i v e to the tank, i is a c h a r a c t e r i s t i c l e n g t h of the tank and p the fluid density.

F o l l o w i n g a s s u m p t i o n s for the full scale ca s e

r g = Z l A l £ V ( B . 7) p_ £ 700 k g / m r * 0.05 m F 3

(for a tank w i t h baffles)

v^ ~ 0.1 m / s F < 3 5 0 * 1 0 m /s ("Bunker C" at +50°C) r 4 give Re^, £ 1 * 1 0 . A n d b e c a u s e v • i> • p r r r (B . 9)

B6

the c o r r e s p o n d i n g v a l u e for a 1:10 scale m o d e l w i t h w a t e r is Re,. k2*10~*.

T h u s v i s c o u s e f f e c t s seem to be n e g l i g i b l e c o m p a r e d to i n e rtial e f f ects for all l i q uids that are s u f f i c i e n t l y fluent to be p u m p e d to and from a c o n t a i n e r o n a road v e h i c l e (500 - 1 0 0 0 c St is said to be a r e a s o n a b l e range for the u p p e r v i s c o s i t y l i m i t ) . T h e r e f o r e v i s c o u s

e f f e c t s do not limit the r e s u l t s to a c e r t a i n g r o u p of liquids if the m o d e l fluid is w a t e r and if the g e o m e t r i c l e n g t h scale is a p p r o x i m a t e l y 1:10.

B .5 C a v i t a t i o n scaling

C a v i t a t i o n occ u r s if the p r e s s u r e p (often c a l l e d

static pressure) at a po i n t in the liq u i d d e c r e a s e s to the c u r r e n t v a p o u r p r e s s u r e p . Thus c a v i t a t i o n m e a n s that the actual p a r t of the fluid c h a n g e s state from l i q u i d to gas (local b o i l i n g ) . This will c h a n g e the q u a l i t a t i v e b e h a v i o u r of the liq u i d and m a y c a u s e i n v alid r e s u l t s if the m o d e l d e v i a t e s from full scale in this aspect.

No g e n eral e x p r e s s i o n (corre s p o n d i n g to R e y n o l d ' s n u m b e r etc) for the ra t i o b e t w e e n i n e r t i a a n d c a v i t a ­ t i o n forces is k n o w n to the author. Thus it seems r e c o m m e n d a b l e to scale the t a n k p r e s s u r e s p r o p e r l y if c a v i t a t i o n is l i k e l y to o c c u r in the m o d e l o r in the full scale case.

If viscosity, c o m p r e s s i b i l i t y etc are n e g l i g i b l e we can q u a n t i f y a c o n d i t i o n for c a v i t a t i o n w i t h the aid of B e r n o u l l i ' s e q u a t i o n

w h e r e the s t a g n a t i o n p r e s s u r e pQ is c o n s t a n t for a p o i n t at a c e r t a i n de p t h from the free s u r f a c e (notice that t he c o n s t a n t p can be m e a s u r e d as p at zero v e l o

-* 0 c

c i t y ) . As no c a v i t a t i o n occ u r s if

p > p v (B.ll)

no p r e s s u r e s c a ling is n e c e s s a r y if the l i q u i d r e l a t i v e m o t i o n s are well b e l o w the

/ (po " PV } * 2 ' V C / p T h e s t a g n a t i o n p r e s s u r e p Q l i q u i d free s u r f a c e is p = p + p r h ^o ^u K w h e r e r is the local v a l u e v e c t o r (including gravity) l i q u i d s u r face a n d w h e r e pi s u r f a c e (ullage p r e s s u r e ) . a n d n o n d i m e n s i o n a l i z a t i o n : c a v i t a t i o n velocity: (B . 12)

at the d i s t a n c e h from the

(B .13) of the tank a c c e l e r a t i o n d i r e c t e d r e c t i l i n e a r to the is the p r e s s u r e at free S u b s t i t u t i o n into (B.12) v,. & pu + rh P V pg£ g lpg£ ( B . 14 )

shows that cavitation scaling is achieved if the Tr-term below is held constant.

= _Ap_ = P u ~ P V

V pg£ pg£ ( B . 15 )

Of course, g e o m e t r i c s i m i l a r i t y and i n ertial s caling m u s t be g r a n t e d as well, in o r d e r to k e e p the r e m a i n i n g t e r m in (B.15) constant.

This d e m a n d on a c o n s t a n t tt^, can be s a t i s f i e d by s e ­ l e c t i n g a m o del fluid w i t h suit a b l e v a p o r p r e s s u r e and by m a n i p u l a t i o n of the u l l a g e p r e s s u r e in the tank.

B8 E x a m p l e : A s s u m i n g that h ^ ^ O . l m (i, = 1 : 1 0 ) , r $ 12 m / s t P u = 10~* N / m ^ and u s i n g w a t e r as the m o d e l l i q u i d ( -*■ P y M ~ 0.023 *10~* N / m t pM = 1 0 0 0 k g / m t g i v e the c a ­ v i t a t i o n v e l o c i t y from (B.12) a nd (B.13) V C M ~ 1 0 5 + 1 0 0 0 * 1 2 * 0 . 1 - 0.023 *10“* 500 » 14 m / s

w h i c h is well above w h a t to exp e c t from l a t e r a l l i q u i d m o t i o n s in a road tan k e r m o d e l . Thus c a v i t a t i o n seems

to be no p r o b l e m in such a model.

