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Graphical analysis of free vibrations of helical springs
GRAPHICAL ANALYSIS OF FREE VIBRATIONS OF HELICAL SPRINGS.* BY KALMAN
J.
DeJUHASZ,
Associate Professor of Engineering Research, The Pennsylvania State College. Member of The Franklin Institute.
Helical springs are used as measuring elements for variable forces in several kinds of engineering instruments such as engine indicators and dynamometers. Their vibration characteristics have an i m p o r t a n t influence on the behavior and accuracy of such instruments. Therefore the following problem is of theoretical and practical importance: Consider a helical spring, with or without concentrated masses attached to its two ends, initially at rest and put under an initial longitudinal strain by means of external constraining forces. Determine the law of motion if the constraining forces are instantaneously removed. The term "helical spring" includes also elastic rods. In a broader sense the above problem applies also to helical springs or elastic rods in torsion by substituting " m o m e n t of inertia" instead of " m a s s , " "tangential strain" instead of "longitudinal s t r a i n " and " c o u p l e s " instead of "forces." This problem has been treated by several authors (Krylow, Holm, T h a m m , Sayre, Ref. IO, I I, 4, I2) under the assumption t h a t the strain at all points of the spring is the same at any given instant of time, which assumption implies an infinite velocity of propagation. The mass of the spring was taken into consideration by either adding a certain calculated fraction of the spring mass to the concentrated mass or by multiplying with a calculated factor the period calculated for the concentrated masses only. Owing to these simplifying assumptions the analytical m e t h o d is not fully satisfying. It is inaccurate because it yields a purely harmonic motion, and it is incomplete because the law of motion for the intermediate points of the spring is not supplied. * R e c e i v e d J u l y , I938. VOL. 227, NO. 1361--45
647
648
KALMAN J. DEJUHASZ.
[J. F. I.
Actually the velocity of propagation of a disturbance in a spring is a finite q u a n t i t y and therefore the strain in a spring element changes n o t only from instant to instant, but also from point to point in the spring. As a result the motion will differ from the pure harmonic as the beautiful photographs of W. Weibull show (Fig. I). This fact has been taken into FIG.
I.
P h o t o g r a p h of free vibrations, as a function of time, of a n unloaded spring, b y W . Weibull.
consideration (Weibull, Ref. 2; Magg, Ref. I) b u t the calculations it involves are impracticably complicated and have been carried out only for some simple particular cases. The graphical m e t h o d used by the a u t h o r in several previous publications (Ref. 5-9) avoids these difficulties and is easy to perform. For a complete description of the phenomenon the (I) velocity v, (2) strain s, and (3) displacement u has to be given for all points x of the spring and for all instants of time. Expressed in analytical terms the following three functions v = fl(t, x) s = f~(t, x ) u = f3(t, x)
are sought. In graphical terms each of these functions can be represented by a tri-dimensional "stereogram," with the v, s and u values as ordinates erected over the t - x plane as base. The problem then resolves itself into finding the data from which these three stereograms can be constructed.
l~fay, I939. ]
FREE
VIBRATIONS
OF H E L I C A L
SPRINGS.
649
T h i s is possible with the aid of the following relationships: (a) T h e location of the d i s t u r b a n c e front can be represented in the t - x d i a g r a m by straight lines having the slope tan (=t= ~), i.e.: progress of d i s t u r b a n c e in a given time element
-
Ax - 4 - a = tan (=t= ~), At
a being the velocity of p r o p a g a t i o n of the d i s t u r b a n c e ; these lines will be referred to as " d i s t u r b a n c e lines." (b) T h e velocity-change is proportional to the corres p o n d i n g strain-change and can be represented in the v-s d i a g r a m by straight lines h a v i n g the slope tan (=~ a), i.e. : change of strain change of velocity
As Av
strain corresponding to stress E velocity of p r o p a g a t i o n I
= - = tan (4- oz), a
which will be referred to as "directrices." (c) T h e d i s p l a c e m e n t u can be d e t e r m i n e d from either the v or s s t e r e o g r a m with the aid of the following relationships: du dt
v
and
du dx
and
u=
Ls,
hence
u = fvdt
r f sdx,
i.e., the d i s p l a c e m e n t can be d e t e r m i n e d by integrating either the v or s stereograms. T h e derivations of (a) and (b) are given in Ref. 6 and 9 and can be o m i t t e d here. T h e m e t h o d will be n o w applied to a n u m b e r of assumed conditions h a v i n g special interest. I.
SPRING WITHOUT CONCENTRATED MASS. (a) One End Fixed, the Other End Free (Fig. 2).
It is a s s u m e d t h a t the spring with the unstressed length L0 is compressed to the length L1, involving an initial strain sl and initial velocity Vl = o, as characterized by point I in
650
KALMAN J. DEJuHASZ.
[J. F. I.
the v-s diagram. At the instant I the unattached end of the spring is instantaneously released, whereby the strain at t h a t end vanishes (s2 = o) and the velocity assumes the value of v2, the new state being defined by point 2 situated on the directrix drawn from point I in the v-s diagram, Fig. 2b. The change of state constitutes a disturbance which proceeds along the spring with the velocity of propagation a and after the elapse of L / a time arrives at the fixed end. Here the velocity is reduced to o and the strain correspondingly changes to s3 defined by point 3 situated on the directrix drawn from point 2 in the v-s diagram. The change from state 2 to state 3 returns towards the free end of the spring with the velocity of propagation and arrives there after the elapse of another L I e time. Here its strain is reduced to zero and its velocity assumes v4 as defined by point 4 in the v-s diagram. Continuing this construction the state of the spring according to points 5, 6, 7, 8 and so on can be determined, each state being valid for the triangular territories marked with the same numbers in the t - x diagram, Fig. 2a. The change of position of the coils, as a function of time, is illustrated on Fig. 2c. The slope of the heavy lines is proportional to the velocity; the relative change of distance between them, as measured on any t = const, ordinate, is proportional to the strain. T h e disturbance lines appear as the sloping lines connecting the two ends of the spring taken at L / a time interval apart. Erecting the v and s coordinates (in the v-s diagram) of the I, 2, 3, 4 " • " points over the t - x plane as base the velocityand strain-stereograms can be built, which are shown in parallel perspective in Figs. 2d and 2e. The displacements Ill
are obtained by forming the u = .~ vdt lines or the u = L J sdx lines over the t - x plane as base, as shown by the u-stereogram, Fig. 2f. These stereograms define the motion of the spring for every point in the spring and for every instant of time. F r o m these the following characteristics of the vibration can be read: (I) The velocity at the fixed end is o, and the strain at the free end is o a t all times.
May, I939.] FREE VIBRATIONS OF T-IELICAL SPRINGS.
651
F I G . 2.
(~
Velocil
~=a
(~P/6Plclcement~-/~x'x ~
~
.
,.~.
~ t,,3/r, .,
~ ,,,
.
........
Unloaded spring fixed at one end, free on the other end. (a) time-disturbance diagram; (b) velocity-strain diagram; (c) displacement vs. time; (d) velocity-stereogram; (e) strain-stereogram: (f) displacement-stereogram; (g) effect of Coulomb friction; (h) effect of periodic damping; (i) aperiodic damping; (j) critical damping.
II, -~_ . . . . . ~ . . . ~ - ~ -~,--.. ~ ,~,.<"
"'~.---~.-.~
~-.-., ._._~'__ ._ ~. ~'.~.'~'-.. - ~_~;~. -.... - -L~.-.~.~._
r-
652
KALMAN J. DEJr:uasz.
[J. F. I.
(2) The motion at the free end is periodic with a period T = 4L/a; it is however not harmonic, b u t has constant positive and negative velocities alternating after the elapse of each consecutive 2L/a interval (cf. Appendix II). (3) T h e motion of the intermediate points of the spring is intermittent, the durations of positive and negative velocities being separated by dwells of zero velocity.
Friction and Damping Friction (Coulomb type) is equivalent to a definite, constant value of strain R which m u s t be exceeded before motion, according to ZXv= (zXs - R) tan a, can result. In the graphical construction friction is represented by two lines in the v-s diagram, parallel to the v-axis drawn at s = -t- R and s = - R distance from it (Fig. 2g). It is seen t h a t the velocity and strain alternate between positive and negative values b u t the absolute magnitude of their consecutive values is not constant b u t decreases in an arithmetic progression. T h e motion stops altogether when the strain drops below the value of friction, i.e., s <
IRI.
Damping (or molecular friction) is equivalent to a strain R which is not constant b u t proportional to the velocity, i.e., R = v tan 0, where tan 0 is the proportionality factor. This value of strain m u s t be exceeded before motion can result, and only the excess of strain can be converted into velocity according to the equation 2~v = (As -- R) tan a = (zXs -- v tan 0) tan c~. In the graphical construction d a m p i n g is represented by a straight line passing through the point of origin of the v-s diagram forming an angle 0 therewith (Fig. 2h). It is seen t h a t the consecutive absolute values of velocity and of strain decrease in geometric progression. If 0 < a the d a m p i n g is periodic and the motion ceases only after the elapse of an infinite n u m b e r of L/a intervals, vibrating on both sides of its
May, I939. ] FREE VIBRATIONS OF I-IELICAL SPRINGS.
653
position of equilibrium with ever-decreasing amplitude. If o < ~- (Fig. 2i) the d a m p i n g is aperiodic and no overshooting the equilibrium position takes place. In the case of critical d a m p i n g p = a (Fig. 2j) the whole potential energy of the spring is consumed by the d a m p i n g in 2L/a period after which the spring comes to a standstill in its position of equilibrium. Other varieties of damping, e.g., in which the d a m p i n g force is n o t a linear function of the velocity, b u t t h a t of a higher power of it, or some other n o t necessarily algebraic function of the velocity, can be represented in the v-s diagram and the graphical construction performed in the m a n n e r shown. (b)
Both E n d s Free
(Fig. 3).
It is assumed t h a t the spring with the unstressed length L0 is compressed to L1 length, the initial state, Vl = o, s = sl FIG. 3.
1:~6" ©
#~#ire
95
' 8 ' ~"
+" :".? " ~ g -
"~...
li
f
Unloaded spring free on both ends. (a) time-disturbance diagram; (b) velocity-strain diagram; (c) displacement vs. time; (d), (e), (f) stereograms of velocity, strain and displacement.
654
KALMAN J. DEJUHASZ.
