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lot. J. Mach. Tool Des. Res. Vol. 11, pp. 31 43. Perga mon Press i971. Printed in Grea t Britain

A N E W D Y N A M O M E T E R P R I N C I P L EC. E. DAVIS*

(Received 15 th Apr i l 1970)Abstraet--A high quality two component dynamometer has been designed and constructed, capable of theaccurate monitoring of forces whose magnitudes range from fractions of a pound to hundreds of pounds.The simplicity of its design makes possible the complete manufacture and assembly in less time than thatrequired to produce a "strain gauge" dynamometer. The load sensitive elements are piezo-electric forcetransducers, in the form of thick washers, which are rapidly finding their way into research and developmentdepartments as standard equipment. The simplicity of the design and the speed with which it can be takenapart, coupled with the multi-purpose use of the transducers makes for an expendable dynamometer. Theprototype has been designed and used as a grinding dynamometer. It must be emphasized, however, that thedynamometer is capable of monitoring much higher forces than those encountered in grinding.

D Y N A M O M E T ER S P E C I F I C A T I O NGENERALLY, forces in grinding do not exceed 60 lbf/in, width of cut during plunge grindingand, since plunge grinding operations are more often performed with wheels less than oneinch wide, a sensitive apparatus is required to monitor these low forces accurately. Sym-pathetic vibrations of a dynamometer result in the recorded force having an excessiveoscillation not necessarily representative of the true cutting force. Vibrations sympatheticto the primary cutting frequency, i.e. vibration due to wheel rotation, can lead to chatter.To avoid the generation of sympathetic vibrations, the frequency of natural vibration of thedynamom eter must be higher than the natural frequency of the machine tool and higherthan the primary cutting frequencies. Secondary cutting frequencies, due to the individualcutting edges, are usually very much higher, especially in grinding, but their magnitude isless and they are absorbed to a great extent in the cutter and workpiece material. Naturalchatter conditions are an exception as these vibrations have large force amplitudes, buttheir cause is associated with the vibration of the whole machine structure.

It is evident that the dynamometer must be stiff to give a high frequency of naturalvibration, and at the same time be very sensitive.

T RA N SD U CE R SE L E CT IO NThe majority of force measur ing devices have a sensitivity which is inversely propo rtion al

to stiffness. Gauges more sensitive than wire strain gauges are available, namely the semi-conductor and piezo-resistive types, which enable a more sensitive system to be constructedwithout loss of stiffness. But semi-conductor gauges require specialized equipment forassembly and piezo-resistive gauges are fragile so tha t great skill and care is required toavoid thermal damage when making electrical connections and physical damage duringassembly. In the event of failure of these types of gauges, there is likely to be a considerabledelay whilst new gauges are fitted and the equipment is re-calibrated.

The accuracy and quality of dynamometers, where the strain of an element is measuredas a function of the applied force, depends largely on the design, but also upon the accuracy

* Special Research Assistant, University of Manchester Institute of Science and Technology.31

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32 C. E. DAVIS

o f m a n u f a c t u r e a n d a s s em b l y . L a r g e c r o s s in t e rf e r en c e b e t w e e n t h e c o m p o n e n t s c a n r e s u l tf r o m i n a c c u r a t e m a c h i n i n g o f st r ut s o r h e x a g o n a l r i n gs a n d f r o m m i s p l a c e m e n t o f t h eg a u g e s . I t i s d e s i r a b le t h a t t h e d e g r e e o f c r o s s i n t e rf e r e n c e b e m i n i m a l i n o r d e r t o a v o i d th en e c e s s i ty o f m a k i n g c o r r e c t i o n s t o t h e i n d i c a t e d f o r c e s. I n d y n a m i c a p p l i c a t i o n s th e c r o s si n t er f e re n c e i s r e q u i re d t o b e s m a l l t o a v o i d s y m p a t h e t i c v i b r a t i o n o f o n e c o m p o n e n t d u et o f o r c e d v i b r a t i o n s o f th e o t h e r . T h i s i s p a r t i c u l a r l y r e l e v a n t w h e r e t h e n a t u r a l f r e q u e n c y o fv i b r a t i o n o f o n e c o m p o n e n t i s t h e s a m e a s o r n e a r to t h a t o f t h e o t h e r c o m p o n e n t .

