Kinematic and Dynamic Design of Four-bar

Simultaneous Minimization of Shaking Moment, Driving Torque, and Bearing Reactions of Complete Force Balanced Linkages Ren-Chung Soong, Hong-Sen Yan

essential steps in the development of mechanisms and are usually treated as separate stages in the design process. Traditionally, the input speed of the driving link is assumed constant. The problem of vibration, noise and fatigue comes mainly from shaking force of the mechanisms, especially in high-speed applications. Therefore, the complete force balancing is normally an expected goal to the designers. Unfortunately, it usually conflicts with other dynamic performance such as shaking moments, bearing reaction forces, and driving torque of the mechanisms. Since, the dynamic balancing performance of mechanisms mentioned above is mainly affected by their inertial properties and kinematic characteristics of the gravities of each moving link. Furthermore, they are dominated by the characteristics of the input link and link dimensions. Therefore, our purpose here is to develop a novel design method that integrates kinematic synthesis, dynamic balancing design, and input speed trajectory design to reach the trade-off of dynamic balancing and satisfy the kinematic requirements and constraints as well for force balanced four-bar linkages by varying the speed of the input link, counterweighing on moving links, and synthesizing the dimensions of each link. Relatively little research has been devoted to the techniques for combining kinematic synthesis and dynamic design. Starr (1973) considered the problem of determining the constrained path of a mechanism member between specified positions such that some quality of dynamic performance is optimized. Conte et al. (1975) synthesized the crank-rocker path generating mechanisms with prescribed timing by optimization based on minimizing shaking force, shaking moment, driving torque, and bearing reactions, respectively. Yan and Soong (2001, 2002, 2004) presented a kinematic synthesis and dynamic balancing integrated design approach for four-bar linkages. Variable input speed mechanisms were seldom discussed in the literature. Tesar and Matthew (1976) derived the motion equation of the follower by considering the case of variable speed cams. Yan

Keywords: variable input speed trajectory, planar linkages, kinematic synthesis, dynamic balancing.

ABSTRACT The present paper deals with an integrated design approach for complete force balanced planar linkages. It combines kinematic synthesis, dynamic balancing design, and the design of speed trajectory of input link in the same design step, satisfies kinematic design requirements, and reaches the trade-off of dynamic balancing with complete force balancing. By properly designing the variable input speed trajectory, the balancing parameters of moving links and link dimensions of the given or desired mechanisms, the expected output motion characteristics, and dynamic balancing performance are obtained. The input motion characteristics are designed with Bezier curves. Optimization is applied to find optimal design parameters for meeting kinematic and dynamic design requirements and constraints. The examples are given to demonstrate the feasibility of the proposed method.

INTRODUCTION Linkages are widely used in mechanical devices owing to their simplicity of structure, ease manufacturing, high reliability, and low cost. Kinematic synthesis and dynamic design are both

*

Associate

Professor,

Department

of

Automation

Engineering, Kao Yuan University, Kaohsiung 82141, TAIWAIN, R.O.C.

** Professor, Department of Mechanical Engineering, National

Cheng

Kung

University,

Tainan

70101,

TAIWAIN, R.O.C.

1

and associates (1996, 1996) contributed the objectives of eliminating the discontinuity in motion characteristics and lowering peak values of the follower acceleration in cam-follower systems by servo control. Yan and Chen (2000) further designed a general input speed trajectory for slider-crank mechanisms that lead to arbitrarily desired output motion. Methods of complete shaking force and moment balancing for planer linkages are developed and documented. Berkof (1973) provided a method for complete force and moment balancing of inline four-bar linkages by utilizing concept of physical pendulum and inertia counterweights. Arakelian and Smith (1999) proposed a new solution to the problem of complete shaking force and moment balancing of linkages by addition of articulated two-link groups to the mechanism to be balanced. Esat and Bahai (1999) shown that if a linkage can be full force balanced using the criterion of Tepper and Lowen, then it can be full force and moment balanced using geared counter-inertias. Arakelian and Dahan (2001) deal with a solution of the shaking force and shaking moment balancing of planar and spatial linkages by the minimization of the root-mean-square value of shaking moment with constant and variable input speed. Some of the balancing methods dealt directly with the bearing reaction magnitudes (Tepper and Lowen, 1975) and input torque magnitudes and fluctuations (Funk, 1993). However, very few methods allow a trade-off among variations and the magnitudes of the shaking force, shaking moment, bearing reactions, and the input torque (Qi and Pennestri, 1991, Tricamo and Lowen, 1983). Besides, there are two major different approaches for optimal balancing of planar linkages. One is obtained by positioning a counterweight on each moving link (Chiou and Bai, 1996, Tricamo and Lowen, 1983). The other one is by mass distribution of each link (Qi and Pennestri, 1991). The present paper is organized in the following way. The design flow chart and design criteria corresponding to a suitable speed trajectory of the input link are presented first. Then, dynamic equations and optimization procedure are provided. Finally, design examples are given to illustrate the design process for gaining the required output motion characteristics and dynamic balancing performance.

