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Dynamic responses of the flexible connecting rod of a slider-crank mechanism with time-dependent boundary effect
PII: !No45-7!M!y~333-I
Compuwrsct Sm~twcs Vol. 63, No. 1. pp. 79-90. 1997 Q 1997 F’ublisbcdby Ebevk Scimcs Ltd Printedin Grcllt Btiuin. AU ri&b wmmed 004s7949p 517.00 + 0.00
DYNAMIC RESPONSES OF THE FLEXIBLE CONNECTING ROD OF A SLIDER-CRANK MECHANISM WITH TIME-DEPENDENT BOUNDARY EFFECT Department of Mechanical Engineering, Chung Yuang Christian University, Chung-Li, Taiwan 32023, Republic of China (Received 5 October 1995)
timedependent boundary effect of the end, which moves reciprocally with slider along the horizontal guide, of the flexible connecting rod on the dynamic responses is studied in this paper. The
Ah&a&-The
constraint of elastic deformation at the joint between the connecting rod and the slider is theoretically derived but not assumed. The Hamilton’s principle is employed to formulate the governing equations of the connecting rod. It is found that the simply-supported assumption at the end is extended and replaced by the time-dependent elastic deformation. The equations of motion are transformed into a set of ordinary differential equations by use of the specific variable transformation and Gakrkin method. Finally, the Rungs-Kutta numerical method is applied to obtain the transient amplitudes. The results are compared for various ratios of the crank radius to the length of the connecting rod. Also, the dynamic responses are compared among Timoshenko and Euler beam theories and those of assuming the ends of connecting rod are simply-supported. 0 1997 Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
radius to the rod length is assumed to be a small parameter. Tadjbakhsh and Younis [8] consider the This paper investigates the dynamic responses of the periodic solution of transverse vibration of the flexible connecting rod of a slider-crank mechanism. Aexible connecting rod and applied Floquet theory to This particular problem has been studied extensively determine those values of the parameters: speed, over the past 30 years, with much of the research input torque, geometry and material properties that going beyond the current paper to include totally constitute the boundaries between the regions of flexible mechanism. stability and instability. An excellent survey of these The steady-state solutions and the elastic stability for both the transverse and longitudinal vibrations of works up to 1992 and related to the topic of flexible mechanism is given by Erdman [9]. Hsieh and the elastic connecting rod in a high-speed slidercrank mechanism were obtained by Jasinski et al. [ 11. Shaw [IO] investigated the dynamic behavior and determined the manner in which this response The response of the system is found by Viscomi and depends on the system parameters with a particular Ayre [2] to be dependent upon the five parameters emphasis on nonlinear analyses of the dynamic classified as the length, mass, damping, external piston force and frequency. Sadler and Sandor [3] response near resonance conditions. In the above references, both ends of connecting developed a method of kineto-elastodynamic analysis. They employed the lumped parameter models to rod are of the simply-supported boundary conditions, i.e. the moment and displacement are assumed to simulate the moving mechanism components. The vanish at each end. However, the realistic operation transient responses in both transverse and longitudinal directions were investigated by Chu and Pan [4] condition is that the crank rotates, and the right hand end of connecting rod moves reciprocally with slider on the basis of the ratios of length of the crank to the along the horizontal guide, shown in Fig. l(a). The length of the connecting rod, rotating speeds of the new view point of Fung [l l] is that the realistic crank, viscous damping, and the natural frequencies. operation condition holds even though the elastic Badlani and Kleinhenz [5] investigated the dynamic deformation is considered, i.e. the point B shown in stability by using the Euler-Bernoulli and the Fig. l(b), is pinned with slider and moves along X Timoshenko beam theories, and showed the new axis. The constraint of elastic deformation at the end regions of instability exist when rotary inertia and point B is derived and introduced into the Hamilton’s shear deformation effects are included. principle to formulate the governing equations of the Badlani and Midha [6] investigated the dynamic connecting rod. behavior with an initially curved connecting rod. Zhu This paper extends the concept of Fung (111 to and Chen [7] studied the stability based on a perturbation technique, in which the ratio of crank model the connecting rod by three separate models: 79
Dynamic responses of
81
connecting rod
the tlcxibk
the Y direction is zero, thus the acceleration of point B can be written as R,,(I, 0,0 = a.(!, r)sec 4i where (I. (I, 1) = R,, (I, 0, r) . Ed Fig. 2. At end point B, the displaament relationshipamong A, U(1. T) and V(I, t).
= -r6,,sin(B+fj+)-r0~cos(fI+f$) + us + 2ur9r + u& - (I + uW: .
relation, I sin 8 = f sin 0, into the above equation, we have ~(1, r) = ~(1, r)tan 4
The kinetic energy of connecting rod can express as
(3) T,=;
which is the flexible deformation constraint of the connecting rod at point B. The above displacement relationship at point B can also be obtained from the geometric plot shown in Fig. 2, which is in dimensionless form. It is seen that ~(1, 1) = u(l, I) = 0 for the assump tion of simply-supported end (Janinski, Lee and Sador [I]; Chu and Pan [4]; Tadjbakhsh and Younis (81) is also included in qn (3). If the axial displacement u(x, 1) is negligible, i.e. the transverse displacement u(x, r) is considered only, the constraint condition at point B becomes ~(1, r) = 0, and it is the simply-supported end. In present work, the axial and transverse displacements are considered simultaneously, ~(1, 1) and ~(1, I) are not independent and related by qn (3). Taking variation of qn (3). we have 6u(l, I) = 6u(f, r)tan f#~.
(6)
pdx
pR,(x,y,r).R,(x,y,r)dY= IY
1 (7)
where T* = !f
{[-d,
sin@ + 4) +
U, +
u&p
+ Ire, cost8 + 4) + U, - 4, (X + u)y) + $h
- tity
+
#$‘I.
The Lagrange strains are L,, = 0,
6,. =uv-yyS.,+fd,
cm = f(u., - $)
(9)
(4)
Substituting qn (4) into the variational qn of (2) and taking x = I, we have
(8)
where the higher-order terms u.,$, y$$,, are neglected in L.,,. The strain energy of connecting rod can be expressed as
6R(I, 0, 1) = &(I, f)q + 6u(l, rk = 6u(l, r)sec t$i. Differentiating qn (2) with respect to time, we get the absolute velocity of the arbitrary point P of connecting rod as R, (x, Y, r) = &ee + (a, - ti, k + u,e,
“,=flc,rldY=[U*dx
(10)
where U+ = f[EA(u, + fu:y + KG&
- $y + H$:]. (11)
-~letx[(~+~-_~,+O1+u)cll
The kinetic energy of crank with mass A& and mass momentum of inertia J1,
= [ - re, SW + 4) + u, - y#, + &o, + u)k + [& cos@ + 4) +u,-d,(x+u-&))a,.
7’2= fkwef
+ fJket
(12)
(5) the kinetic energy of slider is
Recauae the slider moves in the X direction and the component of the acceleration of connecting rod in
T, = &R,
(I, 0,1) . R, (I, 0,1)
(13)
Rong-Fong Fung
82
and the work done by the force and friction force acting at slider is W = [(F - pN)i + Nj] . R(f, 0, t).