[J. F. I.
being defined by the point I in the v - s diagram, Fig. 3b. At the instant I the spring is instantaneously released. The strain at the ends vanishes and the upper end assumes a positive (upward) velocity, the lower end assumes a negative (downward) velocity, as defined by the points 2' and 2 in the v - s diagram. T h u s simultaneously two disturbances are initiated, one from each end of the spring, traveling in opposite directions. After the elapse of ( I / 2 ) ( L / a ) time these meet at the mid-point of the spring and the new state 3 = 3' results. The successive states 4, 6, 8 for the upper free end, 4 r, 6 p, 8' for the lower free end and 5 --- 5 ~, 7 ~ 7' " • " for the mid-point of the spring can be found from the v-s diagram (Fig. 3b), and the t - x zones for which the various states are valid can be found in the triangles and rhomboids formed by the disturbance lines in the t - x diagram (Fig. 3a). The stereograms of velocity, strain and displacement constructed from the d a t a thus obtained are shown in Figs. 3d, e, and f. It is seen t h a t the mid-point is stationary and assumes alternating sl and s3 stresses, while the two ends have zero stress and assume alternating v~. and v4 velocities, the period of motion being T -- 2 L / a (cf. Appendix II). (c) Both Ends Fixed (Fig.4).
It is assumed that the ends of the spring are fixed at a distance equal to the unstressed length L0 of the spring. The mid-point is displaced upward, causing in the upper half a compressional strain corresponding to point I in the v-s diagram, and in the lower half a tensional strain corresponding to point I'. At the instant I the mid-point is instantaneously released. The strain at the mid-point vanishes and it assumes a velocity corresponding to point 2 -- 2' in the v-s diagram which velocity spreads toward the two ends with the velocity of propagation. O n reaching, after the elapse of L/2a time, the two fixed ends the velocity is reduced to o and at the lower end a compressional strain according to point 3', and at the upper end a tensional strain according to point 3 results. The zones having the same v-s values are delimited by the disturbance lines as shown in Fig. 4a. Figures 4d, e and f are the stereograms of velocity, of strain and of displacement respectively which give full information
M a y , I939.]
FREE
VIBRATIONS
OF H E L I C A L
SPRINGS.
655
on the motion of the spring. From these it can be seen that the strain of the mid-point is constant ( = o) at all times, and its velocity alternates between the two values v2 and v4, the period of motion being T = 2L/a (cf, Appendix II). The velocity of the two ends is zero at all times and the strain alternates between the values sl (compression) and Sa (tension). The points intermediate between the mid-point and FIG. 4,
@J~i~plcltement
~
5' .~' ~ 7
S/raia
Unloaded spring fixed at both ends. (a) time-disturbance diagram; (b) velocity-strain diagram; (c) displacement vs. time; (d), (e), (f) stereograms of velocity, strain and displacement.
the ends execute an intermittent motion, the durations of positive and negative velocities being separated by dwells of zero velocity. It is interesting to compare Fig. 3 with Fig. 4. The velocity-stereogram Fig. 3d is identical with the strain stereogram Fig. 4e, and Fig. 3e with Fig. 4 d. The stereogram of displacement Fig. 3f is an equal configuration with Fig. 4f turned by 9o degrees, i.e., with the x- and t-axes interchanged.
656
KALMAN J. DEJUHASZ. n.
[J. F. I.
SPRINGS W I T H CONCENTRATED MASSES.
If a spring force s E F is applied to a mass M during a timeduration a t a velocity change Av is i m p a r t e d to the mass according to the law of m o m e n t u m : s E F A t = MAy,
whence s M I Av - E F A t ' choosing At = L/a, i.e., the duration of a single traversal of a disturbance in the spring, the following formula is obtained: s
Ma
A-? = E F Z
"
Substituting: E = a2p; L F p = m we obtain s MI . . . . . Av m a
M tan~, =--tana m
=-
cr a
as expressing the interrelationship of velocity c h a n g e undergone by the c o n c e n t r a t e d mass when acted upon by the spring force corresponding to the strain s during a single interval L/a. In the v-s d i a g r a m this velocity change can be represented by a straight line having the slope M / m times greater t h a n the As/Av directrix. This result "will be now applied to a spring having (d) One End Fixed, the Other End Attached to a Mass (Figs. 5 and 6).
It is assumed t h a t the spring is compressed to a strain Sl and k e p t at rest Vl = 0 as defined by point I in the v-s diagram, Fig. 5b. At the i n s t a n t I the spring is i n s t a n t a n e o u s l y released. As a result the velocity of the free-end, and of the mass a t t a c h e d to it, increases and the stress decreases along the I-2 directrix and this change spreads towards the fixed end with the velocity of propagation a. D u r i n g the first L / a period the velocity of the free end and of the mass will a t t a i n the value Vl as defined b y the intersection I' of the v axis with the t a n ~, line. D u r i n g the second L / a interval a f u r t h e r increase of velocity takes place according to the t a n ( -- 3,) line and at the end of the 2L/a interval the velocity will be v2 as
M a y , T939.]
FREE VIBRATIONS OF HELICAL ~PRINGS.
657
F r o . 5.
"~.
~
~ ~ ~....-h "-.~_ ..... ~-'~-~.......~....---~
,'~
..,~
,'7
~
Spring fixed at one end. loaded with concentrated mass on the other end. The time interval during which the strain is assumed to be constant is L/a, (a) time-disturbance diagram; (b) v-s diagram; (c) strain at fixed end vs. time; (d) strain and velocity at movable end vs. time; (e) schematic arrangement; (f) effect of periodic damping.
PQ"-L. /I
R=<,io.f
658
KALMAN J. DEJuHASZ.
[J. F. I.
defined b y the point 2 s i t u a t e d on the intersection of the tan a directrix d r a w n from the I point, with the tan ( - +) directrix d r a w n from the I t point. This construction is based on the a s s u m p t i o n t h a t during the first L/a interval the force corresponding to the c o n s t a n t st strain, during the second interval F I G . 6+
--k~,,C)3m~,>+,,i ~'~d f.d +'+'~++~'+'"~'+ '+ ' ',
.++<~,,'+'+ ;,%...~+ I
II
]
'+
P"-.
__.,,w~ '/'~'
'o,+
I
+ P
I
f0o
,,.
....
,
+.,: ' + ' "--~' ' ~" ' + '+,~']~ ' '~s ' "a.~+.+
Spring fixed at one end, loaded with concentrated m a s s on the other end. T h e time-interval d u r i n g which the strain is a s s u m e d to be constant is (I/3)(L/a). (a) schematic a r r a n g e m e n t ; (b) v - s d i a g r a m ; (c) a n d (d) strain a n d velocity history a t the two ends.
the force corresponding to the c o n s t a n t s2 strain acted upon the mass. This is n o t strictly correct, because the strain has been continuously changing during this 2L/a interval, b u t the a p p r o x i m a t i o n is close, because st is greater than the average strain, and s2 is less t h a n the average strain, and the t w o errors
May, '939.1 FREE VIBRATIONS
OF
HELICAL
SPRINGS.
659
fairly well cancel out (see Appendix III). In a similar manner the motion at the end of the second 2L/a interval will be defined by the intersection of the directrix tan a drawn from the 3' point. Continuing this construction the consecutive 3, 4, 5 "'" points are obtained, which, as it is evident from the v-s diagram, lie on a polygon resembling an ellipse. The complete stereograms could be easily drawn, but are omitted here, and only the s = f(t) line for the fixed end (Fig. 5c) and the s = f(t) and v = f(t) lines for the movable end (Fig. 5d) are shown. These show an approximately, but not exactly, harmonic interrelationship, the period of vibration being about I9.5 L/a, showing a good agreement with the value I9.3 L/a calculated from the formula, Appendix III. The deviation is due partly to our assumption of constant instead of the actually varying strains during each L/a interval, and partly to the fact t h a t the actual motion being not truly harmonic the " p e r i o d " is not strictly constant but varies slightly from cycle to cycle. The influence of friction and damping can be taken into consideration in a manner similar to t h a t treated in Figs. 2g-j. In Fig. 5e the v-s diagram is shown assuming the presence of periodic damping as represented by the tan p characteristic line passing through the point of origin, 0 < % It is seen t h a t the consecutive peak values of velocity and strain decrease in an apparently geometric progression, resulting in a "spiral polygon"-shaped v-s diagram. Aperiodic (0 > "Y) and critical (p = -~) damping can be treated in a similar manner. The assumption of constant strain was allowable in this case, because the M/m = 9 ratio was fairly high, and therefore the change of strain during an L/a interval not great. If the M/m ratio is small then this assumption is no longer permissible. In such cases a submultiple of L/a has to be used for the determination of the tan -y directrix. This more accurate construction is illustrated in Fig. 6 in which an M/m = 4 ratio is assumed, and the velocity change of the mass is determined for an (I/3) (L/a) duration; s IM tan Av - 3 m
c~.
660
KALMAN J. DEJUHASZ.
[J. F. I.
Using this directrix for each 2L/a period three points are obtained instead of the one point in the previous example. T h e construction is evident from the drawing. The strain for the fixed end, and the strain and velocity for the movable end, are shown in Figs. 6c and d on a time basis. The period of motion is approximately I3.I L/a which agrees with the I3.1 L/a value obtainable by calculation (Appendix III). It is seen t h a t the first side I-2 of the polygon, representing the change of velocity and strain during the first 2L/a interval is truly a straight line. The other sides of the p o l y g o n are straight lines only in the first approximation (Fig. 5b), become themselves broken lines in a closer approximation (Fig. 6b), and strictly speaking would become curved lines as the z~t interval (during which the strain producing the velocity change is assumed constant) is made to approach zero. The polygons obtained during the second, third and following circ u m a m b u l a t i o n s are not the same b u t differ from each other. It is reasonable to suspect t h a t ultimately the polygon would lose its corners and irregularities and develop into a true ellipse as the stationary condition is attained. (e) Both Ends Free and Attached to Masses (Fig. 7).
It is assumed t h a t the spring is compressed to an initial strain Sl (point ~) and at the instant I instantaneously released. The velocity change of M1 will take place according to the M1/m tan a = tan 7 directrix, t h a t of M2 according to the M2/m tan a = tan 7' directrix. The change of strain spreads from M1 to M2 and from M2 to M1, and can be represented in the v-p diagram by the points of intersection of the tan a and t a n 7 directrices on the one hand, and of the tan and tan 7' directrices on the other hand. T h e motion of the two masses is defined by the two ellipse-like polygons, the larger mass having the smaller m a x i m u m velocity and the smaller mass having the larger m a x i m u m velocity in inverse ratio to the respective masses. The strain and velocity histories are shown in Figs. 7c and d for both masses. T h e period of vibration is approximately I ~ L/a which agrees well with the value IO. 7 L/a obtainable by calculation, Appendix III. At the cost of more labor a more accurate d e t e r m i n a t i o n
M a y , ~939.]
FREE
VIBRATIONS
OF
HELICAL
SPRINGS.
