• 沒有找到結果。

The Role of Lagrangian Strain in the Dynamic Response of a Flexible Connecting Rod

N/A
N/A
Protected

Academic year: 2021

Share "The Role of Lagrangian Strain in the Dynamic Response of a Flexible Connecting Rod"

Copied!
7
0
0

加載中.... (立即查看全文)

全文

(1)

Jen-San Chen

Professor

Kwin-Lin Chen

Graduate Student

Department of Mechanical Engineering, National Taiwan University, Taipei, Taiwan 10617

The Role of Lagrangian Strain in

the Dynamic Response of a

Flexible Connecting Rod

Previous researches on the dynamic response of a flexible connecting rod can be catego-rized by the ways the axial load in the rod is being formulated. The axial load may be assumed to be (1) dependent only on time and can be obtained by treating the rod as rigid, (2) related to the transverse displacement by integrating the axial equilibrium equation, and (3) proportional to linear strain. This paper examines the validity of these formulations by first deriving the equations of motion assuming the axial load to be proportional to the Lagrangian strain. In order for the dimensionless displacements to be in the order of O(1), different nondimensionalization schemes have to be adopted for low and high crank speeds. The slenderness ratio of the connecting rod arises naturally as a small parameter with which the order of magnitude of each term in the equations of motion, and the implication of these simplified formulations can be examined. It is found that the formulations in previous researches give satisfactory results only when the crank speed is low. On the other hand when the crank speed is comparable to the first bending natural frequency of the connecting rod, these simplified formulations overestimate con-siderably the dynamic response because terms of significant order of magnitude are re-moved inadequately. 关DOI: 10.1115/1.1415738兴

Introduction

Dynamic analysis of flexible mechanisms has become an im-portant research topic in recent years, partly due to the increasing precision demand in machines operating at high speed. The sub-jects of interest include transient response, steady state response, and dynamic stability analysis. In general the mathematical com-plexity involved in the analysis is so overwhelming that finite element method may be the only feasible technique to attack the problems关1–3兴. However considerable insight may be obtained by focusing on the detailed mathematical investigation of certain mechanism which retains the important characteristics of dynamic vibration while renders itself to solvability. The simplest mecha-nism of such kind is a slider-crank mechamecha-nism with elastic con-necting rod.

The flexible connecting rod can be considered as a beam un-dergoing large rigid body motion and may deform in both the axial and transverse directions. A complete formulation must take into account the coupling effect of the axial and transverse vibra-tions. Due to the inherent mathematical complexity, it was very common in the past to adopt various assumptions in the formula-tion to simplify the soluformula-tion procedure. It is important to verify the validity of these assumptions with more accurate formulation. In the following, we categorize the common simplifications adopted in the literature based on the ways the axial load in the rod is being formulated.

1 Assume that the axial force is a function of time only and can be obtained by assuming that the connecting rod is rigid. The resulted equation of motion in terms of the transverse displace-ment is an inhomogeneous Mathieu equation. Neubauer et al.关4兴 adopted this simplest approach to investigate the effects of crank speed, length ratio between crank arm and connecting rod, and slider mass on the dynamic response of the connecting rod. The solutions from normal-mode expansion and from finite difference method are compared. Badlani and Midha关5,6兴 also adopted this approach to investigate the dynamic response of a connecting rod

with initial curvature and material damping. Both Euler beam and Timoshenko beam are considered by Badlani and Kleinhenz关7兴. Tadjbakhsh关8兴 first formulated the equations of motion for a gen-eral mechanism and later focused on a slider-crank mechanism. He used Floquet theory to determine the critical values of various dimensionless parameters which may cause instability of the mechanism, see also关9兴. Zhu and Chen 关10兴 used a perturbation technique to study the dynamic stability of the connecting rod.

2 Relate the axial force to the transverse displacement by inte-grating the axial equilibrium equation after neglecting the axial vibration and the relative centrifugal force. This formulation is an improvement of formulation 共1兲, and has been adopted by Vis-comi and Ayre 关11兴, and Hsieh and Shaw 关12兴. Hsieh and Shaw 关12兴 also investigated the foreshortening effect on the parametric resonance of the mechanism.