P i e z o - e l ec t r ic f o r c e t r a n s d u c e r s m e a s u r e f o r c e s d i r e c t l y a n d t h e i r h i g h s ti ff n es s , r e m a r k a b l es e n s ti v i ty a n d l o a d r a n g e h a v e b e e n u s e d t o a d v a n t a g e i n t h e c o n s t r u c t io n o f th i s d y n a m o -m e t e r t o g i v e h ig h s ti ff n es s a n d h i g h s e n s i ti v it y . T h e t r a n s d u c e r s , i n th e f o r m o f t h i c k w a s h e r s ,c a n b e u s e d i n a m u l t i t u d e o f a p p l i c a t i o n s a n d n e e d n o t n e c e s s a r il y b e p e r m a n e n t l y i n s ta l l e di n a d y n a m o m e t e r . T h i s m a k e s p o s s ib l e th e c o n s t r u c t i o n o f a sp e c ia l p u r p o s e , l o w c o s td y n a m o m e t e r f o r u s e o v e r a l im i t e d p e r i o d o f t im e a f t e r w h i c h , i f n e c e s s a r y , t h e t r a n s d u c e r sc a n b e u s e d e l s e w h e r e .

P I E Z O - E L E C T R IC T R A N S D U C E R SI n o r d e r t o b e t t e r a p p r e c i a t e t h e d y n a m o m e t e r d e s i g n , t h e p i e z o - e l e c t r i c e f f e c t , a n d s o m e

l i m i t a t i o n s o f p i e z o - e l e c t r i c t r a n s d u c e r s w i ll b e d e s c r i b e d . T h e p i e z o - e l e c t r i c e f f e c t i s f o u n di n a n u m b e r o f s e m i - c o n d u c t i n g m a t e r i a l s , th e m o s t w i d e l y u se d b e i n g q u a r t z s in c e l a r g ep u r e q u a r t z c r y s t a l s a r e c o m p a r a t i v e l y e a s y t o s y n t h e s iz e . W h e n a q u a r t z c r y s t a l i s s u b j e c t e dt o a u n i - d i r e c t i o n a l f o r c e t h e r e is a m i g r a t i o n o f e l e c tr o n s t o o n e o f th e f a c e s a t w h i c h t h el o a d i s a p p l i e d , t h e a m o u n t o f e le c t r o - s t a ti c c h a r g e b e i n g li n e a r ly r e l a t e d t o t h e a p p l i e dl o a d . A n e l e c t r o - st a t i c c h a r g e a m p l i f ie r is n e c e s s a ry t o m e a s u r e t h e c r y s t a l c h a r g e w i t h o u td r a w i n g c u r r e n t , w h i c h w o u l d d e s t r o y t h e c h a r g e . T h e m a n u f a c t u r e r s o f t h is t y p e o f tr a n s -d u c e r i n c r e a s e th e " p o w e r " b y s a n d w i c h i n g a n u m b e r o f c ry s t a l d i sc s , c o n n e c t e d e l e c t ri c a ll yi n p a r a l l e l w i t h i n t h e t r a n s d u c e r ; t h i s h a s t h e e f f e c t o f r e d u c i n g d r i f t, i . e. t h e l o s s o f c h a r g ed u r i n g a s t a t i c l o a d a p p l i c a t i o n . T h e r e i s n o c r o s s - i n t e r f e r e n c e b e t w e e n t h e a x i s a n d s u r f a c ep l a n e o f t h e l o a d w a s h e r s a s t h e e l e c t r i c a l c o n n e c t i o n s t o t h e c r y s t a l a r e m a d e i n o n e p l a n eo n l y , s o t h a t t h e r e i s n o m e a s u r e m e n t o f c h a r g e s g e n e r a t e d i n o t h e r p l a n e s.

T h e s e t r a n s d u c e rs c a n n o t b e u s e d t o t h e ir b e s t a d v a n t a g e w i t h o u t t a k i n g t h e p r e c a u t i o nt o a v o i d a p p l y i n g s h e a r s t r e s se s t o t h e t r a n s d u c e r s a s t h e s e m a y p r o m o t e f r a c t u r e o f th ec r y s t a l l a m i n a t i o n s .

