Motor Optimization-have I done the math right?

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In summary, the conversation discusses finding the value of the gear ratio R that minimizes the time required to reach a certain angular displacement \theta. The equation for this is derived using the motor torque curve and the equation for torque. The equation is then simplified and substituted with variables a, b, and c. Finally, the equation is solved using the Lambert W function and the derivative of the function is used to find the equation for R. It is recommended to double check for any errors in the substitutions, typos, integration steps, and the final equation for R.
  • #1
jacobi1
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A motor torque curve is given by \(\displaystyle \tau(\omega) = -\frac{\tau_{s}}{\omega_{f}} \omega +\tau_{s}\), where
\(\displaystyle \tau_{s}\) is the stall torque and \(\displaystyle \omega_{f}\) is the free angular speed. Our objective is to
find the value of the gear ratio \(\displaystyle R \) that minimizes the time required to reach a certain angular displacement
\(\displaystyle \theta \).
We begin from the equation \(\displaystyle \tau = I \frac{d \omega}{dt} \). Setting the two torque expressions equal and rearranging, we get

\(\displaystyle \frac{d \omega}{dt} + \frac{ \tau_{s} \omega}{I \omega_{f}} =
\frac{\tau_{s}}{I}\).

The integrating factor is \(\displaystyle \mu = e^{\int \frac{\tau_{s}}{I \omega_{f}} dt} = e^{\frac{\tau_{s} t}{I \omega_{f}}}
\). Multiplying all the terms of the equation by this factor and recognizing that the left side becomes a derivative, we obtain

\(\displaystyle \left ( \omega e^{\frac{\tau_{s} t}{I \omega_{f}}} \right )' = \frac{\tau_{s}}{I}
e^{\frac{\tau_{s} t}{I \omega_{f}}} \).

Integrating both sides from 0 to t and using the fact that \(\displaystyle \omega=0\) when \(\displaystyle t=0\), we obtain after rearrangement

\(\displaystyle \omega(t)= \omega_{f} \left (1-
e^{- \frac{\tau_{s} t}{I \omega_{f}}} \right )\).

Integrating again from 0 to t and using the fact that \(\displaystyle \theta=0\) when \(\displaystyle t=0\), we obtain

\(\displaystyle \theta(t)= t \omega_{f} - \frac{I \omega_{f}^2}{\tau_{s}}
\left (1 + e^{- \frac{\tau_{s} t}{I \omega_{f}}} \right)\).

Making the substitutions \(\displaystyle \tau_{s} \mapsto
\frac{\varepsilon \tau_{s}}{R}\) (where \(\displaystyle \varepsilon\) is efficiency) and
\(\displaystyle \omega_{f} \mapsto \omega_{f} R\), the equation becomes

\(\displaystyle \theta(t)= t \omega_{f} R - \frac{I \omega_{f}^2 R^3}{\varepsilon \tau_{s}}
\left (1 + e^{- \frac{\varepsilon \tau_{s} t}{I \omega_{f} R^2}} \right )\).

Next, we set

\(\displaystyle a= \omega_{f} R\),

\(\displaystyle b= \frac{I \omega_{f}^2 R^3}{\varepsilon \tau_{s}}\), and

\(\displaystyle c= \frac{a}{b}\).

The equation becomes \(\displaystyle \theta= at -b e^{-ct}-b\). We wish to invert this to get \(\displaystyle \theta\) as a function of t.
Adding b to both sides, multiplying by \(\displaystyle e^{ct}\), and combining the \(\displaystyle e^{ct}\) terms, we have
\(\displaystyle (at- \theta -b) e^{ct} = b \). Dividing on both sides by
\(\displaystyle e^{\frac{c(b+ \theta)}{a}}\) and factoring inside the exponential, we have

\(\displaystyle (at- \theta -b) e^{\frac{c}{a}(at- \theta -b)} = b e^{-\frac{c(b+ \theta)}{a}}\).

Now, multiply both sides by \(\displaystyle \frac{c}{a}\) and the equation becomes

\(\displaystyle \frac{c}{a}(at- \theta -b) e^{\frac{c}{a}(at- \theta -b)} = \frac{bc}{a} e^{-\frac{c(b+ \theta)}{a}}\).

The equation is now in the form \(\displaystyle Xe^X=y\), which implies that we can use the Lambert W function, defined
by \(\displaystyle \operatorname{W} \left (Xe^X \right ) = X\). Therefore, applying this function to both sides of the
equation, solving for t, and remembering that \(\displaystyle c= \frac{a}{b}\) gives

\(\displaystyle t= \frac{b}{a} \operatorname{W} \left ( e^{-(1+\frac{\theta}{b})} \right ) + \frac{\theta+b}{a}\).

We seek the value of R that minimizes this equation. The derivative of W is
\(\displaystyle \frac{\operatorname{W}(x)}{x(\operatorname{W}(x)+1)}\). Back-substituting for a and b, using this equation
and the chain rule, setting the result equal to zero, and clearing fractions gives

\(\displaystyle 2 I \omega_{f}^2 R^3 \left ( 1 + \operatorname{W} \left
(e^{-\left (1+\frac{\theta \varepsilon \tau_{s}}{I \omega_{f}^2 R^3} \right )}
\right ) \right )^2 +
\theta \varepsilon \tau_{s} \left (-1+ 2 \operatorname{W} \left
(e^{-\left (1+\frac{\theta \varepsilon \tau_{s}}{I \omega_{f}^2 R^3} \right )} \right ) \right )=0 \)

as the equation to be solved for R.
Do I have any errors?
 
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  • #2
I can point out a few things for you to double check:

1. Make sure all the substitutions are consistent throughout the equation. For example, in the beginning, you substitute \tau_s to \frac{\varepsilon \tau_s}{R}, but later on, you substitute back to \tau_s.

2. Check for any typos or missing terms in the equation.

3. Double check the integration steps to make sure they are correct.

4. Make sure the final equation for R is in the correct form and can be solved for R.

It is always a good idea to have someone else check your work as well to catch any mistakes. Good luck!
 

Related to Motor Optimization-have I done the math right?

1. Can you explain what motor optimization is?

Motor optimization is the process of improving the performance and efficiency of a motor, typically by adjusting its design or operating parameters. It involves evaluating factors such as torque, speed, power, and energy consumption to find the optimal settings for the motor.

2. What are the benefits of motor optimization?

Motor optimization can lead to several benefits, including increased efficiency, reduced energy consumption, improved reliability, and extended motor lifespan. It can also result in cost savings and reduced environmental impact.

3. How do I know if I have done the math right for motor optimization?

To determine if you have done the math right for motor optimization, you should first establish clear goals and performance criteria for the motor. Then, use mathematical equations and calculations to analyze the motor's design and operating parameters and compare them to your goals. It may also be helpful to consult with a motor expert or use simulation software.

4. Can motor optimization be applied to any type of motor?

Yes, motor optimization can be applied to various types of motors, including AC motors, DC motors, and servo motors. However, the specific methods and techniques used may vary depending on the type of motor and its intended application.

5. How often should motor optimization be performed?

The frequency of motor optimization depends on factors such as the motor's usage, operating conditions, and any changes or upgrades made to the motor. In general, it is recommended to regularly review and optimize motor performance to maintain efficiency and minimize potential issues.

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