3 Consider the effect of axial vibration, but assume that the axial force is proportional to the linear axial strain. The two equa-tions of motion involve both the axial and transverse displace-ments. Jasinski et al. 关13,14兴 adopted this approach but ‘‘linear-ized’’ the equations of motion by deleting all the terms involving product of displacements. As a consequence the stiffening effect of the axial load is missing. Chu and Pan关15兴 also adopted this approach but used inadequate shape function for the axial dis-placement in their Galerkin’s discretization procedure. As a con-sequence the stiffening effect of the axial load was also removed. These two papers did not consider the inertia of the slider. Fung and Chen关16兴 considered the steady state response of a Timosh-enko beam with time dependent boundary conditions based on a finite element formulation. They also ‘‘linearized’’ the equations of motion by dropping all the terms containing products of displacements.

Although these above simplifications reduce the complexity of the solution procedure considerably, it has never been examined in the literature to what extent these assumptions are valid. One ob-vious problem with the linear strain formulation, for instance, is when the crank rotates at a speed comparable to the bending natu-ral frequency, it is reasonable to argue that the quadratic compo-nent in the axial Lagrangian strain may not be negligible any more. In the present paper we start the analysis by adopting the

Contributed by the Design Automation Committee for publication in the JOUR

-NAL OFMECHANICALDESIGN. Manuscript received May 1999. Associate Editor: H. Lankarani.

(2)

most rigorous approach assuming that the axial force is propor-tional to the Lagrangian axial strain, which involves both the axial and transverse displacements. The equations of motion are then nondimensionalized by assuming that the dimensionless displace-ments are of order O共1兲. This assumption is checked later by numerical simulation. The slenderness ratio of the connecting rod arises automatically as a small parameter and can be used to de-termine the relative significance of each term in the equations of motion. It is found that the nondimensionalization schemes ought to be different for the high and low crank speeds. With the appro-priate dimensionless equations of motion for different crank speeds, we can then simplify the equations of motion by ignoring those terms with smaller order of magnitude. It is found that the three simplifications give similar result as the Lagrangian strain formulation does when the crank speed is low. On the other hand they induce considerable errors when the crank speed is compa-rable to the first bending natural frequency of the connecting rod because terms of significant order of magnitude are ignored inad-equately.

Equations of Motion

Figure 1 shows a slider-crank mechanism. The rigid crank with length a rotates with constant speed⍀. The length, cross section area, mass density, and Young’s modulus of the elastic connecting rod are L, A,␳, and E, respectively. The mass of the slider is ms.

XOY is an inertial frame with its origin attached at the center of the rotating crank. xAy is a moving frame with x-axis passing through the two ends of the connecting rod. u(x,t) and v(x,t)

denote the axial and transverse displacements of the connecting rod. From the force balance in the axial and transverse directions and the moment balance of a small element of the connecting rod, we obtain the three equilibrium Eqs.关7兴,

P,x⫽␳A关u,tt⫺2v,t␾˙⫺共x⫹u兲␾˙2⫺v␾¨⫺a⍀2cos共⍀t⫺␾兲兴 (1) Q,x⫽␳A关v,tt⫹2u,t␾˙⫺v␾˙ 2⫹共x⫹u兲␾¨⫺a⍀2 sin共⍀t⫺␾兲兴 (2) M,x⫹Q⫺Pv,x⫽0 (3)

P, Q, and M are the axial force, shear force, and bending moment, respectively, at position x on the connecting rod.␾ is the angle between the x-axis and X-axis, and can be obtained from rigid body kinematics,

␾⫽sin⫺1

a sin⍀t

L

(4)

The constitutive laws for an Euler beam are used,

P⫽EA

u,x⫹ 1 2v,x 2

(5) M⫽EIv,xx (6)

I is the area moment of inertia of the connecting rod, I⫽Ar2

where r is the radius of gyration of the cross section. By substi-tuting Eqs. 共3兲,共5兲,共6兲 into Eqs. 共1兲,共2兲 we obtain the following equations of motion, u,tt⫺␾˙2uE ␳ 共u,xx⫹v,xv,xx兲⫺2␾˙v,t⫺␾¨v⫺x␾˙2 ⫺a⍀2cos共⍀t⫺␾兲⫽0 (7) v,tt⫺␾˙2vEI

␳Av,xxxx⫹2␾˙u,t⫹␾¨u

E

u,xxv,x⫹u,xv,xx⫹ 3 2v,x 2 v,xx

⫹x␾¨⫺a⍀2 sin共⍀t⫺␾兲⫽0 (8)

