1. Vibrations in undamped linear 2-dof systems
(last updated )

2. Aim
The aim of this presentation is to give a short review of basic vibration analysis in undamped linear 2 degree-of-freedom (2-dof) systems.
The basic procedure how to analyze such a system, and the typical features of it, will be discussed by looking at a specific example.
For a more comprehensive treatment of the subject, see any book on vibration analysis.

3. A simple example
Let us study some basic phenomena of linear 2 dof vibration analysis (of un-damped) systems, by considering the example shown below!
By making a free body diagram of the (point) masses, we get

4. A simple example; cont.
Let us now express S1 and S2 as functions of x1 and x2 by using superposition of tabulated beam solutions (elementarfall in Swedish)

5. A simple example; cont.

6. A simple example; cont.

7. A simple example; cont.

8. A simple example; cont.
Thus

9. A simple example; cont.
Solving for the forces as functions of the deflections

10. A simple example; cont.
Finally, for the two equations of motion we get
where the matrices (from the left) are referred to as the mass matrix, acceleration matrix, stiffness matrix, displacement matrix and force matrix, respectively.

11. A simple example; cont.
Eigen/self-vibration
In order to study the self vibration of our system, we need to consider the homogeneous problem
Led by the results obtained in the 1 dof-context, we adopt the following ansatz for the mass displacements
Note that we have given both masses the same eigenfrequency and phase angle, which means that they (depending on the sign of the X-factors) either will move in phase or out of phase. Furthermore, it is to be understood that we here consider the solution to the homogeneous problem.

12. A simple example; cont.
Eigen/self-vibration; cont.
As can be seen, this is an eigenvalue problem. By requiring a non-trivial solution we must require that the determinant of the bracket is zero.

13. A simple example; cont.
Eigen/self-vibration; cont.

14. A simple example; cont.
Eigen/self-vibration; cont.

15. A simple example; cont.
Eigen/self-vibration; cont.
We have found the following two solutions to the 2 dof eigenvalue problem
By inserting the actual expressions for the stiffness- and mass components, we get

16. A simple example; cont.
Eigen/self-vibration; cont.
In order to see the vibration (mode) form, we plug in the found eigenfrequencies in the homogeneous form of the equations of motion. For the lowest eigenfrequency we then find
As can be seen, these two equations are idenitical. Thus, we can not get both amplitudes; only their relation!

17. A simple example; cont.
Eigen/self-vibration; cont.
Thus, for the first eigenmode we have the following vibration form
As can be seen, the two masses move in an out of phase manner.

18. A simple example; cont.
Eigen/self-vibration; cont.
For the second eigenmode we instead find
As can be seen, these two equations are idenitical. Thus, we can not get both amplitudes; only their relation!

19. A simple example; cont.
Eigen/self-vibration; cont.
Thus, for the second eigenmode we have the following vibration form
As can be seen, the two masses move in an in phase manner. That this eigenmode is associated with a higher frequency can be understood in that it requires more strain energy to deform the beam like this.

20. A simple example; cont.
Eigen/self-vibration; cont.
Some final comments
The resulting motion will be a combination of the two eigenmodes where the 4 unknown constants are given by the initial conditions for displacement and velocity of the two masses.
In a stationary state ("fortvarighet" in Swedish) the eigenvibrations have vanished, and only the forced vibration remains.

21. A simple example; cont.
Forced vibration
In order to study the forced vibration of our system, we need to consider the non-homogeneous problem
Led by the form of the force, we now adopt the following ansatz for the mass displacements
It is to be understood that we from now on consider the particular solution (valid at stationary conditions). We get

22. A simple example; cont.
Forced vibration; cont.

23. A simple example; cont.
Forced vibration; cont.
Looking at the denominator for the case that the applied load frequency is equal to one of the eigenfrequencies of the system, we get

24. A simple example; cont.
Forced vibration; cont.
As can be seen, the mass amplitudes go to infinity when the applied load frequency approaches any of the eigenfrequencies of the system, i.e. we get resonance!

25. Summary
The following features are common to all undamped linear 2-dof systems
They possess 2 eigenfrequencies
If the applied loading frequency is equal to any of these we will get resonance
To each eigenfrequency there is an associated vibration mode; a so called eigenmode
Corresponding results are valid for discrete n-dof systems, where we find n eigenfrequencies etc