66]
would be possible by using a submultiple of the L / a interval, e.g., tan ~,1 = _I M~ tan c~ 3m
and
tan ~/~ = -I -M~ - tan 3m
as was done in the previous example. Flo. 7.
-044,@. .st=,,>,ooj ~/,,,,~ ~,IM~
z
t. ~
j_
f'~
¢4b.,,.~z
z .."
~ ,,,
J I'.~r:--..
n#~,~=4 ~ ¢ ~ , : , .1 ~ "
.i~
t, /
I "~.
-~. ':.~Can/= # ~no~ :5'
/ \
\
,i?
~
', J-d"Z ,"i~
i~'~
,I \
\
\
I\\ " ~ . ' # ; ~ l r i =
\ ~
I :
s~-.\ '\ t= ,
I~%,-"'
i
!
\,~.~
•
Siroio and ~e/oezly
_~ : ~.\;
,~ ~ \
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iTt
, \ \i, , , '\> .z'~.~l &/ :", \,/ &/
•
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h
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t
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I,'.r'
\~ ~
ps.
.9
."
~.I
]-~ / / " .0"°'-"°"...
,I i ', s ff" ~ ~ . J >o ,., ,,~.,,
Spring loaded with finite concentrated masses at both ends. T h e time-interval daring which the strain is assumed to be constant is Lla. (a) schematic arrangement; (b) v-s diagram; (c) and (d) strain and velocity history at the t w o ends. (f) Both E n d s Fixed, a Finite M a s s Attached to the Mid-point (Fig. 8).
It is assumed that the two ends of the spring are fixed at the unstressed length L0, a mass M attached to the midpoint, and displaced upward, causing an initial compressional
662
KALMAN J. DEJU~ASZ.
[J. F. I.
strain s~ (point T) in the upper half of the spring, and an equal tensional strain s~' (point T') in the lower half of the spring. At the i n s t a n t ~ the mid-point t o g e t h e r with the mass M is instantaneously released• T h e mass will assume a d o w n w a r d FIG. 8. .s ~
,Ytr,Tm at/;aea L .....
~
• ~ - ~
•
"~
~ ~
.2,,
"q'
~'-[F'I"
I0'
[:,,, ,71..y<-:L. / I ~_/ L : o.l'l-~" I ~
' ~7-./ t AAq"7 ~
® I'W
II
~L~/t
/"-
7"~. t I
~ 't ~
Veloc//#" cmd 5/~itl
~ . - .4'o -""~.~" e ~ _s '. ~ ~
~"
t
"k
tl t -'-~Ir'
~.!%
S o Q
",'\
M)
.J" 8
..**~.,
"""0
Spring fixed at both ends, loaded with a concentrated mass at ,the mid-point. (a) schematic arrangement; (b) v-s diagram; (c) and (d) strain and velocity history at the two ends.
directed velocity, and the strain at the two sides of it will decrease, according to the 1-2 and I ' - 2 ' directrices respectively. T h e increase of the velocity is defined b y the directrix t a n 3, d r a w n from the point I and b r o u g h t into intersection with the dlrectrix tan a passing t h r o u g h the I t point. (Same result is obtained if the role of the I and I I points is interchanged.) T h e construction is clearly visible from the dia-
M ay, 1939.]
FREE
VIBRATIONS
OF
HELICAL
SPRINGS.
663
gram. T h e history of strain for the two ends, and the history of velocity for the mass is also shown in Figs. 8c and d. The period of vibration is a b o u t 15. 7 L/a intervals which agrees well with the calculated value T = 15.8 L/a (Appendix III). As a closing example, proving also the validity of this graphical method, the (f) Natural period of mass suspended on a massless spring will be derived on the basis of the graphical procedure. In Fig. 9 the graphical construction of two subsequent stages I FIG. 9.
Velocity-stress diagram for the calculation of motion of a finite mass on a massless spring.
and 2 is shown from which it can be w r i t t e n :
I 3' _ ( s t + s 2 ) a m zlv = (st + s2)tan and zXs = -
(Vl+V2) tanc~ = -
(Vl+V2) I-. a
Denoting the displacement of the spring end by u the following relationships hold: zXu = L~s; u -- Ls, Au &t ~u -
-
&v
~
-
Vl +
v2 L 2 M
ut +
u2 a z m
-
-
-
°
If, now, the mass m of the spring is m a d e to approach o, it is equivalent to the velocity of propagation a approaching infinity; Au/Av will approach du/dv; vt + v 2 ( = ) 2 v and VOL. 227, NO. 1361--46
664
KALMAN J. DEJuHAsz.
Ul "t-- U2 ( ~ " ) 21.t,, a n d
the differential
[J. F. I.
equation o f m o t i o n
can
be written : dv
u m
'
the solution of which is: U 2 --}- V2
- - Uo 2 = O. re\a~
This is the equation of an ellipse, showing t h a t the displacement and the velocity are harmonic functions of time, the frequency being: (.0 =
"M
2'
m
from which the natural period 2
ML
T-o-2rr~--Mma
= 2rr~J-~g.e.d.
The foregoing examples show t h a t the graphical method, is easy to apply and once its simple principles have been mastered it can be carried o u t almost mechanically, resembling in this respect the procedures of graphic statics. It facilitates forming a clear mental picture of the p h e n o m e n a and its results show a satisfactory agreement with the results of calculation. Its accuracy can be increased without difficulty, only at the cost of some additional labor. It is applicable to an extensive group of related phenomena, such as vibrations in liquid columns, gas columns and electric surges. APPENDIX I. I. Symbols and Formulas.
For Rod in Longitudinal
L = F = p = m = aN =
Stress.
length of rod, cross-sectional area of rod, density of rod material, mass of rod = L F p , modulus of longitudinal elasticity,
May, I939. ] FREE VIBRATIONS OF HELICAL SPRINGS.
c = spring
c o n s t a n t = force
necessary
to
66 5
change
the
FE
length by I =
L' a = v e l o c i t y of p r o p a g a t i o n of a d i s t u r b a n c e
L/a = t i m e - i n t e r v a l of single t r a v e r s a l of a d i s t u r b a n c e
For Helical Spring in Longitudinal Stress. L r d p n
= = = = =
l e n g t h of spring, m e a n r a d i u s of coil, wire d i a m e t e r , d e n s i t y of wire m a t e r i a l , n u m b e r of coils, d27r f - - - - c r o s s - s e c t i o n a l a r e a of wire, 4
d47r J = p o l a r m o m e n t of i n e r t i a of wire c r o s s - s e c t i o n = - - , 32 d27r m = m a s s of s p r i n g = 2rTrn - - p ( n e g l e c t i n g slope of helix), 4 G = m o d u l u s of t o r s i o n a l e l a s t i c i t y , c = s p r i n g c o n s t a n t = force n e c e s s a r y t o a l t e r t h e l e n g t h I
J
by I =--G-27r n/"3 ' a = v e l o c i t y of p r o p a g a t i o n of a d i s t u r b a n c e a l o n g t h e s p r i n g axis =
42 c
L,
L/a = i n t e r v a l of single t r a v e r s e of a d i s t u r b a n c e = ~ m . ~C
t = time, T = c o m p l e t e p e r i o d of v i b r a t i o n x = d i s t a n c e m e a s u r e d f r o m one e n d of r o d or spring,
666
KALMAN J. DEJUHASZ.
[J. F. I.
M = c o n c e n t r a t e d mass,
M / m = 0. = r a t i o of c o n c e n t r a t e d m a s s t o s p r i n g mass, u = displacement,
du v - dt - v e l o c i t y , s =-
du = strain, dt
E = stress.
APPENDIX H. Period of Vibration (Calculated).
A c c o r d i n g t o T h a m m ' s d e r i v a t i o n (Ref. 4) t h e p e r i o d of v i b r a t i o n T of t w o m a s s e s M , a n d M2 a t t a c h e d t o t h e e n d s of a s p r i n g h a v i n g t h e m a s s m a n d s p r i n g c o n s t a n t c c a n b e expressed by the equation: T = 2~
2~L
w h e r e t h e f a c t o r e is a f u n c t i o n of t h e r a t i o of s p r i n g m a s s t o the c o n c e n t r a t e d masses and can be determined from the trigonometric equation :
(
,
-
m2 )
M-7-M2
tan e =
m
+ -
~
m
M,
or t
I I ) t a n E = - -I+ - - . I E 0"10"2 0"1 0"2 -
-
T h e f a c t o r ~ h a s a n infinite n u m b e r of v a l u e s , of w h i c h t h e l o w e s t v a l u e c o r r e s p o n d i n g t o t h e g r e a t e s t v a l u e of T is t h e most important. T h e s e l o w e s t v a l u e s of e a n d t h e g r e a t e s t of T will be g i v e n for a n u m b e r of p a r t i c u l a r cases: (a) S p r i n g fixed a t o n e e n d , free a t t h e o t h e r e n d .
47
7r M2=o;e=2;T=4
m
(b) S p r i n g fixed a t b o t h ends. T=2
m
=2-.
L
a
=4
L a -
-
M1 = oo ;
°
M I = oo; M2 = oo; ~ = 7r;
667
May, 1939.1 F R E E \?IBRATIONS OF H E L I C A L SPRINGS.
(c) S p r i n g
free a t
T=2
both
=2-.
ends.
M1 = o; M2 = o;
e = ~r;
g
(d) S p r i n g m a s s negligible in c o m p a r i s o n t o t h e finite m a s s e s M1 a n d M2 a t t a c h e d t o t h e t w o ends. M~>> m ; M2>>
m;~
4(1 1)
=
m
~+M---~_
;
,/ MIM2 ~ M,M2 L T = 2Zr~c(MI_+_M2) = 2re m ( M l + M2) a (e) S p r i n g m a s s is negligible, o n e e n d is fixed. M2 = ~ ; ~ =
M1
'
T
=
271-
1
M1 >> m;
27i-
•
O n e e n d fixed.
M M1 = o o ; - - =
L 4;e = o.48;T = 13.1--.
(g) One e n d fixed.
M M1 = o ~ ; - - =
L 9;e = .325;T = 19.3--
M1 M2 (h) - - = 4; ....
L 9; e = .588; T = l O . 7 - -
(f)
/7¢
(i)
m
m
B o t h e n d s fixed.
a
a
M m
is e q u i v a l e n t t o M
-
12 m a s s in t h e m i d - p o i n t . 6; ~ = .398; T
This
I5.8 L
m
a APPENDIX HI.
ds Change of Velocity of a Mass Assuming a Linear dvv Relationship.
