Before you start the lab experiments, you should study the background materials provided in

Section $1$ and work through the questions in this section.

In Section $1\mathrm{.}1\mathrm{.}3$ we obtained Equation $(26)$ that described the dynamic behavior of the

load shaft speed as a function of the motor input voltage. Use this equation to find the

transfer function $\frac{{}_l(s)}{V_m(s)}.$ Write the transfer function in the form of Equation $(1),$ express the

steady$-$state gain $(K)$ and the time constant $()$ of the model in terms of $J_{eq},B_{eq,v},$ and $A_m$

parameters.

Calculate the following parameters using the system specifications given in the Appendix

$($also available in the SRV$02$ user manual$)$ and equations derived in Section $1.$ Use the

high$-$gear configuration $($see Appendix A$)$ to calculate the parameters.

a$)$ Model parameter $B_{eq,v}$ and $A_m($refer to Equations $(27)$ and $(28)).$

b$)$ Moment of inertia about the motor shaft $J_m=J_{\text{t}\text{a}\text{c}\text{h}}+J_{m,\text{r}\text{o}\text{t}\text{o}\text{r}},$ where $J_{\text{t}\text{a}\text{c}\text{h}}$ and $J_{m,\text{r}\text{o}\text{t}\text{o}\text{r}}$ are

the moment of inertia of the tachometer and the rotor of the SRV$02$ DC motor,

respectively.

c$)$ The load attached to the motor shaft includes a $24-$tooth gear, two $72-$tooth gears, and a

single $120-$tooth gear along with any other external load that is attached to the load shaft.

Refer to Appendix B$.$ Thus, the load inertia is $J_l=J_g+J_{l,\text{e}\text{x}\text{t}},$ where $J_g$ is the gear moment

of inertia and $J_{l,\text{e}\text{x}\text{t}}$ is the external load moment of inertia.

i$.$ Find the total moment of inertia $J_g$ from the gears. Hint: Assume gears are disks,

moment of inertia for a disk of radius $r$ and mass $m$ is defined as: $J_{\text{d}\text{i}\text{s}\text{k}}=m\frac{r^2}{2}.$ First

calculate the moment of inertia of each gear, i$.$e$.,J_{72},J_{120},$ and $J_{24},$ then use the gear

ratios to obtain $J_g,$ i$.$e$.,J_g=J_{72}+J_{120}+(\frac{120}{24}{)}^2{J}_{24}+(\frac{72}{72}{)}^2{J}_{72}.$

ii$.$ Assuming the disk load is attached to the load shaft, calculate the inertia of the disk

load, $J_{l,\text{e}\text{x}\text{t}},$ and the total load moment of inertia, $J_l.$

iii. Evaluate the equivalent moment of inertia $J_{eq}.$ Hint: refer to Equation $(18).$

Calculate the steady$-$state gain $K$ and time constant $$ using the results from Questions $1$

and $2.$ These are the nominal model parameters and will be compared with parameters that

are later found in the experiment.