Experiment #4Dynamic Response of a Vibrating Structure

to Sinusoidal Excitation

Objectives

Perform a standard vibration test to measure the frequency response of a structural system

Solve a differential equation describing the motion of a structure with one degree of freedom under sinusoidal excitation

Calculate the equivalent viscous damping coefficient (ζ) of a single degree of freedom structure

1.

2.

3.

Test Specimen and Test Setup

Load Cell

Specimen

Accelerometer

Exciter

Part I – Frequency Response

Forced Response Free Response

Mechanical Excitation Manual Excitation

Forced Response

Node First Mode – 10.1 Hz

First Resonance – displays one node

Forced Response

Second Mode – 68.8 Hz

Nodes

Second Resonance – displays two nodes

Third Mode – 115 Hz

Nodes

Forced Response

Third Resonance – displays three nodes

Forced Response

Force

Acceleration

Before 68.6 Hz Resonance is small

In phase

Forced Response

Force

Acceleration

Before 68.6 Hz Resonance is small

In phase

Forced Response

Force

Acceleration

At 68.6 Hz Resonance is large

90°phase shift

Forced Response

After 68.6 Hz Resonance

Force

Acceleration

is small

Back in phase

Magnitude Ratio vs. Frequency

30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.00.000E+00

5.000E-04

1.000E-03

1.500E-03

2.000E-03

2.500E-03

3.000E-03

Frequency (Hz)

Mag

nitu

de R

atio

(ft/l

b)

Experimental data indicates that there is a resonance ~ 68 Hz

Phase Angle vs. Frequency

30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 110.0

-200.00

-150.00

-100.00

-50.00

0.00

50.00

Frequency (Hz)

Phas

e A

ngle

(°) Experimental data

indicates that there is a phase shift of 90° at

~68 Hz

Free Response

Acceleration

Decreasing acceleration and

decreasing displacement

Part II – Lumped Parameter Model

The diagram below describes the motion of our beam:

x

F = applied force by the exciter

X = beam displacement

Single DOF Spring-Mass-Damper System

The mathematical model we used to describe the motion of the beam was a Single DOF Spring-Mass-Damper System.

X(t) = displacement

F(t) = applied load

m = mass

k = spring constant

c = damping coefficient

The lumped parameter model can be modeled by a non-homogeneous differential equation:

𝑚�̈�+𝑐 �̇�+𝑘𝑥=𝐹 (𝑡)We developed two particular solutions to this DE:

𝜙= tan−1

−2𝜁 ( 𝜔𝜔𝑛 )1−( 𝜔𝜔𝑛 )

2

- Phase angle between forcing function and the displacement of the beam

𝑋𝐹0

=1

𝑘√[1−( 𝜔𝜔𝑛 )2]

2

+[2𝜁 ( 𝜔𝜔𝑛 )]2

- Magnitude ratio of displacement and applied force

Single DOF Spring-Mass-Damper System

Part III – Equivalent Viscous Damping Coefficient (ζ)

Three Methods for Finding ζ

Half-Power Method

Log Decrement

Method

Best Guess Method

Half-Power Method

𝜁 𝐻𝑃=𝑓 2− 𝑓 1

2 𝑓 𝑛

𝜁 𝐻𝑃=69.1142−68.5295

2∗68.9

𝜻𝑯𝑷=𝟎 .𝟎𝟎𝟒𝟐𝟒𝟗

The half-power method utilizes frequencies on either side of the natural frequency along with the natural frequency to approximate the viscous damping coefficient (ζ).

Half-Power Method

68.3 68.5 68.7 68.9 69.1 69.3 69.50

0.0005

0.001

0.0015

0.002

0.0025

0.003

68.5381, 0.001934307 69.1003, 0.001934307

Frequency (Hz)

Magnitude Ra-tio (ft/lb)

70.7%

Half-Power Points

Resonance

𝑋𝐹0

Log Decrement Method

𝛿= 1𝑛 ln ( 𝑥0

𝑥𝑛 )

The log decrement method utilizes frequencies at different points along the Free Response result in Part I.

- This is the log decrement

The log decrement is then used to find the viscous damping coefficient (ζ):

𝜁 𝐿𝐷=1

√1+( 2𝜋𝛿 )

2𝜁 𝐿𝐷=

1

√1+( 2𝜋0.07597 )

2

𝜻𝑳𝑫=𝟎 .𝟎𝟏𝟐𝟎𝟗

Best Guess Method

The Best Guess Method involved simply picking a value for ζ and then plotting the theoretical curves alongside the experimental data. The correct value of ζ is found when the theoretical curves match the experimental data.

𝜻𝑩𝑮=𝟎 .𝟎𝟏

Comparison of HP and LD ζ Values

Differential error analysis shows that:

Therefore, we conclude that the Log Decrement Method is a much more accurate method of calculating the viscous damping coefficient (ζ).

Frequency Response Function Curves - Magnitude

34.0 44.0 54.0 64.0 74.0 84.0 94.0 104.00

0.0005

0.001

0.0015

0.002

0.0025

0.003

Magni-tude RatioBest GuessHalf PowerLog Decrement

Frequency (Hz)

Mag

nitu

de R

atio

All curves agree as to the location of the resonant frequency

The value of ζ affects both the height of the curve and the

slope leading up to the resonance

68.6

Magnitude Ratio vs. Frequency, ω

35 45 55 65 75 85 95 105-10

10

30

50

70

90

110

130

150

170

190

Phase ShiftBest GuessHalf PowerLog Decrement

Frequency (Hz)

Phas

e Sh

ift (°

)Frequency Response Function Curves –

Phase Angle (φ)

FRF curves don’t correlate well with the

experimental phase shift in this region

All curves indicate that there is a

phase shift of ~90° at 68.6 Hz

Phase Shift, φ vs. Frequency, ω

68.6

Conclusions

We successfully performed experiments to find both the forced and free response of our test specimen.

The magnitude ratio curves produced by our model correlate very well with our experimental data.

Our theoretical phase angle curves fit the experimental data well up until the resonance, and show the expected phase shift.

Recommendations

I recommend that there be a more permanent test setup for the frequency response experiments in such a way that it can’t be altered between lab sessions.