I want to model an AC Servomotor where I assume that a dynamic load is attached to the shaft of AC Servomotor. The link of the paper that i have attached at the beginning of this post has ac servomotor model running without load that's why in equation $5$ $TL(s)$ (the load torque)$=0$ . Now I want to include $TL(s)$ in the model which means that a load is connected to the shaft of the motor. in order to include TL(s) into the system I need to know the formula of the TL(s) because I can not simply include TL(s) in the equation else I will not be able to get the final transfer function ( Theta(s)/E(s) ). So Ii am thinking to put in the components of the TL(s) into the equation such as $T=KI$ or $T=J$(inertia) * a(accelaration).
So May someone plz help me that what can be the correct equation of TL(s) in this case so i can subtitute in? or/and how can I include the load torque (TL(s)) into the motor model.
In the paper you attached, the model assumed is linear and has two inputs $E$ and $T_L$ and one output $\theta$. From such a model you can deduce two transfer functions:
So you cannot have $T_L(t)\ne0, \forall t$ in order to determine the first transfer function, as you say.
Probably you are searching for $T_L(t)$ to do some simulation. But it can be whatever, just as can be $E(t)$. So determine the second transfer function putting $E(t)=0$, $\forall t$ and then add such transfer function in your simulation scheme as a disturbance additive contribution on $\theta$ and feed it with whatever $T_L(t)$ you may think, starting perhaps with a constant function.
There is a conceptual error. The second section of the paper start sayng that the model of the system consists of a morot coupled to a gear box and an intertial load rigidly fixed to the output shaft. In the light of this, fix your attention to the parameters $J$ and $B$ in the eq. $(2)$.
What do they represent?
1.
If they represent only the mechanical parameter of the motor then $T_L$ should have the component terms that you say (load inertia and friction) and one more component that represents the torque demanded by the load during its operation. The load can be an elevator, for instance, and in this case this additional term is a function of the time that is constant to the weight lifted. Or it can be another kind of load, and in this case you can think of whatever function of time you want. So this additional term that is a direct function of the time (and not indirect through velocity or acceleration) describe the very nature of the load atteched to the motor drive. If you do not use it the whole motor drive would be useless, it would drive something useless. Usually this term of the torque is called disturbance, because it tends to perturbate the regime that can be attained by any value of the main input, that is $E$.
But if this is the case, then the author of the paper is wrong in determining the transfer function $\theta/E$, because he should have set to zero only the disturbance, that is all the other input different from $E$.
$T_L(t)=J_L\ddot{\theta}(t)+B_L\dot{\theta}(t) + T_{dist}(t)$
And eq. $(2)$ would have been
$T_c(t)=J\ddot{\theta}(t)+B\dot{\theta}(t) +J_L\ddot{\theta}(t)+B_L\dot{\theta}(t) + T_{dist}(t)$
Form this, one can determine $\left(\frac{\theta}{E}\right)_{T_{dist}=0}$. It does not make sense setting $T_L(t)=0$.
2.
If instead they represent the mechanical parameter of the motor with the load attached (as written at the beginning of the section 2) then what is indicated by $T_L$ is just the disturbance term. In this case setting $T_L(t)=0$ to determine the transfer function is correct. (But note that in this case $\tau_m$ should be called motor drive system time constant, not simply motor time constant).
In the light of this follow my instruction in the earlier answer, to go on.