In t h e g r a p h i c a l c o n s t r u c t i o n of t h e c h a n g e of v e l o c i t y of t h e m a s s it h a s b e e n a s s u m e d t h a t t h e s t r a i n in t h e p o r t i o n of t h e s p r i n g a d j o i n i n g t h e m a s s is c o n s t a n t d u r i n g a n L/a t i m e - i n t e r v a l or d u r i n g a s u b m u l t i p l e ( I / n ) (L/a) t h e r e o f . 1. A s s u m i n g (Fig. IO) a t t h e i n s t a n t of r e l e a s i n g t h e m a s s vt = o, s -- Sl ( p o i n t I in t h e v-s d i a g r a m ) , a n d c o n s i d e r i n g d u r i n g t h e first L/a i n t e r v a l Sl c o n s t a n t a n d d u r i n g t h e s e c o n d
668
KALMAN
J.
DEJUHASZ.
[J. F. I.
L/a i n t e r v a l s3 c o n s t a n t , t h e v e l o c i t y v3 a t t h e e n d of t h e first 2L/a i n t e r v a l will b e : v3 = (sl + s 3 ) ~
-
t a n 3"
c o n s i d e r i n g t h a t sl -
(s, + s ~ ) - , a
s3 = v3 t a n a =
V3 = s t a
I
---To" I
vs/a, it c a n b e w r i t t e n : = slc~Zt.
F I G . IO.
,
®
I
:}
~',,'%~'-'-~
$
1
7 ¢,
g"
I!/',, i';/~o-;~ ItI;,L..':i at~ ij ilzfi4~.'t "iDetermination of velocity change with the value of strain assumed constant for (a) one L]a interval, (b) (~/2)(L/a) interval, and (c) (I/3)(L/a) interval.
2. A m o r e close a p p r o x i m a t i o n will b e a t t a i n e d b y a s s u m ing a v a l u e of s b e i n g c o n s t a n t o n l y f o r (I/2)(L/a) i n t e r v a l , f o r w h i c h t a n 7 ' = 2 t a n 3" = 2 ~ / a a n d AV .
S
Carrying out the summation intervals we obtain: V3 t
~
SlC~
[
I
--
sa
. . . . t a n 3" 2~ for 4 × ( I / 2 )
i)2]
"Jc I
(L/a) = 2L/a
= SlaZ2.
May, 1939.]
FREE VIBRATIONS OF HELICAL SPRINGS.
669
v = SlaZ.
M m
I
I
I v
m M
a--I
2¢--1
* ~.
.
.
.
/ . 3o'--I \ k 3~r. + I ]
Zl=i
-~--@.~ Z ~ = I - -
2
0
2
1.000
1.8I 8
o.556
I.I 56
o.9999
1.667
o.817
1.o16
0.9999
538
0.937
1,000
0.999
i 500
0.960
1.000
o.998
1.428
0.988
0.999
0.996
1.333
I.OOO
0.992
0.982
Z n = I - - £ -2]0"
[ 0.0
o.I
00
lO
0.2
0.3
3.333
I
0.333 o.4
2.5
o.5 o.6
1.667
1.25o
0.992
0.977
o.965
0.667
1.5
1.200
0.980
0.963
0.95 °
o.7
1.428
1.176
0.972
0.955
o.942
o.8
1.25
I.III
o.945
0.931
o.918
o.9
l.III
1.o53
o.918
0.903
o.891
I.O
1.000
1.000
0.89 °
o.875
0.865
o.5
0.667
0.640
0.635
0.632
o.333
o.5oo
0.490
o.488
0.487
0,25
0.400
0.395
0.394
0.393
0,2
0.333
0.33o
o.33o
o.33o
o.167
0.286
0.284
0.285
0.283
7
o.143
0.250
0-249
0.249
0.247
8
O. 1 2 5
0.222
0.222
0.220
0.220
9
O. I I I
0.200
0.200
O.2OO
0.200
O.I
0.I82
0.I80
0.180
0.180
I0
3. A still more close a p p r o x i m a t i o n will be attained b y assuming the value of s being c o n s t a n t only for (1/3) (L/a) interval, for which tan 7" = 3 tan 3' = 3¢/a, and 2xv = sa/3¢. Carrying o u t the s u m m a t i o n for 6 X (1/3) (L/a) = 2L/a intervals
[
I-
670
KALMAN J. DEJUHASZ.
[J. F. I.
4. T h e true value of v3 will be obtained if the time duration
(I In)(L/a) is m a d e to a p p r o a c h zero, i.e., n is m a d e to a p p r o a c h infinity, and the s u m m a t i o n of the velocity increments so o b t a i n e d is carried o u t for the 2n(I/n) (L/a) = 2L/a interval. T h e n it can be w r i t t e n :
[ v3 =
l i m Sla
(no.--I) I
--
n]
n o " --~ I
n=oo
( =
Sla
) I
--
e -21¢
=
SlaZn,
where e = 2.71828, the base of the n a t u r a l logarithm. (Note: T h e value of v~ can be also o b t a i n e d directly from the law of momentum :
FEdt = Mdv whence
dv FE dt M'
considering t h a t
V S = S l- ---; a
E . E.
.
o
.LFo
a 2 -m
LF
LF'
it can be w r i t t e n :
dv
a 0"( L) a ($1
as
v)
I n t e g r a t i n g we o b t a i n :
--f0 ,o nat(°s =v3)
fo ~3a s 1 dv- -
1)
\
whence
aS1
O.
• v~ =
asl(I
-
e-2/").)
T h e values of ZI, Z2, Z3.and Z~ are t a b u l a t e d in the adjoining table which shows t h a t for values of ~ > 2 the Z1 and Z2, and for values ~ > 0.2 Zs, differ less t h a n 2 per cent from the true Z n value.
BIBLIOGRAPHY. I. W. HoRT: Technische Schwingungslehre. Publ. J. Springer, Berlin. 2nd Edition. The pages 5o3-519 treat the surges in springs. Velocityof propagation; an example (according to J. Magg) is given and illustrated.
May, 1939.]
F R E E VIBRATIONS OF H E L I C A L SPRINGS.
671
2. W. WEIBULL: De dynamiska egenskaperna hos spiralfjadrar. (The Dynamic Properties of Helical Springs.) 25 p. 19 figs. Stockholm. I927 . Svenska Bokhandelcentralen A.-B. Formulas for the velocity of disturbance and for the strain in helical springs have been derived as functions of the dimension and of the material. Calculations have been made on the assumptions (I) that one end of the spring has an arbitrary velocity, and (2) that a mass impacts against a free end of a spring with finite velocity. These have been verified experimentally by means of photographing on a revolving film drum the motions of the spring coils in an ingenious and successful manner. 3. WILLY MARTI: Ventilfeder-Schwingungen. (Vibrations in Valve-Springs.) Dissertation T. H. Zurich, Switzerland. I935. 2I p. 33 figs. Publ. Buchdruckerei Winterthur vorm. G. 13inkert A.-G. An analytical and experimental investigation of spring surges; dynamic stresses; vibration of spring after a single lift-motion, as a function of speed ; influence of cam-curve; criterion for resonance; influence of damping, of initial spring-tension, of backlash in the roller. The forces at the two ends of the springs were determined by means of piezo-electric quartz-elements and an oscillograph. 4- I. THAMM: Free Longitudinal Vibrations of Helical Springs Loaded with Concentrated Masses. Jl. of the Hungarian Society of Engineers and Architects, 70, Nos. 31-36. August I6, I936. Pp. 239-244. 6 figs. An analytical treatment of the vibration of helical springs having uniformly distributed mass and concentrated masses, under the assumption of a massless spring and "equivalent" concentrated masses, 5. K. J. DEJUHASZ: Graphical Analysis of Hydraulic Phenomena in Fuel Injection Systems. Paper presented before the Society of Automotive Engineers, June 16-21, 1935. Mimeographed pamphlet, 28 p. 18 figs. Abstracted in S. A. E. Jl., August, I935. 6. K. J. DEJUHASZ: Graphical Analysis of Transient Phenomena in Linear Flow. Jl. Franklin Institute, 223, No. 4, Apr.; No. 5, May; and No. 6, June, I937. Published also in reprint form. 72 p. 36 figs. 7- K. J. DEJUnASZ: Hydraulic Phenomena in Fuel Injection Systems for Diesel Engines. Transactions of the Am. Soc. Mech. Engrs., November, I937. HYD-59-9. Pp. 669-677. 2I figs. 8. K. J. DEJUHASZ: Determination of the Rate of Discharge in Jerk-Pump FuelInjection Systems. Transactions of the Am. Soc. Mech. Engrs., February, I938. OGP-6o-2. Pp. I27-I36. 19 figs. 9 - K . J. DEJUttASZ: Graphical Analysis of Surges in Mechanical Springs. Jl. Franklin Institute, 226, No. 4, October, and 5, November, 1938. Pp. 505-526 and 631-644. 26 figs. The first four articles (5, 6, 7, 8) treat free and forced vibrations of an elastic system in general, with special reference to hydraulic systems; the fifth article deals with the vibration of a helical spring with and without attached mass, assuming one end fixed and to the other end a predetermined motion imparted. All these articles contain exhaustive bibliographies of this group of problems.
672
K A L M A N J. D E J U H A S Z .
[J. F. I.
IO. A. H. KRYLOW: Some Observations on Crushers and Indicators. Trans. of the Russian Imperial Akademy. I9O9. Pp. 623-654. Contains an analytical treatment of indicator spring vibrations assuming a spring with uniformly distributed and concentrated masses. II. O. HOLM: The Influence of Spring-Mass on the Dynamic Characteristics of Indicators. Zeitschrift der Instrumentenkunde, 48, January, 1928. Pp. 14-26. A comparison of the dynamic properties of a beam spring and a helical spring on the basis of their natural frequency of vibration. I2. M. F. SAYRE: Chapter on Springs, Kent's Mechanical Engineers' Handbook. I l t h edition, 3rd Volume, Design, pp. lO-2O. Treats the longitudinal and lateral vibrations of helical springs, and the lateral vibrations of flat and leaf springs carrying attached loads. The natural frequency is given by the formula: k Frequency = d.I2 "W~ q- C1Ws ' in which k = stiffness of spring, lb. per inch of deflection; Ws = weight of spring itself; Wz = weight of load carried by spring; C1 = constant, for helical springs in extension or compression C1 = -~. 13. L. BERGERON: M6thode Graphique Generale de Calcul des Propagation d'Ondes Planes. (General Graphical Method for the Calculation of the Propagation of Plane Waves.) Bulletin of the Socidtd des Ingenieurs Civils de France. July-Aug., 1937, 93 pages, 43 figs. A comprehensive treatment of waves in elastic media. Chapter III treats the motion of a heavy weight dropped on a cord under tension. Chapter IV treats the motion of a rod under tension, with and without concentrated masses attached to its two ends.
