Shock Spectra of Cantilever Beam

UROS Project

School of Engineering

College of Science

Shock Spectra of Cantilever Beam

Zachary Thompson

14511199

Supervisor

Dr Fotios Georgiadis

Contents

Introduction ........................................................................................................................ 2

Damping Ratios of Steel and Aluminium Literature Review ............................................ 2

Damping .......................................................................................................................... 2

Damping of different materials ...................................................................................... 3

Modelling ............................................................................................................................ 6

Equation of motion for the cantilever beam: ................................................................. 6

Results ................................................................................................................................ 15

Conclusion ......................................................................................................................... 23

References ......................................................................................................................... 24

2 Zachary Thompson 14511199

Introduction

The aim of this project is to determine the shock spectra of a cantilever beam; this is

essential for identifying critical or dangerous circumstances in structures. This area of

is fundamental in mechanics in all structures. The project is on theoretical modal

analysis of a cantilever beam using particular geometries and material found in the

literature. Due to the linear nature of the beam system it will be possible to scale the

excitation amplitudes up in order to determine shock spectra in larger structures. This

report outlines the equations of motion for a cantilever beam. The beam was modelled

using the extended Hamilton s Principle and then it was determined the natural

frequencies and the values obtained were compared to experimental results from the

literature. The system was discretised in order to be coded into MATLAB. From the

MATLAB code Maximum values for the Mode shapes were found, as well as their

position on the beam. Completion of this project can lead to further investigation of

more complicated structures that could be effected by shock excitation such as car sub

frames in an accident or buildings in an earthquake. Further work on this project

would include experimental validation of theoretical findings.

Damping Ratios of Steel and Aluminium Literature Review

Damping

Damping is the occurrence by which mechanical energy is dissipated in dynamic

systems. Theoretically, if an object had zero damping and was excited by an external

force the vibration would be continuous without reducing amplitude, nor changing

frequency, as outlined below in figure 1, this also shows the effect of differing damping

ratios. Vibrations in real life applications are almost always unavoidable. Therefore, a

knowledge of the level of damping in a dynamic system is important in utilization,

analysis, and testing of the system. There are several types of damping present in

different mechanical systems, as stated in [1] by H. Mevada and D. Patel. These are;

Internal or material damping, structural damping and fluid damping. This project will

focus on internal or material damping. This type of damping results from mechanical-

energy dissipation within the material due to various microscopic and macroscopic

processes [1]. Different levels of damping are desirable, highly dependent on the


application of the system. For example in turbine blades, crack shafts of many other

applications where a part can resonate at some frequency, high damping would be

desirable. This is because high damping minimizes the amplitude of resonant

frequencies [2]. Therefore, minimising operating stresses that a part may be exposed

to. However low material damping is needed for the correct working of other

applications such as many musical instruments, such as strings on a piano or bells.

Thus, it is plain to see that damping is a property of particular importance in industry

and system design.

Estimating damping in structures of different materials, for example; steel and

aluminium still poses many difficulties [1]. Although there is extensive literature on

vibration in beams damping in materials has been somewhat neglected [3].

Damping of different materials

In [1] H. Mevada and D. Patel calculated the damping of aluminium, brass and mild

steel in cantilever beams using the Half-power Bandwidth Method which is an

experimental data analysis method, because damping properties don t arise from

stiffness or mass. This method calculates the damping ratio is calculated using the

following equation:

= ∙ ∆

Where ∆ denotes bandwidth which is defined as √ of the peak value, and is the

frequency at the peak value; as shown below in figure 1 below.


Figure 1, Half-Power Bandwidth Method courtesy [1].

The values obtained from this method were then compared with experimentally

obtained values, the results can be seen below in figure 2.

Figure 1. Damping Ratios of Mild steel and Aluminium First Mode


One of the main conclusions that can be made from this results chart is that materials

with higher density have higher damping ratios, according to [4], [5] and [6] the

densities of aluminium (2024), mild steel and brass are (average): 2780kg/m3

7,750kg/m3 and 8,520kg/m3 respectively.

F. Poncelet in [7] experimentally determined the frequencies and damping ratios of 7

modes for a steel cantilever beam with a 0.041 meter square cross section and a length

of 0.7 meters.

Poncelet in [7] also states that Proportional damping may also be introduced

in the system by means of the coefficients and . The ensuing damping matrix

used for the simulation is provided by: C = · M + · K (2)


In this case, =2 and = . × −

Figure. 2 Effect of differing damping ratios on vibration and amplitude.

Modelling

Equation of motion for the cantilever beam:

Here a Euler-Bernoulli cantilever beam is modelled as elastic continuum by means of

deriving the equations of motion. It is considered that distributed motion , in

the transversal direction over the beam span.

The extended Hamilton s principle will be used:

∫ − + ̅ 𝑐 = , 𝒓 = , = , , 𝑖 = , , … , 𝑁

Kinetic energy is defined by: = ∫ ∙ 𝑉 (4)

0 5 10 15

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

time(s)

x/x

0

= 0.5 >> Under Damped

= 1 >> Critically Damped

= 1.5 >> Over Damped

= 0 >> No Damping


Where is the velocity of the beam displacement (in time), is the density of the

beam all integrated over the volume of the beam, .

Considering the variation of the kinetic energy gives: = ∫ ∙𝑉 = ∫ 𝑉 (5)

Using integration by parts with respect to time, taking into consideration that the

variations in the initial and final conditions are equal to zero; from the definition of

Hamilton s principle leads to: ∫ = ∫ ∙ = − ∫ ∫ 𝑣 (6)

Therefore the variation of kinetic energy is given by:

∫ = − ∫ ∫ 𝑣 = − ∫ ∫ 𝐴 = − ∫ ∫

(7)

Where A is the cross sectional area of the beam and l is the length of the beam.

The potential energy is given as: = ∫ 𝜎 + 𝜎 + 𝜎 + 𝜎 + 𝜎 + 𝜎 𝑣 (8)

However, due to the isotropic nature of the Euler-Bernoulli beam the following

assumptions can be made: = − ′′ (9a) 𝜎 = − ′′ (9b) 𝜎 = 𝜎 = (9c) = = = = = (9d)

The dash relates to differentiation in space.

From these assumptions, this means that variation of the potential energy is as

follows:


= ∫ 𝜎 𝑉 = ∫ (− ∙ − ′′ ) 𝑉= ∫ ′′ ′′ ∫ 𝐴𝐴 = ∫ ( 𝐼 ′′ ′′ ) = ∫ 𝐼 ′′ ′ = 𝐼 ′′ ′ | − ∫ ( 𝐼 ′′ ′ ′) = 𝐼 ′′ ′ | − 𝐼 ′′′ | + ∫ 𝐼 𝐼𝑉

(10)

The variation of the external work, created by:

a) External force at point ‘d’:

𝑐 = (11)

b) Rayleigh damping distributed force ,Q,:

𝑐 = ∫ = ∫ (12)

Where: = − (13)

is the viscous damping coefficient.

Applying these to the extended Hamilton’s principle: ∫ − + ̅ 𝑐 = , 𝒓 = , = , , 𝑖 = , , … , 𝑁 (14)

∫ ∫ , − 𝐼 𝐼𝑉 − + − − 𝐼 ′′ ∙ ′ |+ ( 𝐼 ′′′ ∙ | ) =

(15)

The above equation should be true for any arbitrary variation, and ′ therefore, each

part should be zero.


Applying the fundamental lemma for calculus we get the equation of motion of a

cantilever beam with external forcing at point d: , + , + 𝐼 𝐼𝑉 − − = (16)

Considering the geometry of the beam, and how it is supported, leads to the boundary

conditions:

, = , → No Displacement at fixed end. (17a)

′ , = , → No rotation at fixed end. (17b)

𝐼 ′′ 𝐿, = , → No bending moment at free end. (17c) 𝐼 ′′′ 𝐿, = , → No shear force at free end (17d)

Discretization

From this the modal equations can be derived using Galerkin s approximation, where

the displacement can be separated in space and time with appropriate functions (one

in space and one in time): , = ∑ 𝑎 ∙ 𝑌= (18)

As well as separated equations describing motion in space and time:

𝐼𝑉 , = ∑ 𝑎 ∙ 𝑌𝐼𝑉= (19a) , = ∑ 𝑎 ∙ 𝑌= (19b) , = ∑ 𝑎 ∙ 𝑌= (19c)

With the associated modal boundary conditions: 𝑌 = , 𝑌′ = 𝑌′′ = , 𝑌′′′ =

(20a-d)

This gives: ∑ 𝑎 ∙ 𝑌= + ∑ 𝑎 ∙ 𝑌= + 𝐼 ∑ 𝑎 ∙ 𝑌𝐼𝑉 − −= = (21)


