vibration analysis of fully and partially filled

Journal of Engineering Science and Technology Vol. 15, No. 5 (2020) 3162 - 3177 © School of Engineering, Taylor’s University

3162

VIBRATION ANALYSIS OF FULLY AND PARTIALLY FILLED SANDWICHED CANTILEVER BEAM

WITH MAGNETORHEOLOGICAL FLUID

N. SRINIVASA, GURUBASAVARAJU T.M., H. KUMAR*, ARUN M.

Department of Mechanical Engineering, National Institute of Technology Karnataka,

Surathkal, Mangalore, 575025, Karnataka, India

*Corresponding Author: [email protected]

Abstract

This paper presents the experimental and computational study on damping

effect of the fully and partially filled sandwich cantilever beams. The sandwich

beams referred as adaptive beams have a core layer filled with

magnetorheological fluid (MRF) between two aluminium face plates. Forced

vibration tests were conducted under different magnetic fields with the

application of external force in the form of sinusoidal sweep excitation using

an electrodynamic shaker. Effect on damping and natural frequency due to

change in MR fluid core thickness of 2 mm, 4 mm and 6 mm for the fully filled

beam and fluid core length of 75 mm, 150 mm and 250 mm for partially filled

beam were investigated. Modal and harmonic analysis of the MR sandwich

beams were carried out using FE analysis. The results indicated that in the case

of the fully filled beam, a reduction in the natural frequency with the increase

in MR fluid core thickness and a better damping at 2 mm fluid core thickness

were observed. Also, in the case of the partially filled beam a reduction in

natural frequency and improvement in damping is found with the increase in

core length and magnetic field. The results of these analyses can be useful in

designing the sandwich beams for structural application.

Keywords: Damping ratio, Magneto rheological fluid (MRF), Modal and harmonic

analysis, Sandwich beam.

Vibration Analysis of Fully and Partially Filled Sandwiched . . . . 3163

Journal of Engineering Science and Technology October 2020, Vol. 15(5)

1. Introduction

In the case of structural applications, it is important to control vibration. This can

be controlled by modifying mechanical properties or by applying passive, semi-

active or active damping inputs. Undesirable vibration in beams can be supressed

by incorporating a damping layer, either over the surface or stacked between two or

more layers of plates or laminates. Use of viscoelastic damping layers like rubber,

shape memory alloys, piezoelectric materials, electrorheological (ER) fluids, MRF

and magnetorheological elastomer (MRE) help in stabilizing the vibration.

Nowadays, MRF is widely used in many applications and it is well known that its

property changes with the influence of external magnetic field. An MR fluid

comprises of soft ferromagnetic particles (0.03-10-micron) such as pure iron,

carbonyl iron powder, cobalt, ceramic metal, or alloys, dispersed in a carrier liquid

namely mineral oil, synthetic oil or silicone oil. The MR particles present in the fluid

then forms a chain like structure along the magnetic field lines.

The MR fluid changes from liquid state to semi-solid state within a fraction of

milliseconds in the presence of magnetic field and retains its fluid form in the absence

of the magnetic field [1]. The semi-solid state is represented as a viscoelastic material

in the pre-yield region in the form of complex shear modulus, and as non-Newtonian

behaviour model in the post yield region [2]. Sun et al. [3] stated that MRF has higher

stiffness values compared to ERF in adaptive structure applications. Yeh et al. [4] and

Yeh and Chen [5] investigated the outcome of the structural stiffness, natural

frequencies and loss factors of the sandwich beam with ER fluid and used FEM to

evaluate the same. Yalcintas and Dai [6] studied the vibration characteristics of ER

and MR fluid filled simply supported beams and found that the beam filled with MR

fluid has better stiffness under magnetic field as compared to the beam with ER fluid

under electric field. Yeh and Chen [7] presented the effect of electric field on damping

of a sandwich beam through experimentation.

Limitations in the use of ERF led some of the researchers to develop a smart

fluid called magnetorheological fluid, where the electric field is replaced by the

use of permanent magnets or electromagnet. Hirunyapruk et al. [8] investigated

the use tuned vibration absorber (TVA) to control the vibration of the system

using magnetorheological fluid layer between the structure. Lara-Prieto et al. [9]

discussed the vibration characteristics of the poly-ethylene terephthalate (PET)

and aluminium sandwiched beam filled with MRF. Rajamohan et al. [10] carried

out studies on vibration responses of multi-layered MRF beam. Rajamohan et al.