The connecting rod is assumed to be simply-supported at two ends. Therefore the boundary conditions for u andv are

v共0,t兲⫽v共L,t兲⫽v,xx共0,t兲⫽v,xx共L,t兲⫽u共0,t兲⫽0 (9) From the force balance of the slider we can obtain the boundary condition for u at x⫽L, u,x⫹ 1 2v,x 2

x⫽L ⫽⫺ 1 EA cos

⫺ 1 3␳AL 2␾¨ sin ␾ ⫹1 2␳ALa⍀ 2 sin␾ sin共⍀t⫺␾兲

⫹ms共⫺a⍀2cos⍀t⫺L␾˙2cos␾⫹a␾¨ sin ⍀t兲

(10) Low Crank Speed

We now investigate the case when the crank rotates at low speed compared to the first bending natural frequency of the con-necting rod. Equations共7兲 and 共8兲 can be nondimensionalized by introducing the following dimensionless quantities,

x*⫽x L, a*⫽ a L, u*⫽ L2 r3u, v*⫽ Lv r2, ␧⫽ r L, ms*⫽ ms ␳AL, P*⫽ P EA, ⍀*⫽ ⍀ ␻b , t*⫽␻bt

bis the lowest bending natural frequency of the connecting rod,

b

␲2

L2

EI ␳A

The ratio between the bending natural frequency and the longitu-dinal natural frequency of the connecting rod is proportional to␧, the slenderness ratio parameter. In the above nondimensionaliza-tion we have assumed that both the dimensionless displacements u*andv*are of the order O共1兲. Whether this assumption is valid can be verified by numerical simulation later. After substituting these relations into Eqs.共7兲, 共8兲, and 共10兲, and dropping the su-perposed asterisks for simplicity, we can rewrite the equations of motion and the inhomogeneous boundary condition in the follow-ing forms,

(3)

␧3共u ,tt⫺␾˙2u兲⫺␧2

2␾˙v,t␾¨v⫹ 1 ␲4v,xv,xx

⫺␧14u,xx⫺x␾˙2⫺a⍀2cos共⍀t⫺␾兲⫽0 (11) ⫺3␧ 4 2␲4v,x 2 v,xx⫹␧3

2␾˙u,t⫹␾¨u⫺ 1 ␲4共u,xxv,x⫹u,xv,xx

⫹␧2

v ,tt⫺␾˙2v⫹ 1 ␲4v,xxxx

⫹x␾¨⫺a⍀2sin共⍀t⫺␾兲⫽0 (12) P共1,t兲 ␧2 ⫽␧u,x⫹␧2 1 2v,x 2

x⫽1 ⫽⫺ ␲ 4 cos␾ 共⌰⫺msa⍀ 2 cos⍀t兲 (13) where ⌰⫽a3␾¨ sin ⍀t⫺1 2a 22sin⍀t sin共⍀t⫺␾兲

⫹ms共⫺␾˙2cos␾⫹a␾¨ sin ⍀t兲

We assume a one-mode approximation for u andv as following

u共x,t兲⫽ f 共t兲sin␲x

2 (14)

v共x,t兲⫽g共t兲sin ␲x (15)

After substituting Eqs.共14兲 and 共15兲 into Eqs. 共11兲 and 共12兲, mul-tiplying Eqs.共11兲 and 共12兲 by sin(␲x/2) and sin ␲x, respectively, and then integrating by parts from x⫽0 to 1 we obtain

␧3共 f¨⫺␾˙2f兲⫺␧2

8 3␲ 共2␾˙g˙⫹␾¨g兲⫺ 7 15␲2g 2

␧ 4␲2f82␾˙ 24 ␲a⍀2cos共⍀t⫺␾兲cos2␾ 共⌰⫺msa⍀2cos⍀t兲 (16) 3␧4 8 g 3⫹␧3

8 3␲ 共2␾˙ f˙⫹␾¨ f 兲⫹ 14 15␲2f g

⫹␧ 2共g¨⫺␾˙2g⫹g兲 ⫽⫺␲ ␾2 ¨ ⫹4 ␲a⍀2sin共⍀t⫺␾兲 (17)

By specifying initial conditions, Eqs.共16兲 and 共17兲 can be solved by Runge-Kutta method. In the following we compare the com-mon simplifications adopted in the literature with Eqs.共16兲 and 共17兲.

1. Linear Strain Simplification. If we start the analysis by assuming linear strain in Eq.共5兲, then we have to modify Eq. 共11兲 by deleting the term containing␧2v

,xv,xx, and modify共12兲 by deleting the term containing ␧4v,x

2

v,xx. As a consequence, Eqs.

