Advanced Dynamics and Vibrations: Numerical methods for vibrating systems
[Jason’s part goes here]
where we can have a different mass as the equation is different. For SHM
If we choose then
Therefore:
Note:
If
Now using the result above
Consider
Note:
![](https://engcourses-uofa.ca/wp-content/uploads/MECE788_CH6P6Final.png)
We can apply this idea to various dynamical systems which are more complicated than the simple spring/mass. Consider an approximation to longitudinal vibration of a bar
![](https://engcourses-uofa.ca/wp-content/uploads/MECE788_CH6P7Final.png)
Consider a beam of total mass and cross section
modulus
which we approximate as
![](https://engcourses-uofa.ca/wp-content/uploads/MECE788_CH6P8Final.png)
if , go to
if
Therefore:
Therefore:
N = 4
To calculate the mode shapes one can start at the free end e.g.:
Assume
Now we use the recursion relationship
Therefore:
For the 4 storey building for the lowest natural frequency is
Therefore:
![](https://engcourses-uofa.ca/wp-content/uploads/MECE788_CH6P9Final.png)
Check
![](https://engcourses-uofa.ca/wp-content/uploads/MECE788_CH6P10Final.png)
Therefore:
if (
is low) then:
for the lower frequencies
where is density
For other boundary conditions, the expression for is determined then the natural frequencies are known. For fixed ends.
![](https://engcourses-uofa.ca/wp-content/uploads/MECE788_ch.6_2-1.jpg)
![](https://engcourses-uofa.ca/wp-content/uploads/MECE788_ch.6_2-1-2.jpg)
Consider the fixed boundaries:
![](https://engcourses-uofa.ca/wp-content/uploads/MECE788_ch.6_2-2.jpg)
Therefore:
Therefore:
Example
![](https://engcourses-uofa.ca/wp-content/uploads/MECE788_ch.6_2-3-1.jpg)
Therefore: