Define nodes

Begin with the origin (0, 0, 0). 

Enter the end of the beam (750, 0, 0).

Note, we will be working with consistent units of mm, N, and MPa, so these locations are in units of mm. 

Define elements

Select "node i" at the the origin (0, 0, 0). 

Select "node j" at the end of the beam (750, 0, 0).

Click Apply and an element will appear between these two nodes. 

Subdivide elements

Because the mass of the element will be lumped anywhere there is a node, we must subdivide the element into many smaller elements. The more the better, but the more elements, the larger the model and the greater the computational effort required. Precision is related to the refinement of the model. 

Define and attach section properties

The cross section of the steel beam used in the experiments was 16 mm by 2 mm. So, the properties are:

• Area, A = 32 mm2

• Weak-axis moment of inertia, Izz = 10.6667 mm4

• Strong-axis moment of inertia, Iyy = 682.6667 mm4

• Polar moment of inertia, J = 693.333 mm4

These values were defined and then attached to all elements in the model. 

Define and attach material properties

The beam used in these experiments was steel. The properties are:

• Modulus of elasticity, E = 210,000 MPa

• Weight-density = 7,700 kg/m3 = 7.7 x 10-5 kg/mm3

These values were defined and then attached to all elements in the model. 

Define support conditions

A cantilever has support conditions (what MASTAN calls "Fixities") that fix all displacement and rotational degrees of freedom at the supported end. We will pick node 1 at the origin and check the boxes indicating fixed conditions for rotations and displacements in the x, y, and z directions. 

Analysis

A Linear Elastic analysis should be run first to ensure the model functions correctly. Examining the Results-Deflected Shape and setting the scale factor to 1 let's us see what we already know, that this beam deflects noticeably under its self weight. These results appear to match reality! 

A Natural Period analysis can now be run. Be sure to check that gravity is acting in the correct direction and that the units are correct. We know that gravity is 9.81 m/s2, or 9,810 mm/s2, since we chose to work in N, mm, and MPa. 

Mode 1

The first, or fundamental, mode of vibration always has the lowest natural frequency (aka longest natural period). MASTAN reports the natural period. In this case it is 0.33665 seconds for a natural frequency of 2.97 Hz. 

Mode 2

The second natural period is 0.053729 seconds for a natural frequency of 18.61 Hz. 

Mode 3

The third natural period is 0.042081 seconds for a natural frequency of 23.76 Hz. Note that the shape looks the same as for the first mode, but I have changed the view to look at the beam from the top. This mode is associated with strong axis, rather than weak axis, bending. In the lab, the beam oscillated laterally rather than vertically. 

Mode 4

The fourth natural period is 0.019192 seconds for a natural frequency of 52.11 Hz. We have returned to a pattern of adding a half wave to the mode shape with bending about the weak axis. 

Mode 5

The fifth natural period is 0.0097958 seconds for a natural frequency of 102.08 Hz. This frequency is outside the range measurable with on-board mobile phone accelerometers (unless you have a phone that samples at more than 200 Hz); it was measured using the portable system from PCB described here.