共16兲 and 共17兲 should be modified by dropping the terms contain-ing␧2g2and␧4g3, respectively, i.e.,

␧3共 f¨⫺␾˙2 f兲⫺␧2 8 3␲ 共2␾˙g˙⫹␾¨g兲⫹ ␧ 4␲2f82␾˙ 24 ␲a⍀2cos共⍀t⫺␾兲 ⫺cos2␾ 共⌰⫺msa⍀2cos⍀t兲 (18) ␧3

8 3␲ 共2␾˙ f˙⫹␾¨ f 兲⫹ 14 15␲2f g

⫹␧ 2共g¨⫺␾˙2 g⫹g兲 ⫽⫺␲ ␾2 ¨ ⫹4a⍀2sin共⍀t⫺␾兲 (19)

Since these ignored terms are small in order of magnitude com-pared to other dominant terms, we expect that at low speed range, the linear strain assumption is a good approximation when the slenderness ratio␧ is small.

2. Axial Load by Integrating Axial Equilibrium Equation. In this approach we rewrite Eq.共11兲 by ignoring the ␧3-terms,

P,x⫽⫺␧2␲4关␧2共2␾˙v,t␾¨v兲⫹x␾˙2⫹a⍀2cos共⍀t⫺␾兲兴 (20) The axial load P can be integrated to

P共x,t兲⫽P共1,t兲⫺

1 x ␧24关␧2共2␾˙v ,t␾¨v兲 ⫹␰␾˙2⫹a⍀2 cos共⍀t⫺␾兲兴d␰ (21)

P(1,t) has been given in Eq. 共13兲. Equation 共21兲 can then be substituted into Eq.共12兲 to obtain

14共Pv,x,x⫹␧3关2␾˙u,t␾¨u兴⫹␧2

v,tt⫺␾˙2v⫹ 1 ␲4v,xxxx

⫹x␾¨⫺a⍀2

sin共⍀t⫺␾兲⫽0 (22)

Equation共22兲 is similar to the equation used by Viscomi and Ayre 关11兴, and can be discretized by the one-mode approximation, Eq. 共15兲, to ␧2

msa⍀ 22 cos␾ cos⍀tg⫹g

⫹␧ 2g

5 4⫹ ␲2 3

␾˙ 2 ⫹␲ 2 2 a⍀ 2cos共⍀t⫺␾兲⫺␲ 2 cos␾

⫹␧ 4␲共␾¨g2⫹2␾˙gg˙兲 ⫽⫺␲ ␾2 ¨ ⫹4a⍀2sin共⍀t⫺␾兲 (23)

It is noted that Eq.共23兲 can also be obtained by expressing f in terms of g after ignoring the␧3-terms in Eq.共18兲, and then sub-stituting this expression into Eq.共19兲. The resulted single equation in terms of g is very similar to Eq.共23兲 except some minor dif-ferences on the coefficients due to slightly different Galerkin’s procedure. Since Eq.共23兲 is different from Eq. 共18兲 only in the high order␧3-terms, we expect this to be a good approximation to the linear strain formulation.

3. Time-Dependent-Only Axial Load Simplification. In this approach we further neglect the right hand side of the axial equi-librium Eq.共20兲 to obtain

P,x⫽0 (24)

This dramatic simplification may be justified by assuming the crank-to-rod ratio a to be small. The angle␾ in Eq. 共4兲 can then be approximated as

␾⬇⫺a sin ⍀t

By dropping the smaller terms containing a2 in the inhomoge-neous boundary condition共13兲 we obtain the axial load as a func-tion of time only

P共t兲⫽␧2␲4msa⍀ 2

cos⍀t (25)

By dropping the terms␧3(2␾˙u

,t⫹␾¨u) and ␧2␾˙2v in Eq.共22兲 and making use of Eq.共25兲, we finally arrive at the single equation

(4)

␧2

v,tt⫹ 1 ␲4v,xxxx⫺msa⍀2cos⍀tv,xx

⫽共1⫺x兲a⍀2 sin⍀t⫹a 2 2 ⍀ 2 sin 2⍀t (26) Equation共26兲 is the one used by Badlani and Midha 关5兴. After Galerkin’s procedure, Eq.共26兲 can be discretized to

␧2共g¨⫹m

sa⍀2␲2cos⍀tg⫹g兲⫽ 2a⍀2

␲ 共sin⍀t⫹a sin 2⍀t兲 (27) Equation 共27兲 is a standard inhomogeneous Mathieu equation 关17兴. Equation 共27兲 can also be obtained from Eq. 共23兲 by ignor-ing the␧4-terms and the terms inside the bracket.