J.
DeJUHASZ,
Associate Professor of Engineering Research, The Pennsylvania State College. Member of The Franklin Institute.
Helical springs are used as measuring elements for variable forces in several kinds of engineering instruments such as engine indicators and dynamometers. Their vibration characteristics have an i m p o r t a n t influence on the behavior and accuracy of such instruments. Therefore the following problem is of theoretical and practical importance: Consider a helical spring, with or without concentrated masses attached to its two ends, initially at rest and put under an initial longitudinal strain by means of external constraining forces. Determine the law of motion if the constraining forces are instantaneously removed. The term "helical spring" includes also elastic rods. In a broader sense the above problem applies also to helical springs or elastic rods in torsion by substituting " m o m e n t of inertia" instead of " m a s s , " "tangential strain" instead of "longitudinal s t r a i n " and " c o u p l e s " instead of "forces." This problem has been treated by several authors (Krylow, Holm, T h a m m , Sayre, Ref. IO, I I, 4, I2) under the assumption t h a t the strain at all points of the spring is the same at any given instant of time, which assumption implies an infinite velocity of propagation. The mass of the spring was taken into consideration by either adding a certain calculated fraction of the spring mass to the concentrated mass or by multiplying with a calculated factor the period calculated for the concentrated masses only. Owing to these simplifying assumptions the analytical m e t h o d is not fully satisfying. It is inaccurate because it yields a purely harmonic motion, and it is incomplete because the law of motion for the intermediate points of the spring is not supplied. * R e c e i v e d J u l y , I938. VOL. 227, NO. 1361--45
647
648
KALMAN J. DEJUHASZ.
[J. F. I.
Actually the velocity of propagation of a disturbance in a spring is a finite q u a n t i t y and therefore the strain in a spring element changes n o t only from instant to instant, but also from point to point in the spring. As a result the motion will differ from the pure harmonic as the beautiful photographs of W. Weibull show (Fig. I). This fact has been taken into FIG.
I.
P h o t o g r a p h of free vibrations, as a function of time, of a n unloaded spring, b y W . Weibull.
consideration (Weibull, Ref. 2; Magg, Ref. I) b u t the calculations it involves are impracticably complicated and have been carried out only for some simple particular cases. The graphical m e t h o d used by the a u t h o r in several previous publications (Ref. 5-9) avoids these difficulties and is easy to perform. For a complete description of the phenomenon the (I) velocity v, (2) strain s, and (3) displacement u has to be given for all points x of the spring and for all instants of time. Expressed in analytical terms the following three functions v = fl(t, x) s = f~(t, x ) u = f3(t, x)
are sought. In graphical terms each of these functions can be represented by a tri-dimensional "stereogram," with the v, s and u values as ordinates erected over the t - x plane as base. The problem then resolves itself into finding the data from which these three stereograms can be constructed.
l~fay, I939. ]
FREE
VIBRATIONS
OF H E L I C A L
SPRINGS.
649
T h i s is possible with the aid of the following relationships: (a) T h e location of the d i s t u r b a n c e front can be represented in the t - x d i a g r a m by straight lines having the slope tan (=t= ~), i.e.: progress of d i s t u r b a n c e in a given time element
-
Ax - 4 - a = tan (=t= ~), At
a being the velocity of p r o p a g a t i o n of the d i s t u r b a n c e ; these lines will be referred to as " d i s t u r b a n c e lines." (b) T h e velocity-change is proportional to the corres p o n d i n g strain-change and can be represented in the v-s d i a g r a m by straight lines h a v i n g the slope tan (=~ a), i.e. : change of strain change of velocity
As Av
strain corresponding to stress E velocity of p r o p a g a t i o n I
= - = tan (4- oz), a
which will be referred to as "directrices." (c) T h e d i s p l a c e m e n t u can be d e t e r m i n e d from either the v or s s t e r e o g r a m with the aid of the following relationships: du dt
v
and
du dx
and
u=
Ls,
hence
u = fvdt
r f sdx,
i.e., the d i s p l a c e m e n t can be d e t e r m i n e d by integrating either the v or s stereograms. T h e derivations of (a) and (b) are given in Ref. 6 and 9 and can be o m i t t e d here. T h e m e t h o d will be n o w applied to a n u m b e r of assumed conditions h a v i n g special interest. I.
SPRING WITHOUT CONCENTRATED MASS. (a) One End Fixed, the Other End Free (Fig. 2).
It is a s s u m e d t h a t the spring with the unstressed length L0 is compressed to the length L1, involving an initial strain sl and initial velocity Vl = o, as characterized by point I in
650
KALMAN J. DEJuHASZ.
[J. F. I.
the v-s diagram. At the instant I the unattached end of the spring is instantaneously released, whereby the strain at t h a t end vanishes (s2 = o) and the velocity assumes the value of v2, the new state being defined by point 2 situated on the directrix drawn from point I in the v-s diagram, Fig. 2b. The change of state constitutes a disturbance which proceeds along the spring with the velocity of propagation a and after the elapse of L / a time arrives at the fixed end. Here the velocity is reduced to o and the strain correspondingly changes to s3 defined by point 3 situated on the directrix drawn from point 2 in the v-s diagram. The change from state 2 to state 3 returns towards the free end of the spring with the velocity of propagation and arrives there after the elapse of another L I e time. Here its strain is reduced to zero and its velocity assumes v4 as defined by point 4 in the v-s diagram. Continuing this construction the state of the spring according to points 5, 6, 7, 8 and so on can be determined, each state being valid for the triangular territories marked with the same numbers in the t - x diagram, Fig. 2a. The change of position of the coils, as a function of time, is illustrated on Fig. 2c. The slope of the heavy lines is proportional to the velocity; the relative change of distance between them, as measured on any t = const, ordinate, is proportional to the strain. T h e disturbance lines appear as the sloping lines connecting the two ends of the spring taken at L / a time interval apart. Erecting the v and s coordinates (in the v-s diagram) of the I, 2, 3, 4 " • " points over the t - x plane as base the velocityand strain-stereograms can be built, which are shown in parallel perspective in Figs. 2d and 2e. The displacements Ill
are obtained by forming the u = .~ vdt lines or the u = L J sdx lines over the t - x plane as base, as shown by the u-stereogram, Fig. 2f. These stereograms define the motion of the spring for every point in the spring and for every instant of time. F r o m these the following characteristics of the vibration can be read: (I) The velocity at the fixed end is o, and the strain at the free end is o a t all times.
May, I939.] FREE VIBRATIONS OF T-IELICAL SPRINGS.
651
F I G . 2.
(~
Velocil
~=a
(~P/6Plclcement~-/~x'x ~
~
.
,.~.
~ t,,3/r, .,
~ ,,,
.
........
Unloaded spring fixed at one end, free on the other end. (a) time-disturbance diagram; (b) velocity-strain diagram; (c) displacement vs. time; (d) velocity-stereogram; (e) strain-stereogram: (f) displacement-stereogram; (g) effect of Coulomb friction; (h) effect of periodic damping; (i) aperiodic damping; (j) critical damping.
II, -~_ . . . . . ~ . . . ~ - ~ -~,--.. ~ ,~,.<"
"'~.---~.-.~
~-.-., ._._~'__ ._ ~. ~'.~.'~'-.. - ~_~;~. -.... - -L~.-.~.~._
r-
652
KALMAN J. DEJr:uasz.
[J. F. I.
(2) The motion at the free end is periodic with a period T = 4L/a; it is however not harmonic, b u t has constant positive and negative velocities alternating after the elapse of each consecutive 2L/a interval (cf. Appendix II). (3) T h e motion of the intermediate points of the spring is intermittent, the durations of positive and negative velocities being separated by dwells of zero velocity.
Friction and Damping Friction (Coulomb type) is equivalent to a definite, constant value of strain R which m u s t be exceeded before motion, according to ZXv= (zXs - R) tan a, can result. In the graphical construction friction is represented by two lines in the v-s diagram, parallel to the v-axis drawn at s = -t- R and s = - R distance from it (Fig. 2g). It is seen t h a t the velocity and strain alternate between positive and negative values b u t the absolute magnitude of their consecutive values is not constant b u t decreases in an arithmetic progression. T h e motion stops altogether when the strain drops below the value of friction, i.e., s <
IRI.
Damping (or molecular friction) is equivalent to a strain R which is not constant b u t proportional to the velocity, i.e., R = v tan 0, where tan 0 is the proportionality factor. This value of strain m u s t be exceeded before motion can result, and only the excess of strain can be converted into velocity according to the equation 2~v = (As -- R) tan a = (zXs -- v tan 0) tan c~. In the graphical construction d a m p i n g is represented by a straight line passing through the point of origin of the v-s diagram forming an angle 0 therewith (Fig. 2h). It is seen t h a t the consecutive absolute values of velocity and of strain decrease in geometric progression. If 0 < a the d a m p i n g is periodic and the motion ceases only after the elapse of an infinite n u m b e r of L/a intervals, vibrating on both sides of its
May, I939. ] FREE VIBRATIONS OF I-IELICAL SPRINGS.
653
position of equilibrium with ever-decreasing amplitude. If o < ~- (Fig. 2i) the d a m p i n g is aperiodic and no overshooting the equilibrium position takes place. In the case of critical d a m p i n g p = a (Fig. 2j) the whole potential energy of the spring is consumed by the d a m p i n g in 2L/a period after which the spring comes to a standstill in its position of equilibrium. Other varieties of damping, e.g., in which the d a m p i n g force is n o t a linear function of the velocity, b u t t h a t of a higher power of it, or some other n o t necessarily algebraic function of the velocity, can be represented in the v-s diagram and the graphical construction performed in the m a n n e r shown. (b)
Both E n d s Free
(Fig. 3).
It is assumed t h a t the spring with the unstressed length L0 is compressed to L1 length, the initial state, Vl = o, s = sl FIG. 3.
1:~6" ©
#~#ire
95
' 8 ' ~"
+" :".? " ~ g -
"~...
li
f
Unloaded spring free on both ends. (a) time-disturbance diagram; (b) velocity-strain diagram; (c) displacement vs. time; (d), (e), (f) stereograms of velocity, strain and displacement.
654
KALMAN J. DEJUHASZ.