However, some full scale fluids have q u i t e a h i g h v a ­ p o r pressure, w h i c h m a y ca u s e c a v i t a t i o n l o c a l l y in a tank w i t h i n e f f i c i e n t b a f f l e s or o t h e r sh a r p edges - e s p e c i a l l y if l o n g i t u d i n a l m o t i o n s (due to braking) are co n s i d e r e d . A p a r t from the s c a l i n g p r o b l e m in t h ese expe r i m e n t s , full scale c a v i t a t i o n m a y c a u s e d a n g e r o u s d a m a g e to the tank walls.

Anyhow, it seems to be m o r e of a scaling p r o b l e m in this case, as tank wall d a m a g e p r o b a b l y o c c u r s o n l y if the flow is m o r e or less continuous. A n d h e r e the h y ­ p o t h e t i c c a v i t a t i o n is a c t i v e o n l y for short ti m e p e r i o d s .

B .6 C h o s e n e x p e r i m e n t a l d e s i g n and scale f a c t o r s

As i n d i c a t e d ab o v e the g e o m e t r i c length s c a l e ratio was s e l e c t e d to be

* =-5^ = 0 . 1 (B. 16a)

r F

w h i c h implies the time, f r e q u e n c y and v e l o c i t y scale ratios a c c o r d i n g to (B.2) - (B.4)

t = /UTT » 0.316 r (B.16b) w = /To1* 3.16 r (B.16c) r = /OTT » 0.316 r ( B .16d)

A sli g h t d e v i a t i o n from the g e n eral scaling t h e o r y is n e c e s s a r y as it w as d e c i d e d to e x c lude l o n g i t u d i n a l m o t i o n s and to i n v e s t i g a t e the s l oshing p r o b l e m s in two

d i m e n s i o n s only. T h e r e f o r e the m o d e l tank c y l i n d e r s c o u l d r e p r e s e n t any r e a s o n a b l e full scale c y l i n d e r l e n g t h (Lp) w i t h o u t c o r r e s p o n d i n g a d j u s t m e n t of the m o d e l l e n g t h (L ). However, the scaling e x p r e s s i o n s (B.5) a n d (B.6) m u s t be m o d i f i e d to i n c lude the tank c y l i n d e r len g t h (L) as w e l l as the c h a r a c t e r i s t i c l e n g t h £ for the cross s e c t i o n (yz-) plane.

Of the same reason, the ir-terms for force and m o m e n t w e r e n o t e x p r e s s e d before. B ut the m e t h o d is not

c h a n g e d from c h a p t e r B.3 if the n o n d i m e n s i o n a l i z a t i o n a n d ir-term f o r m u l a t i o n is b a s e d on force a nd m o m e n t E ^ E - i ^ n g t h unit of the tank (F/L and M/L) . T h e d e m a n d on c o n s t a n t iT-terms: f m

//

=

//

pM ^ (V

pF g <

V

g i v e the ratios b e t w e e n mo d e l and full s cale for forces and m o m e n t :

F

r (B.17b)

( B .18b)

BIO

As w a t e r was s e l e c t e d to be the m o d e l fluid, v i s c o s i t y and c a v i t a t i o n s c a l i n g was r e g a r d e d less i m p o r t a n t -see c h a p t e r s B.4 and B.5.

TABLE Bl A p p r o x i m a t e c o n s t a n t valu e s c h a r a c t e r i z i n g the

m e c h a n i c a l p r o p e r t i e s for some fluids. F r o m Dal z e l l & S i l v e r m a n (1964), A S T M s t a n d a r d s and u n p u b l i s h e d data from k i n d l y s u p p o r t i n g l a b o r a tories. *

Name Dens j kg/m^ Lty at °c Kinemat viscosit; cStX :ic y at °c Vapc pressu] i ^ x kPa }r re at °c Water (H2o) 998 20 1.01 20 2.34 20 Glycerol (c h2o h c h o h c h2o h) 1260 20 1200 20 <<0.01 20 Mercury (Hg) 13550 20 0.117 20 1.6•10-4 20 Oxygen (02) 1140 -184 0.17 -185 99 -189 Nitrogen (N2) 815 -198 0.21 -198 85.8 -199 Ethanol ( C ^ O H ) 790 20 1.58 20 5.9 20 Gasoline 740 Probably 20 40 Probably 20 700-740 16 70-100 38 Diesel Oil (1-D) 875 16 1.4-2.5 38 Diesel Oil (2-D) 919 16 2.0-5.8 38 Diesel Oil (4-D) 960 16 5.8-26.4 38

Fuel Oil (Swedish EOl) 830 20 3.5 20

Fuel Oil (Swedish E03) Sulphur contents A " " B 880 20 r 30 1 9 20 50 920 20 r ~ 11 0 1 28 20 50