Runge-Kutta Simulation. The responses f and g of the con-necting rod at⍀⫽0.1 based on the above four formulations can be calculated by Runge-Kutta method. The parameters used in the calculations are a⫽0.1, ␧⫽0.01, ms⫽0.1. The initial conditions are

f共0兲⫽ f˙共0兲⫽g共0兲⫽g˙共0兲⫽0

The time step used in Runge-Kutta method is 10⫺3. Figure 2 shows the lateral response g from t⫽0 to 200. The results from all four formulations are almost indistinguishable and we plot only the one from Lagrangian strain formulation as line共1兲 in Fig. 2. Figure 3 shows the response f from t⫽0 to 1. Again the results from linear and Lagrangian strain formulations are indistinguish-able. In this small time span from t⫽0 to 1, we observe only the free axial vibration component共homogeneous solution兲 of the re-sponse f. In order to observe the forced rere-sponse component 共par-ticular solution兲 in the axial direction, we plot the envelop of response curve f from t⫽0 to 200 in Fig. 4. Line 共1兲 in Fig. 4 represents the envelope calculated from Lagrangian strain formu-lation. All other three simplified formulations give indistinguish-able envelopes in Fig. 4 and are plotted as line 共2兲. It is also important to observe that both f and g are of the order O共1兲, as was required in examining the order of magnitude of each term in the equations of motion.

As observed from Figs. 2 and 4 at low crank speed, both f and g contain two harmonic components, one corresponds to free vi-bration while the other corresponds to forced response. To predict these two components in closed forms we simplify Eqs.共16兲 and 共17兲 by removing higher order terms in ␧ and a to

␧3 ␧ 4␲2f

2ms⫹ 4 ␲

a⍀2cos⍀t (28) ␧2共g¨⫹g兲⫽2 ␲a⍀2sin⍀t (29)

After applying the initial conditions the solutions f and g of these two uncoupled equations are

f⫽ 4␲ 2a2 ␧共1⫺4␧222

2ms⫹ 4 ␲

冊冋

cos⍀t⫺cos

t 2␧␲

冊册

(30) g2a⍀ 2 ␧2␲共1⫺⍀2兲 共sin⍀t⫺⍀ sin t兲 (31)

Solution 共31兲 is plotted in Fig. 2 as line 共2兲. The envelope of solution共30兲 is plotted in Fig. 4 as line 共3兲. The fact that these two solutions are very good approximation to those from Lagrangian strain formulation confirms the general belief that at low crank speed the coupling effect between the axial and the transverse vibrations are negligible.

Fig. 2 Response g of the connecting rod at ⍀Ä0.1.1 La-grangian strain formulation.2Solution predicted by Eq.31.

Fig. 3 Responsefof the connecting rod at⍀Ä0.1

Fig. 4 Envelope of responsefof the connecting rod at⍀Ä0.1.

1Lagrangian strain formulation.2Linear strain formulation.

(5)

High Crank Speed

We now examine the case when the crank rotates at a speed comparable to the first bending natural frequency of the connect-ing rod. By expectconnect-ing the displacements to be larger than those at low speed we redefine the dimensionless displacements as

u*⫽L

r2u, v*⫽

v

r

The definitions of other parameters remain the same. Again, whether displacements u* andv*are of the order O共1兲 needs to be checked by numerical simulation later. The dimensionless equations of motion and the inhomogeneous boundary condition are rewritten in the following forms,

␧2共u ,tt␾˙2u兲⫺␧共2␾˙v,t␾¨v兲⫺ 1 ␲4共u,xx⫹v,xv,xx⫺x␾˙2⫺a⍀2cos共⍀t⫺␾兲⫽0 (32) ␧2共2␾˙u ,t⫹␾¨u兲⫹␧

v,tt␾˙2v⫹ 1 ␲4

v,xxxx⫺u,xxv,x⫺u,xv,xx ⫺32v,x2v,xx

冊册

⫹x␾¨⫺a⍀2sin共⍀t⫺␾兲⫽0 (33) P共1,t兲 ␧2 ⫽u,x⫹ 1 2v,x 2

x⫽1 ⫽⫺ ␲ 4 cos␾ 共⌰⫺msa⍀ 2cos⍀t兲 (34) We notice from Eq.共32兲 that the terms v,xv,xxand u,xxare of the same order of magnitude and both are dominant terms at high crank speed. Therefore, it will be erroneous to delete a term con-taining product of displacements as in the conventional ‘‘linear-ization’’ procedure. The termv,xv,xx is from the quadratic com-ponent in the Lagrangian strain, while the term u,xx is from the linear component in the strain. Similar observation on Eq. 共33兲 also reveals that the contribution from the quadratic component in the Lagrangian strain is as significant as the one from the linear component at high crank speed.