(14)
C, which are proportional to relative velocities u, , u, and J/, , we obtain the governing equations of system pA[r@ cos(8 + 4) - 24,v, + (x + uw:
2.2. Hamilton’s principle
+r& sin@ + 4) - u,, - o&l - C.94
By using Hamilton’s principle, we can write
(174 0=
“{6T,+bT,-6U,+6T~+6W}dt. s (I
(15)
PA ] - r& cos(0 + 4) + r-6: sin@ + 4) The crank kinetic energy variation, 6T2, is zero, since 0, is prescribed. For the detailed manipulations of j: 6T, dt, see Jou [12]. Substitutingeqns (7, 10, 13, 14) into (15), we obtain
+ 2&u, + v& - v,, + (x + u)&,] - c+,
+ KGA(u,, pZ(4:$
- vx ) = 0
(l7W
+ 6, - $,r) - C,ti, - KGA($
- v.r)
a + Y& (EZ$,) = 0 (17c)
+[!g-qep!._g
and the boundary conditions ~(0, t) = 0,
~(0, t) = 0,
+$($#Wjdsdt
&(I, t) = 0
1
” dx ,, >
-
s[ I= au*
+
au*
au*
=
” [(F -
I (8
(l9a)
- KGA[v,(I,
t) - W,
t)ltan 4
fuf(f, t)][l
+ u,(f, t)tan 41 = 0.
i
pV)sec 4 - iU4a, (I, t)sec’ 41
x &(I, t) dt.
(18a,b,c)
[(F - ~iV)sec 4 - M.+a,U, t&c’ 41
- EA[u, (I, t) + I
du,du+dv,dv+aJI,6~ 1r_odt
,,
&(O, t) = 0
(16)
The varied path coincides with the true path at the two end points tl and t2. It follows that 6v(x, t,) = 6v(x, t2) = 0 6U(X, t,) = 6u(x, tz) = 0, point 6$(x, tl) = S$(x, t*) = 0. The and (x, y) = (0,O) is the common revolute joint point of the rigid crank and the flexible connecting rod, and the values ~(0, t) = 0 and ~(0, t) = 0 are specified, thus we have 6u(O, t) = 0, and 6v(O, t) = 0. The slope angles + at end points x = 0, I of the connecting rod are free, therefore St&O, t) # 0,6$(1, t) # 0. ’ Substituting T* of eqn (8) and U* of eqn (11) into the eqn (16), using the constraint condition (4) at x = I, and introducing the damping terms C, , C, and
(l9b) From the governing eqn (17a,b,c), boundary conditions (18a,b,c; 19a,b) and one constraint condition (3), the following observations are. made: (i) The nonlinearly partial differential equations (17a,b,c) are of the second-order spatial derivative in the all variables u, u and $. The five boundary conditions (18a,b,c; 19a,b) and one constraint condition (3) are satisfied to solve those three equations. (ii) From the physical view point, it is valid to have zero moment (19a) at revolute joint B. However, the boundary condition (19b) and the constraint condition (3) obtained in the present paper are different from those of Jasinski et al. [l] and Badlani and Kleinhenz [5], who assumed zero displacement, u(l, t) = 0, and utilized Newton’s second law to balance the axial, shear loads and inertia force of the piston at x = I. (iii) The investigator (Badlani and Midha [6]; Zhu and Chen [A; Tadjbakhsh and Younis [S]) assumed
84
Rong-Fong Fung
the expression
of (26), we have
I
r @x9
conditions
= +x(0,7) = 0
(30)
0 dx $x(1,7)
1 = 1 + tk(l, f)tan f$ ](F - 0)=
t)sec2c$] -
-M,&(l,
boundary
U(0, 7) = V(0, 7)
‘a
Ax, 0 = PU,t) -
the dimensionless
4
= 0
@la)
F(‘(r)sec4-- SA.,(l,7)~~-~[Yx(l,r)
‘EAu,u,, dx.
--4W,7)1tan~-~]U~(1,7)+~W1,7)1
s1
x [l + Vx(l, 7)tan 41 = 0
2.5. Variable transformation Defining the following dimensionless
parameters
(31b)
as where
A(l,
7T4EI 0: = -dEA d=pAP’ pAP ’ w; = -n4KGA pAP ’
z E2= !!!I 0;’
a, (1, 7)
7) = -
PAW
= - Ae,, sin@ + 4) -
+ UT, +
T= =WTt,
and the dimensionless
F=F-~‘N pAI%:
2 VT9, + V& - (1 + U)& constraint
condition
V(l, 7) = U(l, 7)tan 4.
c, C, c,. “=pAq’ L2=PA*’ c3= pAI2WT and substituting into eqns (17)-(19) and the constraint condition (3), we have the dimensional governing equations
(3W
In order for the application of Galerkin method, it is needed to simplify the boundary conditions to be homogeneous by using the following variable transformation as U(l,T)
-A&
ne: ws(e + 4)
= h(7)
(32a)
U(X, 7) = 0(X, 7) + X/I(r)
(32b)
V(X, 7) = Q’cx, 7) + Xh(7)tan 4.
(324
sin(0 + 4) + V&
-$p*+ V,, + t2 V, -
(294
V,V,,)=O,
net
sin@ + 4) - 2i#~,u, - f#
v
- 4,, (x + u) - le,, c0s(e + 9)
(29b)
- 6:‘k - &j
tixx - & = 0,
(29~)
The physical meaning of h(7) is the nondimensionally axial deformation of the connecting rod at X = 1. Substituting eqn (32a,b,c) into eqn (29a,b,c) and using eqn (31~) one obtains the following equations of motion
Dynamic responses of the flexible connecting rod
where ()’ means derivative with respect to the dimensionless time r. After the variable transformations (32a,b,c), the nonhomogeneous boundary condition (31b,c) are changed to be homogeneous boundary conditions (35) and one additional eqn (36), which describes the axial deformation of the connecting rod at X = 1.
V,+e&-2#,O,-&V++J #x) -&j{v*OXX+h&tand
+(V*X+ f[9
VXX+ 2hVxVmtan 4 + hzVxxtan2 #]
+ ir, VXX+ ~VH} + X{h” tan Q
2.6. Galerkin method Satisfying the homogeneous boundary conditions (34) and (35), the nth mode solutions of D, V and JI as follows:
+ h’[e2tan 4i - 24, (1 - set? 4)] +h[cz&secf+-&ftanc#(l - cL (1 - sa+ #)I+ - me,, cos(e
-2sec’4)
&} = Mf sin@ + 4)
+ +)
hr + $ $1+ &
85
0(X, T) = f f. (r)sin n7r.Y m-1
WW V(cX,r) = f g.(r)sin nnX I-I
(IL - vx:,, - #$ - $z$xx h tan 4
=0=+&j
(33~)
$(X, T) = f
n-1
and boundary
conditions
O(0, r) =
nnx.