tentative linkage. If the result is not acceptable, then the combination design of balancing parameters and the speed trajectory of the input link is the next step. If the design result is still not acceptable, then the integrated design that adds the dimensional synthesis to the former design step to fulfill design requirements and constraints will be done. Figure 1 shows the design flow chart of the proposed approach. Comparing with the traditional design approaches, the concept of integrated design is presented for complete force balanced linkages. A Given Linkage

Design Balancing Parameters

yes

Satisfy Kinematic and Dynamic Design Requirements y and Constraints ? no Design Input Speed Trajectory & Balancing Parameters

yes

Satisfy Kinematic and Dynamic Design Requirements and Constraints ? no Design Input Speed Trajectory, Balancing Parameters & Link Dimensions

yes

Satisfy Kinematic and Dynamic Design Requirements and Constraints ? no

The Feasible Servo Linkage

Adjust Design Requirements and Constraints

Figure 1. Design flow chart

DESIGN APPROACH INPUT SPEED TRAJECTORY The proposed concept of combining kinematic synthesis and dynamic design is based on the consideration that both kinematic and dynamic design requirements and constraints are included in the same design step. The design procedure starts from designing balancing parameters to meet design requirements and constraints, based on a given or

We assume the input link of the planar linkages is a crank. The position trajectory of the crank is defined by an nth order Bezier curve φ (t ) with parameter t as follows:

2

shown in Fig. 2(a). Link elasticity and friction in the joints are neglected. The principle of virtual work and Newtonian dynamic analytical method are used to derive dynamic equations. The conditions of complete force balancing were derived in (Berkof, 1973) for four-bar linkages as follows:

n

φ (t ) = ∑ θ i ⋅ Bi , n (t ),

(1)

i =0

where

Bi , n (t ) =

n! ⋅ t i ⋅ (1 − t ) n − i i!⋅( n − i )!

t ∈ [0,1],

(2)

where φ (t ) is a Bezier curve that represent the angular displacement of the input link defined by control points θ i . Parameter t is regarded as the normalized time from 0 to 1. Since the Bezier curve is nth order differentiable, this guarantees smoothness of the entire motion. Hence, the angular velocity ω (t ) and acceleration α (t ) of the input link can be derived by continuously differentiating equation (1) and (2) with respect to time as follows:

m4 r4 = m3 r3

L4 , L3

(8)

ϕ2 =τ 3 ,

(9)

ϕ4 = ϕ3 + π ,

(10)

where mi is the mass of the ith link including

d 2 Bi , n (t ) d 2φ (t ) n = θ ⋅ , ∑ i dt 2 dt 2 i =0

(4)

counterweight, Li is the length of the ith link, ri is the distance from the fixed or moving pivot to the center of gravity of the ith link, ϕ i is the angle

dBi , n (t ) dt

d 2 Bi , n (t ) dt

(7)

(3)

measured from the center line between the two pivots to the center of gravity of the ith link, τ 3 is the

where

2

L2 , L3

dBi , n (t ) dφ (t ) n = ∑θ i ⋅ , dt dt i =0

ω (t ) =

α (t ) =

m 2 r2 = m3 e3

n! ⋅ t i −1 ⋅ (1 − t ) n − i (i − 1)!⋅(n − i )! n! − ⋅ t i ⋅ (1 − t ) n − i −1 , i!⋅(n − i − 1)!

angle measured from the center line between the pivots A and B to the center of gravity of link 3 counterclockwise, e3 is the distance from pivot B to

=

n! ⋅ t i − 2 ⋅ (1 − t ) n − i (i − 2)!⋅(n − i )! n! − ⋅ t i −1 ⋅ (1 − t ) n −i −1 (i − 1)!⋅(n − i − 1)! n! − ⋅ t i −1 ⋅ (1 − t ) n −i −1 (i − 1)!⋅(n − i − 1)! n! + ⋅ t1 ⋅ (1 − t ) n −i − 2 , i!⋅(n − i − 2)!

the center of gravity of link 3. On the conditions of complete force balancing, by applying the principle of virtual work to the system shown in Fig. 2(a), driving torque TD is obtained as follow:

(5)

=

4

TD = ∑ i =2

(6)

ki

ω2

,

(11)

where

k i = mi a i v i − Teiωi − Fei v ei + I iα i ωi ,

(12)

in which ω i and α i are, respectively, the angular

Based on kinematic analysis of the four-bar linkage by vector loop approach, all the kinematic magnitudes (positions, velocities and accelerations) of moving links and their centers of mass can be obtained as function of the crank motion.

velocity and acceleration of the ith link, Tei and

Fei are, respectively, the working torque and force applied on the ith link, I i is the moment of inertial of ith link about its center of gravity, v i and a i are, respectively, the velocity and acceleration vector of center of gravity of the ith link, v ei is the

DYNAMIC EQUATIONS

velocity vector of point of working force applied on the ith link.