Following the similar one-mode Galerkin’s procedure, we ob-tain the discretized equations

␧2共 f¨⫺␾˙2f兲⫺␧ 8 3␲ 共2␾˙g˙⫹␾¨g兲⫹ 1 ␲2

f 4⫹ 7 15g 2

82␾˙ 24 ␲a⍀2cos共⍀t⫺␾兲cos2␾ 共⌰⫺msa⍀2cos⍀t兲 (35) ␧2 8 3␲ 共2␾˙ f˙⫹␾¨ f 兲⫹␧

␾˙ 2 g⫹g⫹ 14 15␲2f g⫹ 3 8g 3

⫽⫺␲ ␾2 ¨ ⫹4a⍀2sin共⍀t⫺␾兲 (36)

1. Linear Strain Simplification. In the case if we start the analysis by assuming linear strain in Eq.共5兲 we have to modify Eq.共32兲 by deleting the term containing v,xv,xx, and modify共33兲 by deleting the term containing␧v,x

2

v,xx. As a consequence, Eqs.

共35兲 and 共36兲 should be modified by dropping the terms contain-ing g2 and ␧g3, respectively. Since these terms are actually the dominant terms in the equations of motion, deleting them will induce significant errors. Therefore, we expect that linear strain assumption is a poor approximation at high crank speed.

2. Axial Load by Integrating Axial Equilibrium Equation. By following the similar procedure as in the low crank speed range, we arrive at the following equations

␧␲14共Pv,x,x⫹␧2关2␾˙u,t⫹␾¨u兴⫹␧

v,tt⫺␾˙2v⫹ 1 ␲4v,xxxx

⫹x␾¨⫺a⍀2sin共⍀t⫺␾兲⫽0 (37)

msa⍀ 22 cos␾ cos⍀tg⫹g

⫹␧g

⫺ 5 4⫹ ␲2 3

␾˙ 2 ⫹␲ 2 2 a⍀ 2cos共⍀t⫺␾兲⫺␲ 2 cos␾

⫹␧ 2␲共␾¨g2⫹2␾˙gg˙兲 ⫽⫺␲ ␾2 ¨ ⫹4a⍀2sin共⍀t⫺␾兲 (38)

3. Time-Dependent-Only Axial Load Simplification. In this case the equation of motion and the discretized equation are

v,tt⫹ 1 ␲4v,xxxx⫺msa⍀2cos⍀tv,xx

⫽共1⫺x兲a⍀2sin⍀ta 2 2 ⍀ 2 sin 2⍀t (39) ␧共g¨⫹msa⍀2␲2cos⍀tg⫹g兲⫽ 2a⍀2

␲ 共sin⍀t⫹a sin 2⍀t兲 (40) Runge-Kutta Simulation. Figure 5 shows the response g of the connecting rod at⍀⫽0.8. The other parameters are the same as in the low crank speed case. Line共1兲 represents the response from the Lagrangian strain formulation. Line共2兲 is from linear strain simplification. Lines共3兲 and 共4兲 are from Eqs. 共38兲 and 共40兲, respectively. At this high speed the response from the three sim-plified formulations are almost ten times larger than the one pre-dicted from Lagrangian strain formulation. Figure 6 shows the response f from the Lagrangian strain and linear strain formula-tions.

While the three simplified formulations fail to reproduce the results obtained from Lagrangian strain formulation, it is possible to simplify Eqs.共35兲 and 共36兲 directly by neglecting the higher order terms. By ignoring the ␧2- and ␧-terms in Eq. 共35兲 and assuming a to be small, we can express f in terms of g as

Fig. 5 Responseg of the connecting rod at ⍀Ä0.8.1 La-grangian strain formulation.2Linear strain formulation.3Axial load by integrating axial equilibrium equation.4 Time-dependent-only axial load formulation.5Response predicted by Eq.42.

(6)

f⫽⫺28 15g

2⫹8a␲⍀2共␲m

s⫹2兲cos ⍀t (41)

After substituting Eq. 共41兲 into Eq. 共36兲, we obtain the single equation for g,

g¨⫹g⫹112 15 a⍀ 2 cos⍀t

ms⫹ 2 ␲

g

3 8⫺ 392 225␲2

g 3

2a⍀ 2

␲ 共sin⍀t⫹a sin 2⍀t兲 (42)

The response from Eq.共42兲 is plotted in Fig. 5 as line 共5兲, which approximates line共1兲 much better than those from other simplifi-cations. The main difference between Eq.共42兲 and those from the simplified formulations is the cubic term. If this cubic term is ignored, Eq.共42兲 will return to the inhomogeneous Mathieu equa-tion. Since this cubic term is also a dominant term, it will have a substantial effect on the stability property of the Mathieu equa-tion. It is noted that if a approaches zero, on the other hand, Eq. 共42兲 will become a Duffing equation 关18兴.