[J. F. I.
being defined by the point I in the v - s diagram, Fig. 3b. At the instant I the spring is instantaneously released. The strain at the ends vanishes and the upper end assumes a positive (upward) velocity, the lower end assumes a negative (downward) velocity, as defined by the points 2' and 2 in the v - s diagram. T h u s simultaneously two disturbances are initiated, one from each end of the spring, traveling in opposite directions. After the elapse of ( I / 2 ) ( L / a ) time these meet at the mid-point of the spring and the new state 3 = 3' results. The successive states 4, 6, 8 for the upper free end, 4 r, 6 p, 8' for the lower free end and 5 --- 5 ~, 7 ~ 7' " • " for the mid-point of the spring can be found from the v-s diagram (Fig. 3b), and the t - x zones for which the various states are valid can be found in the triangles and rhomboids formed by the disturbance lines in the t - x diagram (Fig. 3a). The stereograms of velocity, strain and displacement constructed from the d a t a thus obtained are shown in Figs. 3d, e, and f. It is seen t h a t the mid-point is stationary and assumes alternating sl and s3 stresses, while the two ends have zero stress and assume alternating v~. and v4 velocities, the period of motion being T -- 2 L / a (cf. Appendix II). (c) Both Ends Fixed (Fig.4).
It is assumed that the ends of the spring are fixed at a distance equal to the unstressed length L0 of the spring. The mid-point is displaced upward, causing in the upper half a compressional strain corresponding to point I in the v-s diagram, and in the lower half a tensional strain corresponding to point I'. At the instant I the mid-point is instantaneously released. The strain at the mid-point vanishes and it assumes a velocity corresponding to point 2 -- 2' in the v-s diagram which velocity spreads toward the two ends with the velocity of propagation. O n reaching, after the elapse of L/2a time, the two fixed ends the velocity is reduced to o and at the lower end a compressional strain according to point 3', and at the upper end a tensional strain according to point 3 results. The zones having the same v-s values are delimited by the disturbance lines as shown in Fig. 4a. Figures 4d, e and f are the stereograms of velocity, of strain and of displacement respectively which give full information
M a y , I939.]
FREE
VIBRATIONS
OF H E L I C A L
SPRINGS.
655
on the motion of the spring. From these it can be seen that the strain of the mid-point is constant ( = o) at all times, and its velocity alternates between the two values v2 and v4, the period of motion being T = 2L/a (cf, Appendix II). The velocity of the two ends is zero at all times and the strain alternates between the values sl (compression) and Sa (tension). The points intermediate between the mid-point and FIG. 4,
@J~i~plcltement
~
5' .~' ~ 7
S/raia
Unloaded spring fixed at both ends. (a) time-disturbance diagram; (b) velocity-strain diagram; (c) displacement vs. time; (d), (e), (f) stereograms of velocity, strain and displacement.
the ends execute an intermittent motion, the durations of positive and negative velocities being separated by dwells of zero velocity. It is interesting to compare Fig. 3 with Fig. 4. The velocity-stereogram Fig. 3d is identical with the strain stereogram Fig. 4e, and Fig. 3e with Fig. 4 d. The stereogram of displacement Fig. 3f is an equal configuration with Fig. 4f turned by 9o degrees, i.e., with the x- and t-axes interchanged.
656
KALMAN J. DEJUHASZ. n.
[J. F. I.
SPRINGS W I T H CONCENTRATED MASSES.
If a spring force s E F is applied to a mass M during a timeduration a t a velocity change Av is i m p a r t e d to the mass according to the law of m o m e n t u m : s E F A t = MAy,
whence s M I Av - E F A t ' choosing At = L/a, i.e., the duration of a single traversal of a disturbance in the spring, the following formula is obtained: s
Ma
A-? = E F Z
"
Substituting: E = a2p; L F p = m we obtain s MI . . . . . Av m a
M tan~, =--tana m
=-
cr a
as expressing the interrelationship of velocity c h a n g e undergone by the c o n c e n t r a t e d mass when acted upon by the spring force corresponding to the strain s during a single interval L/a. In the v-s d i a g r a m this velocity change can be represented by a straight line having the slope M / m times greater t h a n the As/Av directrix. This result "will be now applied to a spring having (d) One End Fixed, the Other End Attached to a Mass (Figs. 5 and 6).
It is assumed t h a t the spring is compressed to a strain Sl and k e p t at rest Vl = 0 as defined by point I in the v-s diagram, Fig. 5b. At the i n s t a n t I the spring is i n s t a n t a n e o u s l y released. As a result the velocity of the free-end, and of the mass a t t a c h e d to it, increases and the stress decreases along the I-2 directrix and this change spreads towards the fixed end with the velocity of propagation a. D u r i n g the first L / a period the velocity of the free end and of the mass will a t t a i n the value Vl as defined b y the intersection I' of the v axis with the t a n ~, line. D u r i n g the second L / a interval a f u r t h e r increase of velocity takes place according to the t a n ( -- 3,) line and at the end of the 2L/a interval the velocity will be v2 as
M a y , T939.]
FREE VIBRATIONS OF HELICAL ~PRINGS.
657
F r o . 5.
"~.
~
~ ~ ~....-h "-.~_ ..... ~-'~-~.......~....---~
,'~
..,~
,'7
~
Spring fixed at one end. loaded with concentrated mass on the other end. The time interval during which the strain is assumed to be constant is L/a, (a) time-disturbance diagram; (b) v-s diagram; (c) strain at fixed end vs. time; (d) strain and velocity at movable end vs. time; (e) schematic arrangement; (f) effect of periodic damping.
PQ"-L. /I
R=<,io.f
658
KALMAN J. DEJuHASZ.
[J. F. I.
defined b y the point 2 s i t u a t e d on the intersection of the tan a directrix d r a w n from the I point, with the tan ( - +) directrix d r a w n from the I t point. This construction is based on the a s s u m p t i o n t h a t during the first L/a interval the force corresponding to the c o n s t a n t st strain, during the second interval F I G . 6+
--k~,,C)3m~,>+,,i ~'~d f.d +'+'~++~'+'"~'+ '+ ' ',
.++<~,,'+'+ ;,%...~+ I
II
]
'+
P"-.
__.,,w~ '/'~'
'o,+
I
+ P
I
f0o
,,.
....
,
+.,: ' + ' "--~' ' ~" ' + '+,~']~ ' '~s ' "a.~+.+
Spring fixed at one end, loaded with concentrated m a s s on the other end. T h e time-interval d u r i n g which the strain is a s s u m e d to be constant is (I/3)(L/a). (a) schematic a r r a n g e m e n t ; (b) v - s d i a g r a m ; (c) a n d (d) strain a n d velocity history a t the two ends.
the force corresponding to the c o n s t a n t s2 strain acted upon the mass. This is n o t strictly correct, because the strain has been continuously changing during this 2L/a interval, b u t the a p p r o x i m a t i o n is close, because st is greater than the average strain, and s2 is less t h a n the average strain, and the t w o errors
May, '939.1 FREE VIBRATIONS
OF
HELICAL
SPRINGS.
659
fairly well cancel out (see Appendix III). In a similar manner the motion at the end of the second 2L/a interval will be defined by the intersection of the directrix tan a drawn from the 3' point. Continuing this construction the consecutive 3, 4, 5 "'" points are obtained, which, as it is evident from the v-s diagram, lie on a polygon resembling an ellipse. The complete stereograms could be easily drawn, but are omitted here, and only the s = f(t) line for the fixed end (Fig. 5c) and the s = f(t) and v = f(t) lines for the movable end (Fig. 5d) are shown. These show an approximately, but not exactly, harmonic interrelationship, the period of vibration being about I9.5 L/a, showing a good agreement with the value I9.3 L/a calculated from the formula, Appendix III. The deviation is due partly to our assumption of constant instead of the actually varying strains during each L/a interval, and partly to the fact t h a t the actual motion being not truly harmonic the " p e r i o d " is not strictly constant but varies slightly from cycle to cycle. The influence of friction and damping can be taken into consideration in a manner similar to t h a t treated in Figs. 2g-j. In Fig. 5e the v-s diagram is shown assuming the presence of periodic damping as represented by the tan p characteristic line passing through the point of origin, 0 < % It is seen t h a t the consecutive peak values of velocity and strain decrease in an apparently geometric progression, resulting in a "spiral polygon"-shaped v-s diagram. Aperiodic (0 > "Y) and critical (p = -~) damping can be treated in a similar manner. The assumption of constant strain was allowable in this case, because the M/m = 9 ratio was fairly high, and therefore the change of strain during an L/a interval not great. If the M/m ratio is small then this assumption is no longer permissible. In such cases a submultiple of L/a has to be used for the determination of the tan -y directrix. This more accurate construction is illustrated in Fig. 6 in which an M/m = 4 ratio is assumed, and the velocity change of the mass is determined for an (I/3) (L/a) duration; s IM tan Av - 3 m
c~.
660
KALMAN J. DEJUHASZ.
[J. F. I.
Using this directrix for each 2L/a period three points are obtained instead of the one point in the previous example. T h e construction is evident from the drawing. The strain for the fixed end, and the strain and velocity for the movable end, are shown in Figs. 6c and d on a time basis. The period of motion is approximately I3.I L/a which agrees with the I3.1 L/a value obtainable by calculation (Appendix III). It is seen t h a t the first side I-2 of the polygon, representing the change of velocity and strain during the first 2L/a interval is truly a straight line. The other sides of the p o l y g o n are straight lines only in the first approximation (Fig. 5b), become themselves broken lines in a closer approximation (Fig. 6b), and strictly speaking would become curved lines as the z~t interval (during which the strain producing the velocity change is assumed constant) is made to approach zero. The polygons obtained during the second, third and following circ u m a m b u l a t i o n s are not the same b u t differ from each other. It is reasonable to suspect t h a t ultimately the polygon would lose its corners and irregularities and develop into a true ellipse as the stationary condition is attained. (e) Both Ends Free and Attached to Masses (Fig. 7).
It is assumed t h a t the spring is compressed to an initial strain Sl (point ~) and at the instant I instantaneously released. The velocity change of M1 will take place according to the M1/m tan a = tan 7 directrix, t h a t of M2 according to the M2/m tan a = tan 7' directrix. The change of strain spreads from M1 to M2 and from M2 to M1, and can be represented in the v-p diagram by the points of intersection of the tan a and t a n 7 directrices on the one hand, and of the tan and tan 7' directrices on the other hand. T h e motion of the two masses is defined by the two ellipse-like polygons, the larger mass having the smaller m a x i m u m velocity and the smaller mass having the larger m a x i m u m velocity in inverse ratio to the respective masses. The strain and velocity histories are shown in Figs. 7c and d for both masses. T h e period of vibration is approximately I ~ L/a which agrees well with the value IO. 7 L/a obtainable by calculation, Appendix III. At the cost of more labor a more accurate d e t e r m i n a t i o n
M a y , ~939.]
FREE
VIBRATIONS
OF
HELICAL
SPRINGS.