[11] investigated vibration analysis using developed FE formulations to find the

effect of partial treatment of MRF sandwich beam and the tests were validated

through experimental studies.

Further, Rajamohan et al. [12] utilized an optimization technique to study the

optimal position of MRF partial treatment of a partially-treated MR sandwich

beam. Li et al. [13] studied the dynamic responses of a rectangular plate with

MRF and isotropic face plates using FEM. Investigations on fully and partially

treated laminated beams with MR fluid were carried out by Rajamohan et al. [14]

and they observed from the experimental investigation that the frequency and the

displacement response of the system is strongly influenced by the applied

magnetic field, pocket position and size of the partial fluid filling. Vibration

parameters like frequency and damping coefficients were investigated for two

types of MR fluids [15].

3164 N. Srinivasa et al.


Rajamohan et al. [16] developed a finite element model to perform vibration

analysis of a sandwich beam filled with MRF. They developed a mathematical

model for the beam and validated the experimental results with the FE method and

Ritz method. Romaszko and Węgrzynowski [17] modelled a three-layered

sandwich beam with MRF core and conducted FEM analysis.

The study carried out by recent researchers mainly emphasizes on experiments

conducted with free vibration and analytical formulations for the MR sandwich beam.

But there is limited study on experiments considering forced vibration. The work

carried out in this paper mainly focuses on the use of MR fluid as a semi-active core

layer to determine its vibration isolation capability. The variation of natural frequency,

vibration amplitude and damping effect with MR fluid thickness, length and their

positions under different magnetic fields were investigated using forced vibration tests

and finite element analysis in ANSYS software.

2. Fabrication of MR Sandwich Beams

2.1. Fabrication of fully filled MR sandwich beam

An MR sandwich cantilever beam consists of three layers namely two solid face

layers (plates) and an MR fluid core layer. The rheology of the fluid and the

behaviour of the beam is controlled by the application of an external magnetic field.

Geometric models of the fully filled MR sandwich beam with different fluid core

thickness are shown in Fig. 1(a). Exploded view of parts of the MR sandwich beam

is shown in Fig. 1(b).

Aluminium was used as face plates as it has a relative permeability equal to

unity and does not affect the distribution of magnetic field. The face plates were

fabricated to a dimension of 290 mm span length and 2 mm thickness. An

aluminium square piece of 20 mm length, 25 mm width and 2 mm thickness were

riveted and glued to the upper and lower face plates at their ends to give a stiff

support and to maintain an equal MRF core layer thickness along the length of the

MR sandwich beam. The MR fluid core length is 250 mm for all the fully filled

beams. The same is done for 4 mm and 6 mm MR fluid core thickness. The width

of the beam is 25 mm and the gaps are sealed with silicone sealant to avoid MR

fluid leaks. A small gap of 5 mm was left on one side of the beam to fill the core

with MR fluid and the gap was properly sealed with silicone sealant. Later the

sealant was set to dry.

2.2. Fabrication of partially filled MR sandwich beam

The fluid core thickness was maintained constant at 2 mm for the partially filled beams

and the position of the fluid core and its length was varied. The fluid core was

considered at three different positions. This is shown in Figs. 2 and 3. The face plates

and the middle plate made of aluminium are glued and riveted together to form a rigid

beam with required fluid core length. During this process, the middle layer was cut in

different dimensions depending upon the fluid core position required. The fluid core

lengths considered are 75 mm and 150 mm with locations named as P1 near clamped

end, P2 for middle and P3 near the free end of the beam.



(a) Geometric model.

(b) Exploded view of parts.

Fig. 1. Geometric models of MR sandwich beam with

different core thickness and fabricated MR sandwich beams.

(a) P1.

(b) P2.

(c) P3.

Fig. 2. Geometric models of partially filled MRF

beam with 75 mm fluid core length at different positions.

4 mm8 mm

250 mm

290 mm

2mm

6 mm

250 mm

290 mm

250 mm

290 mm

6 mm10 mm

MRF Aluminium face plates (2 mm)

2mm

MRFAluminium face plates (2 mm)

6 mm

75 mm

290 mm

P1

6 mm

75 mm

290 mm

P2

2mm

P3

2mm

6 mm

75 mm

290 mm



(a) P1.

