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Structural Dynamics Lab.

Experiment No.-03Vibration Characteristics of 3 DOF system by Accelerometer

I.I.T DELHI

Experiment Report

Submitted by : Akshay Kumar

2014CES2027

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EXPERIMENT 3 – Vibration Characteristics of 3 dof system by Accelerometer

OBJECTIVE:

To study the vibration characteristics of 3 degree of freedom system consisting of shear wall using

accelerometer. Also comparison is done between natural frequencies obtained theoretically and thoseobtained experimentally. The response of the structure is also plotted at the first natural frequency.

EXPERIMENTAL SETUP:

The setup consists of a 3 storey frame model with shear wall (all made of Perspex). The geometric

and material properties of the model is observed and given below. In this model, it is assumed

that beams are rigid with columns supplying all the stiffness for the system and mass lumped at

story level. The frame is modeled as system with three degree of freedom.

Two accelerometers were attached to the structure, one to measure the force and the other tomeasure the acceleration responses which are used to get the response of the structure. The

accelerometers are connected to the Data Acquisition System.

Fig (1): Experimental setup

THEORY:

Damping is inherent property. It is an effect that reduces the amplitude of oscillations in an

oscillatory system. If the physical properties of the system are such that its motion can be described by

Accelerometer

Force sensor

Columns

Shear wall

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a single coordinate and no other motion is possible, then it actually is a SDOF system and the solution of

the equation provides the exact dynamic response. On the other hand, if the structure actually has more

than one possible mode of displacement and it is reduced mathematically to a SDOF approximation by

assuming its deformed shape, the solution of the equation of motion is only an approximation of the

true dynamic behavior. This necessitates the development of dynamic models which consider multiple

degrees of freedom (MDOF).

Fig.(2): Characteristic figure to represent 3 DOF system

The equations of motion can therefore be expressed as:

11 + 11 − 22 − 1 − 1 = 0,

22 + 2(2 − 1) − 33 − 2 − 2 = 0,

33 + 33 − 2 − 3 = 0

In matrix notation, the equation in written as:

Where; [ M ] and [ K ] denote mass and stiffness matrices written as follows:

(Here we are taking the undamped case; hence the damping matrix becomes zero.)

For finding out the natural frequencies the eigen value method is used, putting

− 2 = 0

The flexural stiffness is given by the expression:

=123

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The shear stiffness is given by:

K =GA

L

Where,

E - Modulus of elasticity of members

I - Moment of inertia of the section in bending direction

L-Length of member

G-Modulus of rigidity

A-Area of the section

PROCEDURE:

1. Measure the dimensions with the help of vernier calliper.

2. Accelerometer is attached to the third storey only and one force sensor is also attached

to the same storey near the vibrator.

3.

These accelerometers are connected to data acquisition system, with the help of labview software further analysis is done.

4. Using LabView software the input signal is selected and the accelerometer is selected as

we are using accelerometer to measure the acceleration in the beam.

5. The input data is placed in the software regarding the no. of samples, sensitivity of the

instrument, rate of samples, the maximum and minimum range for the accelerometer

used and plots are obtained by changing RPM values.

6. The data is exported to the MS excel and the graph is plotted in frequency domain.

OBSERVATION AND CALCULATIONS:Density = 1200 kg/m3

Modulus of elasticity E = 3210 MPa

Poisson’s Ratio = µ = 0.39

Modulus of rigidity G = 1155 MPa

Lumped masses of the system in each storey

m1 = 0.484 kg m2 = 0.614 kg m3 = 0.614 kg

Width

(mm)

Depth

(mm)

Length

(mm)

Flexural stiffness

(x106

N/m)

Shear stiffness

(x106 N/m)

Column 12.16 19.1 155 0.1461 -

Shear Wall 12.28 75 155 4.466 6.863

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Total stiffness in each floor = (Flexural stiffness of columns + flexural stiffness of shear wall +

Shear stiffness of shear wall)

= (0.1461 + 4.466 +6.863) x 106 N/m = 11.475 x 10

6 N/m

The mass matrix of the system, M = 0.484 0 0

0 0.614 0

0 0 0.614

The mass matrix of the system, M = 0.484 0 0

0 0.614 0

0 0 0.614

The stiffness matrix of the system, K = − 0− 2 −0

− 2

= 11.59 −11.59 0−11.59 23.18 −11.59

0

−11.59 23.18

× 106

To find the natural frequencies, solve the following characteristic equation (Eigen value

problem),

Det

Solving the above equation we get,

ω1 = 2052.62 rad/s ; f 1 = 325.28 Hz

ω2 = 5659.95 rad/s ; f 2 = 901.31 Hz

ω3 = 7950.08 rad/s ; f 3 = 1264.8 Hz

Forcing frequencies have been considered in the range from 30 to 110 Hz. Hence the

experiment shall be further extended near to the values of Theoretical frequencies.

Figure: Observed natural frequency calculation

0

2

4

6

8

10

12

0 20 40 60 80 100 120

A c c e l e r a t i o n ( m i l i g )

Frequency (Hz)

Fig.(2) : Force vs Frequency graph

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Table (1): Observed acceleration vs frequrncy (Hz)

Frequency (Hz) Absolute maximum response (mili g)

30 2.2532

40 4.5846

50 9.2558

55 9.5

57 10.2

59 10.5

60 10.616

61 10.8

62 10.6

65 10.5

70 10.414

80 9

90 7.4

100 7.2

RESULTS:

1. The theoretical frequency = 322.28 Hz, 901.31 Hz, 1264.8 Hz.

2. The experimental frequency = 61 Hz for fundamental frequency/ first natural frequency.

DISCUSSION AND CONCLUSION:

1. Frequencies are varied only in the range of 30 to 110 Hz. The experiment should be

further extended up to the theoretical frequency.

2.

From the experimental data we found that the theoretical fundamental natural

frequency of the structure and the experimental fundamental natural frequency of the

structure are far apart. The difference in the theoretical and experimental fundamental

natural frequency can be due to following reasons:

The accelerometer and the wires become a part of the vibrating system and may cause

significant interference and their effect have not been considered in the calculations.

We assume supports are exactly fixed in theoretical calculations, but in practice

supports are not exactly rigid.

Lumped mass approximation.

The effect of shear wall and unavailability of an accurate expression for its stiffess is also

a source of approximation.

REFERENCES:1) Structural Dynamics: Theory and Computations, 2nd Ed, Paz, M (2004), CBS Publishers

and Distributors, New Delhi.

2) Dynamics of Structures by Ray W. Clough and J. Penzien, Computers & Structures Inc.,

Berkeley, USA.

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