The general analytical model and inertial properties of each moving link can be represented, as 3

Fe3y

Y L5

ρ3 μ Fe2y

r3

s3

e3

γ

φ3 ψ3 β3

Fe3x

τ3 B

Fe4y

L

A

TD

Fe2x

φ2 L2

s2 r2

Fe4x

r4

ψ2

μ2

F23 y =

β4

μ4

L1

O4

ψ1

O2

ρ2

s4

ψ4

φ4

β2

F23 x =

Te4 L4

Te2

Furthermore, based on the free-body diagrams shown in Fig. 2(b)-2(d), the close-form expressions are obtained for computing reactions of the coupler joints A and B as follows:

Te3

X ρ4

F43 x =

(a) Fe3y

F43 y =

Te3 m3

F23y

( Rv Bx + Sv By − P )v Ay − Qv By

( P − Rv Bx − Sv By )v Ax + Qv Bx V

V

( Rv Bx − P)v Ay + ( Sv Ay − Q)v Bx

Fe3x

where

Fe4y

P = ∑ K i − TD ω 2 ,

(14)

,

( P − Sv By )v Ay − ( Rv Ax − Q )v By

V

(13)

,

V

,

(15)

,

(16)

3

B A

F23x

(17)

i =2

Te4 4

Fe4x

m4

Fe3y Te3 F43y m3

R = m3 a 3 x − Fe 3 x ,

(19)

S = m 3 a 3 y − Fe 3 y ,

(20)

V = v Bx v Ay − v By v Ax ,

(21)

Fe3x in

B

A

m2

Fe2y

F43x

(18)

i =3

O4

(b)

Q = ∑ Ki ,

which

Feix

and

Feiy

are,

respectively,

components of the working force applied on the ith link in x and y direction, Fijx and Fijy are, respectively, components of force exerted on the jth link by the ith link in x and y direction, v Ax and

TD Te2

v Ay are, respectively, components of the velocity of

Fe2x

O2

(c)

point A on link 3 in x and y direction, v Bx and v By

Fe3y

are, respectively, components of the velocity of point B on link 3 in x and y direction, aix and aiy are,

Te4 F43y m3

F23y

respectively, components of the acceleration of the center of gravity of the ith link in x and y direction. By solving the equilibrium equations (13)-(16), respectively, all unknown reaction forces of joints, F12 and F14 are obtained as follows:

Fe3x F43x

B A

F23x (d)

Figure 2. The links counterweighing model of the four-bar linkage 4

F12 x = F23 x + m 2 a 2 x − Fe 2 x ,

(22)

F12 y = F23 y + m 2 a 2 y − Fe 2 y ,

(23)

F14 x = F43 x + m 4 a 4 x − Fe 4 x ,

(24)

F14 y = F43 y + m 4 a 4 y − Fe 4 y ,

OPTIMIZATION

(25)

In order to reach the trade-off of dynamic balancing of four-bar linkages that fulfill the kinematic design requirements, the undetermined control points ( θ 0 ,θ 1,L,θ n −1,θ n ), link dimensions

Once all joint forces are determined, the normalized shaking moments can be formulated as follows:

M sho / o2 = −( F14o y L1o cosφ1 − F14o x L1o sin φ1 + TDo + Teo2 + Teo3 + Teo4 + G1 + H 1 ), M

= −( F L sin φ1 − F

o sh / o4

o 12 x

o 1

o 12 y

L cos φ1 + o 1

T + T + T + T + G 2 + H 2 ), o D

o e2

o e3

o e4

(

(26)

F

(27)

ns

δ =1

(32)

Feo3 x ( Lo2 sin φ 2 + s 2o sin β 2 −

Subject to

(29)

equality constrained equations

L1o sin φ1 ) + Feo4 x s 4o sin β 4 ],

c j (θ1,..,θ n −1, ρ i , μ i , hi , L1 ,., L4 ) = 0

H 1 = [− Feo2 y s 2o cos β 2 + Feo3 y ( Lo2 cos φ 2 + s 2o cos β 2 ) +

inequality constrained equations

g j (θ 1,L,θ n−1, ρ i , μ i , hi , L1 ,K, L4 ) < 0

H 2 = [ Feo2 y ( s 2o cos β 2 − L1o cos φ1 ) + Feo3 y ( Lo2 cos φ 2 + s 2o cos β 2 −

(31)

j = 1,.., n g , (34)

where objkδ and obj dδ are, respectively, kinematic

s cos β 4 ],

o o e4 y 4

and dynamic objective function, wkδ and wdδ are, respectively, kinematic and dynamic weighting factor, ns denotes the number of kinematic and dynamic

in which M sh° / oI is the normalized shaking moment

φi

j = 1,.., nc , (33)

and equality constrained equations (7-10)

(30)

Feo4 y ( L1o cos φ1 + s 4o cos β 4 )],

L cos φ1 ) + F

parameters

f (θ 1,L,θ n −1, ρ i , μ i , hi , L1 ,K, L 4 ) = ∑ wkδ obj kδ + w dδ obj dδ ,

o 4

G 2 = −[ Feo2 x ( s 2o sin β 2 − L1o sin φ1 ) +

o 1

balancing

Minimizing

(28)