(37)
are
V(0,2) =
O(1, T) = V(l,7)
!lJ.(r)cos
=
$x(0, ?) = 0
(34)
$X(1, T) = 0
(35)
Substituting the solutions (37) into eqns (33a,b,c) and (36) and then applying Galerkin method to the resulting equations, one obtains
and (3lb) becomes
+A
+h (2sec2${
l)#+&tan9
+--&kcos’4
1 +2VX(l,r)tan4+ztan24
(-
I)“+‘{h”
+ h’[c, + 24,
tan 41
- h[& (1 - 2 set 4’) + & tan 91)
=-j--;I(--i)m+i+ ilpefc0s(e++)
[ +ne.,sin(e+~)]+~(-lr+‘~~ +f
PX(l,r)tan’$
(3ga)
+ OX(l, r)tan2 4 II
-’ cos2 t#~{t/+tan’ r$ +’rr’e2 s
-A
+ th2 tan2 #[I + VX(l, r)tan 4]}
+lh2
(mrr)‘Pm +
&
[h tan4(mn)%
tan 6(mn)2g, + h(mnYg,,,]
+++s~9[~X(l,T) +
f R(l,
r)][l + Vx(l, r)tan 41 =
+nCc0s(e+))+c5:+$%0s~
Ae,, sin@ + 4) (36)
-2k(l
-sac’4)l+h[c2&zsec29
- $4 tan 4(1 - 2 set’ 9) - &(l
- UK+
4>1}
Rong-Fong Fung
86
=&[(-l)m+t
+ l][M sin@ + 4)
(38b)
-~e,,cos(e+~)l-~(-l)“+‘~,~
and V(f , r), of middle point of connecting rod are obtained, respectively. Following the processes (17)-(38), we can obtain the nth mode equations of the Euler beam theory as $A + Gjz + 24,gLl - 4ffm + &gm
Y; + ; Y; + &j?
(Yv, - (mx)gfW) +&
](m~)*fm+ h tan
4(mrr)*gml
+ & (mx)ZY’, - +f Y, = 0 (38~) +A f g.(nn)(-1) n-1
- t I-
I
Y,(-1)”
1{ +h
(2sec’4-
(- l)m+ ‘{h” + h’[.5,+ 24, tan 41
- WT (1 - 2 set 49 + & tan
41}
I)&
+ le,, sin@ + f$)] + $
(- l)m+l&
+ &, tan f$ + r L co? fj rr4<2s
x
1+2tan4 [
(39a)
f n.(nn)(-l~+~tanid n-1 +-$j
+;
f f g,g”(in)(nrr)(-l)‘+” [ ,=I n-1
+tan2b
ff.(nr)(-lr n-1
[h tan 4(mn)*fn + ;h2 tan &rrn)‘gm
tan2 f$ I
+h(mrr)‘g,l+~(mn)4g~+~(-l)“+l
x{h”
11
+-&~coP~
tan f$ + h’[~~tan f$ - 24, (1 - sec2 4)]
+ h[e24 xc2 4 - 43 tan &l - 2 xc2 4) fh’ tan4 4 + ih2 tan* 4 1 + tan C$.g, [
- &(l
1
s.(na)(- 1)
+$&cos24 fJ+, 11 [ n=l
- sec’4)1> =--$(-lr”
x ]Ae: sin@ + 4) -
+ 11
ke,, c0qe + 4)]
---$-l)“t’& x (-1)“+;
x
1 +tan4 [
f f gig”(in)(nir)(-l)‘+” ,=I n-1
f g”(nn)(-1)” n-1
1
h”+2#,h’tan4+h
(2sec24-1)&+4,tan4~ 1
1
+-&icos2f$
=M,,sin(6+4)
1 +2tanf$
+Ae:cos(e+$)+$:+~Fcos~.
[
(384
The Runge-Kutta numerical method will be used to integrate the above ordinary differential equations for the transient solutions of&, h and Y, which are then substituted into eqn (37) to obtain o(i, t) and p(cf, T) by taking X = f. From eqn (32b,c), the transverse and longitudinal displacements, U(f) 7)
g, ,
(39b)
,
+;
i g.(nlt)(-1) I-I
5 f g,g.(In)(nlr)(-l)‘+” [ I-I I-I
+ tan2 6 5 f.(nn)(-ly
n-l
II
1
tan*4
. -\I I 81
Dynamic responses of the tlexiblc connecting rod x ~0s’ t# fh’ tan’ 4 + fh’ tan’ 4
1
0.008 -
x
1 +tand [
f g.(nn)(-lp “II
0.006 -
II
j
0.004-
2B
0.002~
1
0
I,---..
--. ____ ---- -__.
_____ . . .
l -0m2 -0.004
\
-0.010
x
l+tanQ [
fg#(nn)(-lr “-1
1
5
10
I5
-_- : 82
20
25
.-_ 30
B(W
=&sin(e+ti)
Fig. 3. Transient amplitudes of first and second modes: (-) first mode; (---) second mode.
+Mfms(e+~)+&+~Fcos~. (39d Also, the nth mode equation of (28) which is considered the transverse displacement only, will be obtained. For detailed results see Jou [12]. 3. NUMERICAL
RESULTS AND DI!3CU!3SIONS
In the previous investigations (Zhu and Chen [7]; Tadjbakhsh and Younis [8] and Jou [ 12]), it has been shown that in all cases the second mode response is only a small percentage of the first mode response. The difference is more pronounced for the third mode, fourth mode, etc. Only the iirst mode shape of vibrations is important to practical engineering design problems. The transient amplitudes will be obtained by using RunwKutta method to integrate (38) for the Tiioshenko beam theory, (39) for the Euler beam theory, with these initial conditions:fi (0) -1; (0) = 0, and P,(O)= F{(O)=0 g1(0) = g;(o) = 0, h(0) = h’(0) = 0, and a specific accuracy lo-‘. We consider the crank rotates with constant angular vdocity:
e(o) =
0,
8, (0) = zr = 0.2266,
The increasing in 1 value relates the increasing in the transient amplitudes. The transverse amplitude gI in Fig. S(b) is about 500 times of the longitudinal amplitude f; in Fig. 5(a). The transient rotary angle Y,(r) due to bending is shown in Fig. 5(c). From numerical integration of eqn (39), the longitudinal displacement h(r), is obtained and shown in Fig. 5(d). Moreover, the fi and h in Fig. 5(a,d), respectively, oscillate violently first, and then approach to be stable because of the introducing of damping terms c,, cl and c3. However, in the present work the end point B of the connecting rod moves freely along X axis, both ~(1, t) and u(l, t) are not independent and are related by eqn (3). Therefore, the piston position could be predicted due to the elastic deformation. With the
81
e,, (0) = 0
and the other parameters are S = 0.5, cl = 0.2, c2 = 0.2, e3 = 0.001, C-’ = 96, c-r = 56.61 and F = 0. The transient amplitudes of the connecting rod which has only the transverse displacement are shown in Fig. 3. It can be seen the pronounced difference between the first and second mode responses. Fire 4 shows the transient amplitude of present work is smaller than that of the lumped parameter approach (Sadler and Sandor [3]). The results in Fig. S(a-f) are obtained for various ratios of the crank radius to connecting rod length.