66]
would be possible by using a submultiple of the L / a interval, e.g., tan ~,1 = _I M~ tan c~ 3m
and
tan ~/~ = -I -M~ - tan 3m
as was done in the previous example. Flo. 7.
-044,@. .st=,,>,ooj ~/,,,,~ ~,IM~
z
t. ~
j_
f'~
¢4b.,,.~z
z .."
~ ,,,
J I'.~r:--..
n#~,~=4 ~ ¢ ~ , : , .1 ~ "
.i~
t, /
I "~.
-~. ':.~Can/= # ~no~ :5'
/ \
\
,i?
~
', J-d"Z ,"i~
i~'~
,I \
\
\
I\\ " ~ . ' # ; ~ l r i =
\ ~
I :
s~-.\ '\ t= ,
I~%,-"'
i
!
\,~.~
•
Siroio and ~e/oezly
_~ : ~.\;
,~ ~ \
S,o
iTt
, \ \i, , , '\> .z'~.~l &/ :", \,/ &/
•
I I
h
\
t
~' /
i i
I,'.r'
\~ ~
ps.
.9
."
~.I
]-~ / / " .0"°'-"°"...
,I i ', s ff" ~ ~ . J >o ,., ,,~.,,
Spring loaded with finite concentrated masses at both ends. T h e time-interval daring which the strain is assumed to be constant is Lla. (a) schematic arrangement; (b) v-s diagram; (c) and (d) strain and velocity history at the t w o ends. (f) Both E n d s Fixed, a Finite M a s s Attached to the Mid-point (Fig. 8).
It is assumed that the two ends of the spring are fixed at the unstressed length L0, a mass M attached to the midpoint, and displaced upward, causing an initial compressional
662
KALMAN J. DEJU~ASZ.
[J. F. I.
strain s~ (point T) in the upper half of the spring, and an equal tensional strain s~' (point T') in the lower half of the spring. At the i n s t a n t ~ the mid-point t o g e t h e r with the mass M is instantaneously released• T h e mass will assume a d o w n w a r d FIG. 8. .s ~
,Ytr,Tm at/;aea L .....
~
• ~ - ~
•
"~
~ ~
.2,,
"q'
~'-[F'I"
I0'
[:,,, ,71..y<-:L. / I ~_/ L : o.l'l-~" I ~
' ~7-./ t AAq"7 ~
® I'W
II
~L~/t
/"-
7"~. t I
~ 't ~
Veloc//#" cmd 5/~itl
~ . - .4'o -""~.~" e ~ _s '. ~ ~
~"
t
"k
tl t -'-~Ir'
~.!%
S o Q
",'\
M)
.J" 8
..**~.,
"""0
Spring fixed at both ends, loaded with a concentrated mass at ,the mid-point. (a) schematic arrangement; (b) v-s diagram; (c) and (d) strain and velocity history at the two ends.
directed velocity, and the strain at the two sides of it will decrease, according to the 1-2 and I ' - 2 ' directrices respectively. T h e increase of the velocity is defined b y the directrix t a n 3, d r a w n from the point I and b r o u g h t into intersection with the dlrectrix tan a passing t h r o u g h the I t point. (Same result is obtained if the role of the I and I I points is interchanged.) T h e construction is clearly visible from the dia-
M ay, 1939.]
FREE
VIBRATIONS
OF
HELICAL
SPRINGS.
663
gram. T h e history of strain for the two ends, and the history of velocity for the mass is also shown in Figs. 8c and d. The period of vibration is a b o u t 15. 7 L/a intervals which agrees well with the calculated value T = 15.8 L/a (Appendix III). As a closing example, proving also the validity of this graphical method, the (f) Natural period of mass suspended on a massless spring will be derived on the basis of the graphical procedure. In Fig. 9 the graphical construction of two subsequent stages I FIG. 9.
Velocity-stress diagram for the calculation of motion of a finite mass on a massless spring.
and 2 is shown from which it can be w r i t t e n :
I 3' _ ( s t + s 2 ) a m zlv = (st + s2)tan and zXs = -
(Vl+V2) tanc~ = -
(Vl+V2) I-. a
Denoting the displacement of the spring end by u the following relationships hold: zXu = L~s; u -- Ls, Au &t ~u -
-
&v
~
-
Vl +
v2 L 2 M
ut +
u2 a z m
-
-
-
°
If, now, the mass m of the spring is m a d e to approach o, it is equivalent to the velocity of propagation a approaching infinity; Au/Av will approach du/dv; vt + v 2 ( = ) 2 v and VOL. 227, NO. 1361--46
664
KALMAN J. DEJuHAsz.
Ul "t-- U2 ( ~ " ) 21.t,, a n d
the differential
[J. F. I.
equation o f m o t i o n
can
be written : dv
u m
'
the solution of which is: U 2 --}- V2
- - Uo 2 = O. re\a~
This is the equation of an ellipse, showing t h a t the displacement and the velocity are harmonic functions of time, the frequency being: (.0 =
"M
2'
m
from which the natural period 2
ML
T-o-2rr~--Mma
= 2rr~J-~g.e.d.
The foregoing examples show t h a t the graphical method, is easy to apply and once its simple principles have been mastered it can be carried o u t almost mechanically, resembling in this respect the procedures of graphic statics. It facilitates forming a clear mental picture of the p h e n o m e n a and its results show a satisfactory agreement with the results of calculation. Its accuracy can be increased without difficulty, only at the cost of some additional labor. It is applicable to an extensive group of related phenomena, such as vibrations in liquid columns, gas columns and electric surges. APPENDIX I. I. Symbols and Formulas.
For Rod in Longitudinal
L = F = p = m = aN =
Stress.
length of rod, cross-sectional area of rod, density of rod material, mass of rod = L F p , modulus of longitudinal elasticity,
May, I939. ] FREE VIBRATIONS OF HELICAL SPRINGS.
c = spring
c o n s t a n t = force
necessary
to
66 5
change
the
FE
length by I =
L' a = v e l o c i t y of p r o p a g a t i o n of a d i s t u r b a n c e
L/a = t i m e - i n t e r v a l of single t r a v e r s a l of a d i s t u r b a n c e
For Helical Spring in Longitudinal Stress. L r d p n
= = = = =
l e n g t h of spring, m e a n r a d i u s of coil, wire d i a m e t e r , d e n s i t y of wire m a t e r i a l , n u m b e r of coils, d27r f - - - - c r o s s - s e c t i o n a l a r e a of wire, 4
d47r J = p o l a r m o m e n t of i n e r t i a of wire c r o s s - s e c t i o n = - - , 32 d27r m = m a s s of s p r i n g = 2rTrn - - p ( n e g l e c t i n g slope of helix), 4 G = m o d u l u s of t o r s i o n a l e l a s t i c i t y , c = s p r i n g c o n s t a n t = force n e c e s s a r y t o a l t e r t h e l e n g t h I
J
by I =--G-27r n/"3 ' a = v e l o c i t y of p r o p a g a t i o n of a d i s t u r b a n c e a l o n g t h e s p r i n g axis =
42 c
L,
L/a = i n t e r v a l of single t r a v e r s e of a d i s t u r b a n c e = ~ m . ~C
t = time, T = c o m p l e t e p e r i o d of v i b r a t i o n x = d i s t a n c e m e a s u r e d f r o m one e n d of r o d or spring,
666
KALMAN J. DEJUHASZ.
[J. F. I.
M = c o n c e n t r a t e d mass,
M / m = 0. = r a t i o of c o n c e n t r a t e d m a s s t o s p r i n g mass, u = displacement,
du v - dt - v e l o c i t y , s =-
du = strain, dt
E = stress.
APPENDIX H. Period of Vibration (Calculated).
A c c o r d i n g t o T h a m m ' s d e r i v a t i o n (Ref. 4) t h e p e r i o d of v i b r a t i o n T of t w o m a s s e s M , a n d M2 a t t a c h e d t o t h e e n d s of a s p r i n g h a v i n g t h e m a s s m a n d s p r i n g c o n s t a n t c c a n b e expressed by the equation: T = 2~
2~L
w h e r e t h e f a c t o r e is a f u n c t i o n of t h e r a t i o of s p r i n g m a s s t o the c o n c e n t r a t e d masses and can be determined from the trigonometric equation :
(
,
-
m2 )
M-7-M2
tan e =
m
+ -
~
m
M,
or t
I I ) t a n E = - -I+ - - . I E 0"10"2 0"1 0"2 -
-
T h e f a c t o r ~ h a s a n infinite n u m b e r of v a l u e s , of w h i c h t h e l o w e s t v a l u e c o r r e s p o n d i n g t o t h e g r e a t e s t v a l u e of T is t h e most important. T h e s e l o w e s t v a l u e s of e a n d t h e g r e a t e s t of T will be g i v e n for a n u m b e r of p a r t i c u l a r cases: (a) S p r i n g fixed a t o n e e n d , free a t t h e o t h e r e n d .
47
7r M2=o;e=2;T=4
m
(b) S p r i n g fixed a t b o t h ends. T=2
m
=2-.
L
a
=4
L a -
-
M1 = oo ;
°
M I = oo; M2 = oo; ~ = 7r;
667
May, 1939.1 F R E E \?IBRATIONS OF H E L I C A L SPRINGS.
(c) S p r i n g
free a t
T=2
both
=2-.
ends.
M1 = o; M2 = o;
e = ~r;
g
(d) S p r i n g m a s s negligible in c o m p a r i s o n t o t h e finite m a s s e s M1 a n d M2 a t t a c h e d t o t h e t w o ends. M~>> m ; M2>>
m;~
4(1 1)
=
m
~+M---~_
;
,/ MIM2 ~ M,M2 L T = 2Zr~c(MI_+_M2) = 2re m ( M l + M2) a (e) S p r i n g m a s s is negligible, o n e e n d is fixed. M2 = ~ ; ~ =
M1
'
T
=
271-
1
M1 >> m;
27i-
•
O n e e n d fixed.
M M1 = o o ; - - =
L 4;e = o.48;T = 13.1--.
(g) One e n d fixed.
M M1 = o ~ ; - - =
L 9;e = .325;T = 19.3--
M1 M2 (h) - - = 4; ....
L 9; e = .588; T = l O . 7 - -
(f)
/7¢
(i)
m
m
B o t h e n d s fixed.
a
a
M m
is e q u i v a l e n t t o M
-
12 m a s s in t h e m i d - p o i n t . 6; ~ = .398; T
This
I5.8 L
m
a APPENDIX HI.
ds Change of Velocity of a Mass Assuming a Linear dvv Relationship.
