Slider Crank Mechanism

PROJECT II SLIDER CRANK MECHANISM

Presented by: 1. Lokesh Kumar 2. Majid Hameed 3. Saurabh Sharma

Under the Guidance of

Prof. S K Saha MULTI-BODY SYSTEMS & VIBRATION DESIGN

Objective  Dynamic Modeling of a Slider Crank

Mechanism.  To carry out Inverse Dynamics.  Perform Free and Forced Simulation.

 Verification

Frame Representation Y1

2

X2 X4

Y3

3

1

Z4

X1 X3

Link parameters Y1

X2

2

X4

#2 M2,a2

#1 M1,a1

Y3

3

1 X1

#3 M3,b 4

Z4

X3

Methods used for modeling  Euler Lagrange Independent Generalised

Coordinate Method.  Euler Lagrange Dependent Set of

Generalised Coordinates Method.  Using NOC / DeNOC and Lagrange

Multipliers.

Dependent Set Method

 d L L  T ( . )  ai   i dt  q  qi i

Generalised Equation

     T I q h  g a   ..

I Matrix

H Matrix .

.

.

[( m2 a1a2  1 sin  2  2  (m2 a1a2  sin  2 / 2)] 2 2

.

m2 a1a2 sin  2 

2 1

0 0

G Matrix (m1a1 g cos 1 / 2) 0 0 0

Constraint Matrix

Jacobian J Relations

Matlab Program for Euler Lagrange Dependent set of Coordinates % Initial Values a1=1; a2=3; m1=1; m2=1; m3=1; th0=0; ths=pi/2; T=2; i=1; g=9.81; for t= 0:0.01:2 % Initializing th1(i) = th0 + (((ths-th0)/T)*(t-(((T/(2*pi))*sin(2*pi*t/T))))); dth1(i) = ((ths-th0)/T)*(1-((T/(2*pi))*cos(2*pi*t/T))*(2*pi/T)); ddth1(i) = ((ths-th0)/T^2)*sin(2*pi*t/T)*2*pi; % Relations th2(i)=(2*pi)- th1(i) - asin(2*a1*sin(th1(i))/a2) ; dth2(i)=-(2*a1*cos(th1(i))*dth1(i))/(a2*cos(th1(i)+th2(i))) - dth1(i);

ddth2(i)= -ddth1(i)- [2*a1*[(cos(th2(i)+ th1(i))*((cos(th1(i))*ddth1(i))(sin(th1(i))*dth1(i)*dth1(i)))) + (cos(th1(i))*sin(th2(i)+th1(i))*dth1(i)*(dth2(i)+dth1(i)))]/ (cos(th2(i)+th1(i))*cos(th2(i)+th1(i))*a2)]; th3(i)=(2.5*pi)-(th1(i)+th2(i)); dth3(i)=-(dth1(i)+dth2(i)); ddth3(i)=-(ddth1(i)+ddth2(i)); ddb4(i)=(a1*cos(th1(i))*dth1(i))+(a1*sin(th1(i))*ddth1(i))(a1*cos(th1(i))*tan(th1(i)+th2(i))*ddth1(i))+(sin(th1(i))*tan(th1(i)+th2(i))*(dth1(i)*dth1(i)))(cos(th1(i))*(sec(th1(i)+th2(i))*sec(th1(i)+th2(i))*dth1(i)*(dth1(i)+dth2(i)))); % ddtheta vectors ddq=[ddth1(i);ddth2(i);ddth3(i);ddb4(i)]; % Inertia Matrix i11 = (m1*a1*a1/3)+(m2*a1*a1)+(m2*a2*a2/3)+(m2*a1*a2*cos(th2(i))); i12 = (m2*a2*a2/3)+(m2*a1*a2*cos(th2(i))/2); i13=0; i14=0; i21 = (m2*a2*a2/3)+(m2*a1*a2*cos(th2(i))/2); i22 = m2*a2*a2/3; i23=0; i24=0; i31=0; i32=0; i33=0; i34=0; i41=0; i42=0; i43=0; i44=m3; I=[i11 i12 i13 i14;i21 i22 i23 i24;i31 i32 i33 i34;i41 i42 i43 i44]; % H matrix h1=-(m2*a1*a2*dth1(i)*dth2(i)*sin(th2(i)))-(m2*a1*a2*dth2(i)*dth2(i)*sin(th2(i))/2);

h2=(m2*a1*a2*sin(th2(i))*dth1(i)*dth1(i)/2); h3=0; h4=0; H=[h1;h2;h3;h4]; % G matrix g1=m1*g*a1*cos(th1(i))/2; g2=0; g3=0; g4=0; G=[g1;g2;g3;g4];