0
2
4
6
8
10
Fig. 4. Transient __~ amplitude.^g,(~).co~_wcd
Sadler ad Sandor tsj.
12
4
with that of
88
Rong-Fong Fung
(d) (...) X = 0.2, (- -) X = 0.25, (-) x = 0.3.
I
2
3
0
2
-2 -4
1 fl
h
0
-6
-1
-8
-2
-10
-3
-12
I-
(e)
(b) 0.02 x =
0.2 (- -) x = 0.2
-4 x 10
2
0.015 1
0
w,e
-1
-2
-3
1
_4-.-_-_-.-_--L--. 0
20
40
60
80
loo
120
7
(f) 4
x 10
-4
k (...) A = 0.2, (- -) A = 0.25, (-_I
Fig. 5. (a) Transient longitudinal amplitudefi(r) of connecting rod: (. .) A= 0.2, (---) I = 0.25, (-) L = 0.3; (b) transient transverse amplitude g(r) of connecting rod: (. . .) A= 0.2, (---) I = 0.25, (-) 1 = 0.3; (c) transient rotary angle VI(T) of connecting rod: (. .) I = 0.2, (---) I = 0.25, (-) A = 0.3; (d) transient axis! deformation h(r) of connecting rod at end point B: (. .) I = $2, (---) A = 0.25, (-) ; 18_:;(;) transient transverse amphtude Y(1,7) of connecting rod at end point B: (. . .) L = 0.2, (---) 1= 0.3; (f) transrent honzontal displacement Db of connecting rod at end point E: ( .) . , -) 2.= 0.2, (---) 1 = 0.25, (-) A = 0.3.
140
Dynamic responses of the flexible connecting rod help of Fig. 2, it is convenient to define the horizontal displacement at point B as
(4
89
x 10
IP-----7 2
AA&
0
Using the displacement relationship shown in Fig. 2, we have V(1, T) = h(r)tan 4. The transient transverse amplitude, V( 1, r), is then obtained and shown in Fig. S(e). From eqn (40), the transient horizontal displacement & at end point B along the X axis is obtained and shown in Fig. 5(f). A slider+rank mechanism with the usual proportions (connecting rod longer than crank) has two limiting positions, both occurring when the crank and connecting rod are collinear, i.e. Q = 0. At the limiting positions, we have zero transverse displacement, v(Z, I) = 0, from eqn (3), and the horizontal displacement equals the axial deformation, Db = II(~), from eqn’(40). The results in Fig. 6(a-c) are obtained for comparing among Timoshenko and Euler beam theories and those of assuming the ends of connecting rod are simply-supported. The longitudinal displacement U(1, r), is shown in Fig. 6(a). Using the displacement relationship shown in Fig. 2, we have V(1, T) = h(r)tan 4. The transient transverse amplitude, V( 1, r), is then obtained and shown in Fig. 6(b). From eqn (40), the transient horizontal displacement Db at end point B along the X axis is obtained and shown in Fig. 6(c). Since the effects of rotary inertia and shear deformation are considered in the Timoshenko beam theory, the transient amplitudes shown in Fig. 6(a-c) are larger than those of Euler beam theory. The transient amplitudes are always zero for the assumption of simply-supported ends.
-2 w.9 -4 v
-6
1
t1
lb) -4 x 10 1.5 ,
(-.-.):
Simple suppert sssumpti011.
vu,f)
20 4. CONCLUSIONS This paper studies the time-dependent boundary effect on the dynamic responses of the flexible connecting rod of a slider-crank mechanism. From the formulation of the governing equations, the numerical results and discussion, we conclude that: (1) The significant new contribution provided in this paper is the definition of a revise boundary condition at the joint between the connecting rod and the slider. From the derivation it is found that the boundary condition of the connecting rod moving with slider is not a simply-supported boundary condition but
Fig. 6. (a) Transient axial deformation U(1, T) of connecting rod at end point B: (-.-.) simple-support assumption, (---) Euler beam, (-) Timoshenko beam; (b) transient transverse amplitude V(1,~) of connecting rod at end point 8: (-.-.) simple-support assumption, (---) Euler beam, (-) Tiihenko beam, (c) transient horizontal displacement Db of connecting rod at end point B: (-.-.) simple-support assumption, (---) Euler beam, (-) Timoshcnko beam.
‘5
Cc)
-4
x 10 4
-2 Db -4
_iIi[ 0
-):
(-.-.) : Simple support assumption, (y):Tinyosha&;heam:
ytmun, 20
60 r
80
100
1
120
90
Rong-Fong Fung
(2)
being extended and replaced by V( 1, T) = .!I’(1, r)tan 4 in this paper. Based on the theory developed, the elastic deformation at the slider position could be predicted. Therefore, the accuracy will be improved in the mechanical design. Another contribution of the paper is in the variable transformation from the body-fixed coordinate system to that with simpler boundary condition, i.e. new variables being zero at the boundary. Obtaining the simple boundary condition allows for the choice of simple harmonic functions in the used expansions for the vibration of the beam.
Acknowledgement-Support of this work by the National Science Council of the Republic of China through Contract NSC 842212-E033-009 is gratefully acknowledged.
REFERENCES
I. P. W. Jasinski, H. C. Lee and G. N. Sandor, Vibrations of elastic connecting rod of a high-speed slidercrank mechanism. ASME J. Engng Indust., 636-644 (1971). 2. B. V. Viscomi and R. S. Arye, Nonlinear dynamic response of elastic slider-crank mechanism. ASME J. Engng Indust. 93, 251-262 (1971).
3. J. P. Sadlerand and G. N. Sandor, A lumped parameter approach to vibration and stress analysis -of elastic linkaaes. ASME J. Enann Indusr.. 549-557 (1973). 4, S. CT Chu and K. CT Pan, Dynamic response ‘of a high-speed slider-crank mechanism with an elastic connecting rod. ASME J. Engng Indust.. 542-550 (1975). M. Badlani and W. Kleinhenz, Dynamic stability of 5’ elastic mechanism. ASME J. Mech. Des., 149-153 (1979). 6. M. Badlani and A. Midha, Member initial curvature effects on the elastic slider-crank mechanism response. 7, ASME J. Mech. Des., 159-167 (1982). Z. G. Zhu and Y. Chen, The stability of the motion of a connecting rod. ASME J. Mech., Transmissions Automation Des., 637-640 (1983). 8. I. G. Tgdjbakhsh and C. J. Younis, Dynamic stability of the flexible connecting rod of a slider-crank mechanism. ASME J. Mech., Transmissions Automation Des., 487496 (1986). 9, A. G. Erdman, Modern Kinematics, Developments in the Lost For?y Years. Wiley, New York, Chapter 9 (1993). 10. S-R. Hsieh and S. W. Shaw, The dynamic stability and nonlinear resonance of a flexible connecting rod: single-mode model. J. Sound Vibr. 170(l), 2549 (1994). 11. R. F. Fung, Dynamic analysis of the flexible connecting rod of a slider-crank mechanism. ASME J. Vibr. Acoustics (1995) (submitted). 12. C. H. Jou, Dynamic stability of a high-speed slider-crank mechanism with a flexible connecting rod. Master thesis, University of Chun Yuan, Chung-Li, Taiwan (1992).
