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:

1. the elastic restoring force from the spring;
2. the viscous damping force from the shock absorber and
3. 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.

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Credits:

All equations in this tutorial were created with QuickLatex

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First Order Differential Equations

1. Introduction to Differential Equations - Part 1
2. Differential Equations: Order and Linearity
3. First-Order Differential Equations with Separable Variables - Example 1
4. Separable Differential Equations - Example 2
5. Modelling Exponential Growth of Bacteria with dy/dx = ky
6. Modelling the Decay of Nuclear Medicine with dy/dx = -ky
7. Exponential Decay: The mathematics behind your Camping Torch with dy/dx = -ky
8. Mixing Salt & Water with Separable Differential Equations
9. How Newton's Law of Cooling cools your Champagne
10. The Logistic Model for Population Growth
11. Predicting World Population Growth with the Logistic Model - Part 1
12. Predicting World Population Growth with the Logistic Model - Part 2
13. What's faster? Going up or Coming Down?

First order Non-linear Differential Equations

1. There's a hole in my bucket! Let's turn it into a cool Math problem!
2. The Calculus of Hot Chocolate Pouring!
3. Foxes hunting Bunnies: Population Modelling with the Predator-Prey Equations

Second Order Differential Equations

1. Introduction to Second Order Differential Equations
2. Finding a Basis for solutions of Second Order ODE's
3. Roots of Homogeneous Second Order ODE's and the Nature their Solutions
4. Modelling with Second Order ODE's: Undamped Free Oscillations
5. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 1
6. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 2
7. Modelling Car Suspension with ODE's: Damped Free Oscillations Part 3
8. Non-homogeneous Differential Equations
9. Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 1
10. Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 2
11. Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 3
12. Solving Non-homogeneous ODE's: Method of Undetermined Coefficients Part 4
13. Modelling Forced Oscillations with Second Order ODE's

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