( L sin φ1 + s sin β 4 )], o 1

and

procedure. It is clear that θ 0 and θ n are the boundary conditions for the crank displacement in a cycle. Therefore, θ 0 = 0 ° and θ n = 360 ° must be specified. The objective function and constrained equations are formulated as follows:

G1 = −[ Feo2 x s 2o sin β 2 + o e4 x

),

( ρ i , μ i , hi , i = 2 ~ 4 ) are determined by optimization

where

Feo3 x ( Lo2 sin φ 2 + s 2o sin β 2 ) +

Li , i = 1 ~ 4

is the angular displacement

objective function, nc and n g denote, respectively,

of the ith link, L°i ( = Li / L2 R ) is the normalized

the number of equality and inequality constrained equation. Note that the equality and inequality constraints are defined to meet the desired output motion characteristics and dynamic performance. Up to here, all information for optimization has been derived. The optimization problem in this approach actually can be regarded as a single objective optimization problem with the continuous variables. An optimization program namely MOST (Tseng et al., 1994) which is developed for solving multi-objective optimization problems with mixed continuous and discrete design variables in C language, is used to solve design variables in the present approach. At MOST, a Sequential Quadratic Programming (SQP) is selected as a single objective optimizer for

w.r.t. fixed pivot oi ,

length of the ith link,

s i° ( = s i / L2 R ) is the

normalized distance from the fixed or moving pivot to the applied point of working force on the ith link, F ° = F /(m2ol L2 R ω 22av ) is the normalized force,

M ° = M /( m 2 ol L22 Rω 22av )

is

the

shaking moment, T = T /( m 2 ol L ω °

2 2R

normalized 2 2 av

) is the

normalized torque, m 2 ol is the mass of link 2 of the reference linkage, L2 R is the length of link 2 of the reference linkage, and ω 2 av (=2π) is average angular speed of the crank.

5

the continuous variables for accuracy, reliability and efficiency. The modified branch-and-bound algorithm, which converts discontinuous design space into a continuous one by dropping discontinuous restrictions, is used to solve discrete optimization. For multi-objective optimization, MOST provides Compromise Programming, Goal Programming and the Surrogate Worth Trade-off method for decision maker to find the best compromise solutions.

EXAMPLES AND DISCUSSION

Table 1. Link dimensions and inertial properties of the reference linkage Link Length of link Mass of link (kg) number (m)

1 2 3 4

0.0762 0.0254 0.0508 0.0762

0.04585 0.05317 0.06602

Moment of inertial about center of gravity (kg-m2)

0.6733*10-5 0.3013*10-4 0.6768*10-4

The optimization task is to determine the balancing parameters ( ρ i , μ i , hi , i = 2 ~ 4 ), and minimizing

Here the feasibility of this proposed approach according to the design flow chart as shown in Fig. 1 will be demonstrated by three examples. In all these examples, the goal is focused on reaching the trade-off of dynamic balancing with complete force balancing under the conditions that meet the kinematic requirements and constraints. Therefore, the objective function is defined as follows:

f ( ρ i , μ i , hi ) (36)

=

1 2π

∫

2π

0

⎡ w F ° 2 + F ° 2 + w T ° 2 ⎤ dφ , 41 d2 D ⎥ 2 ⎢⎣ d 1 21 ⎦

Subject to

f (θ 1,..θ n −1, ρ i , μ i , hi , L1,.., L4 ) (35)

=

1 2π

∫

2π

0

⎡w F ° 2 + F ° 2 + w °2 d 2 TD 41 ⎢⎣ d 1 21

⎤ dφ , ⎥⎦ 2

A 10th order Bezier curve (with 11 control points) is used to represent the trajectory of the crank displacement, and the wd 1 and wd 2 are set to be 0.5 in order to reach the trade-off of dynamic balancing in the following examples.

3 4 O3(A) 1

c2 (ρ i, μi, hi ) = m4 r4 − m3 r3 L4 L3 = 0

i = 3 ~ 4,

c 3 ( ρ i , μ i , hi ) = ϕ 2 − τ 3 = 0

i = 2 ~ 3,

c4 (ρi,μi,hi ) = ϕ4 − (ϕ3 + π ) = 0

i = 3 ~ 4,

Table 2. Link dimensions and balancing parameters for all examples

(B)

O2

i = 2 ~ 3,

The constrained equations are in order to satisfy the conditions of complete force balancing. The design results of balancing parameters are shown in Table 2, the comparisons of rms dynamic quantities are shown in Table 3. The comparisons of dynamic balancing performance are shown in Figure 5.

Example 1 Designing the balancing parameters of links of the force unbalanced reference linkage with constant input speed for reaching the trade-off of dynamic balancing with complete force balancing. The force unbalanced reference linkage as shown in Figure 3 has its link dimensions and inertial properties given in Table 1 and running at an average crank speed of 60 rpm. The circular disks with the thickness less than 2.54 cm are introduced for counterweights of the moving links, and the cross section of links is kept the same with the force unbalanced reference linkage.