Compuwrsct Sm~twcs Vol. 63, No. 1. pp. 79-90. 1997 Q 1997 F’ublisbcdby Ebevk Scimcs Ltd Printedin Grcllt Btiuin. AU ri&b wmmed 004s7949p 517.00 + 0.00
DYNAMIC RESPONSES OF THE FLEXIBLE CONNECTING ROD OF A SLIDER-CRANK MECHANISM WITH TIME-DEPENDENT BOUNDARY EFFECT Department of Mechanical Engineering, Chung Yuang Christian University, Chung-Li, Taiwan 32023, Republic of China (Received 5 October 1995)
timedependent boundary effect of the end, which moves reciprocally with slider along the horizontal guide, of the flexible connecting rod on the dynamic responses is studied in this paper. The
Ah&a&-The
constraint of elastic deformation at the joint between the connecting rod and the slider is theoretically derived but not assumed. The Hamilton’s principle is employed to formulate the governing equations of the connecting rod. It is found that the simply-supported assumption at the end is extended and replaced by the time-dependent elastic deformation. The equations of motion are transformed into a set of ordinary differential equations by use of the specific variable transformation and Gakrkin method. Finally, the Rungs-Kutta numerical method is applied to obtain the transient amplitudes. The results are compared for various ratios of the crank radius to the length of the connecting rod. Also, the dynamic responses are compared among Timoshenko and Euler beam theories and those of assuming the ends of connecting rod are simply-supported. 0 1997 Published by Elsevier Science Ltd. All rights reserved.
1. INTRODUCTION
radius to the rod length is assumed to be a small parameter. Tadjbakhsh and Younis [8] consider the This paper investigates the dynamic responses of the periodic solution of transverse vibration of the flexible connecting rod of a slider-crank mechanism. Aexible connecting rod and applied Floquet theory to This particular problem has been studied extensively determine those values of the parameters: speed, over the past 30 years, with much of the research input torque, geometry and material properties that going beyond the current paper to include totally constitute the boundaries between the regions of flexible mechanism. stability and instability. An excellent survey of these The steady-state solutions and the elastic stability for both the transverse and longitudinal vibrations of works up to 1992 and related to the topic of flexible mechanism is given by Erdman [9]. Hsieh and the elastic connecting rod in a high-speed slidercrank mechanism were obtained by Jasinski et al. [ 11. Shaw [IO] investigated the dynamic behavior and determined the manner in which this response The response of the system is found by Viscomi and depends on the system parameters with a particular Ayre [2] to be dependent upon the five parameters emphasis on nonlinear analyses of the dynamic classified as the length, mass, damping, external piston force and frequency. Sadler and Sandor [3] response near resonance conditions. In the above references, both ends of connecting developed a method of kineto-elastodynamic analysis. They employed the lumped parameter models to rod are of the simply-supported boundary conditions, i.e. the moment and displacement are assumed to simulate the moving mechanism components. The vanish at each end. However, the realistic operation transient responses in both transverse and longitudinal directions were investigated by Chu and Pan [4] condition is that the crank rotates, and the right hand end of connecting rod moves reciprocally with slider on the basis of the ratios of length of the crank to the along the horizontal guide, shown in Fig. l(a). The length of the connecting rod, rotating speeds of the new view point of Fung [l l] is that the realistic crank, viscous damping, and the natural frequencies. operation condition holds even though the elastic Badlani and Kleinhenz [5] investigated the dynamic deformation is considered, i.e. the point B shown in stability by using the Euler-Bernoulli and the Fig. l(b), is pinned with slider and moves along X Timoshenko beam theories, and showed the new axis. The constraint of elastic deformation at the end regions of instability exist when rotary inertia and point B is derived and introduced into the Hamilton’s shear deformation effects are included. principle to formulate the governing equations of the Badlani and Midha [6] investigated the dynamic connecting rod. behavior with an initially curved connecting rod. Zhu This paper extends the concept of Fung (111 to and Chen [7] studied the stability based on a perturbation technique, in which the ratio of crank model the connecting rod by three separate models: 79
Dynamic responses of
81
connecting rod
the tlcxibk
the Y direction is zero, thus the acceleration of point B can be written as R,,(I, 0,0 = a.(!, r)sec 4i where (I. (I, 1) = R,, (I, 0, r) . Ed Fig. 2. At end point B, the displaament relationshipamong A, U(1. T) and V(I, t).
= -r6,,sin(B+fj+)-r0~cos(fI+f$) + us + 2ur9r + u& - (I + uW: .
relation, I sin 8 = f sin 0, into the above equation, we have ~(1, r) = ~(1, r)tan 4
The kinetic energy of connecting rod can express as
(3) T,=;
which is the flexible deformation constraint of the connecting rod at point B. The above displacement relationship at point B can also be obtained from the geometric plot shown in Fig. 2, which is in dimensionless form. It is seen that ~(1, 1) = u(l, I) = 0 for the assump tion of simply-supported end (Janinski, Lee and Sador [I]; Chu and Pan [4]; Tadjbakhsh and Younis (81) is also included in qn (3). If the axial displacement u(x, 1) is negligible, i.e. the transverse displacement u(x, r) is considered only, the constraint condition at point B becomes ~(1, r) = 0, and it is the simply-supported end. In present work, the axial and transverse displacements are considered simultaneously, ~(1, 1) and ~(1, I) are not independent and related by qn (3). Taking variation of qn (3). we have 6u(l, I) = 6u(f, r)tan f#~.
(6)
pdx
pR,(x,y,r).R,(x,y,r)dY= IY
1 (7)
where T* = !f
{[-d,
sin@ + 4) +
U, +
u&p
+ Ire, cost8 + 4) + U, - 4, (X + u)y) + $h
- tity
+
#$‘I.
The Lagrange strains are L,, = 0,
6,. =uv-yyS.,+fd,
cm = f(u., - $)
(9)
(4)
Substituting qn (4) into the variational qn of (2) and taking x = I, we have
(8)
where the higher-order terms u.,$, y$$,, are neglected in L.,,. The strain energy of connecting rod can be expressed as
6R(I, 0, 1) = &(I, f)q + 6u(l, rk = 6u(l, r)sec t$i. Differentiating qn (2) with respect to time, we get the absolute velocity of the arbitrary point P of connecting rod as R, (x, Y, r) = &ee + (a, - ti, k + u,e,
“,=flc,rldY=[U*dx
(10)
where U+ = f[EA(u, + fu:y + KG&
- $y + H$:]. (11)
-~letx[(~+~-_~,+O1+u)cll
The kinetic energy of crank with mass A& and mass momentum of inertia J1,
= [ - re, SW + 4) + u, - y#, + &o, + u)k + [& cos@ + 4) +u,-d,(x+u-&))a,.