Fuel Oil (Swedish E04) 940 20 r~450

1 75

20 50

Fuel Oil (Swedish E05) 920 20 r ~ 9 0 0

1 130

20 50

* 1 cSt = 10 6 m 2/s

1 kPa = 103 N/m2 1 0.009869 atm

T A B L E B2 R e p e a t i n g v a r i a b l e s S ymbol S I - u n i t D i m e n s i o n D e s c r i p t i o n l m L T a n k c y l i n d e r inside w i d t h or d i a m e t e r

g

p

A P P E N D I X C

TANK MODEL A N D L I Q U I D F O R C E S

C .1 B r i e f d e s c r i p t i o n o f the l a b o r a t o r y e q u i p m e n t

R e f e r r i n g to f i g u r e Cl each tank scale m o d e l w i t h f i x ­ ture (mass m ) is fitted to a frame (mass M ) w h i c h is

b

D

s u s p e n d e d from the m o t i o n syst e m p l a t f o r m by t h r e e bars AP, BP a nd CQ. No r e l a t i v e m o t i o n b e t w e e n tank, frame and p l a t f o r m is possible.

T he bars AP a n d BP are force t r a n s d u c e r s s e n s i t i v e to the forces F, and F ( + sign for t e n s i l e forces) app- lied at the f r a m e point P in y- and z - d i r e c t i o n s r e s ­ pectively. T h e y w i l l not t r a n s m i t any t o r q u e or d i f f e ­ rent l y d i r e c t e d forces (apart from inertia forces due to their o w n m a s s e s m and m B p) •

T he bar CQ in itself p r e v e n t s a n g u l a r m o t i o n s a r o u n d the y- and z-a x e s for the frame a n d tank. It is

s u pported by ball b e a r i n g s to the shear force t r a n s ­ du c e r C, w h i c h is s e n s i t i v e to v e r t i c a l (± z-directed) forces only. (According to the m a n u f a c t u r e r - AB Bofors, S w e d e n - this c a n t i l e v e r type t r a n s d u c e r sho u l d be

p r a c t i c a l l y i n s e n s i t i v e to the b e n d i n g t o r q u e s that m i g h t oc c u r f r o m s e l f - i n d u c e d l o n g i t u d i n a l s l o s h i n g in this c a s e ) . The m o t i o n s y s t e m p l a t f o r m a nd all c o m p o n e n t s on it are m o v e d a l o n g ra i l s r e c t i l i n e a r l y in the y - d i r e c t i o n by the h y d r a u l i c servo. T he a c c e l e r a t i o n SA is m e a s u r e d by an a c c e l e r o m e t e r m o u n t e d at the platform.

C2

C.2 F orces and m o m e n t s in the m o d e l

The m o d e l liqu i d forces and m o m e n t in the y z - p l a n e r e ­ f e rred to the tank c e n t r e 0 are s y m b o l i z e d by F

and M . T h e i r p o s i t i v e d i r e c t i o n s are s h own in figure Cl p r o v i d i n g they are liq u i d m a s s inertia f orces a c t i n g upon tank, frame etc.

If the h o r i z o n t a l i nertia force c o m p o n e n t of d ue to the m a s s e s ni , m__ , rn ^ , in and in is f__-SA, the

AP BP CQ D E AE

e q u i v a l e n t m a s s f will be e x p r e s s e d by

"AE mAP + fBP f CQ + m D + I1L (C.l)

w h e r e fBp and f ^ are c o r r e s p o n d i n g e q u i v a l e n t m a s s e s act i n g in p o i n t s P and Q. T he f v a l u e a c c o r d i n g to

(C.l) was c h e c k e d from the static o u t p u t of force t r a n s ­ ducer AP a f t e r r o t a t i o n of the w h o l e p l a t f o r m 90° in the yz-plane. Thus

F

yM = fAE SA - FA (C.2)

For c o n v e n i e n c e the th r e e forces b e l o w are d i v i d e d in static a n d d y n a m i c parts w i t h index s and d r e s p e c t i v e l y

F — F ., + F -,,. zM z sM zdM

F = F

+ F

Jim'

T h u s the v e r t i c a l d y n a m i c l i q u i d force c a n be c o m p u t e d as fo l l o w s

F z d M F Bd F Cd (C.7)

If the i n d ices of the c o o r d i n a t e s (y and z) refer to t he m a s s cen t r e l o c a t i o n of the c o r r e s p o n d i n g part, m o ­ m e n t e q u i l i b r u m a b o u t the p o i n t P in the static cas e g i v e s F CS (yp - V ■ ,nC Q - 9 < y p ~ V * m D E ' g ■ - rnHm "3 ‘Yp = (C.8) a n d in t h e d y n a m i c case M O M “ m D E - SA (ZP - ZD E } + f C Q ' SA (ZQ ‘ ZP > + F y M *ZP ~ ~ FzM * YP _ F C (yP _V + m C Q ' g (YP ~ V + + m D E * g ^Y P Y DE' ° ( C . 9 ) S u b s t i t u t i o n s a c c o r d i n g to (C.3), (C.5), (C.6) and

s o l v i n g from (C.8) will give an e x p l i c i t e x p r e s s i o n of M from (C.9):