(b) P2.

(c) P3.

Fig. 3. Geometric models of partially filled MRF

beam with 150 mm fluid core length at different positions.

3. Experimental Setup

Based on the size of the test specimen, an experimental test setup was built to clamp the

beam and mount the permanent magnets as shown in Fig. 4. The setup consists of

various components such as beam fixture, magnet support, electrodynamic shaker,

permanent magnets, power amplifier, data acquisition (DAQ) system, accelerometer

and force sensor. An electrodynamic shaker was used to provide forced excitation to

the beam. Permanent magnets were mounted on upper and lower plates having north

pole and south pole respectively. The magnetic field was varied by changing the

distance between the upper and lower plates. The magnetic field was measured using a

gauss meter of Lakeshore model 410. The magnetic field was applied in the range from

0 T to 0.1 T in steps of 0.025 T. Permanent magnet arrangement for fully and partially

filled MR sandwich beams are shown in Fig. 5.

Fig. 4. Vibration test setup for MR sandwich beam.

2mm

P1MRF

Aluminium face plates (2 mm)

6 mm

150 mm

290 mm

6 mm

290 mm2mm

150 mm

P2

2mm

6 mm

290 mm

150 mm

P3



(a) 250 mm.

(b) 75 mm. (c) 150 mm.

Fig. 5. Permanent magnet positions for fully and partially treated beams.

A stinger of 6 mm diameter was fabricated using brass and was used to

connect the free end of the beam with the shaker. The power amplifier was used

to amplify the voltage output to the electrodynamic shaker. A LabVIEW program

was developed to send and acquire the signals through DAQ cards to the beam.

A National Instruments (NI) DAQ card 9264 was used to supply power to the

amplifier in the form of voltage. The amplified voltage signal was given in the

form of sine sweep ranging from 10 Hz to 100 Hz. Uniaxial accelerometer and

force sensor were placed on the beam and stinger respectively to measure the

frequency response of the beam. These sensor signal data were acquired using

NI DAQ 9234 and fed to the computer through LabVIEW software for

further processing. The forced vibration tests were carried out for different

magnetic fields.

4. Results and Discussions

A set of experiments were conducted on fully and partially filled MR sandwich beams

to study the influence of different magnetic field intensities on displacement response,

natural frequency and the damping ratio. The amplitude and frequency of vibration of

the beam at different magnetic fields were obtained from the data acquired using

LabVIEW software. Initially, a general study was conducted to check the effect of

magnetic field on fully filled MR sandwich beam with 2 mm fluid core thickness. Figs.

6(a) and 6(b) show the time domain and frequency domain plots of the beam before and

after application of magnetic field respectively. From the acquired signals it can be

observed that the magnetic field has a significant effect on the natural frequency and

amplitude of vibration. This is due to the increase in stiffness and damping of MR beam

with the application of magnetic field.

MRF

N NN

S S S Lower support

plate

Upper support

plate

Free

endClamped

end

Magnets

N

S

P1

N

S

P1

N

S S

N

S

P2

N

P2

N

S S

N

P3

S

N

P3

N

S S



(a) Time domain. (b) Frequency domain.

Fig. 6. Free vibration response of the sandwich beam

with 2 mm fluid core thickness without and with magnetic field.

4.1. Fully filled MR sandwich beam

The frequency response for 2 mm, 4 mm and 6mm MR fluid core sandwich beams is

calculated as ratio of the acceleration (output) to the force (input) and are shown in Fig. 7.

The peak amplitude and the corresponding frequencies were obtained from the plot of the

forced vibration test in LabVIEW software. The shift in frequency and improvement in

damping is observed when the magnetic field is gradually increased. Comparing the

frequency shift and amplitude reduction between the off state (0 T) and 0.1 T magnetic

field, the 2 mm MR core shows 9.53% increase in frequency and 72.986% reduction in

amplitude, whereas 4 mm MR core shows 5.77% increase in frequency and 18.18 %

reduction in amplitude. Then finally 6 mm MR core shows 6.65% reduction in frequency

and 59.39% reduction in amplitude.

(a) 2 mm. (b) 4 mm.

