Question

MAT 282 Activity 7: Determining System Parameters Sometimes you need to figure out parameters of a system without having full information. For example, an engineer might need to determine the amount of damping needed to limit the range of motion of a spring system without being certain of the forces that will be applied to the mass. In this activity, let's consider the spring system m +8嗇+ kx-rt), with initial conditions x(0-0,x'(0) =0. Furthermore, it is known that whatever f) is, there is a constant A such that Ol s A for all r. 1) Show that the transforms X(s) and Fs) are related by X) 2) Use the convolution theorem to derive the formular() [F2 r sinßt-rt-t) dr, wheres-* k bT 3) Show tht bls andA 4) Suppose m- 5 kg.k- 3000 N/m, and the force is bounded by A - 10 N. What range of values for the damping constant mk-b. b will ensure a displacement of no more than 1 cm?

Answer 1

1)

Given m d^2 x/dt^2 + b dx/dt + kx = f (t) x (0) = 0 x' (0) = 0 To show: X (s) = 1/m^2 + bs + k F (s) Proof: Equation (1) can be written as mx'' (t) + bx' (t) + kx (t) = f (t) Take Laplace transform on both sides we get L [mx'' (t) + bx' (t) + kx (t)] = L [f (t)] L [mx'' (t)] + L [bx' (t)] + L [kx (t)] = F (s) mL [x'' (t)] + bL [x' (t)] + kL [x (t)] = F (s) m {s^2 X (s) - sx (0) - x' (0)} + b {sX (s) - x (0)} + kX (s) = F (s) Using initial condition x (0) = 0 and x' (0) = 0 we get m {s^2 X (s) - 0 - 0} + b {sX (s) - 0} + kX (s) = F (s) ms^2 X (s) + bsX (s) + kX (s)

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