- cantilever response slide 1 Part 2: Second order systems: cantilever response Goals: Understand the behavior and how to characterize second order measurement systems Learn how to operate: function generator, oscilloscope, lock-in amplifiers, HeNe laser, beam position detector, Understand and characterize the optical beam deflection method for measuring angle and position
- cantilever response slide 2 Second order system: A second order measurement system is a system whose dynamics is described by a second order differential equation. The transfer function assumes the form with ω ο being the eigenfrequency of the system and ζ the attenuation (damping) constant.
- cantilever response slide 3 A cantilever rigidly fixed at one end has a mechanical response similar to a mass/spring combination: ba fixed end l where the spring constant k = ab 3 E/4l 3 and lowest resonance frequency f = 0.16b/l 2 (E/ρ) are determined by the dimensions of the cantilever (a = width, l = length, b = thickness) as well as Young s modulus E and the density ρ of the cantilever material. Young s modulus: Measure for material stiffness E = pressure on material p / relative length change ε ΔL F p = force F per area A L ε = length change ΔL per length L A
- cantilever response slide 4 When the cantilever is subjected to a force F applied to its end, the shape of the cantilever is described by the expression x y Note that this expression implies that a force F at the position x=l causes a displacement y(l) and F as known from the spring equation.
- cantilever response slide 5 With the force equation, the equation of motion of the cantilever becomes: Note: - m* is NOT the cantilever mass but an effective mass λis the viscous coefficient which, for the conditions given in the cantilever experiment, will be determined mostly by air resistance If we substitute: The equation of motion will assume a general form for second order systems Laplace transform both sides of the ODE, with Δx 0 = 0: hence.. General form of transfer function of second order element
- cantilever response slide 6 The cantilever described above represents a 2nd order mechanical system. It has a similar mechanical response as a spring/mass system with finite damping. In the spring mass system, the natural frequency is also given by Like the cantilever, any other real system also has a dissipation or damping mechanism, which causes any free oscillation to eventually decay. The damping of the system is characterized by a damping ratio which is given by where λ is the viscous coefficient in the equation of motion (in the differential equation in front of the velocity term dy/dt. The quality factor (Q) of the system is related to the damping ratio as The Q of the system and its resonance frequency have a fixed relation to the width of the resonance as defined by the frequency range Δf at which the oscillation amplitude near resonance falls to 0.707 of the amplitude on resonance. The width is called the full width at half max (FWHM), as the oscillation power at an amplitude of 0.707 is exactly half of the oscillation power on resonance.
- cantilever response slide 7 The optical beam deflection method (optical lever method) is based upon the detection of the lateral displacement of an optical beam with a position sensitive photo-detector, after it has been reflected from the surface of an object to be measured. In this case, the beam is reflected from the tip of the cantilever. When the cantilever feels a force, it is deflected and the reflected beam at the detector shifts laterally. See the figure below. By measuring the photo-current from a position sensitive photodetector, a linear relation (approximate) between the lateral position of the laser beam and force on the cantilever can be achieved. The optical beam deflection method (or optical lever) can be used to measure very small positional or angular changes of the reflecting surface. The angular sensitivity is improved if the distance from the reflecting surface to the split detector is large (large optical lever effect). y laser Beam position detector x F However, diffraction ultimately limits the sensitivity as this distance becomes large. Measurement of deflections below 1 nm can be easily achieved under the proper conditions by this simple method.
- cantilever response slide 8 In this lab section, the displacement of a ferromagnetic, macroscopic cantilever is measured caused by the magnetic force of an electromagnetic coil. The entire setup can be considered a measurement system with a magnetic field inducing current of the coil as input and the voltage response of the position detector representing the cantilever displacement as output For the steel cantilever used, the following data applies: Density: ρ = 7.86 x 10 3 kg/m 3 Length: l = 14.0 mm Thickness: b = 132 μm Width: a = 2.65 mm Young s Modulus: E = 2.06 x 10 11 Nm -2 Hence, the cantilever is expected to exhibit a spring constant k = 114 N/m and a resonance frequency of f = 551 Hz
- cantilever response slide 9 Principle sketch of the cantilever experiment laser Beam position detector SC10D Tetra lateral position detector cantilever magnet magnet coil Preamp: 3018AC analog-lockin amplifier ITHACO 3941 in ampl out phase out ref in oscilloscope Tektronix TDS1012 Ch.1 Ch.2 Electro magnet driver (voltage to current converter, AX 200) Function generator Stanford Research DS345 R=1Ω Keithley voltmeter (for amplitude display) Keithley voltmeter (for phase display)
- cantilever response slide 10 Explanation of the experimental setup: The function generator produces a voltage that describes the time dependence of the force that is to be applied to the cantilever. The voltage is converted into a current by the magnet driver. The current is directly proportional to the magnetic field produced by the magnet coil and thus, to the force applied to the cantilever. Therefore, force of the cantilever can be measured directly by a coil current measurement. This is done by the 1Ω resistor. The voltage across this resistor represents the magnetic force. As the force moves the cantilever, the laser beam is deflected which leads to a current change in the position detector and thus to a voltage change at the output of the detector preamplifier. This output is fed into the input of the lock-in amplifier and, in parallel, into channel 1 of the oscilloscope. The direct force measurement (resistor voltage) serves as lock-in reference signal which is also in parallel to channel 2 of the oscilloscope. In a sense, the experiment measures the magnetic force in two different ways (i) via the resistor and (ii) via the cantilever. The delay of the resistor can be assumed to be negligible in comparison to the delay of the cantilever. Thus one can use the resistor signal as reference.