% Constraint Matrix [Jacobian] a11=(-a1*sin(th1(i)))-(a2*sin(th1(i)+th2(i))/2); a21=-(a2*sin(th1(i)+th2(i))/2); a31=0; a41=1; a12=(a1*cos(th1(i)))+(a2*cos(th1(i)+th2(i))/2); a22=a2*cos(th1(i)+th2(i))/2; a32=0; a42=0; a13=1; a23=1; a33=1; a43=0; A=[a11 a12 a13 ;a21 a22 a23 ;a31 a32 a33 ;a41 a42 a43]; % J Matrix 'Orthogonal Complement of Constraint Matrix' j11=1; j21=(-1-(2*a1*cos(th1(i))/(a2*cos(th2(i))))); j31= 2*a1*cos(th1(i)) / (a2*cos(th1(i)+th2(i))); j41=((a1*sin(th1(i)))-(a1*cos(th1(i))*tan(th1(i)+th2(i)))); J=[j11 j21 j31 j41]; % Torque matrix Tor(i) = (J*I*ddq) + (J*H) + (J*G)

i=i+1; end t=0:0.01:2 subplot(2,2,1); plot(t,ddb4,'linewidth',4) grid on; %figure(2) subplot(2,2,2); plot(t,th1,t,dth1,t,ddth1) grid on; subplot(2,2,3); plot(t,th2,t,dth2,t,ddth2) subplot(2,2,4); plot(t,th3,t,dth3,t,ddth3) figure(4); plot(t,Tor) pp1=spline(t,Tor); save('td1.mat','pp1')

Euler Lagrange Independent Coordinate Method

Contd….

Matlab Program for Independent Euler Lagrange Method m1=1; m2=1; m3=1; a1=1;a2=3; g=9.81; i=1; q1= 0; q2= (pi/2); T=2; for t= 0:0.01:2 th1(i)= (((q2-q1))/T)*(t-((T/(2*pi))*sin(2*pi*t/T))); thd1(i)=(((q2-q1)/T)*(1-cos(2*pi*t/T))); thdd1(i)= ((q2-q1)/T)*(2*pi/T)*(sin(2*pi*t/T)); th2(i)=asin(-2*a1*sin(th1(i))/a2)-th1(i); thd2(i)=(-1-2*a1*cos(th1(i))/(a2*cos(th1(i)+th2(i))))*thd1(i); thdd2(i)=(2*a1*(sin(th1(i))*(thd1(i))*(thd1(i))cos(th1(i))*(thdd1(i)))/a2+sin(th1(i)+th2(i))*((thd1(i))+thd2(i))*((thd1(i))+thd2(i)))/cos(th1(i) +th2(i))-(thdd1(i)); bd(i)=a1*sin(th1(i))*thd1(i)+a2*sin(th1(i)+th2(i))*(thd1(i)+thd2(i))/2; bdd(i)=a1*sin(th1(i))*thdd1(i)+a1*cos(th1(i))*thd1(i)*thd1(i)+a2*sin(th1(i)+th2(i))*((thdd1(i ))+thdd2(i))/2+a2*cos(th1(i)+th2(i))*(thd1(i)+thd2(i))*(thd1(i)+thd2(i)); s(i)=m1*a1*a1/3+m2*a1*a1+m2*a1*a2*cos(th2(i))/22*m2*a1*a2*cos(th1(i))/(3*cos(th1(i)+th2(i)))m2*a1*a1*cos(th1(i))*cos(th2(i))/cos(th1(i)+th2(i)); u(i)=m2*a1*a2*cos(th2(i))/2-2*m2*a1*a2*cos(th1(i))/(3*cos(th1(i)+th2(i)));