(c) 6 mm.

Fig. 7. Frequency response of the MR sandwich

beams with varying core thickness at different magnetic fields.



To determine damping ratio at different magnetic fields for the MR sandwiched

beam, half-power bandwidth method was used. Damping ratio is calculated using

Eq. (1):

n

2

12

(1)

The damping ratio at 0.1T increased by 676.36%, 11.11 % and 231.48% MR

for sandwich beams with 2 mm, 4 mm and 6 mm MRF core thickness respectively,

as compared to that obtained at 0 T.

4.2. Partially filled MR sandwich beam

The study was carried out for the partially filled beam with 2 mm fluid core

thickness and fluid core length of 75 mm and 150 mm at different positions. The

test results for partially treated beams are presented in Table 1. For the 75 mm MR

beam at the P1 position, there is a small shift in natural frequency and a significant

increase in damping with the increase in magnetic field. MR sandwich beam with

75 mm core length at P3 position shows a negative shift in frequency, which is due

to change in stiffness and the location of the fluid core. Since the stiffness may vary

at any point on the beam, the frequency may shift in positive nature or in negative

nature [18].

Table 1. Natural frequency and damping ratio

at different core lengths and positions under different magnetic fields.

Partially

filled MRF

sandwich

beam

Magnetic

Field (T)

Natural frequency

(Hz) Damping ratio

P1 P2 P3 P1 P2 P3

75 mm

0 29.66 27.99 36.33 0.011 0.010 0.012

0.025 30.66 34.66 35.99 0.020 0.018 0.017

0.05 27.66 28.66 34.99 0.023 0.025 0.032

0.075 30.42 29.99 35.34 0.042 0.034 0.031

0.1 30.33 34.66 30.66 0.027 0.052 0.039

150 mm

0 24.66 27.99 28.66 0.020 0.023 0.013

0.025 24.99 26.99 28.66 0.024 0.029 0.027

0.05 25.33 29.33 28.99 0.034 0.033 0.027

0.075 25.36 27.99 28.99 0.048 0.043 0.035

0.1 26.33 28.99 29.33 0.061 0.082 0.039

Therefore, as the fluid core length gets reduced and is placed farther away from the

fixed end, the frequency of the MR beam reduces. The next test was conducted for the

sandwich beams with 150 mm fluid core length. As the fluid core length increased, an

increase in frequency was observed. Results obtained for 150 mm fluid core length

shows increase in damping effect with the increase in magnetic field.

From the study, it was observed that there was a maximum shift in frequency

and increase in damping at P2 position for both 75 mm and 150 mm fluid core

lengths and the frequency shift was not profound at P1 and P3 positions. The

damping ratio at 0.1 T increased by 144%, 406.3% and 224.25% for the MR

sandwich beams with 75 mm core length at P1, P2 and P3 positions respectively as



compared to those at 0 T. In case of MR sandwich beam with 150mm fluid core

length, the damping ratio at 0.1 T increased by 195.64%, 258.69% and 210.08% at

P1, P2 and P3 positions respectively as compared to those at 0 T. It is evident that

the MR sandwich cantilever beam with 150 mm fluid core length gives better

damping at all the magnetic fields than the beam with 75 mm core length.

4.3. Effect of magnetic field on frequency for different fluid core

lengths considered from centre of beam

The effect of magnetic field on the frequency with increasing magnetic field and

fluid core length was studied. The partially filled fluid core length of 75 mm and

150 mm at P2 position were considered along with fully filled beam. It was

observed that as the fluid core length increased, the frequency of the beam tends to

decrease as shown in Fig. 8.

Fig. 8. Effect of fluid core

length on frequency for different magnetic fields.

This clearly shows that the magnetic field and fluid core length of the fully filled

and partially filled MR beam plays a significant effect on frequencies. Also, the

frequency increases with increase in the applied magnetic field which is in

agreement with the results of earlier studies of fully filled and partially filled

sandwich MR fluid beams. The results indicate that the damping ratio for the fully

filled beam is higher than those of the partially filled MR sandwich beam

irrespective of different configurations.

5. Finite Element Analysis

MRF sandwich cantilever beam is modelled in ANSYS software by creating a

geometry for aluminium face plates with corresponding cores for MR fluid layer

for fully filled and partially filled beams of different lengths and positions. The

shear stress and shear modulus of the MR fluids are highly influenced with the

application of magnetic fluid [19].