7’2= fkwef
+ fJket
(12)
(5) the kinetic energy of slider is
Recauae the slider moves in the X direction and the component of the acceleration of connecting rod in
T, = &R,
(I, 0,1) . R, (I, 0,1)
(13)
Rong-Fong Fung
82
and the work done by the force and friction force acting at slider is W = [(F - pN)i + Nj] . R(f, 0, t).
(14)
C, which are proportional to relative velocities u, , u, and J/, , we obtain the governing equations of system pA[r@ cos(8 + 4) - 24,v, + (x + uw:
2.2. Hamilton’s principle
+r& sin@ + 4) - u,, - o&l - C.94
By using Hamilton’s principle, we can write
(174 0=
“{6T,+bT,-6U,+6T~+6W}dt. s (I
(15)
PA ] - r& cos(0 + 4) + r-6: sin@ + 4) The crank kinetic energy variation, 6T2, is zero, since 0, is prescribed. For the detailed manipulations of j: 6T, dt, see Jou [12]. Substitutingeqns (7, 10, 13, 14) into (15), we obtain
+ 2&u, + v& - v,, + (x + u)&,] - c+,
+ KGA(u,, pZ(4:$
- vx ) = 0
(l7W
+ 6, - $,r) - C,ti, - KGA($
- v.r)
a + Y& (EZ$,) = 0 (17c)
+[!g-qep!._g
and the boundary conditions ~(0, t) = 0,
~(0, t) = 0,
+$($#Wjdsdt
&(I, t) = 0
1
” dx ,, >
-
s[ I= au*
+
au*
au*
=
” [(F -
I (8
(l9a)
- KGA[v,(I,
t) - W,
t)ltan 4
fuf(f, t)][l
+ u,(f, t)tan 41 = 0.
i
pV)sec 4 - iU4a, (I, t)sec’ 41
x &(I, t) dt.
(18a,b,c)
[(F - ~iV)sec 4 - M.+a,U, t&c’ 41
- EA[u, (I, t) + I
du,du+dv,dv+aJI,6~ 1r_odt
,,
&(O, t) = 0
(16)
The varied path coincides with the true path at the two end points tl and t2. It follows that 6v(x, t,) = 6v(x, t2) = 0 6U(X, t,) = 6u(x, tz) = 0, point 6$(x, tl) = S$(x, t*) = 0. The and (x, y) = (0,O) is the common revolute joint point of the rigid crank and the flexible connecting rod, and the values ~(0, t) = 0 and ~(0, t) = 0 are specified, thus we have 6u(O, t) = 0, and 6v(O, t) = 0. The slope angles + at end points x = 0, I of the connecting rod are free, therefore St&O, t) # 0,6$(1, t) # 0. ’ Substituting T* of eqn (8) and U* of eqn (11) into the eqn (16), using the constraint condition (4) at x = I, and introducing the damping terms C, , C, and
(l9b) From the governing eqn (17a,b,c), boundary conditions (18a,b,c; 19a,b) and one constraint condition (3), the following observations are. made: (i) The nonlinearly partial differential equations (17a,b,c) are of the second-order spatial derivative in the all variables u, u and $. The five boundary conditions (18a,b,c; 19a,b) and one constraint condition (3) are satisfied to solve those three equations. (ii) From the physical view point, it is valid to have zero moment (19a) at revolute joint B. However, the boundary condition (19b) and the constraint condition (3) obtained in the present paper are different from those of Jasinski et al. [l] and Badlani and Kleinhenz [5], who assumed zero displacement, u(l, t) = 0, and utilized Newton’s second law to balance the axial, shear loads and inertia force of the piston at x = I. (iii) The investigator (Badlani and Midha [6]; Zhu and Chen [A; Tadjbakhsh and Younis [S]) assumed
84
Rong-Fong Fung
the expression
of (26), we have
I
r @x9
conditions
= +x(0,7) = 0
(30)
0 dx $x(1,7)
1 = 1 + tk(l, f)tan f$ ](F - 0)=
t)sec2c$] -
-M,&(l,
boundary
U(0, 7) = V(0, 7)
‘a
Ax, 0 = PU,t) -
the dimensionless
4
= 0
@la)
F(‘(r)sec4-- SA.,(l,7)~~-~[Yx(l,r)
‘EAu,u,, dx.
--4W,7)1tan~-~]U~(1,7)+~W1,7)1
s1
x [l + Vx(l, 7)tan 41 = 0
2.5. Variable transformation Defining the following dimensionless
parameters
(31b)
as where
A(l,
7T4EI 0: = -dEA d=pAP’ pAP ’ w; = -n4KGA pAP ’
z E2= !!!I 0;’
a, (1, 7)
7) = -
PAW
= - Ae,, sin@ + 4) -
+ UT, +
T= =WTt,
and the dimensionless
F=F-~‘N pAI%:
2 VT9, + V& - (1 + U)& constraint
condition
V(l, 7) = U(l, 7)tan 4.
c, C, c,. “=pAq’ L2=PA*’ c3= pAI2WT and substituting into eqns (17)-(19) and the constraint condition (3), we have the dimensional governing equations
(3W
In order for the application of Galerkin method, it is needed to simplify the boundary conditions to be homogeneous by using the following variable transformation as U(l,T)
-A&
ne: ws(e + 4)
= h(7)
(32a)
U(X, 7) = 0(X, 7) + X/I(r)
(32b)
V(X, 7) = Q’cx, 7) + Xh(7)tan 4.
(324
sin(0 + 4) + V&
-$p*+ V,, + t2 V, -
(294
V,V,,)=O,
net
sin@ + 4) - 2i#~,u, - f#
v
- 4,, (x + u) - le,, c0s(e + 9)
(29b)
- 6:‘k - &j
tixx - & = 0,
(29~)
The physical meaning of h(7) is the nondimensionally axial deformation of the connecting rod at X = 1. Substituting eqn (32a,b,c) into eqn (29a,b,c) and using eqn (31~) one obtains the following equations of motion
Dynamic responses of the flexible connecting rod
where ()’ means derivative with respect to the dimensionless time r. After the variable transformations (32a,b,c), the nonhomogeneous boundary condition (31b,c) are changed to be homogeneous boundary conditions (35) and one additional eqn (36), which describes the axial deformation of the connecting rod at X = 1.
V,+e&-2#,O,-&V++J #x) -&j{v*OXX+h&tand
+(V*X+ f[9
VXX+ 2hVxVmtan 4 + hzVxxtan2 #]
+ ir, VXX+ ~VH} + X{h” tan Q
2.6. Galerkin method Satisfying the homogeneous boundary conditions (34) and (35), the nth mode solutions of D, V and JI as follows:
+ h’[e2tan 4i - 24, (1 - set? 4)] +h[cz&secf+-&ftanc#(l - cL (1 - sa+ #)I+ - me,, cos(e
-2sec’4)
&} = Mf sin@ + 4)
+ +)
hr + $ $1+ &
85
0(X, T) = f f. (r)sin n7r.Y m-1
WW V(cX,r) = f g.(r)sin nnX I-I
(IL - vx:,, - #$ - $z$xx h tan 4
=0=+&j
(33~)
$(X, T) = f
n-1
and boundary
conditions
O(0, r) =
nnx.
