Viniamin Tokarchuk
Average Power Calculations
The point of all these derivations is to come up with a way to accurately calculate the profile average power and to be able to plot instantaneous power via excel or matlab. We begin with equation 1 which references figure 1 in order to solve for α2(t) equal to a constant α.
(1)
Figure 1 Profile of Guide Rollers
Where Vfis final velocity in either RPM or rad/sec or m/s depending on what is desired. The ramp time is tr. β is the percentage of Jerk that is used for the portion of the ramp time. If β=0.5, for 25% of the ramp time a positive Jerk will be used and 25% a negative Jerk value will be used to create the curved profile (see figure 1 Jerk waveform). We would like to solve for the steady state acceleration α during the no Jerk period of the ramp up profile to setup all the equations in terms of α. Equation 1 can be simplified in terms of α using equations 2.
(2)
(3)
We can substitute the result in 3 into equation 1 to get α1(t) in terms of α, β, and tr.
(4)
By definition the acceleration is constant once there is no Jerk applied.
(5)
α3(t) can be solved for by substituting for
(6)
Substituting the corresponding acceleration equations into equation 1 provides 7 to be solved for α.
(7)
With a good amount of algebra, α can be easily solved for in terms of Vf, β, and tr. Units don’t matter for this equation, but Vf was normally given in RPM so α is in RPM/sec.
(8)
The definition of average power is given below assuming the units for velocity are radians and T is the torque. Since the profile changes so much, different velocity and torque apply at different times, therefore this expression is not so easily solved for. However, once the equations are setup, a Matlab script can easily be written to plot the profile with the instantaneous power.
(9)
For the ramp up portion of the curve, the following equations apply for the torque. The numbers correspond to each of the segments of the curve in figure 1. The variable Iis the inertia in kgm2.
(10)
The 2π/60 is simply a conversion of the α which is assumed to be in rpm/sec! The next set of equations are the velocity equations at each of the 3 ramp up segments. The definitions are given but the algebra will not be shown.
(11)
As a check, plugging would yield Vf= which proves that the functions represent the profile correctly. Plugging the equations in 10 and 11 into 9 would yield only part of the problem since the rest of the steady-state portion and deceleration are missing. However, that can be easily found by negating certain terms. The steady state torque and velocity are given in 12 and 13.
(12)
(13)
These equations assume the time starts at the beginning of the steady-state flat velocity portion and not at the beginning of the profile.
For the ramp down portion, the equations in 10 can be modified easily if the start time of the ramp down is set at t=0. The Jerk values would simply be negated relative to their counterparts thus negating the acceleration equations α4(t)= -α1(t), α5(t)= -α2(t), α6(t)= -α3(t) .
(14)
The fall time tfis used instead of trin case the ramp down time is different. This would also change alpha as described in 8. The assumption is that Jerk percentage remains the same, though that could easily be changed. For the velocity equations all that changes is that the initial starting point is Vfwhich drops down to zero.
(15)
Plugging in tfinto V6(t) provides an RPM of 0 so the equations are correct. The equations can be plugged into equation 9 to provide the average power of the profile.
(16)
The equation is quite a monster and I would not recommend solving it analytically, though technically it is possible. A matlab script can easily be implemented with these equations to solve for the average power. The variable T is simply equal to the ramp up, steady-state, and ramp down times tr,ts, and tf. It is extremely important to keep in mind the units need to be rad/sec and Nm to get proper results!
However, before you go and do this the hard way, the Pavg can be approximated to a very accurate degree by assuming constant acceleration in the ramp up and down portions, therefore the area under the curve becomes very trivial. Alpha simply becomes Vf/tr(convert to rad/sec^2 with 2π/60).
(17)
The equation is valid if Vf is in rad/sec and α in rad/sec2. Using a matlab script the long equation provides the same results within a few decimal places. Therefore, the derivation simply proved that percentage Jerk does not influence average power, but it does influence peak power.
Using matlab, the instantaneous acceleration, velocity, torque, and power could be plotted. At 900 RPM, tr=7.5 sec, ts=20 sec, tf=7.5 sec, β=0.5, V, and I=0.8 kgm2 here are the plots.
The plots agree with what makes sense logically at each of the velocity segments. One interesting discovery was that increasing the percentage Jerk decreased peak power significantly. At 100% Jerk (β=1), the peak power was 3.39 kW. At a very small percentage, the peak power was 3.87 kW.
At low amounts of Jerk, the power curve is very linear unlike at higher percentages. Finally, an examination of efficiency for the RDD-B21529 was done.
From motion analyzer motoring average power and peak power were obtained. They were compared to the theoretical power needed and the efficiency was found to be pretty low. It shows that the DS-264 currently is probably in the very low efficiency percentages currently.