D E V E L O P M E N T T O W A R D S T H E E D G E P R I N C I P L EI n F i g . 1 A i s c o n s t r a i n e d t o m o v e p a r a l l e l t o B . T h e t r a n s d u c e r e x p e r i e n c e s a s h e a r f o r c e

d u e t o W , w h i c h m a y r e s u l t i n p h y s i c a l d a m a g e , b u t d o e s n o t c a u s e a n e l e c t r i c a l r e s p o n s e .W a l s o r e s u l t s i n a b e n d i n g m o m e n t b e i n g a p p l i e d t o t h e t r a n s d u c e r w h i c h c a n b e s e e n a s a

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A New Dynam ome ter Pr incip le 33c r o s s- i nt e r fe r e n c e. T h e b e n d i n g m o m e n t o n t h e t r a n s d u c e r c a n b e r e d u c e d b y i n cr e a s in gL a n d p o s i t i o n i n g t h e tr a n s d u c e r a t th e p o i n t o f m i n i m u m b e n d i n g b u t n e c e s s ar i ly L w o u l db e l a r g e .

T h e m o s t o b v i o u s d e si g n f o r a tw o c o m p o n e n t d y n a m o m e t e r w o u ld p r o b a b l y b e to h a v ea l o a d c a r r y i n g b l o c k s u p p o r t e d o n t h r e e t ra n s d u c e r s ; o n e t o m e a s u r e h o r i z o n t a l f o rc e sa n d t w o , t h e i r e l e c tr i c c h a r g e b e i n g s u m m e d , t o m e a s u r e v e r t i ca l f o r c e s ; a s s h o w n i n F i g . 2 .

W " 1I I

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T h i s d e s i g n i s n o t s a t i s f a c t o r y d u e t o c e r t a i n n o n - l i n e a r i ti e s :T h e s e n s i ti v i ty o f t r a n s d u c e r c i s a f u n c t i o n o f th e h e i g h t a b o v e t h e b a s e a t w h i c h a

h o r i z o n t a l c o m p o n e n t o f fo r c e is a p p li e d . T h i s i s b e c a u s e t h e l o a d s e n s it iv e d y n a m o m e t e rb l o c k A is r i g id l y f ix e d t o th e d y n a m o m e t e r f r a m e B b y th e t r a n s d u c e r s a a n d b . T h e t o t a lr e a c t i o n t o a h o r i z o n t a l f o r c e is d i s p r o p o r t i o n a t e l y d i v i d e d b e t w e e n t h e tr a n s d u c e r c a n dt h e d y n a m o m e t e r b a s e a t t he a n c h o r a g e o f t r a n s d u c e r s a a n d b . T h e n e a r e r t he b a s e t h eh o r i z o n t a l f o r c e i s a p p l i e d , t h e g r e a t e r s h a r e o f th e t o t a l f o r c e is t a k e n a t t h e b a s e a n c h o r a g eo f t r a n s d u c e r s a a n d b . C l e a r l y th i s is a d i s a d v a n t a g e s i n c e w h e n a w o r k p i e c e i s r e d u c e d i nh e i g h t th e r e is an a p p a r e n t r e d u c t i o n i n t h e h o r i z o n t a l c o m p o n e n t o f c u t t in g f o rc e .

S i m i l a r ly , t h e s u m m e d o u t p u t o f t r a n s d u c e r s a a n d b i s r e d u c e d t h e n e a r e r a v e r ti c a lf o r c e is a p p l i e d t o t r a n s d u c e r c . T h i s i s n o t o n l y o f d i s a d v a n t a g e i n t h e m o n i t o r i n g o f f o r c e sb u t a l s o r e s u l t s i n a l a r g e c r o s s s e n s i t i v i t y b e t w e e n t h e h o r i z o n t a l a n d v e r t i c a l d y n a m o m e t e ra x is . T h i s is b e c a u s e a h o r i z o n t a l f o r c e r e s u l t s i n a m o m e n t o n t h e d y n a m o m e t e r l o a d s e n s it i v eb l o c k A a n d u n l e s s t h e s e n s i ti v i ty o f t r a n s d u c e r a m a t c h e s e x a c t l y t h a t o f b th e s u m m a t i o no f t h e t r a n s d u c e r o u t p u t s w i l l n o t b e z e r o . A v e r t i c a l f o r c e w il l t h e r e f o r e b e a p p a r e n t w h e no n l y a h o r i z o n t a l f o r c e is a p p l i e d . F u r t h e r m o r e , t h e t r a n s d u c e r s a r e a ll su b j e c t e d t o s i m p l es h e a r s t r e s s e s w h i c h , a s in d i c a t e d a b o v e , is u n d e s i r a b l e .

