第六章 結論與未來展望
6.2 未來展望
針對本研究之相關主題，提出以下可進一步探討之建議方向：
1. 擴展動力模型至滑塊或其他複合連桿組，以拓展應用層面。
2. 搭配可得全域最佳解之最佳化方法。
3. 探討權重的最佳配置方法，以得最合適的動力性能指標。
4. 建立質量、質心距、質心角、轉動慣量間的關聯性，可列為限制條件，確保 解集合具實務上之可行性，以及建立將質量性質轉換為桿件幾何設計之系統，
使其更便於實務應用。
5. 本研究可擴展至其他調整形式的可調連桿組，使可調連桿組之動態平衡探討 更臻完備。
參考文獻
[1] Tao, D. C. and Krishnamoorthy, S., “Linkage mechanism adjustable for variable coupler curves with cusps,” Mechanism and Machine Theory, Vol.13, No. 6, pp.
577-583, 1978.
[2] Tao, D. C. and Krishnamoorthy, S., “Linkage mechanism adjustable for variable symmetrical coupler curves with a double point,” Mechanism and Machine Theory, Vol.13, No. 6, pp. 585-591, 1978.
[3] Ahmad, A. and Waldron, K. J., “Synthesis of adjustable planar 4-bar mechanisms,”
Mechanism and Machine Theory, Vol.14, No. 6, pp. 405-411, 1979.
[4] Shimojima, H., Ogawa, K., Fujiwara, A. and Sato, O., “Kinematic synthesis of adjustable mechanisms: Part 1, path generators,” Bulletin of the JSME, Vol. 26, No.
214, pp. 627-632, 1983.
[5] Shimojima, H. and Ogawa, K., “Kinematic synthesis of adjustable mechanisms:
Part 2, function generators,” Bulletin of the JSME, Vol. 27, No. 227, pp. 1025-1030, 1984.
[6] Shimojima, H., Iida, K., and Kuwabara, M., “Kinematic synthesis of adjustable mechanisms: Part 3, 6-link dwell mechanisms,” Bulletin of the JSME, Vol. 29, No.
254, pp. 2718-2723, 1986.
[7] Naik, D. P. and Amarnath, C., “Synthesis of adjustable four bar function generators through five bar loop closure equations,” Mechanism and Machine Theory, Vol.24, No. 6, pp. 523-526 ,1989.
[8] Zhou, H., “Synthesis of adjustable function generation linkages using the optimal pivot adjustment,” Mechanism and Machine Theory, Vol.44, No. 5, pp. 983-990, 2009.
[9] Zhou, H., “Dimensional synthesis of adjustable path generation linkages using the optimal slider adjustment,” Mechanism and Machine Theory, Vol.44, No. 10, pp.
1866-1876, 2009.
[10] Chang, C.-F., “Synthesis of adjustable four-bar mechanisms generating circular arcs with specified tangential velocities,” Mechanism and Machine Theory, Vol.36, No. 3, pp. 387-395, 2001.
[11] Zhou, H. and Ting, K.-L., “Adjustable slider-crank linkages for multiple path generation,” Mechanism and Machine Theory, Vol.37, No. 2, pp. 499-509, 2002.
[12] 林進水，
平面連桿組可調整高階暫停機構之探討
，碩士論文，國立台灣大學 機械工程學研究所，台北，1994。[13] Wang, S. J. and Sodhi, R. S., “Kinematic synthesis of adjustable moving pivot four-bar mechanisms for multi-phase motion generation,” Mechanism and
Machine Theory, Vol.31, No. 4, pp. 459-474, 1996.
[14] Hong, B. and Erdman, A. G., “A method for adjustable planar and spherical four-bar linkage synthesis,” Journal of Mechanical Design, Vol.127, No. 3, pp.
456-463, 2005.
[15] Pennock, G. R. and Israr, A., “Kinematic analysis and synthesis of an adjustable six-bar linkage,” Mechanism and Machine Theory, Vol.44, No. 2, pp. 306-323, 2009.
[16] Zero-Max: Adjustable Speed Drives,
http://www.zero-max.com/adjustable-speed-drives-c-21-l-en.html
[17] Arakelian, V. H. and Smith, M. R., “Shaking force and shaking moment balancing of mechanisms: A historical review with new examples,” Journal of Mechanical
Design, Vol.127, No. 2, pp. 334-339, 2005.