(37)
are
V(0,2) =
O(1, T) = V(l,7)
!lJ.(r)cos
=
$x(0, ?) = 0
(34)
$X(1, T) = 0
(35)
Substituting the solutions (37) into eqns (33a,b,c) and (36) and then applying Galerkin method to the resulting equations, one obtains
and (3lb) becomes
+A
+h (2sec2${
l)#+&tan9
+--&kcos’4
1 +2VX(l,r)tan4+ztan24
(-
I)“+‘{h”
+ h’[c, + 24,
tan 41
- h[& (1 - 2 set 4’) + & tan 91)
=-j--;I(--i)m+i+ ilpefc0s(e++)
[ +ne.,sin(e+~)]+~(-lr+‘~~ +f
PX(l,r)tan’$
(3ga)
+ OX(l, r)tan2 4 II
-’ cos2 t#~{t/+tan’ r$ +’rr’e2 s
-A
+ th2 tan2 #[I + VX(l, r)tan 4]}
+lh2
(mrr)‘Pm +
&
[h tan4(mn)%
tan 6(mn)2g, + h(mnYg,,,]
+++s~9[~X(l,T) +
f R(l,
r)][l + Vx(l, r)tan 41 =
+nCc0s(e+))+c5:+$%0s~
Ae,, sin@ + 4) (36)
-2k(l
-sac’4)l+h[c2&zsec29
- $4 tan 4(1 - 2 set’ 9) - &(l
- UK+
4>1}
Rong-Fong Fung
86
=&[(-l)m+t
+ l][M sin@ + 4)
(38b)
-~e,,cos(e+~)l-~(-l)“+‘~,~
and V(f , r), of middle point of connecting rod are obtained, respectively. Following the processes (17)-(38), we can obtain the nth mode equations of the Euler beam theory as $A + Gjz + 24,gLl - 4ffm + &gm
Y; + ; Y; + &j?
(Yv, - (mx)gfW) +&
](m~)*fm+ h tan
4(mrr)*gml
+ & (mx)ZY’, - +f Y, = 0 (38~) +A f g.(nn)(-1) n-1
- t I-
I
Y,(-1)”
1{ +h
(2sec’4-
(- l)m+ ‘{h” + h’[.5,+ 24, tan 41
- WT (1 - 2 set 49 + & tan
41}
I)&
+ le,, sin@ + f$)] + $
(- l)m+l&
+ &, tan f$ + r L co? fj rr4<2s
x
1+2tan4 [
(39a)
f n.(nn)(-l~+~tanid n-1 +-$j
+;
f f g,g”(in)(nrr)(-l)‘+” [ ,=I n-1
+tan2b
ff.(nr)(-lr n-1
[h tan 4(mn)*fn + ;h2 tan &rrn)‘gm
tan2 f$ I
+h(mrr)‘g,l+~(mn)4g~+~(-l)“+l
x{h”
11
+-&~coP~
tan f$ + h’[~~tan f$ - 24, (1 - sec2 4)]
+ h[e24 xc2 4 - 43 tan &l - 2 xc2 4) fh’ tan4 4 + ih2 tan* 4 1 + tan C$.g, [
- &(l
1
s.(na)(- 1)
+$&cos24 fJ+, 11 [ n=l
- sec’4)1> =--$(-lr”
x ]Ae: sin@ + 4) -
+ 11
ke,, c0qe + 4)]
---$-l)“t’& x (-1)“+;
x
1 +tan4 [
f f gig”(in)(nir)(-l)‘+” ,=I n-1
f g”(nn)(-1)” n-1
1
h”+2#,h’tan4+h
(2sec24-1)&+4,tan4~ 1
1
+-&icos2f$
=M,,sin(6+4)
1 +2tanf$
+Ae:cos(e+$)+$:+~Fcos~.
[
(384
The Runge-Kutta numerical method will be used to integrate the above ordinary differential equations for the transient solutions of&, h and Y, which are then substituted into eqn (37) to obtain o(i, t) and p(cf, T) by taking X = f. From eqn (32b,c), the transverse and longitudinal displacements, U(f) 7)
g, ,
(39b)
,
+;
i g.(nlt)(-1) I-I
5 f g,g.(In)(nlr)(-l)‘+” [ I-I I-I
+ tan2 6 5 f.(nn)(-ly
n-l
II
1
tan*4
. -\I I 81
Dynamic responses of the tlexiblc connecting rod x ~0s’ t# fh’ tan’ 4 + fh’ tan’ 4
1
0.008 -
x
1 +tand [
f g.(nn)(-lp “II
0.006 -
II
j
0.004-
2B
0.002~
1
0
I,---..
--. ____ ---- -__.
_____ . . .
l -0m2 -0.004
\
-0.010
x
l+tanQ [
fg#(nn)(-lr “-1
1
5
10
I5
-_- : 82
20
25
.-_ 30
B(W
=&sin(e+ti)
Fig. 3. Transient amplitudes of first and second modes: (-) first mode; (---) second mode.
+Mfms(e+~)+&+~Fcos~. (39d Also, the nth mode equation of (28) which is considered the transverse displacement only, will be obtained. For detailed results see Jou [12]. 3. NUMERICAL
RESULTS AND DI!3CU!3SIONS
In the previous investigations (Zhu and Chen [7]; Tadjbakhsh and Younis [8] and Jou [ 12]), it has been shown that in all cases the second mode response is only a small percentage of the first mode response. The difference is more pronounced for the third mode, fourth mode, etc. Only the iirst mode shape of vibrations is important to practical engineering design problems. The transient amplitudes will be obtained by using RunwKutta method to integrate (38) for the Tiioshenko beam theory, (39) for the Euler beam theory, with these initial conditions:fi (0) -1; (0) = 0, and P,(O)= F{(O)=0 g1(0) = g;(o) = 0, h(0) = h’(0) = 0, and a specific accuracy lo-‘. We consider the crank rotates with constant angular vdocity:
e(o) =
0,
8, (0) = zr = 0.2266,
The increasing in 1 value relates the increasing in the transient amplitudes. The transverse amplitude gI in Fig. S(b) is about 500 times of the longitudinal amplitude f; in Fig. 5(a). The transient rotary angle Y,(r) due to bending is shown in Fig. 5(c). From numerical integration of eqn (39), the longitudinal displacement h(r), is obtained and shown in Fig. 5(d). Moreover, the fi and h in Fig. 5(a,d), respectively, oscillate violently first, and then approach to be stable because of the introducing of damping terms c,, cl and c3. However, in the present work the end point B of the connecting rod moves freely along X axis, both ~(1, t) and u(l, t) are not independent and are related by eqn (3). Therefore, the piston position could be predicted due to the elastic deformation. With the
81
e,, (0) = 0
and the other parameters are S = 0.5, cl = 0.2, c2 = 0.2, e3 = 0.001, C-’ = 96, c-r = 56.61 and F = 0. The transient amplitudes of the connecting rod which has only the transverse displacement are shown in Fig. 3. It can be seen the pronounced difference between the first and second mode responses. Fire 4 shows the transient amplitude of present work is smaller than that of the lumped parameter approach (Sadler and Sandor [3]). The results in Fig. S(a-f) are obtained for various ratios of the crank radius to connecting rod length.