I n o r d e r t o a v o i d s h e a r o n t h e t r a n s d u c e r s a n d t o r e d u c e c r o s s - i n t e r f e r e n c e t o a s m a l lv a l u e w i t h a s h o r t s t r u t l e n g t h L , t h e p r i n c i p l e u s e d in F i g . 3 c a n b e a d o p t e d . I t s h o w s as i m i l a r c o n f i g u r a t i o n t o t h a t o f F ig . 2 , d i f fe r i n g o n l y in t h a t t h e t r a n s d u c e r s a r e m o u n t e d o n" e d g e s " . W i t h t h i s a r r a n g e m e n t t h e re c a n b e v e r y li tt tl e s h e a r f o r c e t ra n s m i t t e d t h r o u g h at r a n s d u c e r a n d n o b e n d i n g m o m e n t . O b v i o u s l y , i f t h e f o rc e W w e r e t o r e s u lt in l a rg e m o v e -m e n t s o f a r e l a t iv e t o b t h e r e w o u l d b e a n o t i c e a b l e c h a n g e i n p r e - l o a d o n t h e t r a n s d u c e r .A s s u m i n g , h o w e v e r , t h a t t h e m o v e m e n t i s r e s t r a i n e d t h e c r o s s- i n t e r f e re n c e o f t h i s a r r a n g e -m e n t w i ll b e v e r y s m a l l . H e n c e t h e d e s i g n s h o w n i n F i g . 2 b e c o m e s t h e b a s i s f o r a g r i n d i n gd y n a m o m e t e r , F i g . 3 .

T h e g e o m e t r y o f t he e d g e s h a s to b e c a r e fu l ly c o n s i d e r e d s o a s t o p r e v e n t p r e m a t u r eb r e a k d o w n d u r i n g o p e r a t i o n a s c y c l ic l o a d s c o u l d g i v e r is e t o f a i l u r e b y f a t i g u e a n d f r e t ti n g .T h e m o v e m e n t o f th e l o a d - c a r r y i n g b l o c k i s v e r y sm a l l s o t h a t t h e w e d g e a n g l e o f t h e

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e d g e s c a n b e h i g h t o g iv e m a x i m u m s t r e n g th . H o w e v e r , t o o h i g h a w e d g e a n g l e w i llr e s u l t in a l a r g e e d g e c o n t a c t a r ea a n d t h e b e n e f i t c o n f e r r e d b y u s i n g e d g e s w o u l d b e l o s t.A w e d g e a n g l e o f 1 7 0 h a s b e e n f o u n d s a t i s fa c t o r y f o r b e a r i n g s u r fa c e s o f h a r d e n e d h i g hc a r b o n s t e e l .

E D G E L I F E A N D S T I F F N E S ST h e s ti ff n e ss a n d f a t ig u e l i fe o f t h e e d g e s c a n b e c a l c u l a t e d b y u s i n g H e r t z e x p r e s s i o n s f o r

s u r f a c e d e f l e c t i o n s . I t i s a s s u m e d t h a t :( 1 ) T h e l i n e o f c o n t a c t i s n o t a t r u e e d g e b u t c a n b e r e p r e s e n t e d b y a r o l l e r o f d i a m e t e r

d . ( Se e F i g . 5 ) .( 2 ) T h e r o l l e r r a d i u s i s 0 . 0 6 2 5 i n . a f t e r i n i t i a l e l a s t i c d e f o r m a t i o n . ( T h i s f i g u r e i s b a s e d

o n o b s e r v a ti o n ) ./ \

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A New Dynam ometer Pr incip le 35