[18] Hertrich, F. R., “How to balance high-speed mechanisms with minimum-inertia
counterweights,” Machine Design, Vol. 35, No. 6, pp. 160-164, 1963.
[19] Demeulenaere, B., Verschuure, M., Swevers, J. and De Schutter, J., “A general and numerically efficient framework to design sector-type and cylindrical
counterweights for balancing of planar linkages,” Journal of Mechanical Design, Vol. 132, No. 1, pp. 011002.1-011002.10, 2010.
[20] Chaudhary, H. and Saha, S. K., “Balancing of four-bar linkages using maximum recursive dynamic algorithm,” Mechanism and Machine Theory, Vol. 42, No. 2, pp.
216-232, 2007.
[21] Chaudhary, H. and Saha, S. K., “Balancing of shaking forces and shaking moments for planar mechanisms using the equimomental systems,” Mechanism and
Machine Theory, Vol. 43, No. 3, pp. 310-334, 2008.
[22] Guo, G., Morita, N. and Torii, T., “Optimum dynamic design of planar linkage using genetic algorithms,” JSME International Journal, Series C: Mechanical
Systems, Machine Elements and Manufacturing, Vol. 43, No. 2, pp. 372-377, 2000.
[23] Rigelman, G. A. and Kramer, S. N., “A computer-aided design technique for the synthesis of planar four bar mechanisms satisfying specified kinematic and dynamic conditions,” Journal of Mechanisms, Transmissions, and Automation in
Design, Vol. 110, No. 3, pp. 263-268, 1988.
[24] Yan, H.-S. and Soong, R.-C., “Kinematic 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.
[25] Yan, H.-S. and Soong, R.-C., “An integrated design approach of four-bar linkages with variable input speed,” JSME International Journal, Series C: Mechanical
Systems, Machine Elements and Manufacturing, Vol. 47, No. 1, pp. 350-362, 2004.
[26] Berkof, R. S. and Lowen, G. G., “New method for completely force balancing
simple linkages,” Journal of Engineering for Industry, Vol. 91, No. 1, pp. 21-26, 1969.
[27] Balasubramanian, S. and Bagci, C., “Design equations for the complete shaking force balancing of 6R 6-bar and 6-bar slider-crank mechanisms,” Mechanism and
Machine Theory, Vol. 13, No. 6, pp. 659-674, 1978.
[28] Arakelian, V. and Dahan, M., “Partial shaking moment balancing of fully force balanced linkages,” Mechanism and Machine Theory, Vol.36, No. 11-12, pp.
1241-1252, 2001.
[29] Arakelian, V. H., Dahan, M. and Smith, M. R., “Complete shaking force and partial shaking moment balancing of planar four-bar linkages,” Proceedings of the
Institution of Mechanical Engineers, Part K: Journal of Multi-body Dynamics, Vol.
215, No.1, pp. 31-34, 2001.
[30] Esat, I. and Bahai, H., “A theory of complete force and moment balancing of planer linkage mechanisms,” Mechanism and Machine Theory, Vol. 34, No. 6, pp.
903-922, 1999.
[31] 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.
[32] Rao, S. S. and Kaplan, R. L., “Optimal balancing of high-speed linkages using multiobjective programming techniques,” Journal of Mechanisms, Transmissions,
and Automation in Design, Vol. 108, No.4 , pp. 454-460, 1986.
[33] Lee, T. W. and Cheng, C., “Optimum balancing of combined shaking force, shaking moment, and torque fluctuations in high-speed linkages,” Journal of
Mechanisms, Transmissions, and Automation in Design, Vol.106, No. 2, pp.
242-251, 1984.
[34] Conte, F. L., George,G. R., Mayne, R. W. and Sadler, J. P., “Optimum mechanism design combining kinematic and dynamic-force considerations,” Transactions of
ASME Journal of Engineering for Industry, Vol. 95, No. 2, pp. 662-670, 1975.
附錄 程式碼
(a) 最佳化主程式：Opt_multi.m
% Optimum Dynamic Balancing of Adjustable Planar Series-Linkage
% Need State_multi.m & KDSL_multi.m
% Teng, Chiao-Mei, 2011.
clear all;close all;clc format long
tic
global WeightALL weight c_mm_lb mass_g_lb Ic_gmm2_lb c_mm_ub mass_g_ub Ic_gmm2_ub WeightALL=[1/3 1/3 1/3]; % 各調整狀態的目標函數權重
mass_g_0=[105.57 607.03 572.48 105.57 392.52 113.62 105.57]; % 各桿質量[g],即 m2,m3...