Applying the weighted residual method leads to multiplying the equations with mode

shape 𝑌 and integration over length as shown below:

∫ 𝑌𝑥=𝐿𝑥= ∑ 𝑎 ∙ 𝑌= + ∫ 𝑌𝑥=𝐿

𝑥= ∑ 𝑎 ∙ 𝑌= + 𝐼 ∫ 𝑌𝑥=𝐿

𝑥= ∑ 𝑎 ∙ 𝑌𝐼𝑉 −= ∫ 𝑌𝑥=𝐿𝑥= − =

(22)

The natural frequency for the cantilever beam is [9]:

= √ 𝐸𝐼 𝑎 / (23)

If two mode shapes are considered, r and s, which correspond to the frequencies r and

s respectively then the equations can be written as: ∙ 𝑌 − 𝐼𝑌𝐼𝑉 = (24a) ∙ 𝑌 − 𝐼𝑌𝐼𝑉 = (24b)

Multiplying the first of the previous two equations with mode shape 𝑌 and

integrating over length gives: 𝑌 ∙ 𝑌 − ∫ 𝐼𝑌 ∙ 𝑌𝐼𝑉 𝑥=𝑥= = (25)

Multiplying the second of the previous two equations with mode shape 𝑌 and

integrating over length gives: 𝑌 ∙ 𝑌 − ∫ 𝐼𝑌 ∙ 𝑌𝐼𝑉 𝑥=𝑥= = (26)

This means that the system is self adjoint.

Subtracting the previous equations gives:


𝑌 ∙ 𝑌 − ∫ 𝐼𝑌 ∙ 𝑌𝐼𝑉 𝑥=𝑥=− 𝑌 ∙ 𝑌 − ∫ 𝐼𝑌 ∙ 𝑌𝐼𝑉 𝑥=

𝑥== − ∫ 𝑌 ∙ 𝑌 𝑥=𝑥= =

(27)

This means that in the case of two differing natural frequencies: − ≠ (28)

Which leads to:

∫ 𝑌 ∙ 𝑌 𝑥=𝑥= = , ≠ (29)

And in the case that the natural frequencies are equal:

∫ 𝑌 . 𝑌 𝑥=𝑥= = , = (30)

For = :

∫ 𝑌 𝑌 𝑎 + 𝑌 𝑎 𝑥=𝑥= + ∫ 𝑌 𝑌 𝑎 + 𝑌 𝑎 𝑥=

𝑥= + 𝐼 ∫ 𝑌 𝑌𝐼𝑉𝑎 + 𝑌𝐼𝑉𝑎 𝑥=𝑥=− ∫ 𝑌 ∙ − 𝑥=

𝑥= =

(31)

Leads to:

∫ 𝑌 𝑌 𝑎 + 𝑌 𝑎 𝑥=𝑥= = ∫ 𝑌 𝑌 𝑎 +𝑥=𝑥= 𝑌 𝑌 𝑎 (32)

Where:

∫ 𝑌 𝑌 = , ∫ 𝑌 𝑌 = (33)


Or:

∫ 𝑌 𝑌 = (34)

Therefore:

∫ 𝑌 𝑌 𝑎 + 𝑌 𝑎 𝑥=𝑥= = 𝑎 (35)

Similarly:

∫ 𝑌 𝑌 𝑎 + 𝑌 𝑎 𝑥=𝑥= = ∫ 𝑌 𝑌 𝑎 + 𝑌 𝑌 𝑎 𝑥=𝑥= = 𝑐 𝑎 (36)

And 𝐼 ∫ 𝑌 𝑌𝐼𝑉𝑎 + 𝑌𝐼𝑉𝑎 𝑥=𝑥= = 𝐼 ∫ 𝑌 𝑌𝐼𝑉𝑎 + 𝑌 𝑌𝐼𝑉𝑎 𝑥=𝑥= = 𝑎 (37)

And

∫ 𝑌 ∙ − 𝑥=𝑥= = 𝑌 (38)

Because m=1 this leaves the equation: 𝑎 + 𝑐 𝑎 + 𝑎 − 𝑌 = (39)

For = :

∫ 𝑌 𝑌 𝑎 + 𝑌 𝑎 𝑥=𝑥= + ∫ 𝑌 𝑌 𝑎 + 𝑌 𝑎 𝑥=

𝑥= + 𝐼 ∫ 𝑌 𝑌𝐼𝑉𝑎 + 𝑌𝐼𝑉𝑎 𝑥=𝑥=− ∫ 𝑌 ∙ − 𝑥=

𝑥= =

(40)