w h e r e

( 3) T h e m a t e r i a l s o f t h e c o n t a c t i n g s u r f a c e s h a v e t h e s a m e p h y s i c a l p r o p e r t i e s .T h e w i d t h o f c o n t a c t , b , b e t w e e n a r o l l e r a n d p l a n e s u r f a c e i s g i v e n b y th e H e r t z r e l a t i o n :

b = 1 "6 / p a l ( I - - v~ q _ I ~ v ~ ]E1 E2 J - \ (0p i s t h e l o a d p e r u n i t l e n g t h o f e dg e , a n dv i s P o i s s o n ' s r a t i o , v = ( E / 2 G ) - - 1 .

T h e p r o p e r t i e s o f t h e h a r d e n e d h i g h c a r b o n s te e l b e a r i n g s u r f a c e s a r e :E = 30 x 106 lbf / in .Y = 260 ,000 lb f / in . 2 (y ie ld s t r e s s ) .G = 0 .39 Ev = (1 /0 .78 ) - - 1 = 0 .28 .

F r o m a s s u m p t i o n 3 e x p r e ss i o n (1 ) r e d u c e s to :b = 2 " 2 6 / p d l - - v 2E

a n d s u b s t i t u t i n g f o r v , d a n d E g i v e s:b = 1 .38 x lO- 4V 'p . (2)

I t i s a l s o g i v e n t h a t t h e e d g e p l a n e s u r f a c e d e f l e c t i o n u n d e r l o a d ,a = 2p ( -k In ( 2 d / b ) ) ( ( 1 - - v 2 ) f i r E ) (3 )

a n d s u b s t i t u t i n g v a l u e s f o r v , d , E a n d b r e d u c e s ( 3 ) t o :8 = 1.86 x 10 s ( -3 In (25 x 102/1 "38 . V / p ) ) p in . (4)

A p l o t o f ( 5 ~-~ p ) , F i g . 5 , s h o w s t h e t h e o r e t i c a l j o i n t s t if f n e ss , o f t h e " e d g e " a n d p l a n es u r f a c e , o v e r a r a n g e o f a p p l i e d l o a d s . T h e j o i n t i s m o s t s t i f f a t h i g h l o a d s a n d s o a h i g hp r e - l o a d w o u l d a p p e a r t o b e b e n e f i c i a l , b u t f a t i g u e r e s i s t a n c e i s r e d u c e d a t h i g h l o a d s .

T h e l i m i t to w h i c h a m a t e r i a l c a n b e s u b j e c t e d t o c y c l i c l o a d s b e f o r e t h e r e i s a p o s s i b i li t yo f f a t ig u e f a i lu r e c a n b e a s se s se d a p p r o x i m a t e l y b y a p p l y i n g t h e S o d e r b e r g c r i t er io n . F r o mt a b l e s t h e f a t ig u e l i m i t o f a m a t e r i a l c a n b e f o u n d f o r c o n d i t i o n s o f c o m p l e t e l y r e v e r si b l es t re s s. C o n d i t i o n s o f c y c li c l o a d i n g w h i c h d o n o t r e s u l t in c o m p l e t e s t r e s s r e v e r s a l s c a n r e s u l ti n a m a t e r i a l f a i l u r e b y f a t i g u e b y l o a d i n g a b o v e t h e y i e l d p o i n t . T h e S o d e r b e r g l i n e is al in e d r a w n f r o m t h e l i m i t i n g r e v e r s i b le s tr e s s f o r t h e m a t e r i a l b e f o r e p o s s i b l e fa t i g u e f a i lu r eo n t h e S v a x i s, t o t h e y i e l d s t re n g t h o f t h e m a t e r i a l o n t h e S m a x i s , s e e F i g . 6 . M a t e r i a ls t re s s c o n d i t i o n s r e p r e s e n t e d b y p o i n t s o u t s i d e t h e S o d e r b e r g l i n e w il l e v e n t u a l l y r e s u l t int h e m a t e r i a l f a i l u r e a n d t h e f u r t h e r t h e p o i n t is f r o m t h e l in e t h e m o r e r a p i d w i ll b e t h efa i lu r e .