2

c1 ( ρ i , μ i , hi ) = m 2 r2 − m3 e3 L2 L3 = 0

O4

Figure 3. The force unbalanced reference linkage 6

Linkage number L1 (cm) L2 (cm) L3 (cm) L4 (cm) ρ2 (cm) ρ3 (cm) ρ4 (cm) h2 (cm) h3 (cm) h4 (cm) μ2 (°) μ3 (°) μ4 (°)

1 (Example 1) 7.62 2.54 5.08 7.62 1.95 1.3 1.6 1 1 1 180.01 180.04 179.96

2 (Example 2) 7.62 2.54 5.08 7.62 1.82 1.17 1.71 1 1 1 186.48 238.43 144.79

3 (Example 3) 7.60 1.38 6.24 7.62 1.59 1.64 1.63 1 0.63 1 179.56 177.08 182.26

Table 3.

Comparison of the dynamic quantities between all examples

Linkage 1 number r.m.s shaking 2.061 force r.m.s bearing reaction force 2.2157 to fixed pivot o2 r.m.s bearing reaction force 0.8842 to fixed pivot o4 r.m.s shaking moment to 2.338 fixed pivot o2 r.m.s shaking moment to 5.300 fixed pivot o4 r.m.s driving 0.8628 torque Remark

2

3

4

0

0

0

1. 5812 (29%)

1. 3727 (38%)

0. 6709 (70%)

1. 5812 (-79%)

1. 3727 (-55%)

0. 6709 (24%)

2. 3585 (-44%)

3. 0918 (-32%)

1. 8456 (21%)

4. 3585 (36%)

2. 0918 (42%)

2. 8456 (65%)

0. 7828 (9%)

0. 1878 (78%)

0. 1557 (82%)

f (θ 1,..θ n −1, ρ i , μ i , hi ) (37) 2π

0

⎡w F ° 2 + F ° 2 + w T °2 41 d2 D ⎢⎣ d 1 21

3 32.66 64.33 125.77 129.21 165.26 231.51 242.67 293.71 327.34 Example 2

4 34.83 67.11 101.82 136.8 173.66 211.17 248.39 287.77 325.16 Example 3

Example 3 Designing the input speed trajectory, balancing parameters and link dimensions of a desired linkage in which point B in link 4 passes three precision points B1 (0.0381m, 0.066m), B2 (0.0248m, 0.0563m) and B3 (0.0159m, 0.0466m) with the same as the force unbalanced referenced linkage for the trade-off of dynamic balancing with complete force balancing. The optimization task is to determine the control points ( θ1 ,...,θ n −1 ), balancing parameters

and constrained equations become as follows: Minimizing

∫

i = 3 ~ 4,

Linkage number θ1(°) θ2(°) θ3(°) θ4(°) θ5(°) θ6(°) θ7(°) θ8(°) θ9(°) Remark

μ i , hi , i = 2 ~ 4 ), and the objective function

1 2π

c6 (ρi, μi, hi ) = ϕ4 − (ϕ3 + π ) = 0

Table 4. Control points of trajectory of driving link for all examples

Example 2 Designing the input speed trajectory and balancing parameters of moving links of the force unbalanced referenced linkage for the trade-off of dynamic balancing with complete force balancing respectively. The optimization task is to determine the control points ( θ 1 ,...,θ n −1 ) and balancing parameters

=

i = 2 ~ 3,

The first two constrained equations are in order to have continuous angular velocity and acceleration of the input link in two consecutive cycles, and the others are for complete force balancing. The design results of optimal control points and balancing parameters are shown in Tables 4 and 2, respectively, and the comparisons of rms dynamic quantities are shown in Table 3. The corresponding input and output motion characteristics and the comparisons of dynamic balancing performance are shown in Figs. 4 and 5, respectively.

Example 1 Example 2 Example 3 The value in parentheses denotes percent improvement over the corresponding rms value of the linkage number 1.

( ρi ,

c 5 ( ρ i , μ i , hi ) = ϕ 2 − τ 3 = 0

⎤ dφ , ⎥⎦ 2

( ρ i , μ i , hi , i = 2 ~ 4 )

and

link

dimensions

( Li , i = 1 ~ 4 ). The objective function and constrained equations are as follows:

Subject to

Minimizing Equation (35), subject to

c1 (θ 1 , K , θ 9 ) = ω 2 (0) − ω 2 (1) = 0,

c1 (θ 1 , K , θ 9 ) = ω 2 (0) − ω 2 (1) = 0,

c 2 (θ 1 , K , θ 9 ) = α 2 (0) − α 2 (1) = 0,

c 2 (θ 1 , K , θ 9 ) = α 2 (0) − α 2 (1) = 0,

c3 ( ρ i , μ i , hi ) = m2 r2 − m3 e3 L2 L3 = 0

i = 2 ~ 3,

c4 (ρ i, μi, hi ) = m4 r4 − m3 r3 L4 L3 = 0

i = 3 ~ 4,

7

c9 (ρi, μi, hi ) = m2 r2 − m3 e3 L2 L3 = 0

i = 2 ~ 3,

c10 (ρi, μi, hi ) = m4 r4 − m3 r3 L4 L3 = 0

i = 3 ~ 4,

0.07

8

Example 3

Example 1 Example 2 Example 3

Crank Angular Acceleration(rad/sec2)