v(i)=m3*a1*sin(th1(i))-m3*a1*cos(th1(i))*sin(th1(i)+th2(i))/cos(th1(i)+th2(i)); w(i)=-m2*a1*a2*sin(th2(i))*thd2(i)*(thd1(i)/2+thd2(i)/2a1*cos(th1(i))*thd1(i)/(a2*cos(th1(i)+th2(i)))); x(i)=(cos(th1(i))*sin(th1(i)+th2(i))*(thd1(i)+thd2(i))-cos(th1(i)+th2(i))*sin(th1(i))*thd1(i))*(m2*a1*a1*cos(th2(i))*thd1(i)2*m2*a1*a2*(thd1(i)+thd2(i))/3)/(cos(th1(i)+th2(i))*cos(th1(i)+th2(i))); y(i)=m3*bd(i)*(a1*cos(th1(i))*thd1(i)a1*cos(th1(i))*(thd1(i)+thd2(i))/(cos(th1(i)+th2(i))*cos(th1(i)+th2(i)))+a1*sin(th1(i))*sin(th1 (i)+th2(i))*thd1(i)/cos(th1(i)+th2(i))); z(i)=m2*a1*a2*(1+2*a1*cos(th1(i))/(a2*cos(th1(i)+th2(i))))*sin(th2(i))*(thd1(i)*thd1(i)+thd 1(i)*thd2(i))/2-m1*g*a1*cos(th1(i))/2; tor(i)=s(i)*thdd1(i)+u(i)*thdd2(i)+v(i)*bdd(i)+w(i)+x(i)+y(i)-z(i); i=i+1; end t= 0:0.01:2 figure(11) plot(t,tor); pp=spline(t,tor) save ('torquedata.mat',' pp')

Pendode Function function dy=pendodes(t,y) load torquedata.mat; tor=ppval(pp,t); dy=zeros(2,1) m1=1; m2=1; m3=1; a1=1; a2=3; g=9.81; th2=-y(1)-asin((2*a1*sin(y(1)))/a2); thd2=-(1+(2*a1*cos(y(1)))/(a2*cos(y(1)+th2)))*y(2); bd=a1*sin(y(1))*y(2)+a2*sin(y(1)+th2)*(y(2)+thd2)/2; i=m1*a1*a1/3+m2*a1*a1+m2*a1*a2*cos(th2)/2-2*m2*a1*a2*cos(y(1))/(3*cos(y(1)+th2))m2*a1*a1*cos(y(1))*cos(th2)/cos(y(1)+th2); j=(m2*a1*a2*cos(th2)/2-2*m2*a1*a2*cos(y(1))/(3*cos(y(1)+th2)))*(-12*a1*cos(y(1))/(a2*cos(y(1)+th2)))*(a1*sin(y(1))-(a1*cos(y(1))*tan(y(1)+th2))); k=m3*a1*sin(y(1))-m3*a1*cos(y(1))*sin(y(1)+th2)/cos(y(1)+th2); p=2*a1/a2*y(2)*(sin(y(1))*y(2)-cos(y(1))*tan(y(1)+th2)*(y(2)+thd2)); q=a1*cos(y(1))*y(2)*y(2)a1*cos(y(1))*y(2)*(y(2)+thd2)/(cos(y(1)+th2)*cos(y(1)+th2))+a1*sin(y(1))*y(2)*y(2)*tan(y( 1)+th2);

w=-m2*a1*a2*sin(th2)*thd2*(y(2)/2+thd2/2-a1*cos(y(1))*y(2)/(a2*cos(y(1)+th2))); x=(cos(y(1))*sin(y(1)+th2)*(y(2)+thd2)-cos(y(1)+th2)*sin(y(1))*y(2))*(m2*a1*a1*cos(th2)*y(2)-2*m2*a1*a2*(y(2)+thd2)/3)/(cos(y(1)+th2)*cos(y(1)+th2)); u=m3*bd*(a1*cos(y(1))*y(2)a1*cos(y(1))*(y(2)+thd2)/(cos(y(1)+th2)*cos(y(1)+th2))+a1*sin(y(1))*sin(y(1)+th2)*y(2)/co s(y(1)+th2)); z=m2*a1*a2*(1+2*a1*cos(y(1))/(a2*cos(y(1)+th2)))*sin(th2)*(y(2)*y(2)+y(2)*thd2)/2m1*g*a1*cos(y(1))/2; thdd1=(tor-(p+q+w+x+u-z))/(i+j+k); dy(1)=y(2); dy(2)=thdd1;

Matlab Program for Solution using Lagrange Multipliers and NOC clc;clf;clear; a1=1; a=1; a2=3; b=3; g=9.81; m1=1; m2=1; m3=1; i=1; a0=4; q1= 0; q2=(.5*pi); T=2; for t1= 0:0.01:2 th1(i)= (((q2-q1))/T)*(t1-((T/(2*pi))*sin(2*pi*t1/T))); thd1(i)=(((q2-q1)/T)*(1-cos(2*pi*t1/T))); thdd1(i)= ((q2-q1)/T)*(2*pi/T)*(sin(2*pi*t1/T)); th2(i)=-th1(i)-asin((2*a1*sin(th1(i)))/a2); thd2(i)=-(1+(2*a1*cos(th1(i)))/(a2*cos(th1(i)+th2(i))))*thd1(i);