Motoring Avg. Power (kW) / Motor Peak Power (kW) / Profile Avg. Power (kW) / Profile Peak Power (kW) / Efficiency %900 RPM 300 Cuts / 2.59 / 3.94 / 2.296 / 3.51 / 88.6
900 RPM 30 Cuts / 0.369 / 1.4 / 0.230 / 1.30 / 62.3
716 RPM 300 Cuts / 2.06 / 2.96 / 1.826 / 2.65 / 88.6
716 RPM 300 Cuts / 0.271 / 0.935 / 0.183 / 0.860 / 67.5
Matlab Script
%% Viniamin Tokarchuk- Guide Roller Profile Efficiency Analysis
% Program Inputs
clearall
V=900; % roller speed in RPM
Tff=3.1; % approximate frictional force in Nm
tr=7.5; % ramp up time in sec
tf=7.5; % ramp down time is sec
ts=20; % steady time in sec
dt=0.001; % delta time
br=.5; % percentage Jerk applied at acceleration on ramp up (cannot be zero or more than 1)
bf=.5; % percentage Jerk applied at acceleration on ramp down (cannot be zero or more than 1)
I=0.8; % inertia of the guide rollers in kgm^2
%% Basic Setup
cf=2*pi/60; % conversion factor
t1=0:dt:tr; % ramp up segment
t2=0:dt:ts; % steady segment time
t3=0:dt:tf; % ramp down segment time
ar=V/((tr*(1-br/2)));
af=V/((tf*(1-bf/2)));
Tr=zeros(1,length(t1));
Vr=zeros(1,length(t1));
accr=zeros(1,length(t1));
Pr=zeros(1,length(t1));
Ts=Tff.*ones(1,length(t2));
Vs=V*ones(1,length(t2));
as=zeros(1,length(t2));
Ps=Ts.*Vs*cf;
Tf=zeros(1,length(t3));
Vf=zeros(1,length(t3));
accf=zeros(1,length(t1));
Pf=zeros(1,length(t3));
% ramp up portion
for n=1:length(t1)
if t1(n) <=0.5*br*tr
accr(n)=2*ar*t1(n)/(br*tr);
Tr(n)=2*I*ar*t1(n)*cf/(br*tr)+Tff;
Vr(n)=ar*t1(n)^2/(br*tr);
Pr(n)=Tr(n)*Vr(n)*cf;
elseif t1(n) <=tr-0.5*br*tr
accr(n)=ar;
Tr(n)=ar*I*cf+Tff;
Vr(n)=ar*t1(n)-ar*br*tr/4;
Pr(n)=Tr(n)*Vr(n)*cf;
else
accr(n)=-2*ar*(t1(n)-tr)/(br*tr);
Tr(n)=-2*ar*I*cf*(t1(n)-tr)/(br*tr)+Tff;
Vr(n)=ar*tr-ar*br*tr/2-ar*tr/br-ar*t1(n)^2/(br*tr)+2*ar*t1(n)/br;
Pr(n)=Tr(n)*Vr(n)*cf;
end
end
% ramp down portion
for n=1:length(t3)
if t3(n) <=0.5*bf*tf
accf(n)=-2*af*t3(n)/(bf*tf);
Tf(n)=-2*I*af*t3(n)*cf/(bf*tf)+Tff;
Vf(n)=V-af*t3(n)^2/(bf*tf);
Pf(n)=Tf(n)*Vf(n)*cf;
elseif t3(n) <=tf-0.5*br*tf
accf(n)=-af;
Tf(n)=-af*I*cf+Tff;
Vf(n)=V-af*t3(n)+af*bf*tf/4;
Pf(n)=Tf(n)*Vf(n)*cf;
else
accf(n)=2*af*(t3(n)-tf)/(bf*tf);
Tf(n)=2*af*I*cf*(t3(n)-tf)/(bf*tf)+Tff;
Vf(n)=V-af*tf+af*bf*tf/2+af*tf/bf+af*t3(n)^2/(bf*tf)-2*af*t3(n)/bf;
Pf(n)=Tf(n)*Vf(n)*cf;
end
end
ac=[accr as accf];
Vc=[Vr Vs Vf];
Tc=[TrTsTf];
Pc=[Pr Ps Pf];
t=[t1 (tr+t2) (t3+tf+ts)];
Pavg1=sum(Pc.*dt)/(tr+ts+tf) % numerical method with Jerk incorporated
Pmax=max(Pc)
a1=V/tr;
a2=V/tf;
Pavg2=(0.5*V*cf*(a1*cf*I+Tff)*tr+Tff*V*cf*ts+0.5*V*cf*(-a2*cf*I+Tff)*tf)/(tr+ts+tf) % simplified method without Jerk incorporated
%% Plotting
figure(1)
plot(t,ac,'r') % acceleration
title('Acceleration Profile')
ylabel('Radial Acceleration (RPM/sec)')
xlabel('time (sec)')
figure(2)
plot(t,Vc,'m') % velocity
title('Velocity Profile')
ylabel('Radial Velocity (RPM)')
xlabel('time (sec)')
ylim([0 1000])
figure(3)
plot(t,Tc,'g') % Torque
title('Load Torque Profile')
ylabel('Torque (Nm)')
xlabel('time (sec)')
figure(4)
plot(t,Pc,'k') % Power
title('Instantaneous Power')
ylabel('Power (W)')
xlabel('time (sec)')