Moving Trajectory of Couple Poinr in Y Axis (m)

Precision points

0.06

4

0

0.05

-4

0.02

0.02

0.03

0.03

0.04

0

0.04

0.2

Moving Trajectory of Couple Point in X Axis (m)

0.4 0.6 Normalized Time

0.8

1

(d) Figure 4. Output and input motion characteristics

(a) Example 1

6.0

– Examples 1, 2 and 3

Example 2 Example 3

Reference Linkage Example 1

Normalized Bearing Reaction Force to Fixed Pivot o2

Crank Angular Displacement (rad)

4.00

4.0

2.0

0.0 0.0

0.2

0.4

0.6

0.8

Example 2 Example 3

3.00

2.00

1.00

1.0

Normalized Time 0.00

(b)

0.00

0.20

0.40

0.60

0.80

1.00

0.80

1.00

Normalized Time

(a) Reference Linkage Example 1

Normalized Bearing Reaction Force to Fixed Pivot o4

Crank Angular Velocity(rad/sec)

6.4

6 Example 1 Example 2 Example 3

Example 2

3.00

Example 3

2.00

1.00

5.6 0

0.2

0.4 0.6 Normalized Time

0.8

1

0.00 0.00

(c)

0.20

0.40

0.60

Normalized Time

(b) 8

Reference Linkage

c11 ( ρ i , μ i , hi ) = ϕ 2 − τ 3 = 0

i = 2 ~ 3,

c12 (ρi, μi, hi ) = ϕ4 − (ϕ3 +π ) = 0

i = 3 ~ 4,

Example 1

Normalized Shaking Moment to Fixed pivot o2

Example 2

4.00

Example 3

c 3 (θ 1 , K , θ 9 , L1, , K , L 4 ) = B1 x − 0.038 m = 0, 0.00

c 4 (θ1 , K, θ 9 , L1, , K, L4 ) = B1 y − 0.066m = 0, c 5 (θ 1 , K , θ 9 , L1, , K , L4 ) = B2 x − 0.0248m = 0,

-4.00

c 6 (θ 1 , K , θ 9 , L1, , K , L 4 ) = B 2 y − 0.0563 m = 0, -8.00 0.00

0.20

0.40

0.60

0.80

c 7 (θ 1 , K , θ 9 , L1, , K , L 4 ) = B 3 x − 0.0159 m = 0,

1.00

Normalized Time

(c)

c8 (θ 1 , K , θ 9 , L1, , K , L4 ) = B3 y − 0.0466m = 0,

Reference Linkage Example 1 Example 2

Normalized Shaking Moment to Fixed pivot o4

4.00

c13(L1,.........., L4 ) = Lmin − L2 = 0 ,

Example 3

g k +1 ( L1, , K , L4 ) = ( Lmin + Lmax ) − ( L P + Lq ) < 0,

0.00

where Bix and Biy are x and y components of the coordinates of point B on the ith precision point respectively, Lmin and Lmax are the link length of the

-4.00

shortest and longest link in the four-bar linkage respectively, L p and Lq are the link length of the -8.00

0.00

0.20

0.40

0.60

0.80

1.00

0.80

1.00

other two links of the four-bar linkage. The first six constrained equations have the same purpose as Example 2, the seventh to the twelfth are for fulfilling the kinematic design requirements, and the last two are the Grashof criteria for a crank-rocker mechanism. The optimal control points, balancing parameters and link dimensions are shown in Tables 4 and 2, respectively, and the comparisons of rms dynamic quantities are shown in Table 3. The corresponding input and out motion characteristics and the comparisons of dynamic balancing performance are shown in Figs. 4 and 5, respectively. The dynamic balancing performance of a linkage is mainly dominated by its inertial properties such as mass and mass moment of inertial and kinematic characteristics of the mass center such as position, velocity and acceleration, especially, the latter two terms. Designing the balancing parameters is in order to satisfy the complete shaking force balance conditions and regulate inertial properties and positions of the mass center of each moving link. The design of the input speed trajectory is for tuning velocity and acceleration of the mass center of each moving link. Dimensional synthesis is able to satisfy kinematic design requirements and constraints and adjust the inertial properties and kinematic