UiVl

M 0 M SA ^m DE ^ZP ZDE^ fCQ ^ZQ ZP ^ F y M * ZP +

+ F zd M * Y P + F Cd (YP " Y0 ) (C.10)

C.3 F o r c e s and m o m e n t s in full scale

R e f e r r i n g to a p p e n d i x B the c o n s t a n t s K 1 P P F_ M

•g

•g

C4 K 2 P P F_ M

•g

•g

c an be u s e d for t r a n s f o r m a t i o n of forces (K^) and m o ­ m e n t s (I^) f r o m m o d e l scale for full scale. Thu s

1 = K • F

y(F) *1 yM (C . 13)

1

zd(F) 1 zdM (C.14)

b (F) = K2 ' M0M (C . 15 )

w h e r e (C.2), (C.7) and (C.10) e x p r e s s the m o d e l scale v a r i a b l e s .

C .4 E q u i p m e n t data

C o n s t a n t s a nd their v a l u e s i n d e p e n d e n t of tank type a re lis t e d in table Cl. T a b l e C2 shows data for the t h r e e d i f f e r e n t tanks i l l u s t r a t e d in figure 3.7 that w e r e u s e d in this investi g a t i o n . The tank l a b e l s

e l l i p t i c and s u p e r e l l i p t i c do n o t m e a n that the c r o s s s e c t i o n ar e a s c o r r e s p o n d m a t h e m a t i c a l l y to t h e s e c o n ­ cepts. The labe l s are o n l y u s e d as a short d e s c r i p t i o n of the p e r i p h e r i e s in figure 3.7.

E P O R T N O . 1 3 8 A O ui

Fig u r e Cl. T a n k mo d e l (E) fitted to a frame (D) s u s p e n d e d by force t r a n s d u c e r s (AP, BP and C) from the m o t i o n s y s t e m platform.

C6 T A B L E C l . T a n k i n d e p e n d e n t c o n s t a n t s fAE_mE (kg) “d (kg) fCQ (kg) ZQ'ZP (m) V yQ (m) 3.03 2.19 0.35 0.017 0.422 T A B L E C 2 . C o n s t a n t s d e p e n d i n g o n t a n k type Tank label “e (kg) “de (kg) fAE (kg) ZDE (m) ZP (m) ZDE ZP (m) yP (m) (m) (m2) VM (m3) Circular 4.250 6.440 7.30 -0.055 -0.094 0.039 0.221 0.2744 0.02835 0.00778 "Elliptic" 2.540 4.730 5.57 -0.061 -0.094 0.033 0.223 0.3045 0.02756 0.00839 "Super-elliptic” 2.215 4.405 5.23 -0.052 -0.070 0.018 0.224 0.3008 0.02297 0.00691 VTI R E P O R T NO 1 3 8 A

A P P E N D I X D M A T H E M A T I C A L D E S C R I P T I O N O F THE S I M U L A T I O N M O D E L S D.l G e n e r a l c o m m e n t s R e f e r r i n g to fig 3.3 f o l l o w i n g pa r t s will be d e s c r i b e d h e r e : - S p e c i f i c a t i o n of the lateral m a n o e u v r e a c c o r d i n g to DAVIS. See s e c tion D.9.

- T h e v e h i c l e mod e l ( s ) . See s e c t i o n D.4-D.8.

- E v a l u a t i o n and r e c o r d i n g s of results, i n c l u d i n g c a l c u l a t i o n of the o v e r t u r n i n g r i s k factor. See s e c t i o n D.3.

T h e v e h i c l e m o d e l s are v e r y sim p l e w i t h o n l y o ne or two d e g r e e s of freedom, i.e. l a t e r a l t r a n s l a t i o n a nd roll. T h i s a l l o w e d s i m u l t a n e o u s s i m u l a t i o n of several v e h i c l e m o d e l s in the a v a i l a b l e a n a l o g u e c o m p u t e r w i t h o u t u s i n g a m a g n e t i c tape r e c o r d e r as a buffer. (However, a f t e r the first series of e x p e r i m e n t s - p h a s e II in fig 3.1 - a d i g i t a l c o m p u t e r s y s t e m has b e e n i n t e r f a c e d to the analogue. Thus tape b u f f e r i n g is m o r e f a v o u r a b l e now, due to the r e p e t i t i o n f e a s i b i l i t y of the hyb r i d c o m p u ­ ter) .

T h e s i m p l i c i t y of the v e h i c l e m o d e l s c o n f o r m s to the l i m i t a t i o n s of the tank mo d e l and its m o t i o n system. As a m a t t e r of fact th e s e s i m p l i f i c a t i o n s c l a r i f i e d the study r e g a r d i n g its m a j o r p r o b l e m - i.e. lat e r a l s l o s h i n g e f f e c t on the v e h i c l e - w i t h o u t n e g l e c t i o n of the m o s t i m p o r t a n t p henomena. U n f o r t u n a t e l y there is one e x c e p t i o n to this, n a m e l y that the c o u p l i n g b e t ­ w e e n roll m o t i o n s a nd s l o s h i n g c o u l d not be st u d i e d

(see s e c tion D.7).