- cantilever response slide 11 Overview about cantilever experiment laser power supply oscilloscope voltmeter magnet driver lock-in voltmeter function generator laser detector magnet+cantilever setup (hidden) lateral translation stage
- cantilever response slide 12 Carry out the following tasks: (i) Turn on the HeNe laser. Lasers will be discussed later in class and can be considered here as black boxes which are used as light sources for the experiment. (ii) For your safety, absolutely obey the laser safety rules discussed in class! (iii)turn on the lock-in amplifier, the voltmeters, the oscilloscope, the detector amplifier and the magnet driver. Familiarize yourself with these instruments. If necessary, read the manuals. (iv)measure the sensitivity of the beam deflection system which means determine how the position of the beam translates into output voltage of the detector amplifier. To do this, translate the photodetector across the HeNe laser beam, and measure the output signal as a function of position (micrometer reading) in absence of any magnetic field at the cantilever.
- cantilever response slide 13 Carry out the following tasks: (v) With the sensitivity measured, adjust the translation stage such that the voltage offset of the position detector is minimized in absence of any magnetic field. (vi)use the geometry of the beam deflection system and the equations for the cantilever provided in the previous section to determine the static relation between the voltage output of the photodetection system and the force on the cantilever. Note that the beam deflection is caused by two effects: (i) the positional change y of the reflecting surface. (ii) the slope change dy/dx of the reflecting surface. From the equation on slide 2 one can deduce the latter by calculation of the derivative Consider the two effects separately and determine which one can be neglected. Use only the dominant effect for your calculations.
- cantilever response slide 14 Carry out the following tasks: (vii) With an applied current of 0.5 A rms in the coil at 100 Hz, determine the absolute magnitude of the cantilever vibration amplitude using the relations found in (vi). Given the expected spring constant as determined from the cantilever geometry (see slide 5), give an estimate of the magnetic force on the cantilever. (viii) Find the first resonance frequency of the cantilever by sinusoidally driving the magnetic coil (actuator). Suggest an explanation for the difference between the values from the cantilever geometry and the experimental results.
- cantilever response slide 15 Carry out the following tasks: (ix)measure the amplitude and phase of the cantilever using the optical deflection system as a function of drive frequency. As with the 1 st order system, drive the system with a constant amplitude signal generator and compare the drive signal and system response using an oscilloscope and lock-in amplifier. Plot the amplitude versus frequency on a (log-log) graph out to a frequency of 2 khz. Plot the phase versus frequency on a (linearlog) graph out to a frequency of 2kHz. You may assume that the drive source and coils have a flat frequency and phase response out to and beyond 10 khz. Note that for the lock-in amplifier measurements, two Keithley voltmeters are used as displays. For the measurement of the amplitude, the lock-in gain setting determines what the measured voltage means whereas the chosen full range translates into 10V (example: If the 500mV range was chosen, there are 500mV at the input when 10V are read at the output). For the phase, a conversion factor of 10mV/degree is given which means at a maximum of 180º phase difference, 1.8V can be measured at the phase output. To get optimal noise levels, set the lock-in time constant reasonably long, e.g. τ = 1s but note, the longer the time constant, the longer you have to wait for the system to stabilize until a measure value can be read.
- cantilever response slide 16 Carry out the following tasks: (x) Determine the damping ratio and quality factor (Q) of the system by measuring the width of the resonance at FWHM and using the equation which relates these quantities. (xi)measure the response of the system to a step excitation (<1 Hz square wave). Determine the resonance frequency, damping ratio and quality factor of the cantilever system based upon this measurement, using the time domain equations given in the text. Compare your results with those from part (viii) to (x). Explain any discrepancies. Note that one should choose a frequency for this experiment whose multiple do not match the power grid frequency of 60Hz in order to prevent beat oscillations from the room light and the power supplies to couple into the experiment. Choose for instance something uneven such as 0.83Hz.