measurement of stiffness and natural frequency of a vibrating beam

7/31/2019 Measurement of Stiffness and Natural Frequency of a Vibrating Beam

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Measurement of Stiffness and Natural Frequency of a Vibrating Beam

Review of one degree of freedom system

Let us first consider a one degree of freedom system as shown in fig. 1. Assume small

displacement and friction forces.

Figure 1

The only external force acting on the block is the spring force in the opposite direction of

motion. Therefore the equation of motion is given by

or , Therefore (1)

Equation (1) can be solved for a given set of initial conditions at t=0.

(2)The solution of the above system is given by

(3)

Where is the natural frequency of the above system.Equivalent System

Vibrational properties of the cantilever beam shown in fig. 2 are very similar to the one shown in

fig. 1.

m

kx


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Figure 2

The equation of motion for the vibration (y direction) of the tip mass is given by

(4)Where meq and are the equivalent mass and stiffness of the cantilever beam. Therefore thesystem shown in fig. 2 has a natural frequency of .

Equivalent Stiffness of a Cantilever Beam(keq)

The stiffness of a cantilever beam is determined based on the force displacement relationship.

Consider the beam shown in fig. 3.

Figure 3

Displacement of a cantilever beam subjected to a force Pat its free end is given by

The equivalent stiffness of the beam shown in fig. 3 is then determined by

(6)

y

P

E,I

L

keq

Tip

E, I

L

L0


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The natural frequency of the cantilever beam will be measured with an accelerometer, an impact

hammer,an amplifier, and a data acquisition system (see fig. 5). As its name suggests, the

accelerometer measures the acceleration of the beam at its attachment point. The impact

hammer is used to excite the beam and provide a controlled input force. The output of thesetransducers is connected to an amplifier and a data acquisition system, which processes the

transducer signals and outputs the frequency response function between the input and the output.

The data acquisition system includes a data acquisition box (DAQ) and a host computer which

displays the data in real-time and provides a graphical-user interface (Labview software).

Figure 5: Test setup for the cantilever beam experiment.

L

Hammer excitation Accelerometer (Piezoelectric)

Amplifier

Amplifying signals

DAQRaw data


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Figure 6: Actual test setup for the cantilever beam experiment.

DAQ

Amplifie

Labview

Acceleromet

Beam

Hammer

Tip mass


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The main window of Labview software with three different plot displays is shown in fig. 7.

The input force (Lb) and output acceleration ( in/sec2) responses are shown on the top

row of the main display. The frequency response function is shown on the bottom row of

the main display.

(see fig. 7).

Figure 7: Force, acceleration and frequency responses in the main window of Labview

software

Force Response Accel. Response

Freq. Response

measurement of stiffness and natural frequency of a vibrating beam

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