D2

D . 2 A p p r o x i m a t i o n s and a s s u m p t i o n s for the v e h i c l e m o d e l s

D . 2.1 Li§£§i’§i_??2tions_are_grimarY

The lateral a c c e l e r a t i o n (SA) of the tank m o d e l s was a l w a y s used as input to the v e h i c l e m o d els. T h i s is the p r i m a r y v a r i a b l e for e x c i t a t i o n of lateral s l o s h i n g a nd for c a l c u l a t i o n of the o v e r t u r n i n g factor. T h e tank m o d e l m o t i o n sys t e m has o n l y this deg r e e of freedom.

D.2.2 Roll j n o t i o n s

T he v e h i c l e m o d e l w i t h roll m o t i o n w as not c o m p a t i b l e w i t h the ph y s i c a l t a n k m o d e l s a c c o r d i n g to the e x p l a ­ n a t i o n in s e c tion D.7. Anyway, it was i n t r o d u c e d for c o m p a r i s o n s of the i n f l u e n c e s on the o v e r t u r n i n g ris k from d i f f e r e n t roll r e s i s t a n c e numbers. It w as also n e c e s s a r y for e v a l u a t i o n of the r o l l i n g f a c t o r - see e x p r e s s i o n (3.5) - a nd for e s t i m a t i v e c o m p a r i s o n s of e f f e c t s from a n t i - r o l l bars to effects from tank b a f f ­ les .

The roll axis was a s s u m e d to be in a fixed, h o r i z o n t a l p o s i t i o n in the vehicle. T h e roll r e s i s t a n c e (kcp) and the roll d a m p i n g (c^) w e r e c o n s t a n t s i n d e p e n d e n t of roll an g l e and roll a n g l e v e locity. (Of course, n o n ­ l i n e a r roll r e s i s t a n c e a nd roll d a m p i n g c a n e a s i l y be i n t r o d u c e d by u s i n g f u n c t i o n g e n e r a t o r s i n s t e a d of p o ­ t e n t i o m e t e r s in the c o m p u t e r ) . C a l c u l a t i o n of the g r a ­ v i t y force m o m e n t w as b a s e d on the a s s u m p t i o n of small roll a n g l e s (sin cp =« cp a n d cos cp « 1) .

D.2.3 X § w _ m g t i o n s _ n e g l e c t e d

In spite of the a b s e n c e of y a w m o t i o n for the tanks, s p o n t a n e o u s l i q u i d y a w m o t i o n w as s o m e t i m e s o b s e r v e d d u r i n g t he expe r i m e n t s . I n t e r e s t i n g e f f e c t s m a y o c c u r a nd s t u dies of the i n f l u e n c e from t a n k l e n g t h will be p o s s i b l e if y a w m o t i o n of the tank and v e h i c l e m o d e l is introduced. However, the wo r s t cases and l a r g e s t o v e r ­ t u r n i n g risk v a l u e s in t h e s e n o n - y a w i n g e x p e r i m e n t s are p r o b a b l y r e p r e s e n t a t i v e enough.

E r r o r s f r o m this n e g l e c t i o n are sm a l l e s t at h i g h speed m a n o e u v r e s , b e c a u s e the y a w m o t i o n s are s m a l l e r than at l o w speed for a c e r t a i n lateral a c c e l e r a t i o n time h i s t o r y .

Of course, y a w i n s t a b i l i t i e s and o s c i l l a t i o n s due to s k i d d i n g c o u l d not be s t u d i e d a p p r o p r i a t e l y w i t h o u t y a w motion.

D .2.4 L2n2i t u d i n a l _ a n d _ v e r t i c a l _ m o t i o n s _ n e g l e c t e d

T h o u g h l o n g i t u d i n a l l i q u i d m o t i o n is an i m p o r t a n t and a p p a r e n t p h e n o m e n o n for road tankers, it w as e x c l u d e d from this study. This does not affect the v a l i d i t y and a c c u r a c y of the r e s u l t s for pure l a t eral m a n o e u v r e s .

V e r t i c a l m o t i o n s w e r e not r e g a r d e d as i m p o r t a n t for th e s e studies and have been neglected.

D . 3 D e f i n i t i o n of the o v e r t u r n i n g risk f actor

The a c c e l e r o m e t e r and force t r a n s d u c e r o u t p u t s from the m o d e l w e r e u s e d as v e h i c l e lateral a c c e l e r a t i o n and in the c a l c u l a t i o n of liquid forces a c c o r d i n g to a p p e n d i x C. W i t h t h e s e inputs to the v e h i c l e m o d e l in

D4

the a n a l o g u e computer, the o v e r t u r n i n g (risk) f a c t o r (R) was evalu a t e d . W h e n the a b s o l u t e v a l u e of R e x c e e d s u n i t y (1.0) o v e r t u r n i n g has started.

