Modal Analysis of Cantilever Beam

INDIAN INSTITUTE OF TECHNOLOGY ROPAR

Modal Analysis of Cantilever Beam MEL 417 Noise and Vibration Submitted By: Tejas Shah P2009ME1076 Abhishek Ghosh P2009ME1074 Neeharika Kushwaha P2009ME1075

June 12, 2012

Analysis of Cantilever Beam

Table of Contents Abstract .......................... ........................................ ............................ ............................ ............................ ............................ ........................... ....................... ........................ ............................ ......................3 ........3 Introduction....................................... Introduction......................... ............................ ............................ ............................ ............................ ............................ ........................ ........................ ........................... .................3 ....3 Modal analysis ........................... ......................................... ............................ ............................ ............................ ............................ ........................... ........................ ......................... ....................3 ......3 Mode shapes ............................ .......................................... ............................ ............................ ........................... ........................... ............................ ........................ ........................ ......................4 ........4 Setup .......................... ........................................ ............................ ............................ ............................ ............................ ............................ .......................... ......................... ........................... ........................4 ..........4 Observation ............................ .......................................... ............................ ........................... ........................... ............................ ............................ ....................... ....................... ............................ ...............7 .7 t=1mm

t=2mm....................................................................................................................................12

Length: 165mm ............................ .......................................... ............................ ............................ ........................... ........................... ............................ ....................... .......................12 ..............12 Length: 250mm ............................ .......................................... ............................ ............................ ........................... ........................... ............................ ....................... .......................15 ..............15 Length 320mm ........................... ......................................... ........................... ........................... ............................ ............................ ............................ ....................... ....................... ................17 ..17 Discussion ........................... ......................................... ............................ ........................... ........................... ............................ ............................ ........................ ........................ ............................ ...............20 .20 Reference……………………………………………………………………………………………………..22

2

Analysis of Cantilever Beam

Abstract To find the modal frequencies and mode shapes of a cantilever plate an experiment study was  carried out. Finite element modeling was used to calculate modal frequencies and at those  frequencies different mode shapes were studied for a cantilever plate. The system was closed loop  for accurate result. The first four mode shapes observed from experimental analysis and the one  from finite element analysis matches which shows accuracy of the experiment. The first four mode  shapes were easily observed as it is easy to see modes at lower frequencies.

Introduction A cantilever is a beam supported at one end and carries load at the other. Cantilevers are often used in architecture, as they allow for overhanging structures that do not need to be supported from the outside. In many cases, partially-constructed buildings create cantilevers. Airplane wings are also treated as cantilevers, as they are usually only supported from the body of the plane. They have been used for storage units, furniture, micro-electrical structures and many other things. In this project we have used a cantilever plate of mild steel to find the mode shapes and modal frequency. The density is 7850 kg/m 3 and elasticity 205 GPa with length 16.5cm and width 8.5cm and thickness 2mm.

Modal analysis Modal analysis is the study of the dynamic properties of structures under vibration excitation. Modal analysis is a method to describe a structure in terms of its natural characteristics which are the frequency, damping and mode shapes –its dynamic properties. properties. Modal analysis involves process of determining the modal parameters of a structure to construct a modal model of the response. It gives us modal frequencies or natural frequencies i.e. the frequency at which a system naturally vibrates once it has been set into motion and mode shape. The modal parameters maybe determined by analytical means, such as finite element analysis, and one of the common reasons for experimental modal analysis is the verification/ correction of the results of the analytical approach (model updating). Often, though analytical model does not exist and the modal parameters determined experimentally serve as the model for future evaluations such as structural modifications. In this project finite element method (FEM) was used to perform this analysis because, like other calculations using the FEM, the object being analyzed can have arbitrary shape and the results of the calculations are acceptable. The system is solved and the eigenvalues and eigenvectors are 3

Analysis of Cantilever Beam calculated which represent modal frequencies and mode shapes of the system. Then animation of the system can be seen at different modal frequencies. In this project single input multiple output approach was used i.e. giving excitation at one point and doing analysis at multiple points. We did shaker modal testing giving base excitation at the fixed end of the cantilever plate. A shaker is a device that excites the object or structure according to its amplified input signal. Cantilever plate was mounted on the shaker and it was excited using the vibrating instrument in lab. It was excited at modal frequencies calculated using finite element method and mode shapes were studied.

Mode shapes Modes are inherent properties of a structure, and are determined by the material properties (mass, damping and stiffness) and boundary conditions of the structure. Each mode is defined by a natural frequency, modal damping, and a mode shape (i.e. the “modal parameters”). If either the material properties or the boundary conditions of a structure change, its modes will change. Modes are associated with structural resonances. Most of the structures can resonate i.e. vibration with excessive motion. Mode shapes are useful because they represent the shape that the system will vibrate in free motion. These same shapes tend to dominate the motion during an earthquake (or windstorm). By understanding the modes of vibration, we can better design the building to withstand earthquakes. Vibration nodes of a mode shape are the points of the system that do not have any motion. In the first mode, the only node that exists is at the base. In the second mode, a distinct point can be seen that does not move laterally during the modal vibration. The third mode has two of these distinct points; the fourth has four and so on. For the j th mode shape, there will be j vibration nodes.

