2. Accelerometer (30 pts): An accelerometer's behavior can be modeled by a second order differential equation with a natural frequency (fn) of 120 Hz, damping ratio of 0.1, and a static sensitivity of 5 mV/g. The accelerometer is used in a feedback system on a shaker and is measuring the shaker's acceleration. The shaker table is initially at rest and then a sinusoidal excitation is turned on with an acceleration level of 4 m/sec² peak- to-peak and frequency 12 Hz. You may assume there are no bias voltages. (A) Write down the differential equation that relates output voltage, V(t), to input accelerations a(t). The coefficient of the differential equation should have numerical values. Please make sure the units are consistent. (B) Calculation the steady-state response of the accelerometer to the 12 Hz input excitation described above. Assume the steady state response is Vss(t) = A · sin(?t) + B · cos(?t) where ? = 2?f and f is the frequency of the excitation. Substitute the above representation into the differential equation and solve for A and B.
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Substituting the given values, we get: m(d^2V/dt^2) + 2ζωn(dV/dt) + ωn^2V = 5*9.81*a(t) where ζ = 0.1, ωn = 2πf = 2π*120 = 753.98 rad/s, and 9.81 is the acceleration due to gravity in m/s^2. B) The steady-state response of the accelerometer to the 12 Hz input Show more…
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