W h e n the w h e e l loads are equal on b o t h sides, the w h e e l s on o ne side are loa d e d by the v e r t i c a l force P (static

s load) P s (m + m 1 ) g - F zd 2 (D.l) w h e r e is t h e v e r t i c a l d y n a m i c liquid force a c c o r d i n g to eq (D.6) below. If P^ is the d y n a m i c w h e e l l o a d on o n e (the left) side of the vehicle, the o v e r t u r n i n g

f a c t o r is d e f i n e d by

s

In t h e f o l l o w i n g R will be i n d e x e d by

0 for a n o n r o l l i n g v e h i c l e

9 for a r o l l i n g v e h i c l e

S (Sloshing load) w i t h l i q u i d forces from t h e m o d e l ' s t r a n s d u c e r s

R (Rigid load) w i t h l i q u i d forces c a l c u l a t e d from the a c c e l e r o m e t e r o u t p u t as if the load was c o m p l e t e l y f i x e d to the tank in its static position.

D .4 N o n r o l l i n g v e h i c l e w i t h r i g i d load

If m^ is the m a s s of the rigid load, m o m e n t e q u i l i b r i u m r e f e r r e d to the p o i n t A in fig Dl ( <p= 0 for this s e c ­ tion) is e x p r e s s e d by

P^*2c = g (m + m ^ ) •c - S A • [m' (e + u') + m (e + u " ) +

+ m1 (e + u 1 )] (D . 3 )

As F , = 0 w i t h ri g i d load, t he o v e r t u r n i n g f a c t o r can be o b t a i n e d from the e q u a t i o n s (D.l), (D.2) a nd (D.3), s u m m a r i z e d by

R 0 R = S A • K D4 (D . 4 )

w h e r e

rn'Ce + u") + m(e + ) + m . (e + u,)

Y —

D4 c (m + m ^ ) • g (D.4a)

T h e o v e r t u r n i n g l i m i t is d e f i n e d as the ste a d y state l a t e r a l a c c e l e r a t i o n c o r r e s p o n d i n g to R = 1 SA = 1 O R L I M Kd4 (D.4b) D.5 N o n r o l l i n g v e h i c l e w i t h s l o s h i n g load In this case e q u a t i o n (D.3) c o r r e s p o n d s to P ^ * 2 c = g * m * c - S A [ m ' ' ( e + u'') + m (e + u " ) ] - M_. + F (e + u ) - F • c 0 y o' z (D.5) By d i v i d i n g F in a static (F ) and a d y n a m i c z z s p a r t a c c o r d i n g to (Fz d } F = F + F j z zs zd w h e r e (D.6a) F = - nu *g zs 1 ^ the c o r r e s p o n d e n c e to (D.4) will be (D.6b) R 0S " Dd7 N D7 (D . 7)

D6 T h e d e n o m i n a t o r a b o v e is d e f i n e d by D D ^ = c (m + m-jj • g - c • F ^ (D.7a) and the n u m e r a t o r by Nd7 = SA[itT (e + u ') + m (e + u " )] + Mq - F y (e + uq ) (D.7b) D. 6 R o l l i n g v e h i c l e w i t h r i g i d load The c o m p a r i s o n s b e t w e e n n o n r o l l i n g a nd r o l l i n g v e h i c l e s are s u pposed to b e v a l i d if the t r a j e c t o r y of the

w h e e l s (the u n s p r u n g m a s s and the roll axis) is i d e n ­ tical and r e g a r d e d as the r e f e r e n c e input for b o t h k i nds of vehicles. Thus, a c c o r d i n g to fig Dl

SA = SA (D .8)

-

L

i

S A Ocp = S A L " U o Cp + gCp (D-9)

w h e r e S A ^ i s e v a l u a t e d for c o m p a r i s o n s w i t h the s l o s h i n g load case. (SA^ is r e g a r d e d as the a p p r o ­ p r i a t e a c c e l e r a t i o n input to the tank model, if c l o s e d loop s i m u l a t i o n w i t h s l o s h i n g load w e r e p o s s i b l e - see next s e c t i o n ) .

M o m e n t e q u i l i b r i u m for the sprung m a s s and load r e f e r r ­ ed to the roll axis L in fig Dl is e x p r e s s e d by

J Lcp

(

SAt

+ gcp) • (m'u' + m, u, ) - c cp - k cp

L

1 1

cp

cp

if the roll a n g l e s are s u p p o s e d to be small (coscp « 1

a nd sincp^cp ) . The c o n s t a n t

kdio

F i g u r e D l . R o l l i n g v e h i c l e seen from the rear. For n o n r o l l i n g v e h i c l e s 9 = 0 and the lat e r a l a c c e l e r a t i o n SA is equal for all p o i n t s

(SA = S A L = SA0cp) .

is u s e d b e l o w for conven i e n c e .

