Equivalent constant engine torque is clearer than Power

I have some online car and moto transmission calculators and comparators that simulate the acceleration times etc.

After many years of trying to find a replacement for max power to characterize the engine with a single value of torque that is more precise than power, I defined the Weighted average torque. But using it in the simulation did not confirm it as an equivalent (using that torque instead of the engine torque curve did not result in the same distance covered in the same time while using a single gear ratio on all rpm range).

Then I tried the empiric way and noticed there is an Equivalent constant engine torque that fulfills that condition as it can be seen in this video.

I did not had the necessary mathematical skills to define a way to calculate it rather than finding it with multiple simulation tries.

Can it be calculated (and how)?

I will not get into more details for now because there are many, but I will gladly reply if any questions arise.

UPDATE:

Please read all comments and analyze the links before posting so we can keep the discussion on point. This is not a question about power, nor a discussion about it. It is only about torque. Why torque and not power is another discussion that for me started in 2017 and will go on many years in the future (sadly) ...

UPDATE 2

Gear ratio 9.820: Gear ratio 10:

Input data for last picture (the above last one):

{"calcinputname":"Tesla SP 600Nm - web info ~265.51Nm 10","valuea":"245","valueb":"45","valuea1":"","valueb1":"","valuec":"19","valuec1":"","rap1":"1","rap2":"1","rap3":"1","rap4":"1","rap5":"1","rap6":"1","rapf":"10","Pmax":"311","PRPM":"8709","Mmax":"599","MRPM":"4952","M":"2112","schimb1":"15435","schimb2":"15435","schimb3":"15435","schimb4":"15435","schimb5":"15435","RPMNm1":"1350","NmRPM1":"600","RPMNm2":"4952.36","NmRPM2":"599.06","RPMNm3":"5080.59","NmRPM3":"590.64","RPMNm4":"5970.07","NmRPM4":"500.78","RPMNm5":"6681.05","NmRPM5":"447.42","RPMNm6":"7599.41","NmRPM6":"394.07","RPMNm7":"8695.51","NmRPM7":"344.46","RPMNm8":"9213.94","NmRPM8":"306.08","RPMNm9":"9821.24","NmRPM9":"269.58","RPMNm10":"10443.36","NmRPM10":"238.69","RPMNm11":"11169.16","NmRPM11":"207.8","RPMNm12":"11865.33","NmRPM12":"184.4","RPMNm13":"12902.19","NmRPM13":"156.32","RPMNm14":"14101.98","NmRPM14":"131.05","RPMNm15":"15435","NmRPM15":"109.52","airdensity":"1.225","refarea":"2.53","dragcoeff":""}

vs

{"calcinputname":"Tesla SP 600Nm - web info EQ 392Nm 10 ratio","valuea":"245","valueb":"45","valuea1":"","valueb1":"","valuec":"19","valuec1":"","rap1":"1","rap2":"1","rap3":"1","rap4":"1","rap5":"1","rap6":"1","rapf":"10","Pmax":"311","PRPM":"8709","Mmax":"599","MRPM":"4952","M":"2112","schimb1":"15435","schimb2":"15435","schimb3":"15435","schimb4":"15435","schimb5":"15435","RPMNm1":"1350","NmRPM1":"392","RPMNm2":"4952.36","NmRPM2":"392","RPMNm3":"5080.59","NmRPM3":"392","RPMNm4":"5970.07","NmRPM4":"392","RPMNm5":"6681.05","NmRPM5":"392","RPMNm6":"7599.41","NmRPM6":"392","RPMNm7":"8695.51","NmRPM7":"392","RPMNm8":"9213.94","NmRPM8":"392","RPMNm9":"9821.24","NmRPM9":"392","RPMNm10":"10443.36","NmRPM10":"392","RPMNm11":"11169.16","NmRPM11":"392","RPMNm12":"11865.33","NmRPM12":"392","RPMNm13":"12902.19","NmRPM13":"392","RPMNm14":"14101.98","NmRPM14":"392","RPMNm15":"15435","NmRPM15":"392","airdensity":"1.225","refarea":"2.53","dragcoeff":""}

Analog is valid for different masses.

ANSWER:

I ended up implementing 2nd solution (not so accurate as first one but simpler to code):

Edit 2: Alternative way to get the info using only one simulation.

There is a way to find all the information you need using only one simulation.

using the simulation, find min speed, max speed, time taken, distance traveled, as well as torque and acceleration at minimum RPM.

In this case, torque at minimum RPM = 600Nm, and acceleration (found in the second graph) is 7.93N/kg = 7.93m/s^2. Call these values minS, maxS, t1, d1, T, A, respectively.

We can compute t2 = (maxS - minS)/(A/T * 100). Then compute d2 = (maxS + minS)/2 * t2. Then, just use the formula torque = 100 * (d2 - maxS*t2)/(d1 - t1*maxS) as explained earlier.

Overall, this is the closed form formula:

100 * ((maxS + minS)/2 * (maxS - minS)/(A/T * 100) - maxS*(maxS - minS)/(A/T * 100))/(d1 - t1*maxS)

It is very ugly, but at least you can cancel out the 100s, and you just get:

torque = ((maxS + minS)/2 * (maxS - minS)/(A/T) - maxS*(maxS - minS)/(A/T))/(d1 - t1*maxS)

This gave 391.53 Nm so it is acceptable.

Thank you @ChickenCoding123

And to answer you, this is the reason for this: one value comparison - note that 6 gears EQT for Audi is 389.3 vs 1 gear EQT 348.77 Nm

(Full list of examples can be seen in the home page of the calculator - interesting study)

Result without air friction:

I started this journey to understand power in 2015, and now, after 9 years it is complete.

There is now a replacement for max power in engines' regard: THE EQUIVALENT CONSTANT ENGINE TORQUE!