Experiment 3

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Experiment 3

• Part A Instrumented Beam as a Harmonic
  Oscillator
• Part B RC and RLC Circuits Examples of AC
  Circuits

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Part A

• Harmonic Oscillators
• Analysis of Cantilever Beam Frequency Measurements

3
Macroscopic Examples of Harmonic Oscillators

• Spring-mass combination
• Violin string
• Wind instrument
• Clock pendulum
• Playground swing
• LC or RLC circuits
• Others?

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Large Scale Examples

• Petronas (452m) and CN (553m) Towers

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Harmonic Oscillator

• Equation
• Solution x Asin(?t)
• x is the displacement of the oscillator while A
  is the amplitude of the displacement

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Spring

• Spring Force
• F ma -kx
• Oscillation Frequency
• This expression for frequency hold for a massless
  spring with a mass at the end, as shown in the
  diagram.

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Spring Model for the Cantilever Beam

• Where l is the length, t is the thickness, w is
  the width, and mbeam is the mass of the beam.
  Where mweight is the applied mass and a is the
  length to the location of the applied mass.

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Spring Model -- Continued

• For a beam loaded with a mass at the end, a is
  equal to l. For this case
• where E is Youngs modulus of the beam.
• Since real beams have finite mass, it is
  necessary to use the equivalent mass at the end
  that would produce the same frequency response.
  This is 0.23mbeam.

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Spring Model -- Continued

• Now we can apply the expression for the ideal
  spring mass frequency to the beam.
• Where the mass used now is given by

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Atomic Force Microscopy -AFM

• This is one of the key instruments driving the
  nanotechnology revolution
• Dynamic mode uses frequency to extract force
  information

Note Strain Gage
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AFM on Mars

• Redundancy is built into the AFM so that the tips
  can be replaced remotely.

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AFM on Mars

• Soil is scooped up by robot arm and placed on
  sample. Sample wheel rotates to scan head. Scan
  is made and image is stored.

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AFM Image of Human Chromosomes

• There are other ways to measure deflection.

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AFM Optical Pickup

• On the left is the generic picture of the beam.
  On the right is the optical sensor.

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Our Experiment

• For our beam, we must deal with the beam mass,
  the extra mass of the magnet and its holder (for
  the magnetic pick up coil), and any extra load we
  add to the beam to observe how its performance
  depends on load conditions.

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Our Experiment

• For simplicity, call m the combination of 0.23
  times the beam mass plus the magnet and its
  holder.
• To obtain a good measure of k and m, we will make
  4 measurements of oscillation frequency using 3
  extra masses.

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Our Experiment

• We will then have 4 equations and 2 unknowns.
• One we obtain values for k and m we can plot
  to see how we did.

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Our Experiment

• One method is to guess values for k and m and
  then keep adjusting them until we obtain a good
  fit, obtaining something like

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Our Experiment

• How to guess values for k and m?
• Estimate the mass by measuring the beam
  dimensions and assuming the beam, magnet and
  holder are made from something reasonable.
• Use your guess for m with your largest load mass
  to obtain a guess for k.
• Plot f as a function of load mass
• Change values of k and m until your function and
  data match.

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Our Experiment

• Can you think of other ways to more
  systematically determine k and m?
• Experimental hint make sure you keep the center
  of any mass you add as near to the end of the
  beam as possible. It can be to the side, but not
  in front or behind the end.

C-Clamp
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MEMS Accelerometer
Note Scale

• An array of cantilever beams can be constructed
  at very small scale to act as accelerometers.

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Hard Drive Cantilever

• The read-write head is at the end of a
  cantilever. This control problem is a remarkable
  feat of engineering.

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More on Hard Drives

• A great example of Mechatronics.

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Modeling Damped Oscillations

• v(t) A sin(?t)

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Modeling Damped Oscillations

• v(t) Be-at

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Modeling Damped Oscillations

• v(t) A sin(?t) Be-at Ce-atsin(?t)

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Finding the Damping Constant

• Choose two maxima at extreme ends of the decay.

