Determining the Damping Ratio of a Vibrating Cantilever

Aim/Introduction

In the experiment you will use the amplifier and bridge circuit to determine the damping ratio, damped frequency and the natural frequency of vibrations for a cantilever.

Two strain gauges have been attached to a steel beam and you will use them in a half bridge circuit to investigate the harmonic response of the vibrating cantilever.

From the readings you take you can determine the damped frequency of the vibrations and, by looking at successive amplitudes, the damping ratio that exists. From both of these you can then determine the natural frequency that the cantilever has.

The aims of this experiment are to determine the damping ratio, damped frequency and the natural frequency of vibrations for a cantilever.

Keywords

Amplification, differential amplifier, Wheatstone Bridge, operational amplifier (Op-Amp), strain gauge, gain, zero-set, damping, frequency, logarithmic decrement.

Risk Assessment

This is a low risk experiment and in addition to the normal laboratory work rules:

• When cutting wires always wear goggles and ensure people around you are wearing goggles, always cut away from you and downwards making sure your fingers are not near the sharp edges of the wire or cutters
• The edges of the wires and components will be sharp, avoid those ends
• If you do cut yourself, report this to the nearest FEPS staff member who will advise you on the next course of action after inspecting the injury
• Avoid touching the circuit while the voltage source is connected to show good practice when working with electrical circuits
• Ensure no one hurts themselves on the beam by making people aware of its presence.

Equipment

A PCB amplification and bridge circuit

Steel beam with strain gauges attached

PicoScope 2000 and USB connection cable

Electrical connection wires

Computer with “Picoscope” software (you can download your own copy from https://www.picotech.com/downloads and select the PicoScope 2208 B from the list, downloading for the operating system you use)

Micrometer and/or Vernier callipers

G-clamp

Metre rule

Methodology

1. Measure the width (b) and depth (d) of the beam at several points along the beam
2. Clamp the beam approximately at right angles to the edge side of the bench with the strain gauges closest to the bench but not touching it, as in figure 1
3. Measure the free length of the beam (the part from the clamping point to the free end, L)
4. Connect the yellow wires to the Wheatstone Bridge input Y-SG and the black wires to SG-A and SG-B (use figures 2 and 3 to help)
5. Connect the PicoScope probe to the A input on the PicoScope and the O/P and GRD connectors with wires on the amplifier (for a refresher on PicoScopes look at week 9 PEAC: Seminar: PicoScope Basics (Web view))
6. Make sure the power supply is switched off, all dials are turned down to zero and connect the power supply to +5, -5 and earth/ground, see figure 4, but make the current limiter 100 mA
7. Connect the power supply to the amplifier with red, black and blue wires then turn on the voltage supply
8. Ensure you are reading scope A in DC
9. Balance the Wheatstone Bridge using the variable resistor
10. Deflect the cantilever by about 10 mm (to prevent overstraining) and observe the trace on the “PicoScope” software; if the trace isn’t suitable you can offset it by using the “PicoScope” software and you can also change the time-base and vertical gain so you can see a trace similar to Figure 5
11. Save the file as psdata for several traces for future analysis after the laboratory session.

Figure 1, showing the correct set up for the beam

Figure 2, completed differential amplifier board

Figure 3, circuit diagram of the PCB

Figure 4 showing the variable power supply current and voltage settings with connecting wires (note the wires leading off the image all have crocodile clips on the ends).

Figure 5 – A damped harmonic response (after the stepped input)

Theory

If the vibrations were purely simple harmonic (without any damping) then the amplitude would remain constant and the frequency of the oscillations would be the natural frequency (f_n).

However, when damping takes place the response becomes more complex.

The response of the system depends upon the damping ratio (z = ratio of actual damping to critical damping).

For the light damping in this case, the damping ratio is less than unity; notice that the response decays with time.

By measuring the heights of successive peaks the logarithmic decrement of the response can be determined.

Figure 6 – measuring successive heights on a damped response

By measuring the ratios of successive heights (X₁ and X₂) the logarithmic decrement (d) can be determined

(1)

This can be repeated with any successive pair of heights.

It can be shown that the damping ratio, z, is related to the logarithmic decrement, d, by

z                      (2)

The damped frequency can be determined from the time between successive peaks, T_d

(3)

Remember that frequency (f) is related to angular frequency (w) by

Finally, the natural frequency of the vibrations, w_n, is related to the damped frequency, w_d, and the damping ratio,z, by

(4)

It can also be shown that the natural frequency of a vibrating cantilever is given by

(don’t worry, 2^nd year work in mechanical engineering)

(5)

Where            E = Young’s modulus of elasticity » 205 GPa

https://www.engineeringtoolbox.com/young-modulus-d_417.html

I = second moment of area of the cantilever

=

b = breath (width) of beam

d = depth of beam

r = density of steel » 7850 kg m^-3.

https://www.engineeringtoolbox.com/metal-alloys-densities-d_50.html

A = area of beam = b x d

L = length of beam

Report

In Microsoft word format and about 1200 words (word counts in excess are in danger of losing marks)

This report is worth 25% of the overall mark for the module “Further Applied Engineering”

You are required to write a report for this laboratory, to include:

1. Tables of the measurements of the successive heights of the amplitudes from the trace obtained, to then include the measurements of the logarithmic decrements calculated and the period between traces. (Probably a copy of your excel tables)

As there is some noise on the trace and the frequency is around 15 Hz, it is suggested you look at every 5^th amplitude and measure the amplitude as best you can.

In doing this the logarithmic decrement will become

Where X₁ = the first amplitude

X₂ = amplitude of 5^th peak

n = 5 (number of peaks)

Also measure the time between the 1^st and 5^th peak (four time periods), working out the time for 1 cycle and hence the frequency

Keep repeating this process along the wave – average all the results obtained to perform your final calculation. (Continue measuring until the ratios have settled down)

2. A plot of the trace you took from the experiment: PicoScope data fully annotated, with labels, titles and units.
3. A value for the damping ratio (z, from equation 2) of the beam and the damped frequency (f_d, from equation 3) of vibration. Give only one calculation based on your best estimation of the average logarithmic decrement from averaged results.
4. An estimation for the natural frequency (f_n) of the beam, together with calculations of the estimation of the natural frequency from measurements (equation 5) – again just one value required based on the best estimation available.
5. An uncertainty analysis based on equation 5
6. A discussion of the results. Points to include are:-

Does the theory (equation 5) predict the natural frequency? Why?

(Compare the results from the two different calculations)

1. Assess if the logarithmic decrement remains constant and if the time for 4 oscillations remains constant; does it vary from one set of 5 results to the other? Discuss!
2. Does the periodic time for one cycle vary from one point to another?
3. How does your uncertainty analysis affect your results?
4. Where could this analysis be important in an engineering situation?
7. Appendices (for detailed calculations and any other material that would stop the “natural flow” of the report)

The report should include a front sheet (including the name of the report, module and student ID) and an index page showing where each section of the report can be found.

You do not need to include any method (unless it varies from that given here), theory or introduction.

All reports will be checked for plagiarism by Turn-it-in.

Additional information required

How to use “Picoscope”2208 (or Dr Daq) and using the “PicoScope” software.