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Unformatted text preview: BIM 281 Homework 04
Problem 1 Due date: Friday, 06/06/2008 Consider the sketched technique for calibrating the spring constant of an AFM cantilever (AFM: atomic force microscope). A motorized translator is coupled to the cantilever tip via an ideal spring whose spring constant ks is known. As the translator deflects the tip, the measured deformation of this spring reports the force f ks as a function of time. At the same time, the cantilever deflection (displacement measured at the tip) reports the elastic restoring force f kb of the cantilever. We describe this arrangement as a linear system, taking f ks = fin ( t ) as the “input signal” and f kb = f out ( t ) as the “output signal”. In the majority of biologically motivated force measurements, the cantilever is submerged in an aqueous buffer solution. In this case, a drag force f ηb resists cantilever movement through the fluid. We model this drag force as a simple friction force that is proportional to the tip velocity, with proportionality constant ηb . a. Sketch the equivalent combination of springs and dashpots for this setup. b. Find the differential equation describing the behavior of this system in terms of forces, i.e., of f in ( t ) and f out ( t ) . (Hint: You will have to express the drag force in terms of f out ( t ) .) c. Derive the transfer function H(f ) of this system using the differential equation found in b. Calculate the magnitude and phase of this transfer function. (Note: “f ” denotes frequency here, whereas fin and fout are forces.) d. Predict the output f out ( t ) of the system for an input force that varies sinusoidally as fin ( t ) = A sin ( πf 0t ) . (No derivation required, but look carefully at the input.) e. Consider also the simple RC-circuit sketched on the right. Given a cantilever spring constant of kb = 10 pN nm , a friction coefficient of ηb = 10-5 N×s m , and a 1 μF capacitor, what resistance R gives the same time constant for both the above mechanical setup as well as this circuit? (Hint: In the proper combination, 1s, 1V, and 1A give 1F.) Consider now the following two periodic input functions: 1 a triangle wave (consisting of linear force ramps at rates of ±100 pN/s over a total range of 200 pΝ, plotted above on the left), and a purely sinusoidal wave (covering the full 200 pN range; here it is the maximum force rate that reaches ±100 pN/s; plotted above on the right). f. Express each input function as a trigonometric Fourier series. (You may look up the Fourier series of a triangle wave, but make sure you properly scale and shift any series you may find so that it matches the given input.) Plot graphs of your expressions (truncate infinite sums in a sensible manner) to make sure that they are indeed correct representations of the above input functions. g. For each input, give the generic mathematical expression of the system’s output function, and plot three examples of predicted bead trajectories for (sensible) time constants of your choice. Comment on your result. Problem 2 You previously derived the output of a linear, shift-invariant system (with a given transfer function) to iθ f an input that was a pure sine function. Given again the transfer function H f = H f e ( ) , derive () () step-by-step the output for an input of the form gin ( x ) = B cos ( 2πf 0 x ) . Problem 3 Give a step-by-step proof of the identity F ⎡( g * h ) ( x ) ⎤ = G ( f ) H ( f ) ⎣ ⎦
for arbitrary functions g ( x ) and h ( x ) . 2 ...
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- Spring '08