T h e m o m e n t e q u a t i o n c o r r e s p o n d i n g to eq. (D.10) for the u n s p r u n g m a s s y ields

D8

S • e + SA •m"' u + c„ • 9 + k • 9 + P , • c -

L 9 9 d

- (2Pg - P d ) -c = 0 (D.ll)

w h e r e the side force is

S = S A L (m + m 1 ) - 9r *Kd1Q (D.12)

and

m = m' + m"' (D.12a)

A c c o r d i n g to eq. (D.2) the o v e r t u r n i n g f a c t o r in this case - c o m p a r e the e q u a t i o n s (D.4) and (D.7) - is e x p r e s s e d by

V ' H T m + Tip - g fSAL [(m + m 1 )-e + m " u " ]

‘ '’V KD10'e + i c <P+''’V The o v e r t u r n i n g limit - c o m p a r e (D.4b) - is o b t a i n e d by e l i m i n a t i o n of 9 in (D.10) k e e p i n g 9 = 9 = 0 for s t e a d y - s t a t e c o n d i t i o n s - _ s r l i m'k d i o

™

k , -

-KD U

D .7 R o l l i n g v e h i c l e w i t h s l o s h i n g load

As the refe r e n c e a c c e l e r a t i o n (SA = SA ) d i f f e r s from ±j

the tank centre a c c e l e r a t i o n (~SA ) in a r o l l i n g v e ­ hicle, an e x p e r i m e n t a l c o n f i g u r a t i o n sim i l a r to fig D2 was i n i t i a l l y o u tlined. Unfo r t u n a t e l y , the a d j u s t ­ m e n t s of the h y d r a u l i c servo c h a r a c t e r i s t i c s w e r e not

s u f f i c i e n t to br i n g t he c o n t r o l error d o w n to an a c c e p ­ ta b l e amount. T h e r e f o r e the c o n f i g u r a t i o n in fig 3.3 was selected, and i n t e r a c t i o n s b e t w e e n v e h i c l e r o l l i n g a n d load sloshing c o u l d not be i n v e s t i g a t e d properly.

SA O'P A c c e l e r o m e t e r o u t p u t T A N K M O DEL F o r c e s S U B T R A C T IO N O F T A N K IN E R ­ T I A FO R C E S R E F E R E N C E TO T A N K C E N T R E . S C A L IN G F o r c e s " D A V I S " S A M ANOEUVRE J-l V E H IC L E S P E C I F I C A T I O N M O D E L i F it =

_V

or

Figu r e D2. E x p e r i m e n t a l c o n f i g u r a t i o n sug g e s t e d for i n v e s t i ­ g a t i o n of the i n t e r a c t i o n b e t w e e n v e h i c l e r o l l i n g and load sloshing.

DIO

D. 8 V e h i c l e and load dat a

D.8.1 Y g h i c l e _ d a t a

The v e h i c l e c o n f i g u r a t i o n is supposed to be s i m i l a r to a t r a i l e r w i t h t wo axles (according to S w e d i s h l e g i ­ s l a t i o n the g r o s s w e i g h t m u s t not exceed 2 0 0 0 0 kg in this c a s e ) .

F o l l o w i n g c o n s t a n t s were r e g a r d e d i n d e p e n d e n t of load a nd tank co n f i g u r a t i o n , and their v a l u e s (table Dl) w e r e a s s e s s e d a c c o r d i n g to e x p e r i e n c e and d a t a from the i n v e s t i g a t i o n r e p o r t e d by Nordstrom, M a g n u s s o n and S t r a n d b e r g (1972):

the e f f e c t i v e t r a c k w i d t h (2c); the roll axis p o s i t i o n ab o v e road level (e) ; the u n s p r u n g mass ( m " ) a n d its c e n t r e of g r a v i t y p o s i t i o n a b o v e the roll axis (u ); the sprung m a s s w i t h o u t l o a d ( m " ) , its c.g. p o s i t i o n

(u') a nd its roll m o m e n t of i n e r t i a ab o u t the roll axis (J'); roll r e s i s t a n c e (k ) and roll d a m p i n g (c ) .

J_i 't' cp

The sprung m a s s roll m o m e n t of inertia and its c e n t r e of g r a v i t y l o c a t i o n w e r e e v a l u a t e d w i t h f o l l o w i n g a d d i t i o n a l a s s u m p t i o n s

Frame (with c e r t a i n tubes a nd pipes etc) weight: 1100 kg

Frame c.g. d i s t a n c e ab o v e the roll axis: 0.6 m

Frame roll m o m e n t of i n e rtia a b o u t the roll axis: 580 kg m^

T a n k (with accessories) weight: 1700 kg

Tank c.g. d i s t a n c e ab o v e the roll axis: 1.5 m

2

T a n k roll m o m e n t of i n e rtia a b o u t its c e n t r e (920 kg m ) was c a l c u l a t e d by r e g a r d i n g 60% of the m a s s as a c i r ­ cular c y l i n d e r w i t h radius 0.95 m. T he r e m a i n i n g part w as r e g a r d e d as a c o n c e n t r a t e d mass in the c e n t r e of the t a n k (1.5 m a b o v e the roll axis)

Dll

T h u s J£ = 580 + 920 + 1700 • 1 . 52 » 5300 kg m2

and u' = (1700 • 1.5 + 1100 • 0 . 6 ) /2800 » 1.2 m

The influence from estimation errors in these masses, their location and moment of inertia is relatively small compared to corresponding load data. Therefore, the data above were assesssed to the vehicles with

"elliptic" and "superelliptic" tanks as well. Much more important is the estimation of tank centre location mentioned below, because the liquid load forces are

referred to the tank centre (0).