0
2
4
6
8
10
Fig. 4. Transient __~ amplitude.^g,(~).co~_wcd
Sadler ad Sandor tsj.
12
4
with that of
88
Rong-Fong Fung
(d) (...) X = 0.2, (- -) X = 0.25, (-) x = 0.3.
I
2
3
0
2
-2 -4
1 fl
h
0
-6
-1
-8
-2
-10
-3
-12
I-
(e)
(b) 0.02 x =
0.2 (- -) x = 0.2
-4 x 10
2
0.015 1
0
w,e
-1
-2
-3
1
_4-.-_-_-.-_--L--. 0
20
40
60
80
loo
120
7
(f) 4
x 10
-4
k (...) A = 0.2, (- -) A = 0.25, (-_I
Fig. 5. (a) Transient longitudinal amplitudefi(r) of connecting rod: (. .) A= 0.2, (---) I = 0.25, (-) L = 0.3; (b) transient transverse amplitude g(r) of connecting rod: (. . .) A= 0.2, (---) I = 0.25, (-) 1 = 0.3; (c) transient rotary angle VI(T) of connecting rod: (. .) I = 0.2, (---) I = 0.25, (-) A = 0.3; (d) transient axis! deformation h(r) of connecting rod at end point B: (. .) I = $2, (---) A = 0.25, (-) ; 18_:;(;) transient transverse amphtude Y(1,7) of connecting rod at end point B: (. . .) L = 0.2, (---) 1= 0.3; (f) transrent honzontal displacement Db of connecting rod at end point E: ( .) . , -) 2.= 0.2, (---) 1 = 0.25, (-) A = 0.3.
140
Dynamic responses of the flexible connecting rod help of Fig. 2, it is convenient to define the horizontal displacement at point B as
(4
89
x 10
IP-----7 2
AA&
0
Using the displacement relationship shown in Fig. 2, we have V(1, T) = h(r)tan 4. The transient transverse amplitude, V( 1, r), is then obtained and shown in Fig. S(e). From eqn (40), the transient horizontal displacement & at end point B along the X axis is obtained and shown in Fig. 5(f). A slider+rank mechanism with the usual proportions (connecting rod longer than crank) has two limiting positions, both occurring when the crank and connecting rod are collinear, i.e. Q = 0. At the limiting positions, we have zero transverse displacement, v(Z, I) = 0, from eqn (3), and the horizontal displacement equals the axial deformation, Db = II(~), from eqn’(40). The results in Fig. 6(a-c) are obtained for comparing among Timoshenko and Euler beam theories and those of assuming the ends of connecting rod are simply-supported. The longitudinal displacement U(1, r), is shown in Fig. 6(a). Using the displacement relationship shown in Fig. 2, we have V(1, T) = h(r)tan 4. The transient transverse amplitude, V( 1, r), is then obtained and shown in Fig. 6(b). From eqn (40), the transient horizontal displacement Db at end point B along the X axis is obtained and shown in Fig. 6(c). Since the effects of rotary inertia and shear deformation are considered in the Timoshenko beam theory, the transient amplitudes shown in Fig. 6(a-c) are larger than those of Euler beam theory. The transient amplitudes are always zero for the assumption of simply-supported ends.
-2 w.9 -4 v
-6
1
t1
lb) -4 x 10 1.5 ,
(-.-.):
Simple suppert sssumpti011.
vu,f)
20 4. CONCLUSIONS This paper studies the time-dependent boundary effect on the dynamic responses of the flexible connecting rod of a slider-crank mechanism. From the formulation of the governing equations, the numerical results and discussion, we conclude that: (1) The significant new contribution provided in this paper is the definition of a revise boundary condition at the joint between the connecting rod and the slider. From the derivation it is found that the boundary condition of the connecting rod moving with slider is not a simply-supported boundary condition but
Fig. 6. (a) Transient axial deformation U(1, T) of connecting rod at end point B: (-.-.) simple-support assumption, (---) Euler beam, (-) Timoshenko beam; (b) transient transverse amplitude V(1,~) of connecting rod at end point 8: (-.-.) simple-support assumption, (---) Euler beam, (-) Tiihenko beam, (c) transient horizontal displacement Db of connecting rod at end point B: (-.-.) simple-support assumption, (---) Euler beam, (-) Timoshcnko beam.
‘5
Cc)
-4
x 10 4
-2 Db -4
_iIi[ 0
-):
(-.-.) : Simple support assumption, (y):Tinyosha&;heam:
ytmun, 20
60 r
80
100
1
120
90
Rong-Fong Fung
(2)
being extended and replaced by V( 1, T) = .!I’(1, r)tan 4 in this paper. Based on the theory developed, the elastic deformation at the slider position could be predicted. Therefore, the accuracy will be improved in the mechanical design. Another contribution of the paper is in the variable transformation from the body-fixed coordinate system to that with simpler boundary condition, i.e. new variables being zero at the boundary. Obtaining the simple boundary condition allows for the choice of simple harmonic functions in the used expansions for the vibration of the beam.
Acknowledgement-Support of this work by the National Science Council of the Republic of China through Contract NSC 842212-E033-009 is gratefully acknowledged.
REFERENCES
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3. J. P. Sadlerand and G. N. Sandor, A lumped parameter approach to vibration and stress analysis -of elastic linkaaes. ASME J. Enann Indusr.. 549-557 (1973). 4, S. CT Chu and K. CT Pan, Dynamic response ‘of a high-speed slider-crank mechanism with an elastic connecting rod. ASME J. Engng Indust.. 542-550 (1975). M. Badlani and W. Kleinhenz, Dynamic stability of 5’ elastic mechanism. ASME J. Mech. Des., 149-153 (1979). 6. M. Badlani and A. Midha, Member initial curvature effects on the elastic slider-crank mechanism response. 7, ASME J. Mech. Des., 159-167 (1982). Z. G. Zhu and Y. Chen, The stability of the motion of a connecting rod. ASME J. Mech., Transmissions Automation Des., 637-640 (1983). 8. I. G. Tgdjbakhsh and C. J. Younis, Dynamic stability of the flexible connecting rod of a slider-crank mechanism. ASME J. Mech., Transmissions Automation Des., 487496 (1986). 9, A. G. Erdman, Modern Kinematics, Developments in the Lost For?y Years. Wiley, New York, Chapter 9 (1993). 10. S-R. Hsieh and S. W. Shaw, The dynamic stability and nonlinear resonance of a flexible connecting rod: single-mode model. J. Sound Vibr. 170(l), 2549 (1994). 11. R. F. Fung, Dynamic analysis of the flexible connecting rod of a slider-crank mechanism. ASME J. Vibr. Acoustics (1995) (submitted). 12. C. H. Jou, Dynamic stability of a high-speed slider-crank mechanism with a flexible connecting rod. Master thesis, University of Chun Yuan, Chung-Li, Taiwan (1992).