thdd2(i)=sec(th1(i)+th2(i))*[2*a1/a2]*(sin(th1(i))*thd1(i)*thd1(i)cos(th1(i))*thdd1(i))+sec(th1(i)+th2(i))*(sin(th1(i)+th2(i))*(th1(i)+th2(i))*(th1(i)+th2(i)))thdd1(i); b4(i)=a0-a1*cos(th1(i))-0.5*a2*cos(th1(i)+th2(i)) bd4(i)=(a1*sin(th1(i))-a1*cos(th1(i))*sin(th1(i)+th2(i))/cos(th1(i)+th2(i)))*thd1(i); bdd4(i)=(a1*sin(th1(i))(a1*cos(th1(i))*sin(th1(i)+th2(i)))/cos(th1(i)+th2(i)))*thdd1(i)+(a1*cos(th1(i))*thd1(i)*thd1(i ))(a1*cos(th1(i))*thd1(i)*(thd1(i)+thd2(i)))/(cos((th1(i)+th2(i)))*cos((th1(i)+th2(i))))+a1*thd1(i )*sin(th1(i))*tan(th1(i)+th2(i))*thd1(i); i11(i)=(m1*a*a/3)+m2*(a*a+a*b*cos(th2(i))+b*b/3); i12(i)=m2*((b*b/3)+0.5*a*b*cos(th2(i))); i21(i)=m2*((b*b/3)+0.5*a*b*cos(th2(i))); i22(i)=m2*b*b/3; h1(i)=-m2*a*b*sin(th2(i))*thd1(i)*thd2(i)-m2*(a/2)*b*sin(th2(i))*thd2(i)*thd2(i); h2(i)=m2*(a/2)*b*sin(th2(i))*thd1(i)*thd1(i); g1(i)=m1*g*a/2*cos(th1(i))+m2*g*(a*cos(th1(i))+b/2*cos(th1(i)+th2(i))); g2(i)=m2*g*b/2*cos(th1(i)+th2(i));

s11(i)=(i11(i)*thdd1(i)+i12(i)*thdd2(i)+h1(i)+g1(i)); s12(i)=(a1*sin(th1(i))+a2*.5*sin(th1(i)+th2(i)))* m3*bdd4(i); s13(i)=((a1*cos(th1(i))+a2*.5*cos(th1(i)+th2(i)))*2*(i21(i)*thdd1(i)+i22(i)*thdd2(i)+h2(i)+g2(i)+a2* .5*sin(th1(i)+th2(i))*m3*bdd4(i))/(a2*cos(th1(i)+th2(i)))); s1(i)=s11(i)+s12(i)+s13(i) i=i+1; end t1=0:0.01:2 figure(1) plot(t1,s1) figure(2) plot(t1,th1,'-',t1,thd1,':',t1,thdd1,'.') legend('1st link pos','1st link velocity','1st link accelaratio') figure(3) plot(t1,th2,'-',t1,thd2,':',t1,thdd2,'.') legend('2nd link pos','2nd link velocity','2nd link accelaratio') figure(4) plot(t1,bd4,':',t1,bdd4,'.') legend('slider position','slider velocity','slider accelaration') pp1=spline(t1,s1); save('torquedata1.mat','pp1')

Pendode Function function dy=pendode(t,y) dy=zeros(2,1) m1=1;m2=1;m3=1;a1=1;a2=3;a3=1;g=9.81; th1=y(1); thd1=y(2); T1=0; th2=-th1-asin((2*a1*sin(th1))/a2); thd2=-(1+(2*a1*cos(th1))/(a2*cos(th1+th2)))*thd1; thdd1=3*cos(th1+th2)^2/a1^2/(cos(th1+th2)^2*m1+3*cos(th1+th2)^2*m2 -6*m2*cos(th1+th2)*cos(th2)*cos(th1)+4*cos(th1)^2*m2 +6*cos(th1)*sin(th1+th2)*m3*cos(th1+th2)*sin(th1) -3*cos(th1)^2*sin(th1+th2)^2*m3 -3*cos(th1+th2)^2*m3*sin(th1)^2)*(T1+m2*a1*a2*sin(th2)*thd1*thd2 +1/2*m2*a1*a2*sin(th2)*thd2^2-1/2*m1*g*a1*cos(th1)-m2*g*(a1*cos(th1) +1/2*a2*cos(th1+th2)))-3/a2*cos(th1+th2)*(2*a1*cos(th1) +a2*cos(th1+th2))/a1^2/(cos(th1+th2)^2*m1+3*cos(th1+th2)^2*m2 -6*m2*cos(th1+th2)*cos(th2)*cos(th1) +4*cos(th1)^2*m2+6*cos(th1)*sin(th1+th2)*m3*cos(th1+th2)*sin(th1) -3*cos(th1)^2*sin(th1+th2)^2*m3 -3*cos(th1+th2)^2*m3*sin(th1)^2)*(-1/2*m2*a1*a2*sin(th2)*thd1^2 -1/2*m2*g*a2*cos(th1+th2))+1/a1*(-3*m2*cos(th1+th2)*cos(th2) +4*m2*cos(th1)+3*sin(th1+th2)*m3*cos(th1+th2)*sin(th1)

