How to Calculate and Solve for Time Constant | Second Order Instruments | Dynamic Characteristics of Instruments

The image above represents time constant | second order instruments. To calculate time constant| second order instruments, two essential parameters are needed and these parameters are Dumping Ratio (ε) and Undefined Natural Frequency (ω_o).

The formula for calculating time constant:

u = ^2ε / _ωo

Where:

u = Time Constant
ε = Dumping Ratio
ω_o = Undefined Natural Frequency

Let’s solve an example;
Find the time constant when the dumping ratio is 42 and the undefined natural frequency is 2.

This implies that;

ε = Dumping Ratio = 42
ω_o = Undefined Natural Frequency = 2

u = ^2ε / _ωo
u = ^{2 x 42} / ₂
So, u = ⁸⁴ / ₂
u = 42

Therefore, the time constant is 42.

Calculating for the Dumping Ratio when the Time Constant and the Undefined Natural Frequency are Given

ε = ^{u x w_o} / ₂

Where;

ε = Dumping Ratio
u = Time Constant
ω_o = Undefined Natural Frequency

Let’s solve an example;
Find the dumping ratio when the time constant is 14 and the undefined natural frequency is 10.

This implies that;

u = Time Constant = 14
ω_o = Undefined Natural Frequency = 10

ε = ^{u x w_o} / ₂
Then, ε = ^{14 x 10} / ₂
ε = ¹⁴⁰ / ₂
ε = 70

Therefore, the dumping ratio is 70.

Read more: How to Calculate and Solve for Transfer Function (Sensitivity) | Second Order Instruments | Dynamic Characteristics of Instruments

Calculating for the Undefined Natural Frequency when the Time Constant and the Dumping Ratio are Given

ω_o = ^2ε / _u

Where;

ω_o = Undefined Natural Frequency
u = Time Constant
ε = Dumping Ratio

Let’s solve an example;
Find the undefined natural frequency when the time constant is 20 and the dumping ratio is 80.

This implies that;

u = Time Constant = 20
ε = Dumping Ratio = 80

ω_o = ^2ε / _u
That is, ω_o = ²⁽⁸⁰⁾ / ₂₀
ω_o = ¹⁶⁰ / ₂₀
ω_o = 8

Therefore, the undefined natural frequency is 8.

Read more: How to Calculate and Solve for Static Sensitivity | First Order Instruments | Dynamic Characteristics of Instruments

How to Calculate Time Constant | Second Order Instruments With Nickzom Calculator

Nickzom Calculator – The Calculator Encyclopedia is capable of calculating the time constant | second Order Instrument.

To get the answer and workings of the time constant | second Order Instrument using the Nickzom Calculator – The Calculator Encyclopedia. First, you need to obtain the app.

You can get this app via any of these means:

Web – https://www.nickzom.org/calculator-plus

To get access to the professional version via web, you need to register and subscribe for NGN 2,000 per annum to have utter access to all functionalities.
You can also try the demo version via https://www.nickzom.org/calculator

Android (Paid) – https://play.google.com/store/apps/details?id=org.nickzom.nickzomcalculator
Android (Free) – https://play.google.com/store/apps/details?id=com.nickzom.nickzomcalculator
Apple (Paid) – https://itunes.apple.com/us/app/nickzom-calculator/id1331162702?mt=8
Once, you have obtained the calculator encyclopedia app, proceed to the Calculator Map, then click on Materials and Metallurgical under Engineering.

Now, Click on instrumentation under Materials and Metallurgical

Now, Click on Dynamic Characteristics of Instruments under Instrumentation

So, Click on Time Constant| Second Order Instrument under Dynamic Characteristics of Instruments

The screenshot below displays the page or activity to enter your values, to get the answer for the time constant | second Order Instrument according to the respective parameters which is the Dumping Ratio (ε) and Undefined Natural Frequency (ω_o).

Now, enter the values appropriately and accordingly for the parameters as required by the Dumping Ratio (ε) is 42 and Undefined Natural Frequency (ω_o) is 2.

Finally, Click on Calculate

As you can see from the screenshot above, Nickzom Calculator– The Calculator Encyclopedia can calculate the time constant | second Order Instrument and presents the formula, workings and steps too.