Normalized Time

(d) Reference Linkage

2.00

Example 1 Example 2

Normalized Driving Torque

Example 3

0.00

-2.00

0.00

0.20

0.40

0.60

Normalized Time

(e) Figure 5. Dynamic balancing performance - Examples 1, 2 and 3

9

characteristics of the desired linkage simultaneously. Only designing the balancing parameters can have appropriate improvement of dynamic balancing performance except shaking moment to fixed pivot o2 and bearing reaction force to fixed pivot o4 comparing with force unbalanced reference linkage according to the results of Example 1 as shown in Table 3 and Figs. 5, respectively. Because the dynamic performance of the desired linkage is mainly influenced by their kinematic characteristics and inertial properties simultaneously, in those examples, firstly the complete force balanced conditions, which conflict with the rest of dynamic balancing performance, have to be satisfied. Therefore, designing the balancing parameters only may not enough to obtain more obvious improvement of dynamic balancing performance for complete force balanced linkages. As expected, the same variations of dynamic balancing performance in magnitudes and fluctuations with Example 1 but more substantial improvement comparing with the force unbalanced reference linkage, particularly in driving torque, from the results of Example 2 as shown in Table 3 and Figs. 5 are gained, respectively, if the design of the input speed trajectory and balancing parameters is implemented in the same design step simultaneously. Owing to the dynamic performance is also dominated by the kinematic characteristics of each moving link, and the kinematic characteristics of each moving link are functions of the kinematic characteristics of the input link, varying the speed of the input link is an effective way to modify the dynamic performance of desired linkages. The obvious improvement of dynamic balancing performance not only in magnitude but also in variations comparing with the force unbalanced reference linkage from the results of Example 3 shown in Table 3 and Figs. 5, respectively, is reached, the r.m.s of driving torque, shaking moments and bearing reactions can be reduced by about 82%, 65% and 70%, respectively, if the integrated design, which the input speed trajectory, balancing parameters and link dimensions are involved, is applied. Since synthesizing the link dimensions could adjust the kinematic characteristics and inertial properties simultaneously, it increases the much room to adjust the dynamic balancing performance. Therefore, it is an effective way to combine kinematic synthesis in the beginning of designing a force balanced four-bar linkage for the trade-off of dynamic balancing.

CONCLUSIONS The main contribution of this paper is to propose a new design approach that combines kinematic synthesis, dynamic balancing design and the design of input speed trajectory in the same design step for 10

designing the complete force balanced four-bar linkages. The efficiency and feasibility of the suggested method is proved by examples. By properly designing the input speed trajectory, balancing parameters and link dimensions of a desired linkage, its kinematic design requirements and constraints and the trade-off of dynamic balance can be reached simultaneously. The input speed trajectory is designed by using Bezier curves with undetermined control points. Furthermore, this design concept can be extended to the linkages that have more than four bars or other types of mechanisms. From the results of the examples for designing the complete force balanced four-bar linkages, the following conclusions can be made: 1.Only designing the balancing parameters may not enough to obtain the trade-off of dynamic balancing performance. 2.Designing the input speed trajectory and balancing parameters simultaneously, the substantial improvement of dynamic balancing performance will be obtained, especially for driving torque. 3.Integrating the input speed trajectory, balancing parameters, and kinematic synthesis in the same design step, the obvious improvement of dynamic balancing performance will be obtained. The inertial properties and kinematic characteristics of mechanisms will vary with the deformations of links, if the consideration of link elasticity is taken. The conclusions, which have been made in this approach, may need to some modifications. Therefore, the kinematic and dynamic integrated design of mechanisms containing elastic links will be a worthy subject to be studied.

REFERENCES Arakelian, V. H. and Smith, M. R., “Complete shaking force and shaking moment balancing of linkages,” Mechanism and Machine Theory, Vol. 34, No. 8., pp. 1141-1153 (1999). Arakelian, V. H. and Smith, M. R., “Partial shaking moment balancing of full force balanced linkages,” Mechanism and Machine Theory, Vol. 36, No. 11-12., pp. 1241-1252 (2001). Berkof, R. S., “Complete force and moment balancing of inline four-bar linkages,” Mechanism and Machine Theory, Vol. 8, No. 3, pp. 397-410 (1973). Funk, W., “Complete balancing of the inertia input torque for planar mechanisms,” Archive of Applied Mechanics, Vol. 63, No. 4-5, pp. 353-360 (1993). Conte, F. L., George, G. R., Mayne, R. W., and Sadler, J. P., “Optimum mechanism design