Setup The experiment was carried out in the following manner First the plate was formed by cutting a scrap of metal found in workshop junk which was subsequently grinded at ends to form a proper rectangular plate of dimensions 21.5cm by 8.5cm. The plate length was intentionally kept 5cm long so that it could be fixed in the clamps used to fix it on the shaker. Using the clamps the plate was fixed on the shaker so that the length of 5cm was rigidly fixed and the rest of 16.5cm length had 1DOF. This can be treated as a one dimension beam problem in FEM and the analysis can thus be carried out considering a one dimension problem. After the setup on the shaker was done the shaker was set up at first modal frequency and the mode shape was observed using the stroboscope. The shaker was then set to corresponding second modal frequency where the motion using stroboscope and sand was observed. The pattern formed by the sand particles clearly could determine the stationary plane. 4

Analysis of Cantilever Beam The last mode was then observed at the third modal frequency.

P.S: The amplitude for the three modes for the 1 st  type of bar was not sufficient enough so that it  could be captured using a camera, thus only the second mode was captured with sand whereas  for the other plate the video was taken and snaps were taken using sand.

Figure 1 Setup of the Shaker.

5

Analysis of Cantilever Beam

Figure 2 Setup of the Shaker Controller.

Figure 3 Side view of the setup of the 165mm plate.

6

Analysis of Cantilever Beam

Figure 4 Top view of the setup of the 320mm plate.

Observation After having observed the modes of vibration experimentally by using the values found by doing the Finite Element Analysis on Abaqus we now move to the step 2 where we simulate and compare the mode shapes at different modes by changing the length and thickness of the plate. The analysis carried out on Abaqus gives us the comparison in mode shapes when the length of the plate is increased and with the change in thickness. Following observations are made out of the analysis done on Abaqus: The table gives the values of modal frequencies corresponding to the length and thickness of the plate.

7

Analysis of Cantilever Beam Length ngth Thickness 1mm

2mm

165mm

250mm

320mm

30.903 144.5 193.56 .56 458.99 .99

13.143 82.35 .358 230.59 .59 296.49 .49

8.02 50.26 .269 140.75 .75 231.73 .73

60.336 360.66 .66 377.9 1057.3

26.284 164.68 .68 290.57 .57 460.96 .96

16.042 100.52 .52 281.43 .43 472.21 .21

Following images could be taken using sand:

Figure 5 Mode shape at frequency 377.9Hz for the plate with length 165mm and  thickness 2mm.

8

Analysis of Cantilever Beam

Figure 6 Mode shape at frequency 82Hz for the plate with length 250mm and  thickness 1mm.

9

Analysis of Cantilever Beam Figure 7 Mode shape at frequency 230Hz for the plate with length 250mm and  thickness 1mm.

Figure 8 Mode shape at frequency 100Hz for the plate with length 320mm and  thickness 2mm.

10

Analysis of Cantilever Beam

Figure 9 Mode shape at frequency 281Hz for the plate with length 320mm and  thickness 2mm.

11

Analysis of Cantilever Beam Figure 10 Mode shape at frequency 472Hz for the plate with length 320mm and  thickness 2mm.

Following we show the images of mode shapes at lengths 165, 250 and 320 mm respectively and thicknesses of 1mm and 2mm. It is to be noted that these are displacement profiles and the red region shows maximum displacement whereas the blue region shows the fixed positions or zero displacement.

t=1mm

t=2mm

Length: 165mm  Mode shape 1:

12

Analysis of Cantilever Beam

 Mode Shape 2:

13

Analysis of Cantilever Beam  Mode Shape 3:

14

Analysis of Cantilever Beam  Mode Shape 4:

Length: 250mm  Mode Shape 1:

15

Analysis of Cantilever Beam  Mode Shape 2:

 Mode Shape 3:

 Mode Shape 4:

16

Analysis of Cantilever Beam Length 320mm  Mode Shape 1:

 Mode Shape 2:

17

Analysis of Cantilever Beam  Mode Shape 3:

 Mode Shape 4:

Here we observe that the change in thickness from 1mm to 2mm results in change of modal frequencies, although the first three mode shapes do not change the mode shape starts to change from the 4th mode. We observe that in variation to length the 2 nd mode shape at 165mm occurs at the 4 th mode of 250mm and 320 mm beam for the 1mm thick plate and on the 3 rd and 4th mode of the 250mm and 320 mm plates respectively for the 2mm thick plates. The third mode shape for the 165mm long plate occurs at the second mode for the 250mm and 320mm long plates of both the thicknesses. In general we can thus observe that the second and the third mode of the plate interchange as the length is increased and more sinusoidal modes are introduced.

18

Analysis of Cantilever Beam

Graph of mode shape 2 of 165mm plate and variation with increase in length.

Graph for mode shape 3 of 165mm plate and its variation with increase in length. 19

Analysis of Cantilever Beam

Discussion In the experiment carried out we had also calculated the 1 st modal frequency in each condition using the analytical formula to make a better comparison of our work done.

Where

- modal frequency E – Youngs Modulus I – Area Moment of Inertia  – Density wt – Cross Section Area L - Length

20

Analysis of Cantilever Beam

Frequencies and Mode Shapes for Cantilever Beam

21

Analysis of Cantilever Beam We observed the following accuracy in our simulation: Analytical values

3 0 .3 1

1 3 .2 2

8.06

6 0 .6 2

2 6 .4 4

16.12

Simulated Values

30.903

1 3 .1 4 3

8.02

6 0 .3 3 6

2 6 .2 8 4

16.042

Initially the values were not consistent with those found analytically but after working upon the meshing part we were able to reach these specific values. The edges were finely meshed and the other parts were coarsely meshed so that accuracy and speed both were gained.

Reference •

Dynamics of Structures, Humar, J.L., Prentice-Hall, 1990, ISBN 0-13-222068-7

22