Table 2 shows the comparison of first natural frequency obtained from the

experimental and FEA modal analysis as well as their percentage deviation. The

material properties of aluminium and MR fluid [20] are specified in Table 3. Modal

analysis helps in calculating dynamic behaviour parameters like natural frequencies

and mode shapes of the structure. This is considered as a basis for transient and

harmonic analysis.

Table 2. Comparison of natural frequencies computed from FEA

with experimental tests for fully and partially filled MR sandwich beams.

Fluid core

thickness

Experimental

(Hz)

FEA

(Hz)

Percentage deviation

(%)

2 mm 26.88 28.07 4.24

4 mm 25.6 25.74 0.54

6 mm 24.32 25.36 4.1

Fluid core length

75 mm

P1 43.52 54.28 19.82

P2 48.64 55.34 12.11

P3 53.76 56.2 4.34

150 mm

P1 34.56 41.35 16.42

P2 38.4 43.92 12.57

P3 43.52 47.47 8.32

Table 3. Material properties of

Aluminium and MRF in MRF sandwich beam.

Properties Aluminium MRF

Density (kg/m3) 2700 3500

Young’s Modulus (Pa) 69x109 -

Poisson’s Ratio 0.3 0.3

Shear Modulus (Pa) 26.53x109 0.5x106

5.1. Harmonic analysis

Once the dominant natural frequencies and their corresponding mode shapes were

obtained, finding the steady state response of the structures for a sinusoidal varying

load is the main objective of the harmonic analysis. The force applied to the

structure varies sinusoidally for different frequencies and the structural response

also shows a variation in the similar manner. Using this analysis, the natural

frequency of the structure can be computed by plotting the amplitudes at any point

on the structure as a function of forcing frequency. In this case, 1 N force was

applied at the free end of the beam for a harmonic frequency range varying between

1 to 100 Hz.

The mesh properties and the boundary conditions are defined for the geometric

model of MR sandwich beam. The number of elements of MR sandwich beam is

7059 and the mesh type is hexa-mesh. The corresponding conditions are shown in

Fig. 9.



(a) Mesh model. (b) Boundary conditions.

Fig. 9. Mesh model and boundary

conditions of the MR sandwiched cantilever beam

5.1.1. Fully filled MR sandwich beam

The MR fluid layer is considered as a viscoelastic material. The structure is modelled

similar to that as the fully filled fabricated beams of different fluid core thicknesses.

The magnetic field is indirectly specified as an input in terms of shear modulus for

the MR fluid layer in the ANSYS software. This shear modulus is calculated based

on the relation between the storage modulus and magnetic field and the loss modulus

and magnetic field. The shear modulus for the MR fluid is found out by using the

equation developed by Manoharan et al. (2016) as given in Eqs. (2) to (4).

𝐺∗(𝐵) = 𝐺′(𝐵) + 𝑖𝐺"(𝐵) (2)

8.858355.42805035.0 2 BBBG (3)

35.848105.452057.0 2 BBBG (4)

where, B is the magnetic flux intensity in Gauss, G'(B) is the storage modulus and

G"(B) is the loss modulus of MR fluid. Substituting Eqs. (3) and (4) in Eq. (2) gives

the complex shear modulus G*(B) for the MR fluid. The shear modulus is computed

by taking the magnitude of the complex value. The computed first natural frequency

at different magnetic fields are compared with that of the experimental ones as shown

in Table 4.



Table 4. Natural frequency for fully filled sandwich MR

beam with different fluid core thickness by experiment and FEA.

Natural frequencies (Hz)

2 mm 4 mm 6 mm

Magnetic Field (T) Experiment FEA Experiment FEA Experiment FEA

0 20.33 27.05 17.33 24.01 19.99 21.49

0.025 20.33 27.96 18.66 25.13 17.99 22.86

0.05 20.33 28.72 18.66 26 16.99 23.81

0.075 20.99 29.39 17.99 26.76 17.33 24.66

0.1 22.99 29.98 18.66 27.43 18.66 25.4

From harmonic analysis, it is observed that for all the beams with different fluid

core thickness, the frequency increases with increase in the magnetic field as

compared to the beam without magnetic field and the same can also be observed from

experimental study.

