1. Kjell Simonsson 1 Vibrations in beams (last updated 2011-09-14)

2. Kjell Simonsson 2 Aim As an example of vibrations in systems with a continuously distributed mass, we will here look at vibrations in beams (assumed linearly elastic and without damping). The basic procedure of how to analyze vibrating beams, and the typical features of their behavior, will be discussed by looking at a specific example. The main features, such as an infinite number of eigenfrequencies and associated eigenmodes, are also valid for other continuous systems. We will not consider stationary conditions, but know that we are to avoid loading frequencies close to the eigenfrequencies (as we otherwise will get resonance). For a more comprehensive treatment of the subject, see any book on vibration analysis.

3. Kjell Simonsson 3 A simple example Let us study undamped eigenvibrations of linear beams, by considering the specific example shown below! We now do the same steps as usual, i.e. i) make a free body diagram of a mass particle dm in a deformed configuration, and ii) set up its equation of motion Note that

4. Kjell Simonsson 4 A simple example; cont. Now, iii) get rid of the interaction force by expressing it as a function of the deflection w(x,t) ! In order to do this, we treat the force dF as a distributed loading, acting over the distance dx, i.e. where, according to elementary beam theory Thus

5. Kjell Simonsson 5 A simple example; cont. On the previous slide, we found the following partial differential equation that govern the behavior of the vibrating beam. Now, excluding wave propagation, we focus on so called synchronous vibrations, where all mass points of the beam move up and down in a coordinated fashion. That is, we look for solutions of the form Thus function of x function of t constant Now, solve the 2 equations separately!

6. Kjell Simonsson 6 A simple example; cont. Characteristic equation As usual!

7. Kjell Simonsson 7 A simple example; cont. Characteristic equation as usual; see previous slide for convenience

8. Kjell Simonsson 8 A simple example; cont. cont. On the previous slide we got That is

9. Kjell Simonsson 9 A simple example; cont. To summarize, the beam deflection w(x,t) is given by where

10. Kjell Simonsson 10 A simple example; cont. For the hyperbolic functions we note that they in some aspects behave as their corresponding trigonometric functions, which of course is the reason for their names even functionodd function These functions are in a way easier than the corresponding trigonometric functions since no minus sign appears in the derivatives!

11. Kjell Simonsson 11 A simple example; cont. Boundary conditions The prevailing boundary conditions will say something about C1 to C4 ! Let us look at our specific example

12. Kjell Simonsson 12 A simple example; cont. Boundary conditions and the requirement of a non-trivial solution By putting the results from the previous slide in matrix form we get A non-trivial solution requires det[..]=0, which does NOT seem to be a nice task to handle! However, luckily, we see directly that Leading to Finally, a non-trivial solution implies

13. Kjell Simonsson 13 A simple example; cont. Non-trivial solution and the eigenfrequencies for the simply supported beam We have from the previous slide Since we get it is not an acceptable solution Furthermore, since we must have Finally, since The eigenfrequencies of a simply supported beam and

14. Kjell Simonsson 14 A simple example; cont. Final solution We note that Thus, the general solution to the vibration problem for the simply supported beam will look as follows, where a superposition of the contributions from each frequency has been done

15. Kjell Simonsson 15 A simple example; cont. Eigenmodes From the solution for the simply supported beam govern the motion up and down, while the factors it can be seen that the factors govern the mode shape. As an example we have First eigenmode Third eigenmode Second eigenmode

16. Kjell Simonsson 16 Summary and final comments We have found that the simply supported beam has infinitely many eigenfrequencies and to each of these there is an associated eigenmode the same kind of analyses can be made for other boundary conditions, even though the mathematical handling of the final equation for the eigenfrequencies might get more complicated we will get resonance if an applied loading frequency is equal to any of the eigenfrequencies (not shown here) all the above results are valid for a continuous system with distributed mass (not shown here)