Leads to:

∫ 𝑌 𝑌 𝑎 + 𝑌 𝑎 𝑥=𝑥= = ∫ 𝑌 𝑌 𝑎 +𝑥=𝑥= 𝑌 𝑌 𝑎 (41)


Where:

∫ 𝑌 𝑌 = , ∫ 𝑌 𝑌 = (42a,b)

Or:

∫ 𝑌 𝑌 = (43)

Therefore:

∫ 𝑌 𝑌 𝑎 + 𝑌 𝑎 𝑥=𝑥= = 𝑎 (44)

Similarly:

∫ 𝑌 𝑌 𝑎 + 𝑌 𝑎 𝑥=𝑥= = ∫ 𝑌 𝑌 𝑎 + 𝑌 𝑌 𝑎 𝑥=𝑥= = 𝑐 𝑎 (45)

And 𝐼 ∫ 𝑌 𝑌𝐼𝑉𝑎 + 𝑌𝐼𝑉𝑎 𝑥=𝑥= = 𝐼 ∫ 𝑌 𝑌𝐼𝑉𝑎 + 𝑌 𝑌𝐼𝑉𝑎 𝑥=𝑥= = 𝑎 (46)

And

∫ 𝑌 ∙ − 𝑥=𝑥= = 𝑌 (47)

This leaves the equation: 𝑎 + 𝑐 𝑎 + 𝑎 − 𝑌 = (48)


These two equations are now decoupled meaning they are independent from each

other and can be written in the form:

[𝑎𝑎 ] + [ 𝑐 𝑐 ] [𝑎𝑎 ] + [ ] [𝑎𝑎 ] − {𝑌𝑌 } [ ] = (49)

Where: = (50)

Therefore: 𝑎 + 𝑎 + 𝑎 − 𝑌𝑖 = (51)

And:

= ( 𝜂 √ 𝐸𝐼 )𝜋 = 𝜔𝜋 (52)

Where values are:

Mode 𝜼𝒍

1 1.8751

2 4.6941

3 7.8548

4 10.9955

5 14.1372

6 17.2788

7 20.4204

Table 1. values for modes 1-7


Results

Youngs modulus: = 𝑎

Length of the beam: = .

Second moment of area:

𝐼 = 𝑎 = . = . × −9 (53)

Distributed mass of the beam using values from [8] as the model uses distributed mass

over the span length: = 𝐴 = × . = . / (54)

This leads to:

Theoretical Frequency

(Hz)

Experimental

Frequency (Hz)

[7]

% difference Damping Ratio

[7]

= . 23.59 +0.3% 0.69 = . 147.83 +0.3% 0.20 = . 414.30 +0.2% 0.29 = . 814.02 -0.1% 0.53 = . 1353.12 -0.1% 0.86 = . 2038.21 -1.4% 1.29 = . 2845.50 -1.4% 1.79

Table 2. theoretical frequencies with comparison to experimental frequencies from [7]


From this we can find: = × . × . × = . 𝑎 / = × . × . × = . 𝑎 / = × . × . × = . 𝑎 / = × . × . × = . 𝑎 / = × . × . × = . 𝑎 / = × . × . × = . 𝑎 / = × . × . × = . 𝑎 /

(55a-g)

Using order reduction in order to be able to code into MATLAB using the ode45

function: 𝑎 = , 𝑎 = 𝑎 = , 𝑎 = 𝑎 = , 𝑎 =

(56)

Substituting and rearranging this gives: = (57a) = (𝑌 − − ) (57b) = (57c) = (𝑌 − − ) (57d) ⋮


However, we will consider 7 this leads to:

[𝑎⋮𝑎 ] + [ ⋱ ] [𝑎⋮𝑎 ] + [ ⋱ ] [𝑎⋮𝑎 ] − {𝑌 ⋮𝑌 } [ ] = (58)

Using the mode shape equation: 𝑌 = √ ∙𝐼𝑟 × sin − sinh + √ ∙𝐼𝑟 × 𝐶𝐿 cos − cosh (59)

Where in [9] it is defined the coefficient in order that the mode shapes would be

orthonormal:

𝐼 = − sin − + sinh + 𝐶𝐿 [ + sin ] + 𝐶𝐿 [ + sinh ]− [cosh ∙ sin − cos ∙ sinh ] + 𝐶𝐿 [sin ]− 𝐶𝐿 [− cos ∙ cosh + sin ∙ sinh + ]− 𝐶𝐿 [cos ∙ cosh + sin ∙ sinh − ] + 𝐶𝐿 [sinh ]− 𝐶𝐿 [cosh ∙ sin − cos ∙ sinh ]

(60) 𝐼 1.29896 𝐼 0.67485 𝐼 0.70109 𝐼 0.69995 𝐼 0.70000 𝐼 0.70055 𝐼 1.06198

Table 3. 𝐼 values 1-7


And:

𝐶𝐿 = 𝑐 𝜂 +c s 𝜂s 𝜂 −s 𝜂 (61)

Implemented into MATLAB plotting Yi against x this gives the following mode shape

outputs:

Figure 3. mode shape Y1 with max point at (0.7, 1.933)


Figure 4. mode shape Y2 with max point at (0.7, -1.933)









The 1st and 2nd derivative of 𝑌 in space were also found: 𝑌′ = √ ∙𝐼𝑟 ( cos − cosh ) + √ ∙𝐼𝑟 × 𝐶𝐿( − sin − sinh ) (62)

𝑌′′ = √ ∙𝐼𝑟 ( −sin − sinh ) + √ ∙𝐼𝑟 × 𝐶𝐿( − cos − cosh ) (63)

This was them implemented into MATLAB and plotted Yi against x in order to find the

minimum and maximum Yi:

Mode Position, maximum value 𝑌′ . , . 𝑌′ . , .


𝑌′ . , . 𝑌′ . , . 𝑌′ . , . 𝑌′ . , . 𝑌′ . , . 𝑌′′ . , . 𝑌′′ . , . 𝑌′′ . , . 𝑌′′ . , . 𝑌′′ . , . 𝑌′′ . , . 𝑌′′ . , .

Table 4. Yi and Yi maximum positions

Whereas based on stress definition the maximum modal value of 𝑌′′ indicates since

the modal amplitude is the same (for all positions) the maximum stress, 𝜎, for this

mode on the beam.

Conclusion

A model of a cantilever beam was developed, this model was then discretized by

projecting the PDE s to infinite base of linear mode shapes. The theoretical model and

results were compared with experimental frequencies. In addition to this the

maximum values and positions were identified using MATLAB code. Furthermore, a

software was written for further examination of shock spectra.

Future work on this project would include further theoretical examination of shock

spectra, with the possibility of a lab setup in order to verify theoretical results

experimentally.


References

[1]H. Mevada and D. Patel, "Experimental Determination of Structural Damping of

Different Materials", Procedia Engineering, vol. 144, pp. 110-115, 2016.

[2]A. Piersol, T. Paez and C. Harris, Harris' shock and vibration handbook. New York:

McGraw-Hill, 2010.

[3] Syed Ayesha Dynamic characterstic estimation of structural materials by modal

analysis using ansys, International Journal of Advance Research In Science And

Engineering, IJARSE, Vol. No.3, Issue No.7, July 2014,ISSN-2319-835

[4] "Aluminum 2024-O", Matweb.com, 2017. [Online]. Available:

http://matweb.com/search/DataSheet.aspx?MatGUID=642e240585794f0ab91428aa78c

27b4e. [Accessed: 16- Aug- 2017].

[5]"Overview of materials for AISI 4000 Series Steel", Matweb.com, 2017. [Online].

Available:

http://matweb.com/search/DataSheet.aspx?MatGUID=210fcd12132049d0a3e0cabe7d09

1eef. [Accessed: 16- Aug- 2017].

[6] "Overview of materials for Brass", Matweb.com, 2017. [Online]. Available:

http://matweb.com/search/DataSheet.aspx?MatGUID=d3bd4617903543ada92f4c101c2a

20e5. [Accessed: 16- Aug- 2017].

[7] F. Poncelet, "Experimental Modal Analysis using Blind Source Separation

Techniques", Ph.D, University of Liège, 2010.

[8] Prof. G. Kerschen, "Kindly requested for response about details of Fabien's

(Poncelet) thesis", 2017.

[9] Dr. F. Georgiadis, centre of excelence modern composites applied in aerospace

and surface transport infrastructure (cemcast) – 3rd monthly Report

Documents

Shock Spectra of Cantilever Beam