TABLE Dl. Vehicle data independent of tank and load

configuration P a r a m e t e r s y m b o l c Ccp1} e k 1 ) 2 ) <P m' m " u ' P F U n i t m Nm s r a d m k g m2 Nm r a d k g k g m m k g m3 V a l u e 0.9 7.0-104 0.5 5300 2800 2200 1.2 0 . 0 920 1) O n l y u s e d w i t h rigid load

2) I n f l u e n c e from the two d i f f e r e n t v a l u e s c an be c o m p a r e d in table F 4 .

It w as a s s u m e d that the b o t t o m s of the c i r c u l a r and the "elliptic" tanks n o r m a l l y have to be a p p r o x i m a t e l y at the same h e i g h t a b o v e the roll axis. T h i s c a u s e d the t a n k cen t r e of the "elliptic" tank (u q = 1.3 m) to be 0.2 m b e l o w the c i r c u l a r tank (u = 1 . 5 m) . W i t h the same c o n d i t i o n t he " s u perelliptic" tank c e n t r e will be e v e n lower, but due to its p o s s i b l e need for a s u p p o r ­ ting frame stru c t u r e it was a s s u m e d to h a v e the same c e n t r e h e i g h t as the "elliptic" tank (i.e. u q = 1.3 m ) .

Di 2

D.8.2 § l o s h i n g _ l o a d _ d a t a

T he p o i n t of a c t i o n for the l i q u i d forces was d e f i n e d in the p r e v i o u s s e c t i o n by the u q distance. See table D2 for th e s e and f o l l o w i n g values.

The load w e i g h t (m^) was a s s u m e d to be m a x i m u m p e r ­ m i s s i b l e (15000 kg as the u n l o a d e d v e h i c l e w e i g h t was

5000 kg) at 0.75 tank volume. T h e load w e i g h t at half, 75% and full t a n k v o l u m e w e r e equal for all tank c o n ­ f i g u r a t i o n s . As the full scale d e n s i t y ( p ) and the

r

tank scale ratio (i, / ¿ M ) w e r e p redefined, the full sc a l e tank l e n g t h (L ) had to be c a l c u l a t e d as a

secon-r d a r y variable.

= 15000

F A F .pF -0.75 (D.16)

w h e r e A is the full scale tank c r o s s - s e c t i o n area [m ] . The load d e n s i t y w as a s s e s s e d to pF = 920 kg/in c o r r e s p o n d i n g to fuel oil see ta b l e B l . F r o m the m o

-3

del liqu i d d e n s i t y (water: pM = 1 0 0 0 k g / m ) it was p o s s i b l e to c a l c u l a t e the fo r c e a nd m o m e n t s caling c o n s t a n t s K-^ and a c c o r d i n g to e q u a t i o n s (C.ll) and

(C .12) .

D.8.3 Ri9i d _ l o a d _ d a t a

W i t h the load b e i n g r i g i d and fixed to the tank its d a t a w e r e p a r t l y d e f i n e d a c c o r d i n g to the p r e v i o u s section. B ut in o r d e r to m a k e the c a l c u l a t i o n of load i n e r t i a forces p o s s i b l e from the l a t eral ac c e l e r a t i o n , the load c e n t r e of g r a v i t y l o c a t i o n and roll m o m e n t of inertia h ad to be d e f i n e d as well.

The c.g. d i s t a n c e a b o v e the tank c e n t r e was simp l y e v a l u a t e d for e a c h t a n k and load p e r c e n t a g e by

the w e l l - k n o w n "c a r t o o n b a l a c i n g " m e t h o d (the f i l l e d area c o n t o u r s w e r e cut o ut from a stiff p a p e r and the pa p e r was b a l a n c e d on an e d g e ) .

T he roll m o m e n t of inertia w as c a l c u l a t e d for the c i r ­ c u l a r tank in the f o l l o w i n g way.

A s s u m e that the t a n k in fig D3 is filled to x% . The load area will be d e f i n e d in the f o l l o w i n g two w a y s

A = x 1 0 0 ttR (D . 17) A — A -. + A ~ x xl x2 (d .i s ; w h e r e A , R2 xl 2 ; D .18a) A 0 = R sini * R cosí x2 ( D .18b) Thus 2 4> + sin 2<J> = (0 . 0 2 x - 1) • it (D . 19) 4 2 - _{PL 4} R a _{PL A 1}„T , R F i g u r e D 3 . D e f i n i t i o n s of v a r i a b l e s for e v a l u a t i o n of ri g i d load m o m e n t of inertia.