In t h e g r a p h i c a l c o n s t r u c t i o n of t h e c h a n g e of v e l o c i t y of t h e m a s s it h a s b e e n a s s u m e d t h a t t h e s t r a i n in t h e p o r t i o n of t h e s p r i n g a d j o i n i n g t h e m a s s is c o n s t a n t d u r i n g a n L/a t i m e - i n t e r v a l or d u r i n g a s u b m u l t i p l e ( I / n ) (L/a) t h e r e o f . 1. A s s u m i n g (Fig. IO) a t t h e i n s t a n t of r e l e a s i n g t h e m a s s vt = o, s -- Sl ( p o i n t I in t h e v-s d i a g r a m ) , a n d c o n s i d e r i n g d u r i n g t h e first L/a i n t e r v a l Sl c o n s t a n t a n d d u r i n g t h e s e c o n d
668
KALMAN
J.
DEJUHASZ.
[J. F. I.
L/a i n t e r v a l s3 c o n s t a n t , t h e v e l o c i t y v3 a t t h e e n d of t h e first 2L/a i n t e r v a l will b e : v3 = (sl + s 3 ) ~
-
t a n 3"
c o n s i d e r i n g t h a t sl -
(s, + s ~ ) - , a
s3 = v3 t a n a =
V3 = s t a
I
---To" I
vs/a, it c a n b e w r i t t e n : = slc~Zt.
F I G . IO.
,
®
I
:}
~',,'%~'-'-~
$
1
7 ¢,
g"
I!/',, i';/~o-;~ ItI;,L..':i at~ ij ilzfi4~.'t "iDetermination of velocity change with the value of strain assumed constant for (a) one L]a interval, (b) (~/2)(L/a) interval, and (c) (I/3)(L/a) interval.
2. A m o r e close a p p r o x i m a t i o n will b e a t t a i n e d b y a s s u m ing a v a l u e of s b e i n g c o n s t a n t o n l y f o r (I/2)(L/a) i n t e r v a l , f o r w h i c h t a n 7 ' = 2 t a n 3" = 2 ~ / a a n d AV .
S
Carrying out the summation intervals we obtain: V3 t
~
SlC~
[
I
--
sa
. . . . t a n 3" 2~ for 4 × ( I / 2 )
i)2]
"Jc I
(L/a) = 2L/a
= SlaZ2.
May, 1939.]
FREE VIBRATIONS OF HELICAL SPRINGS.
669
v = SlaZ.
M m
I
I
I v
m M
a--I
2¢--1
* ~.
.
.
.
/ . 3o'--I \ k 3~r. + I ]
Zl=i
-~--@.~ Z ~ = I - -
2
0
2
1.000
1.8I 8
o.556
I.I 56
o.9999
1.667
o.817
1.o16
0.9999
538
0.937
1,000
0.999
i 500
0.960
1.000
o.998
1.428
0.988
0.999
0.996
1.333
I.OOO
0.992
0.982
Z n = I - - £ -2]0"
[ 0.0
o.I
00
lO
0.2
0.3
3.333
I
0.333 o.4
2.5
o.5 o.6
1.667
1.25o
0.992
0.977
o.965
0.667
1.5
1.200
0.980
0.963
0.95 °
o.7
1.428
1.176
0.972
0.955
o.942
o.8
1.25
I.III
o.945
0.931
o.918
o.9
l.III
1.o53
o.918
0.903
o.891
I.O
1.000
1.000
0.89 °
o.875
0.865
o.5
0.667
0.640
0.635
0.632
o.333
o.5oo
0.490
o.488
0.487
0,25
0.400
0.395
0.394
0.393
0,2
0.333
0.33o
o.33o
o.33o
o.167
0.286
0.284
0.285
0.283
7
o.143
0.250
0-249
0.249
0.247
8
O. 1 2 5
0.222
0.222
0.220
0.220
9
O. I I I
0.200
0.200
O.2OO
0.200
O.I
0.I82
0.I80
0.180
0.180
I0
3. A still more close a p p r o x i m a t i o n will be attained b y assuming the value of s being c o n s t a n t only for (1/3) (L/a) interval, for which tan 7" = 3 tan 3' = 3¢/a, and 2xv = sa/3¢. Carrying o u t the s u m m a t i o n for 6 X (1/3) (L/a) = 2L/a intervals
[
I-
670
KALMAN J. DEJUHASZ.
[J. F. I.
4. T h e true value of v3 will be obtained if the time duration
(I In)(L/a) is m a d e to a p p r o a c h zero, i.e., n is m a d e to a p p r o a c h infinity, and the s u m m a t i o n of the velocity increments so o b t a i n e d is carried o u t for the 2n(I/n) (L/a) = 2L/a interval. T h e n it can be w r i t t e n :
[ v3 =
l i m Sla
(no.--I) I
--
n]
n o " --~ I
n=oo
( =
Sla
) I
--
e -21¢
=
SlaZn,
where e = 2.71828, the base of the n a t u r a l logarithm. (Note: T h e value of v~ can be also o b t a i n e d directly from the law of momentum :
FEdt = Mdv whence
dv FE dt M'
considering t h a t
V S = S l- ---; a
E . E.
.
o
.LFo
a 2 -m
LF
LF'
it can be w r i t t e n :
dv
a 0"( L) a ($1
as
v)
I n t e g r a t i n g we o b t a i n :
--f0 ,o nat(°s =v3)
fo ~3a s 1 dv- -
1)
\
whence
aS1
O.
• v~ =
asl(I
-
e-2/").)
T h e values of ZI, Z2, Z3.and Z~ are t a b u l a t e d in the adjoining table which shows t h a t for values of ~ > 2 the Z1 and Z2, and for values ~ > 0.2 Zs, differ less t h a n 2 per cent from the true Z n value.
BIBLIOGRAPHY. I. W. HoRT: Technische Schwingungslehre. Publ. J. Springer, Berlin. 2nd Edition. The pages 5o3-519 treat the surges in springs. Velocityof propagation; an example (according to J. Magg) is given and illustrated.
May, 1939.]
F R E E VIBRATIONS OF H E L I C A L SPRINGS.
671
2. W. WEIBULL: De dynamiska egenskaperna hos spiralfjadrar. (The Dynamic Properties of Helical Springs.) 25 p. 19 figs. Stockholm. I927 . Svenska Bokhandelcentralen A.-B. Formulas for the velocity of disturbance and for the strain in helical springs have been derived as functions of the dimension and of the material. Calculations have been made on the assumptions (I) that one end of the spring has an arbitrary velocity, and (2) that a mass impacts against a free end of a spring with finite velocity. These have been verified experimentally by means of photographing on a revolving film drum the motions of the spring coils in an ingenious and successful manner. 3. WILLY MARTI: Ventilfeder-Schwingungen. (Vibrations in Valve-Springs.) Dissertation T. H. Zurich, Switzerland. I935. 2I p. 33 figs. Publ. Buchdruckerei Winterthur vorm. G. 13inkert A.-G. An analytical and experimental investigation of spring surges; dynamic stresses; vibration of spring after a single lift-motion, as a function of speed ; influence of cam-curve; criterion for resonance; influence of damping, of initial spring-tension, of backlash in the roller. The forces at the two ends of the springs were determined by means of piezo-electric quartz-elements and an oscillograph. 4- I. THAMM: Free Longitudinal Vibrations of Helical Springs Loaded with Concentrated Masses. Jl. of the Hungarian Society of Engineers and Architects, 70, Nos. 31-36. August I6, I936. Pp. 239-244. 6 figs. An analytical treatment of the vibration of helical springs having uniformly distributed mass and concentrated masses, under the assumption of a massless spring and "equivalent" concentrated masses, 5. K. J. DEJUHASZ: Graphical Analysis of Hydraulic Phenomena in Fuel Injection Systems. Paper presented before the Society of Automotive Engineers, June 16-21, 1935. Mimeographed pamphlet, 28 p. 18 figs. Abstracted in S. A. E. Jl., August, I935. 6. K. J. DEJUHASZ: Graphical Analysis of Transient Phenomena in Linear Flow. Jl. Franklin Institute, 223, No. 4, Apr.; No. 5, May; and No. 6, June, I937. Published also in reprint form. 72 p. 36 figs. 7- K. J. DEJUnASZ: Hydraulic Phenomena in Fuel Injection Systems for Diesel Engines. Transactions of the Am. Soc. Mech. Engrs., November, I937. HYD-59-9. Pp. 669-677. 2I figs. 8. K. J. DEJUHASZ: Determination of the Rate of Discharge in Jerk-Pump FuelInjection Systems. Transactions of the Am. Soc. Mech. Engrs., February, I938. OGP-6o-2. Pp. I27-I36. 19 figs. 9 - K . J. DEJUttASZ: Graphical Analysis of Surges in Mechanical Springs. Jl. Franklin Institute, 226, No. 4, October, and 5, November, 1938. Pp. 505-526 and 631-644. 26 figs. The first four articles (5, 6, 7, 8) treat free and forced vibrations of an elastic system in general, with special reference to hydraulic systems; the fifth article deals with the vibration of a helical spring with and without attached mass, assuming one end fixed and to the other end a predetermined motion imparted. All these articles contain exhaustive bibliographies of this group of problems.
672
K A L M A N J. D E J U H A S Z .
[J. F. I.
IO. A. H. KRYLOW: Some Observations on Crushers and Indicators. Trans. of the Russian Imperial Akademy. I9O9. Pp. 623-654. Contains an analytical treatment of indicator spring vibrations assuming a spring with uniformly distributed and concentrated masses. II. O. HOLM: The Influence of Spring-Mass on the Dynamic Characteristics of Indicators. Zeitschrift der Instrumentenkunde, 48, January, 1928. Pp. 14-26. A comparison of the dynamic properties of a beam spring and a helical spring on the basis of their natural frequency of vibration. I2. M. F. SAYRE: Chapter on Springs, Kent's Mechanical Engineers' Handbook. I l t h edition, 3rd Volume, Design, pp. lO-2O. Treats the longitudinal and lateral vibrations of helical springs, and the lateral vibrations of flat and leaf springs carrying attached loads. The natural frequency is given by the formula: k Frequency = d.I2 "W~ q- C1Ws ' in which k = stiffness of spring, lb. per inch of deflection; Ws = weight of spring itself; Wz = weight of load carried by spring; C1 = constant, for helical springs in extension or compression C1 = -~. 13. L. BERGERON: M6thode Graphique Generale de Calcul des Propagation d'Ondes Planes. (General Graphical Method for the Calculation of the Propagation of Plane Waves.) Bulletin of the Socidtd des Ingenieurs Civils de France. July-Aug., 1937, 93 pages, 43 figs. A comprehensive treatment of waves in elastic media. Chapter III treats the motion of a heavy weight dropped on a cord under tension. Chapter IV treats the motion of a rod under tension, with and without concentrated masses attached to its two ends.