Ic_gmm2_0=[140610 4356334 2927150 140610 1086465 176897 140610];
% 各桿慣性矩[g-mm^2],即 Ic2,Ic3...
mass_g_ub=[10 5 5 8 5 8 8].*mass_g_0;
Ic_gmm2_ub=[12 8 8 12 8 12 12].*Ic_gmm2_0;
coef_0=[phi_deg_0/360 (c_mm_0-c_mm_lb)./(c_mm_ub-c_mm_lb) (mass_g_0-mass_g_lb)./(mass_g_ub-mass_g_lb)...
(Ic_gmm2_0-Ic_gmm2_lb)./(Ic_gmm2_ub-Ic_gmm2_lb)];
% 依序為 phi_deg,c_mm,mass_g,Ic_gmm2 的係數 coef_lb=zeros(1,4*(n-1));
options=optimset('Algorithm','interior-point','MaxFunEvals',300000);
(b) 最佳化子程式：State_multi.m
% States of Series-Linkage
% Under Opt_multi.m
% Teng, Chiao-Mei, 2011.
function ObjALL=State_multi(coef) global WeightALL
s=3; % 可調整狀態數量
pivotALL_mm=[336.1 485.5;83.8 27.4;...
309.9 532.1;-64.2 -31.4;...
274.2 520.7;-41.9 -37.5]; % 調整軸位置[mm]
% [State1x1 State1x2...;State1y1 State1y2...;State2x1 State2x2;...;State2y1 State2y2...;...]
Obj=zeros(1,s); % 定義子目標函數的維度
for k=1:s
pivot_mm=pivotALL_mm(2*k-1:2*k,:);
Obj(k)=KDSL_multi(pivot_mm,coef);
end
ObjALL=sum(WeightALL.*Obj);
(c) 最佳化子程式：KDSL_multi.m
% Kinematics and Dynamics of Series-Linkage
% Under State_multi.m
% Teng, Chiao-Mei, 2011.
function objval=KDSL_multi(pivot_mm,coef)
global weight c_mm_lb mass_g_lb Ic_gmm2_lb c_mm_ub mass_g_ub Ic_gmm2_ub
%% Parameter Setting n=8; % 總連桿數 phi_deg=360*coef(1:n-1); % 質心在對應坐標系的夾角[deg],即 phi2,phi3...
c_mm=coef(n:2*n-2).*(c_mm_ub-c_mm_lb)+c_mm_lb; % 質心與對應參考點距離[mm],即 c2,c3...
mass_g=coef(2*n-1:3*n-3).*(mass_g_ub-mass_g_lb)+mass_g_lb; % 各桿質量[g],即 m2,m3...
Ic_gmm2=coef(3*n-2:end).*(Ic_gmm2_ub-Ic_gmm2_lb)+Ic_gmm2_lb;
fid=fopen('8bar_State1_T8unit.txt','r');
fclose(fid);
F3=280*[ones(1,81) zeros(1,459) ones(1,181)]; % 270~40deg 有外力 psiF3_deg=230.5*[ones(1,81) zeros(1,459) ones(1,181)]; % 270~40deg 有外力 elseif pivot_mm(1)==309.9
fid=fopen('8bar_State2_T8unit.txt','r');
T8=50*fscanf(fid,'%g')';
fclose(fid);
F3=280*[ones(1,21) zeros(1,479) ones(1,221)]; % 250~10deg 有外力 psiF3_deg=200.5*[ones(1,21) zeros(1,479) ones(1,221)]; % 250~10deg 有外力 else
fid=fopen('8bar_State3_T8unit.txt','r');
T8=40*fscanf(fid,'%g')';
fclose(fid);
F3=280*[zeros(1,520) ones(1,201)]; % 260~360deg 有外力 psiF3_deg=198*[zeros(1,520) ones(1,201)]; % 260~360deg 有外力 end
Tex=[zeros(6,step+1);T8]; % 各桿件的外加扭力[N-m]