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Finding the Damping Constant

• Assume (t0,v0) is the starting point for the
  decay.
• The amplitude at this point,v0, is C.
• v(t) Ce-atsin(?t) at (t1,v1)
  v1 v0e-a(t1-t0)sin(p/2)
  v0e-a(t1-t0)
• Substitute and solve for a v1 v0e-a(t1-t0)

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Part B

• LC and RLC Circuits
• Intro to AC Circuits

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Inductors

• An inductor is a coil of wire through which a
  current is passed. The current can be either AC
  or DC.

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Inductors

• This generates a magnetic field, which induces a
  voltage proportional to the rate of change of the
  current.

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Inductors

• Inductors are also electromagnets.
• The magnetic field produced by the inductor
  current can do work.
• There is, therefore, energy stored in the
  magnetic field of the inductor.

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Combining Inductors

• Inductances add like resistances
• Series
• Parallel

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Inductor Impedance

• Note that this behavior is exactly the opposite
  of capacitors.

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Filters

• Even simple circuits can pass some frequencies
  and reject others. Thus, by selecting the right
  kind of circuit, certain frequencies can be
  filtered out.

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Oscillator Analysis

• Spring-Mass
• W KE PE
• KE kinetic energy½mv²
• PE potential energy(spring)½kx²
• W ½mv² ½kx²

• Electronics
• W LE CE
• LE inductor energy½LI²
• CE capacitor energy½CV²
• W ½LI² ½CV²

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Oscillator Analysis

• Take the time derivative

• Take the time derivative

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Oscillator Analysis

• W is a constant. Therefore,
• Also

• W is a constant. Therefore,
• Also

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Oscillator Analysis

• Simplify

• Simplify

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Oscillator Analysis

• Solution

• Solution

Vc Asin(?t)
x Asin(?t)
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Using Conservation Laws

• Please also see the write up for Experiment 3 for
  how to use energy conservation to derive the
  equations of motion for the beam and voltage and
  current relationships for inductors and
  capacitors.
• Almost everything useful we know can be derived
  from some kind of conservation law.

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Thevenin Voltage Equivalents

• Recall that the function generator (aka waveform
  generator) has an internal impedance of 50 Ohms.
  Thus, the circuit representation of this device
  looks like

Required for HW 3
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Thevenin

• Such a simple combination of a voltage source and
  resistance is called a Thevenin source.
• It is generally possible to simplify any complex
  source into such a combination.
• In fact, the function generator is much more
  complex, but it can be most simply represented
  this way.

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Thevenin

• Example

Load Resistor
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Thevenin

• We might also see a circuit with no load
  resistor, like the voltage divider.

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Thevenin Method
A
B

• Find Vth (open circuit voltage)
• Remove load if there is one so that load is open
• Find voltage across the open load
• Find Rth (Thevenin resistance)
• Set voltage sources to zero (current sources to
  open) in effect, shut off the sources
• Find equivalent resistance from A to B

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Thevenin Method Example

• Remove Load

A
A
B
B
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Thevenin Method Example

• Let Vo12, R12k, R24k, R33k, R41k

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Thevenin Method Example

• Short out the voltage source (turn it off)
  redraw the circuit for clarity.

A
A
B
B
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Thevenin Method Example

• First find the parallel combinations of R1 R2
  and R3 R4.
• Then find the series combination of the results.

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Thevenin Method Example

• The Thevenin Voltage Source is Then

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Thevenin Method

• Note
• When a short goes across a resistor, that
  resistor is replaced by a short.
• When a resistor connects to nothing, there will
  be no current through it and, thus, no voltage
  across it.

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Thevenin Applet (see webpage)

• Test your Thevenin skills using this applet from
  the links for Exp 3

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Thevenin

• To confirm that the Thevenin method works, add a
  load and check the voltage across and current
  through the load to see that the answers agree
  whether the original circuit is used or its
  Thevenin equivalent.
• If you know the Thevenin equivalent, the circuit
  analysis becomes much simpler.

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Thevenin Method Example

• Checking the answer with PSpice
• Note the identical voltages across the load.