ME513 Homework 6 Problem 1 a) First the sprung mass and the front and the rear axle loads are calculated. W_s= 16451N W_f=9998N W_r=7856N Then the sprung mass center of gravity position is calculated. a*=0.432m b*=0.568m h*=0.506m Tilting moment and sum of roll stiffness’s are calculated. K=8780Nm C=18569Nm/rad We calculate the roll angle from the formula: 𝑲𝑲 ∗ 𝒂𝒂𝑳𝑳 𝚽𝚽 = � � = 𝟐𝟐𝟐𝟐. 𝟓𝟓𝟓𝟓° 𝑪𝑪 − 𝑾𝑾𝒔𝒔 𝒉𝒉𝒄𝒄 b) Backwards calculation is applied and the antiroll bar stiffness is calculated from the above formula as: C_roll = 61620Nm/rad c) Steady state roll moment is calculated from the formula: 𝑼𝑼𝟐𝟐 𝑴𝑴𝒓𝒓 = 𝒉𝒉∗ 𝑾𝑾𝒔𝒔 � � = 𝟐𝟐𝟐𝟐𝟐𝟐𝟐𝟐 𝑵𝑵𝑵𝑵 𝒓𝒓𝒓𝒓
Problem 2 Input
a)
b)
c)
The lateral acceleration values are just above the max assumption limit 0.3g for a short period only, we can say that the bicycle model is valid considering this values.
d)
Tire slip angle values for both rear and front tires are in the linear assumption range 0-4 degrees. So the validity of the bicycle model correct. Front tire slip angle is always greater than the rear one so the handling behavior of the car can be said as UNDERSTEER.
e)
In the first question reasonable roll angle 3.5 degrees is given as example to solve. It can be concluded that the roll angle values in our case are reasonable and good values for a standard vehicle.
f)
MATLAB CODE %%ME 513 HW6 %Mümün YILDIZ 2193670 %a------------------------------------------------------------clear all; clc; W=1820; W_uf=67; W_ur=76; w_base=2.427; h_c=0.532; r_t=0.324; k_f=16180; k_r=22420; dist_f=0.56; dist_r=0.44; m=0.106; n=0.106; W_s=(W-W_uf-W_ur)*9.81 W_f=W*9.81*dist_f W_r=W*9.81*dist_r a_s=(W_r-W_ur*9.81)/W_s b_s=(W_f-W_uf*9.81)/W_s hs=(h_c*W*9.81-r_t*(W_uf+W_ur)*9.81)/W_s h_star=hs-((a_s*n+b_s*m)/w_base) K_i=W_s*h_star+(W_uf*0.324+W_ur*0.324)*9.81 f1=0.235; f2=0.310; b=3.59; d=0.665; a=3.69; Cjf=2*(b*d/a)^2*(f1/f2)^2*k_f; Cjr=2*(b*d/a)^2*(f1/f2)^2*k_r; Cj=Cjr+Cjf roll=(K_i*0.5)/(Cj-W_s*h_star)*180/pi
%b-----------------------------roll2=3.5; C_barr=(K_i*0.5)/(roll2/180*pi)+W_s*h_star-Cj; double(C_barr) %c-------------------------------umps=90/3.6; r=185; M_roll=h_star*W_s*(umps^2/r/9.81)
%%Problem 2 ME513 HW6 %----------------------------------------------------------clc; clear all; m=1180;
I_z=2040; a=1.11; kroll=48000; C_f=-48500; m_s=875; I_x=336; b=1.48; croll=5500; C_r=-74550; h_s=0.56; U=90/3.6; D_1=[]; for t=0:0.05:10 deldat=2*sin(pi*(t)/(5)*2); deldat(t<0)=0; deldat(t>5)=0; D_1=[D_1 deldat]; end deltar=D_1.*pi/180; plot(0:0.05:10,D_1) xlabel('Time (s)','fontsize',12,'fontweight','b') ylabel('Steering angle (deg)','fontsize',12,'fontweight','b') M=[m 0 m_s*h_s 0;0 I_z 0 0 ; m_s*h_s 0 I_x 0; 0 0 0 1]; A_i=[(C_f+C_r)/U (a*C_f-b*C_r)/U-m*U 0 0;(a*C_f-b*C_r)/U (a^2*C_f+b^2*C_r)/U.... 0 0; 0 -m_s*h_s*U -croll m_s*9.81*h_s-kroll;0 0 1 0]; B_i=[-C_f;-a*C_f;0;0]; A=inv(M)*A_i; B=inv(M)*B_i; C=eye(4); D=[0;0;0;0]; sys=ss(A,B,C,D); [x,T]=lsim(sys,deltar,0:0.05:10); v=x(:,1); ss=x(:,1)/U; r=x(:,2); figure(2) plot(0:0.05:10,ss,'linewidth',2) xlabel('Time (s)','fontsize',12,'fontweight','b') ylabel('Vehicle Side Slip Angle (^o)','fontsize',12,'fontweight','b') title('Vehicle Side Slip Angle (^o)','fontsize',12,'fontweight','b') grid on figure(3) plot(0:0.05:10,r/pi*180,'linewidth',2) xlabel('Time (s)','fontsize',12,'fontweight','b') ylabel('Yaw Velocity (^o/s)','fontsize',12,'fontweight','b') title('Yaw Velocity vs Time','fontsize',12,'fontweight','b') grid on figure(4) lat=A*x'+B*deltar; g=(lat(1,:)+U*r')/9.81; plot(0:0.05:10,g,'linewidth',2) xlabel('Time(s)','fontsize',12,'fontweight','b') ylabel('Lateral Acceleration (g)','fontsize',12,'fontweight','b') title('Lateral Acceleration vs Time','fontsize',12,'fontweight','b') grid on figure(5)
alfaf=(v+a*r)/U-deltar'; alfar=(v-b*r)/U; plot(0:0.05:10,alfaf/pi*180,'b',0:0.05:10,alfar/pi*180,... 'r--','linewidth',2) legend('Front','Rear','Location','NorthWest') xlabel('Time (s)','fontsize',12,'fontweight','b') ylabel('Tire Slip Angles (^o)','fontsize',12,'fontweight','b') title('Tire Slip Angles vs Time','fontsize',12,'fontweight','b') grid on fi=x(:,4)*180/pi; figure(6) plot(0:0.05:10,fi,'linewidth',2) xlabel('Time(s)','fontsize',12,'fontweight','b') ylabel('Roll Angle (^o)','fontsize',12,'fontweight','b') title('Roll Angle (^o) vs Time','fontsize',12,'fontweight','b') grid on figure(7) lamda = cumtrapz(r)*0.01; xdot = U*cos(lamda)-v.*sin(lamda); ydot = v.*cos(lamda)+ U*sin(lamda); xt = cumtrapz(xdot)*0.01; yt = cumtrapz(ydot)*0.01; plot(xt,yt,'b-','linewidth',2) axis ij xlabel('X [m]','fontsize',12,'fontweight','b') ylabel('Y [m]','fontsize',12,'fontweight','b') title('Steering Trajectory','fontsize',12,'fontweight','b') grid on