5.1.2. Partially filled MR sandwich beam

The geometry for harmonic analysis is modelled similar to that of the partially filled

fabricated beam. The analysis is performed for partially filled MR sandwich beam of

different fluid core lengths of 75 mm and 150 mm at P1, P2 and P3 positions. The

results are represented in Fig. 10, which shows vibration amplitude in terms of

acceleration vs frequency.

From the plots, it is evident that the frequency change is not that significant as

compared to that obtained from experimental study. Though there is minor shift in

the frequency, the magnitude of the harmonic response tends to reduce with increase

in the magnetic field. The shift in frequency is notable with increasing fluid core

length, i.e., from 75 mm to 150 mm. When the fluid core is increased to 250 mm, it

can be observed that the frequency shift is more. From the plots of 75 mm and 150

mm fluid core length at P2 position and 250 mm fully treated beams, it is also

observed that as the fluid core length is increased, the natural frequency of the

sandwich MR beam is decreased. This is because of change in properties of the MR

sandwich beam. A similar trend is observed from the experimental study. The

damping values are increasing with increasing fluid core length for the beam partially

filled at the P2 position of the MR sandwich beam.

(a) 75 mm, P1 position. (d) 150 mm, P1 position.



(b) 75 mm, P2 position. (e) 150 mm, P2 position.

(c) 75 mm, P3 position. (f) 150 mm, P3 position.

Fig. 10. Harmonic response for partially filled beams at

different magnetic fields for different fluid core lengths and positions.

5.2. Effect of magnetic field on frequency for different fluid core length

considered from centre of the beam

Figure 11 shows the effect of magnetic field intensity for fluid core length of 75

mm, 150 mm and 250 mm (fully filled) considered from centre of beam. It can

be observed that for 75 mm fluid core length there is no change in natural

frequency with increase in magnetic field intensity. This may be due to lower MR

fluid core length and the rest of the core layer being covered with aluminium. For

150 mm MR fluid core length, it can be observed that there is a slight increase in

frequency with increase in magnetic field. This increase in frequency is observed

only at 0.025 T and 0.1 T, but there is no change at 0.05 T and 0.075 T. For fully

filled MR beam, it is observed that there is increase in frequency with increase in

the magnetic field intensity. Also, there is a decrease in frequency with increase

in fluid core length because of increase in mass of the structure which is more

evident to the effect of stiffness.



Fig. 11. Effect of fluid core length on frequency

for different magnetic fields from harmonic analysis.

6. Conclusions

This research study presents vibration analyses of fully filled and partially filled

MR fluid aluminium sandwich beam of different fluid core thickness, fluid core

lengths and their positions under different magnetic fields. Based on the

experimental tests and FE analyses performed in ANSYS software, following

conclusions were drawn.

The MR fluid in the sandwich beam has a better ability of reduction in the

magnitude of the peak response. The MR beam has better stiffening effect with

the application of an external magnetic field.

The frequencies and the damping ratios of the MR sandwich beam are mainly

affected by varying the fluid core size both in terms of thickness and length. It was

observed that the frequency decreased with increase in MR fluid core thickness.

With the advantage of design possibilities for partially filled beams, different

configurations were considered with 75 mm and 150 mm fluid core length at

different positions. It was seen that the fluid core length and its position have a

significant effect on frequencies and damping ratios with the increase in

magnetic field.

Finite element analysis study was performed in ANSYS software for the fully

filled and partially filled sandwich MR beams. The analysis shows a similar

trend of the shift in natural frequency, reduction in the magnitude of peak

response and damping with increase in magnetic field when compared with the

experimental results.

Nomenclatures

G' Storage modulus, Pa

G" Loss modulus, Pa

G* Complex shear modulus, Pa

B Magnetic field, T



Greek Symbols

ζ Damping ratio

Natural frequency, Hz

Abbreviations

DAQ Data Acquisition

ER Electrorheological

ERE Electrorheological Elastomer

ERF Electrorheological Fluid

FEM Finite Element Method

MR Magnetorheological

MRE Magnetorheological Elastomer

MRF Magnetorheological Fluid

NI National Instruments

PET Polyethylene Terephthalate

TVA Tuned Vibration Absorber

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