A p p l y i n g t h e S o d e r b e r g c r i t e r i o n t o t h e " e d g e " - p l a n e s u r f a c e j o i n t , F i g . 6, i t c a n b e s e e nt h a t t h e r e i s a w i d e r a n g e o f p r e - l o a d s w h i c h s a t i s f y a c o n d i t i o n o f h i g h r e s is t a n c e t o f a t i q u ef a il u re w h e n a m a x i m u m c y c li c lo a d o f 5 0 l b f is a p p li e d .

I t i s e v i d e n t f r o m F i g s. 5 a n d 6 t h a t t h e v a l u e o f t h e a p p l i e d p r e - l o a d i s n o t c r i ti c a l a n d aw i d e r a n g e o f p r e - l o a d s c o u l d b e a p p l i e d t o t h e " e d g e " j o i n t s w i t h o u t a d v e r s e e f fe c ts . A

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p r e - l o a d o f f r o m 2 0 0 t o 3 00 l b f w o u l d g i v e t h e g r e a t e s t r e s i st a n c e t o f a t i g u e f a i lu r e , a n d a tt h e s e p r e - l o a d s t h e r e i s a d e q u a t e e d g e j o i n t s ti ff n es s .

T H E D Y N A M O M E T E R D E S I G NB y u s i n g t h e p r i n c i p l e o f " e d g e s " t h e s e n s i ti v it ie s o f a a n d b a r e f o u n d t o b e v e r y n e a r l y

t h e s a m e a n d b y a d j u s ti n g t h e p r e - l o a d o n o n e o r t h e o t h e r a p e r f e c t m a t c h i n g o f t h ei rs e n s i t i v i t i e s c a n b e o b t a i n e d .

T h e a p p a r e n t c h a n g e i n t r a n s d u c e r s e n s it i v it y w i t h p r e - l o a d i s d u e t o a c h a n g e o f j o i n ts t if fn e s s w i t h i n t h e t r a n s d u c e r a s s e m b l y . T h e f o r c e a c t i n g o n t h e t r a n s d u c e r a s s e m b l y , F ,i s l a r g e r t h a n t h a t o n t h e t r a n s d u c e r i ts e lf , f , w h e r e :

f = F . k / ( k 4 - k 1 ) (5 )a n d k 1 a n d k a r e t h e p r e - l o a d s c r e w a n d j o i n t s t if f n e s s r e s p e c t i v e l y . S e e F i g . 9 . J o i n t s t i f f n e ssi s s o m e f u n c t i o n o f l o a d , h e n c e , f , i n c r e a se s a s p r e - l o a d i n c r e a s es , c a u s i n g a n a p p a r e n ti n c r e a s e i n s e n s i t i v i t y w i t h r e s p e c t t o t h e c o n s t a n t l o a d F .

A p r o t o t y p e d e s ig n , F i g . 8 , w a s b u i l t a n d t e s t e d b u t w a s f o u n d t o h a v e t w o p r i n c i p a lf a u lt s . F i r st ; t h e d y n a m o m e t e r b o d y d i s t o r t e d d u e t o t h e r m a l e x p a n s i o n r e l e a si n g t h e l o a do n t r a n s d u c e r C w h i c h w a s s e e n, o n t h e r e c o r d i n g a p p a r a t u s , a s a r a p i d d r i f t f r o m t h e d a t u m .S e c o n d ; i t w a s f o u n d i m p o s s i b l e t o b a l a n c e t h e p r e - l o a d o n t r a n s d u c e r s a a n d b a c c u r a t e l yu s i n g t h e i n di r e ct m e t h o d o f p r e -l o a d a p p l i c a t i o n s h o w n , w h i c h w a s a d o p t e d t o f a c il it a tes m a l l m o v e m e n t s o f th e t r a n s d u c e r a s s e m b l ie s to c o m p e n s a t e f o r m a n u f a c t u r i n g i n a c c u ra c i es .A s a r e su l t , t h e d y n a m o m e t e r s e n s it i v it y v a r i e d a l o n g th e w o r k i n g l e n g t h w i t h u n a c c e p t a b l ec r o s s i n t e r fe r e n c e b e t w e e n t h e a x e s.

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