combining kinematic and dynamic-force considerations,” ASME Transactions, Journal of Engineering for Industry, Series B, Vol. 95, No. 2, pp. 662-670 (1975). Chiou, S. T., Bai, G. J., ”Optimum balancing design of four-bar linkages with adding disk counterweights,” Journal of the Chinese Society of Mechanical Engineers (TAIWAN), Vol. 18, pp. 43-54 (1997). Esat, I. and Bahai, H., “A theory of complete force and moment balancing of planar linkage mechanisms,” Mechanism and Machine Theory, Vol. 34, No. 6., pp. 903-922 (1999). Qi, N. M. and Pennestri, E., “Optimum balancing of four-bar linkages,” Mechanism and Machine Theory, Vol. 26, No. 3, pp. 337-348 (1991). Starr, P. J., ”Dynamic Synthesis of Constraint Paths,” ASME Transactions, Journal of Engineering for Industry, Series B, Vol. 95, No. 2, pp. 624-628 (1973). Tesar, D. and Matthew, G. K., The Dynamic Synthesis, Analysis and Design of Modeled Cam Systems, Lexington Books (1976). Tepper, F. R. and Lowen, G. G., “General theorems concerning full force balancing of planar linkages by internal mass redistribution,” ASME Transactions, Journal of Engineering for Industry, Vol. 94, No. 3, pp. 789-796 (1972). Tricamo, S. J. and Lowen, G. G., Sep. “Simultaneous optimization of dynamic reactions of a four-bar linkage with prescribing maximum shaking force,” ASME Transactions, Journal of Mechanisms, Transmissions, and Automation in Design, Vol. 105, pp. 520-525 (1983). Tseng, C. H., Liao, W. C., and Yang, T. C., Most Users’ Manual, Department of Mechanical Engineering, National Chiao Tung University, Hsinchu, Taiwan, R.O.C (1994). Yan, H. S. and Soong, R. C., “Kinmatic and dynamic design of four-bar linkages by links counterweighing with variable input speed,” Mechanism and Machine Theory, Vol. 36, No. 9, pp. 1051-1071 (2001). Yan, H. S. and Soong, R. C., “Kinmatic and dynamic design of four-bar linkages variable input speed and external applied loads,” Transactions of the Canadian Society for Mechanical Engineering, Vol. 26, No. 3, pp. 281-310 (2002). Yan, H. S. and Soong, R. C., “An Integrated Design Approach of four-bar linkages variable input speed and external applied loads,” JSME International Journal, Series C, Vol. 47, No. 1, pp. 350-362 (2004). Yan, H. S., Tsai, M. C., and Hsu, M. H., “A variable-speed method for improving motion characteristics of cam-follower systems,” 11

ASME Transactions, Journal of Mechanical Design, Vol. 118, No. 1, pp. 250-258 (1996). Yan, H. S., Tsai, M. C., and Hsu, M. H., “An experimental study of the effects of cam speed on cam-follower systems,” Mechanism and Machine Theory, Vol. 31, No.4, pp. 397-412 (1996). Yan, H. S. and Chen, W. R., “On the output motion characteristics of variable input speed servo-controlled slider-crank mechanisms,” Mechanism and Machine Theory, Vol. 35, No.4, pp. 541-561 (2000).

Nomenclature Li Length of the ith link L°i Normalized length of the ith link

φi Angular position of the ith link ω i Angular velocity of the ith link α i Angular acceleration of the ith link li Distance to fixed or moving pivot measured from applied point of working forces on the ith link l Normalized Distance to fixed or moving pivot ° i

measured from applied point of working forces on the ith link β i The angle measured from X axis to the center line between applied point of working forces on the ith link and fixed or moving pivots counterclockwise ρ i Radius of the ith link disk counterweight

ri The distance from the fixed or moving pivot to

ϕi

the center of gravity of the ith link The angle measured from the center line between

the two pivots to the center of gravity of the ith link s i The distance to the fixed or moving pivot measured from the applied point of working force on the ith link ° si The normalized distance to the fixed or moving

τ3

pivot measured from the applied point of working force on the ith link The angle measured from the center line between

e3

the pivots A and B to the center of gravity of link 3 counterclockwise The distance measured from pivot B to the center

μi

of gravity of link 3 The angle measured from X axis to the center line between center of the ith link disk

counterweight and fixed or moving pivot counterclockwise Thickness of the ith link disk counterweight

同時最小化完全力平衡連 桿組之搖振力矩、驅動轉矩 及軸承反作用力

hi mi The mass of the ith link including counterweight Feix The component of the working force applied on

the ith link in x direction Feiy The component of the working force applied on

宋仁羣

the ith link in y direction

Fijx The component of force exerted on the jth link

高苑科技大學自動化工程系

by the ith link in x direction Fijy The component of force exerted on the jth link

國立成功大學機械工程學系

顏鴻森

by the ith link in y direction °

F Normalized force T ° Normalized torque m2ol The mass of the link 2 W Average angular speed of crank M sh° / oi Normalized shaking moment w. r. t. fixed

摘要 本文針對完全力平衡連桿組提出一整合運動 合成、動力設計及輸入轉速軌跡設計於同一設計步 驟之連桿組設計方法。藉由適當地設計輸入桿轉速 軌跡、動力平衡參數及各桿之桿長，以達到既滿足 運動設計需求與限制，又同時最小化完全力平衡連 桿組之搖振力矩、驅動轉矩及軸承反作用力之目 的。利用 Bezier 曲線設計輸入桿轉速軌跡，以最佳 化的方法找出最佳之設計參數，最後以設計實例驗 證本設計方法之可行性。

pivot oi

TD Driving torque Tei Working torque applied on the ith link I i Moment of inertial of ith link about its center of gravity v i Velocity vector of center of gravity of the ith link

a i Acceleration vector of center of gravity of the ith link

v ei Velocity vector of point of working force applied on the ith link v Ax The component of the velocity of point A on link 3 in x direction

v Ay The component of the velocity of point A on link 3 in y direction v Bx The component of the velocity of point B on link 3 in x direction

v By The component of the velocity of point B on link 3 in y direction

12