-3*sin(th1+th2)^2*m3*cos(th1))/(cos(th1+th2)^2*m1+3*cos(th1+th2)^2*m2 -6*m2*cos(th1+th2)*cos(th2)*cos(th1)+4*cos(th1)^2*m2 +6*cos(th1)*sin(th1+th2)*m3*cos(th1+th2)*sin(th1)-3*cos(th1)^2*sin(th1+th2)^2*m3 3*cos(th1+th2)^2*m3*sin(th1)^2)*(a1*sin(th1)*thd1+1/2*a2*sin(th1+th2)*(thd1+thd2)*thd 1 +1/2*a2*sin(th1+th2)*(thd1+thd2)*thd2)-3*cos(th1+th2)*m3/a1*(-cos(th1+th2)*sin(th1) +sin(th1+th2)*cos(th1))/(cos(th1+th2)^2*m1+3*cos(th1+th2)^2*m2 -6*m2*cos(th1+th2)*cos(th2)*cos(th1)+4*cos(th1)^2*m2 +6*cos(th1)*sin(th1+th2)*m3*cos(th1+th2)*sin(th1)-3*cos(th1)^2*sin(th1+th2)^2*m3 3*cos(th1+th2)^2*m3*sin(th1)^2)*((a1*cos(th1)*thd1+1/2*a2*cos(th1+th2)*(thd1+thd2))*t hd1 +1/2*a2*cos(th1+th2)*(thd1+thd2)*thd2); dy(1)=y(2); dy(2)=thdd1;

clc;clf;clear; syms b s a m1 m2 m3 a1 a2 b4 th1 th2 g T c d thd1 thd2 S=[(m1*a1*a1/3)+m2*(a1*a1+a1*a2*cos(th2)+a2*a2/3) 2*((a2*a2/3)+0.5*a1*a2*cos(th2)) 0 (a1*sin(th1)+a2*.5*sin(th1+th2)), -(a1*cos(th1)+a2*.5*cos(th1+th2)); m2*((a2*a2/3)+0.5*a1*a2*cos(th2)), m2*a2*a2/3, 0, -a2*.5*sin(th1+th2), -2*0.5*cos(th1+th2)); 0, 0 , m3 , -1, 0; (a1*cos(th1)+a2*0.5*cos(th1+th2)), a2*0.5*cos(th1+th2), 0 , 0, 0; (-a1*sin(th1)-a2*0.5*sin(th1+th2)), -(a2*0.5*sin(th1+th2)), -1, 0 , 0]; c=[T-(-m2*a1*a2*sin(th2)*thd1*thd2-m2*(a1/2)*a2*sin(th2)*thd2*thd2)(m1*g*a1/2*cos(th1)+m2*g*(a1*cos(th1)+a2/2*cos(th1+th2))); -(m2*(a1/2)*a2*sin(th2)*thd1*thd1)-m2*g*a2/2*cos(th1+th2); 0; a1*sin(th1)*thd1+a2*0.5*sin(th1+th2)*(thd1+thd2)*thd1)+(a2*0.5*sin(th1+th2)*(thd1+thd) *thd2

((a1*cos(th1)*thd1+a2*0.5*cos(th1+th2)*(thd1+thd2))*thd1)+(a2*0.5*cos(th1+th2)*(thd1+t hd2)*thd2)]; d=inv(S)*c

RESULTS  INVERSE DYNAMICS FREE SIMULATION FORWARD DYNAMICS

PLOTS FOR LINK #1

PLOTS FOR LINK #2

PLOTS FOR SLIDER

TORQUE OBTAINED

FREE SIMULATION

Forward Dynamics 10 link1 pos link2 vel 8

6

4

2

0

-2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Verification 6 link1 pos link2 vel

4

2

0

-2

-4

-6

0

0.2

0.4

0.6

0.8

1

1.2

M2=0; m3=0; a2=0;

1.4

1.6

1.8

2

Verification 5 link1 posi link1 vel

4 3 2 1 0 -1 -2 -3 -4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

ROBO ANALYSER Comparison of Results obtained from Matlab and Robo-Analyser for RP manipulator System.