Modelling Forced Oscillations with Second Order ODE's
Figure 1 below shows a schematic of the mass-spring-damper model to include an applied external force, F.
Figure 1.Mass-Spring-Damper system with external force
In post #5 of this series on solving 2nd order, linear ODE's, we began by constructing a free-body diagram of a typical mass-spring-damper mechanism, to derive the equation of motion for the free oscillating system - equation (1)...
We derive the equation of motion to this system in exactly the same way as in post #5.
From equation (2), we can see that the sum of the internal forces consisting of:
- the elastic restoring force from the spring;
- the viscous damping force from the shock absorber and
- the inertial force from the motion of the mass...
...are in equilibrium with the external force.
In the this and the next few posts, we are going to look at how the system responds, that is how the mass moves, when the external force is an oscillating force. That is...
Thus the equation of motion (2) becomes...
So we have a 2nd order, linear, non-homogeneous differential equation for which we can solve using the method of undetermined coefficients.
We know from equation (3) in post #8, the solution has the form...
Let's find the particular solution to the non-homogeneous equation first.
Particular Solution to the non-Homogeneous Equation
From the table 1 in post #12 above, we choose...
...then...
...and...
Sub these into (3), we get...
Equating the coefficients on both sides, we have...
Thus from (6)...
...and substituting this result into (5)...
From post #4, we found that the natural frequency of an undamped mechanism can be expressed as...
So let's express the coefficients A and B in terms of the natural frequency on an undamped system. Why? we'll find out in the next post...
Finally, the particular solution to the non-homogeneous equation is...
Alright, in the next post, we'll look at a special case - forced oscillations on an undamped system creating resonance.
Credits:
All equations in this tutorial were created with QuickLatex
First Order Differential Equations
- Introduction to Differential Equations - Part 1
- Differential Equations: Order and Linearity
- First-Order Differential Equations with Separable Variables - Example 1
- Separable Differential Equations - Example 2
- Modelling Exponential Growth of Bacteria with dy/dx = ky
- Modelling the Decay of Nuclear Medicine with dy/dx = -ky
- Exponential Decay: The mathematics behind your Camping Torch with dy/dx = -ky
- Mixing Salt & Water with Separable Differential Equations
- How Newton's Law of Cooling cools your Champagne
- The Logistic Model for Population Growth
- Predicting World Population Growth with the Logistic Model - Part 1
- Predicting World Population Growth with the Logistic Model - Part 2
- What's faster? Going up or Coming Down?
First order Non-linear Differential Equations
- There's a hole in my bucket! Let's turn it into a cool Math problem!
- The Calculus of Hot Chocolate Pouring!
- Foxes hunting Bunnies: Population Modelling with the Predator-Prey Equations
Second Order Differential Equations
- Introduction to Second Order Differential Equations
- Finding a Basis for solutions of Second Order ODE's
- Roots of Homogeneous Second Order ODE's and the Nature their Solutions
- Modelling with Second Order ODE's: Undamped Free Oscillations
- Modelling Car Suspension with ODE's: Damped Free Oscillations Part 1
- Modelling Car Suspension with ODE's: Damped Free Oscillations Part 2
- Modelling Car Suspension with ODE's: Damped Free Oscillations Part 3
- Non-homogeneous Differential Equations
- Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 1
- Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 2
- Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 3
- Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 4
- Modelling Forced Oscillations with Second Order ODE's
Please give me an Upvote and Resteem if you have found this tutorial helpful.
Feel free to ask me any math question by commenting below and I will try to help you in future posts.
Visit my YouTube Channel at: https://www.youtube.com/masterwumathematics
Tip me some DogeCoin: A4f3URZSWDoJCkWhVttbR3RjGHRSuLpaP3
Tip me at PayPal: https://paypal.me/MasterWu
This post has been voted on by the SteemSTEM curation team and voting trail in collaboration with @curie.
If you appreciate the work we are doing then consider voting both projects for witness by selecting stem.witness and curie!
For additional information please join us on the SteemSTEM discord and to get to know the rest of the community!