Accuracy of One-Mode Approximation

Up to this point we have assumed that the one-mode approxi-mation can well represent the essence of responses u andv. While

this is usually the case from previous experience, it is desirable to check the accuracy of this assumption. For more accurate approxi-mation we can assume multiple-mode expansion as

u共x,t兲⫽

n⫽1 N fn共t兲sin 共2n⫺1兲␲x 2 (43) v共x,t兲⫽

n⫽1 N gn共t兲sin n␲x (44)

where N is the number of modes used in the expansion. After substituting Eqs.共43兲 and 共44兲 into Eqs. 共32兲 and 共33兲, multiply-ing Eqs.共32兲 and 共33兲 by sin关(2n⫺1)␲x/2兴 and sin n␲x, respec-tively, and integrating by parts from x⫽0 to 1 we obtain 2N equations for fnand gn. Runge-Kutta method can then be used to calculate the solution. For a two-mode approximation N⫽2, lines 共2兲 and 共3兲 in Fig. 7 represent g1 and g2, respectively when

⍀⫽0.8. From the observation that g2is much smaller than g1in magnitude, we confirm that the first mode is much more important than the second one in approximating the solutionv. The solution

g from the one-mode approximation by solving Eqs.共16兲 and 共17兲 is also plotted in Fig. 7 as line共1兲 for comparison. Since line 共1兲

is fairly close to line共2兲, we conclude that the one-mode approxi-mation can indeed retain the essence of the solutions u andv.

Conclusions

In this paper we point out for the first time that the Lagrangian strain formulation and the axial constitutive relation are required in calculating the response of a flexible connecting rod with crank speed comparable to the first bending natural frequency of the connecting rod. The conventional approach by assuming either the axial load is dependent on time only or is proportional to the linear axial strain gives satisfactory result only when the crank speed is low.

The first potential impact of this conclusion is that this could be the general case in all flexible mechanisms. Therefore, in devel-oping design software for dynamic analysis of general flexible mechanisms based on more powerful technique such as finite el-ement method, it is crucial to retain the coupling effect between the axial and the transverse vibrations. Secondly, as pointed out in Eq.共42兲 the cubic term from Lagrangian strain formulation may have a substantial effect on the parametric instability of the Mathieu equation obtained from conventional simplifications. Therefore, the critical crank speed predicted by the conventional formulation关7,12兴 may need a detailed examination. These two possible research directions should be undertaken in the future in order for us to better understand the dynamic response of flexible mechanisms operating at high speed.

Acknowledgment

The results presented here were obtained in the course of re-search supported by a grant from the National Science Council of the Republic of China.

References

关1兴 Midha, A., Erdman, A. G., and Frohrib, D. A., 1978, ‘‘Finite Element

Ap-proach to Mathematical Modeling of High Speed Elastic Linkages,’’ Mech. Mach. Theory, 13, pp. 603– 618.

关2兴 Shabana, A. A., 1989, Dynamics of Multibody Systems, John Wiley and Sons,

New York.

关3兴 Nagarajan, S., and Turcic, D. A., 1990, ‘‘General Methods of Determining

Stability and Critical Speeds for Elastic Mechanism Systems,’’ Mech. Mach. Theory, 25, No. 2, pp. 209–223.

关4兴 Neubauer, A. H., Cohen, R., and Hall, A. S., 1966, ‘‘An Analytical Study of the

Dynamics of an Elastic Linkage,’’ ASME J. Eng. Ind., 88, pp. 311–317.

关5兴 Badlani, M., and Midha, A., 1982, ‘‘Member Initial Curvature Effects on the

Elastic Slider-Crank Mechanism Response,’’ ASME J. Mech. Des., 104, pp. 159–167.

关6兴 Badlani, M., and Midha, A., 1983, ‘‘Effect of Internal Material Damping on

Fig. 6 Response f of the connecting rod at ⍀Ä0.8.1

La-grangian strain formulation.2Linear strain formulation. Fig. 7tions atComparison of one-mode and two-mode approxima-⍀Ä0.8.1gfrom one-mode approximation.2g1and

(7)

the Dynamics of a Slider-Crank Mechanism,’’ ASME J. Mech., Transm., Au-tom. Des., 105, pp. 452– 459.