Fex=[zeros(1,step+1);F3;zeros(5,step+1)]; % 各桿件的外力[N]
psiF_deg=[zeros(1,step+1);psiF3_deg;zeros(5,step+1)]; % 各桿件外力的絕對角度[deg]
% Tex=zeros(n-1,step+1); % 各桿件的無外加扭矩[N-m]
% Fex=zeros(n-1,step+1); % 各桿件無外力[N]
% psiF_deg=zeros(n-1,step+1); % 各桿件外力的絕對角度[deg]
%% Unit Conversion
inshaft=inshaft_mm/1000; % [m]
outshaft=outshaft_mm/1000; % [m]
r2=r2_mm/1000; % [m]
r3=r3_mm/1000; % [m]
r4=r4_mm/1000; % [m]
pivot=pivot_mm/1000; % [m]
rad2deg=180/pi; % rad 轉換成 deg 時所乘係數 gamma=gamma_deg/rad2deg; % [rad]
omega_in=omega2_rpm*2*pi/60; % [rad/s]
phi=phi_deg/rad2deg; % [rad]
c=c_mm/1000; % [m]
mass=mass_g/1000; % [kg]
psiF=psiF_deg/rad2deg; % [rad]
e=e_mm/1000; % [m]
psi=psi_deg/rad2deg; % [rad]
%% Kinematic Analysis
m=(n-2)/2; % 總串聯四連桿組數 r1C1=[pivot(1,:) outshaft(1)]-[inshaft(1) pivot(1,:)];
% 各個固定桿長的 x 分量,即 r1*cos(theta1)[d1x d2x d3x ...][m]
r1S1=[pivot(2,:) outshaft(2)]-[inshaft(2) pivot(2,:)];
% 各個固定桿長的 y 分量,即 r1*sin(theta1)[d1y d2y d3y ...][m]
[~,r1]=cart2pol(r1C1,r1S1); % 各個固定桿長[m]與絕對坐標角[rad]
num_data=step+1; % 輸入軸轉360 度切割後的數據總數 theta_in=linspace(0,2*pi,num_data); % 輸入軸角度陣列[rad]
armdx=cumsum(r1C1,2); % 軸承力對於輸入軸的x 方向力臂[d1x d1x+d2x d1x+d2x+d3x ...][m]
armdy=cumsum(r1S1,2); % 軸承力對於輸入軸的y 方向力臂[d1y d1y+d2y d1y+d2y+d3y ...][m]
% 定義維度
theta2=[theta_in;zeros(m-1,num_data)]; % 指定存放串聯的各個四連桿中輸入角的維度[rad]
theta3=zeros(m,num_data); % 耦桿角的維度[rad]
alpha=zeros(n-1,num_data); % 所有桿件角加速度的維度[rad/s^2],動力源等角速度輸入 aAx=theta3; % A 點(接頭 23,45,67...)加速度 x 分量的維度[m/s^2]
r4y=theta3; % 輸出桿長y 分量的維度[m]
C=r1(i)^2+r2(i)^2+r4(i)^2-r3(i)^2-2*(r1C1(i)*r2x(i,:)+r1S1(i)*r2y(i,:));
discriminant=B.^2-C.^2+A.^2;
if sum(discriminant(:)<0)>0
disp('error, (B.^2-C.^2+A.^2)<0') return
end
theta4(i,:)=2*atan((-B+sigma(i)*sqrt(discriminant))./(C-A));
C4=cos(theta4(i,:));
S4=sin(theta4(i,:));
r4x(i,:)=r4(i)*C4; % 計算動力分析會用到的 x,y 方向分量 r4y(i,:)=r4(i)*S4; % 計算動力分析會用到的 x,y 方向分量
% 計算角速度
omega4(i,:)=omega2(i,:).*(C3.*r2y(i,:)-S3.*r2x(i,:))./D;
omega3(i,:)=(omega4(i,:).*r4y(i,:)-omega2(i,:).*r2y(i,:))./r3y(i,:);
% 計算角加速度
omega2sq=omega2(i,:).^2;
omega3sq=omega3(i,:).^2;
omega4sq=omega4(i,:).^2;
aAx(i,:)=-alpha2(i,:).*r2y(i,:)-omega2sq.*r2x(i,:);
aAy(i,:)=alpha2(i,:).*r2x(i,:)-omega2sq.*r2y(i,:);
