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15.1 Seismometer Response The frequency response of a seismograph (meaning the entire seismometer and recording system) is primarily governed by the seismometer (inertial pendulumn) simply because it is the most non-linear part of the system. Amplifiers can be designed to give a flat or constant gain response over a wide frequency range, but the mechanical properties of an seismometer preclude equal detection (sensitivity) over a wide frequency range unless they employ some complicated mechanical feedback networks. Inertial Seismometer Response: Consider a simple suspended mass on a spring with dashpot damping . We will consider the inertial properties of the mass M and the relative displacement between the ground and the mass. For small displacements of the mass, (x'-x), in terms for small angles, (x'-x) ≈ l φ. Now summing the mechanical moments: 1) Torque due to angular acceleration of the mass, M, is: 2 ΓM = M a l = M l d x' dt2 15.1.1 3 but since x' - x = l φ, then x' = x + l φ, so the torque due to the mass can be written in terms of x and φ giving: 2 d2φ ΓM = M l d x + l dt2 dt2 15.1.2 2) Torque due to spring: The restoring force F is proportional to the displacement (x'-x): Force = -k(x'-x) , where k is the spring constant , giving: Γs = - k l (x'-x) = -k l2φ (a torque due to the spring ) 15.1.4 3) Torque due to the damping mechanism for a dashpot: For small velocities the force on a viscous body is proportional to the velocity, d(x'-x) , then: dt FD = -η velocity = -η dφ d(x'-x) = -η l dt dt 15.1.5 where η is the damping coefficient. Then the total torque due to the return action of the dashpot is: ΓD = -η l2 dφ dt 15.1.6 Now according to Newton's second law, the sum of the moments will equal the angular acceleration of the mass, giving: ΓM = Γ s + ΓD 15.1.7 d2 φ dφ 2 Ml d x + l = -k l2 φ − η l2 dt dt2 dt2 15.1.8 or after rearranging terms 15.1.8 becomes: 4 M l2 d 2φ dt2 + l2 η 2 dφ + k l2 φ = −M l d x dt dt2 15.1.9 And defining common terms: 2 k, Ωο = m η= η c , 2 β Ωο = m 2 M Ωo where Ωo is the natural frequency of the undamped system. If we substitute L for l then 1/L is referred to as the "reduced" pendulum length. Substituting for the above terms and dividing out common terms in 15.1.15 gives an important inhomogeneous differential equation for the seismometer response. φ + 2 β Ωο φ + Ωο φ = −1 x L 2 a frequency and time dependent equation. 15.1.10 This is the basic differential equation of a simple harmonic oscillator driven by an external force, such as ground motion. The coupling to the driving force term, on the right side of 15.1.10 is: x L In addition, if there is a moving coil transducer, i.e., a coil moving in a magnetic field which is responsive to velocity because of Lenz's law, then an additional term is added to the right side of 15.1.10 giving: 2 φ + 2 β Ωο φ + Ω ο φ = x - G I L 15.1.11 where I is the current from an external source, and G is a constant. Possible external current sources are: a) pure resistance load -- prospecting type seismometers. b) galvanometer -- a recording device that has its own differential equation of motion. We now want to solve 15.1.10 for φ as a function of frequency, the spectral response, and for t, the time response due to a steady state input and a transient input. 5