关7兴 Badlani, M., and Kleninhenz, W., 1979, ‘‘Dynamic Stability of Elastic

Mecha-nism,’’ ASME J. Mech. Des., 101, pp. 149–153.

关8兴 Tadjbakhsh, I. G., 1982, ‘‘Stability of Motion of Elastic Planar Linkages With

Application to Slider Crank Mechanism,’’ ASME J. Mech. Des., 104, pp. 698 – 703.

关9兴 Tadjbakhsh, I. G., and Younis, C. J., 1986, ‘‘Dynamic Stability of the Flexible

Connecting Rod of a Slider Crank Mechanism,’’ ASME J. Mech., Transm., Autom. Des., 108, pp. 487– 496.

关10兴 Zhu, Z. G., and Chen, Y., 1983, ‘‘The Stability of the Motion of a Connecting

Rod,’’ ASME J. Mech., Transm., Autom. Des., 105, pp. 637– 640.

关11兴 Viscomi, B. V., and Ayre, R. S., 1971, ‘‘Nonlinear Dynamic Response of

Elastic Slider-Crank Mechanism,’’ ASME J. Eng. Ind., 93, pp. 251–262.

关12兴 Hsieh, S. R., and Shaw, S. W., 1994, ‘‘The Dynamic Stability and Nonlinear

Resonance of a Flexible Connecting Rod: Single Mode Model,’’ J. Sound Vib., 170, pp. 25– 49.

关13兴 Jasinski, P. W., Lee, H. C., and Sandor, G. N., 1970, ‘‘Stability and

Steady-State Vibrations in a High Speed Slider-Crank Mechanism,’’ ASME J. Appl. Mech., 37, pp. 1069–1076.

关14兴 Jasinski, P. W., Lee, H. C., and Sandor, G. N., 1971, ‘‘Vibrations of Elastic

Connecting Rod of a High Speed Slider-Crank Mechanism,’’ ASME J. Eng. Ind., 93, pp. 636 – 644.

关15兴 Chu, S. C., and Pan, K. C., 1975, ‘‘Dynamic Response of a High Speed

Slider-Crank Mechanism With an Elastic Connecting Rod,’’ ASME J. Eng. Ind., 97, pp. 542–550.

关16兴 Fung, R.-F., and Chen, H.-H., 1997, ‘‘Steady-State Response of the Flexible

Connecting Rod of a Slider-Crank Mechanism With Time-Dependent Bound-ary Condition,’’ J. Sound Vib., 199, pp. 237–251.

关17兴 Bolotin, V. V., 1964, The Dynamic Stability of Elastic Systems, Holden-Day,

San Francisco.

关18兴 Nayfeh, A. H., and Mook, D. T., 1979, Nonlinear Oscillations, Wiley, New

數據

Figure 1 shows a slider-crank mechanism. The rigid crank with length a rotates with constant speed ⍀
Figure 3 shows the response f from t ⫽0 to 1. Again the results from linear and Lagrangian strain formulations are  indistinguish-able
Fig. 5 Response g of the connecting rod at ⍀Ä 0.8. „ 1 … La- La-grangian strain formulation

參考文獻

相關文件

好了既然 Z[x] 中的 ideal 不一定是 principle ideal 那麼我們就不能學 Proposition 7.2.11 的方法得到 Z[x] 中的 irreducible element 就是 prime element 了..

volume suppressed mass: (TeV) 2 /M P ∼ 10 −4 eV → mm range can be experimentally tested for any number of extra dimensions - Light U(1) gauge bosons: no derivative couplings. =>

For pedagogical purposes, let us start consideration from a simple one-dimensional (1D) system, where electrons are confined to a chain parallel to the x axis. As it is well known

The observed small neutrino masses strongly suggest the presence of super heavy Majorana neutrinos N. Out-of-thermal equilibrium processes may be easily realized around the

incapable to extract any quantities from QCD, nor to tackle the most interesting physics, namely, the spontaneously chiral symmetry breaking and the color confinement.. 

(1) Determine a hypersurface on which matching condition is given.. (2) Determine a

• Formation of massive primordial stars as origin of objects in the early universe. • Supernova explosions might be visible to the most

The difference resulted from the co- existence of two kinds of words in Buddhist scriptures a foreign words in which di- syllabic words are dominant, and most of them are the