alpha4(i,:)=(r3(i)*omega3sq-omega4sq.*(C3.*r4x(i,:)+S3.*r4y(i,:))-aAx(i,:).*C3-aAy(i,:).*S3)./D;
alpha3(i,:)=(aAx(i,:)-omega3sq.*r3x(i,:)+(omega4sq.*r4x(i,:)+alpha4(i,:).*r4y(i,:)))./r3y(i,:);
alpha((2*i):(2*i+1),:)=[alpha3(i,:);alpha4(i,:)];
% 計算質心加速度
c3x(i,:)=c(2*i)*cos(theta3(i,:)+phi(2*i));
c3y(i,:)=c(2*i)*sin(theta3(i,:)+phi(2*i));
ac3x(i,:)=aAx(i,:)-alpha3(i,:).*c3y(i,:)-omega3sq.*c3x(i,:);
ac3y(i,:)=aAy(i,:)+alpha3(i,:).*c3x(i,:)-omega3sq.*c3y(i,:);
c4x(i,:)=c(1+2*i)*cos(theta4(i,:)+phi(1+2*i));
c4y(i,:)=c(1+2*i)*sin(theta4(i,:)+phi(1+2*i));
ac4x(i,:)=-alpha4(i,:).*c4y(i,:)-omega4sq.*c4x(i,:);
ac4y(i,:)=alpha4(i,:).*c4x(i,:)-omega4sq.*c4y(i,:);
e3x(i,:)=e(2*i)*cos(theta3(i,:)+psi(2*i));
e4x(i,:)=e(1+2*i)*cos(theta4(i,:)+psi(1+2*i));
e4y(i,:)=e(1+2*i)*sin(theta4(i,:)+psi(1+2*i));
armFx(2*i,:)=e3x(i,:)+r2x(i,:)+(armdx(i)-r1C1(i));
armFy(2*i,:)=e3y(i,:)+r2y(i,:)+(armdy(i)-r1S1(i));
armFx(2*i+1,:)=e4x(i,:)+armdx(i);
armFy(2*i+1,:)=e4y(i,:)+armdy(i);
%% Dynamic Analysis
U=-Tex(2:end,:)+diag(Ic(2:end))*alpha(2:end,:); % [-T3+Ic3*alpha3;-T4+Ic4*alpha4...][N-m]
[Fexx,Fexy]=pol2cart(psiF,Fex); % 各桿件外力的x,y 分量[N]
for j=m:-1:1
P=U(2*j-1,:)+Fexx(2*j,:).*(e3y(j,:)-r3y(j,:))-Fexy(2*j,:).*(e3x(j,:)-r3x(j,:))...
-mass(2*j)*(ac3x(j,:).*(c3y(j,:)-r3y(j,:))-ac3y(j,:).*(c3x(j,:)-r3x(j,:)));
if j==m
f43x(j,:)=f32x(j,:)-Fexx(2*j,:)+mass(2*j)*ac3x(j,:);
if j==m
f14x(j,:)=f43x(j,:)-Fexx(2*j+1,:)+mass(2*j+1)*ac4x(j,:);
f14y(j,:)=f43y(j,:)-Fexy(2*j+1,:)+mass(2*j+1)*ac4y(j,:);
else
f14x(j,:)=f43x(j,:)-f32x(j+1,:)-Fexx(2*j+1,:)+mass(2*j+1)*ac4x(j,:);
f14y(j,:)=f43y(j,:)-f32y(j+1,:)-Fexy(2*j+1,:)+mass(2*j+1)*ac4y(j,:);
end end
f12x=-f32x(1,:)-Fexx(1,:)+mass(1)*ac2x; % 輸入軸軸承力 x 分量[N]
f12y=-f32y(1,:)-Fexy(1,:)+mass(1)*ac2y; % 輸入軸軸承力 y 分量[N]
[~,f12]=cart2pol(f12x,f12y); % 輸入軸軸承力[N]
[~,f14]=cart2pol(f14x,f14y); % 非輸入軸軸承力[N]
Td=-Tex(1,:)+f32x(1,:).*r2y(1,:)-f32y(1,:).*r2x(1,:)+Fexx(1,:).*e2y-Fexy(1,:).*e2x;
% 輸入扭矩 Driving Torque[N-m],動力源等角速度輸入
% Normalization
norf=m2_0*r2(1)*omega_in^2; % 力的無因次化因子normalized factor nort=norf*r2(1); % 力矩的無因次化因子normalized factor
f12n=f12/norf; % 無因次化的輸入軸軸承力
f14n=f14/norf; % 無因次化的非輸入軸軸承力
Tdn=Td/nort; % 無因次化的輸入扭力
objval=sum(weight(1)*sqrt(f12n.^2+sum(f14n.^2,1))+weight(2)*sqrt(Tdn.^2))/num_data;