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Tutorial on Gravitational Pendulum Theory Applied
to Seismic Sensing of Translation and Rotation
by Randall D. Peters
Abstract Following a treatment of the simple pendulum provided in Appendix A,
a rigorous derivation is given first for the response of an idealized rigid compound
pendulum to external accelerations distributed through a broad range of frequencies. It
is afterward shown that the same pendulum can be an effective sensor of rotation, if
the axis is positioned close to the center of mass.
Introduction
When treating pendulum motions involving a noniner- tial (accelerated) reference frame, physicists rarely consider the dynamics of anything other than a simple pendulum. Seismologists are concerned, however, with both instruments more complicated than the simple pendulum and how such instruments behave when their framework experiences accel- eration in the form of either translation or rotation. Thus, I look at the idealized compound pendulum as the simplest approximation to mechanical system dynamics of relevance to seismology. As compared to a simple pendulum, the prop- erties of a compound pendulum can be radically modified according to the location of its axis relative to the center of mass.
Pendulum Theory
The Simple Pendulum
The theory of the simple (mathematical) pendulum is provided in Appendix A.
Theory of an Exemplary Compound Pendulum When an external force is applied to an extended object whose shape is invariant, it generally causes two responses: a linear acceleration of the center of mass and a rotational acceleration around the center of mass. For this system, New- ton’s laws of translation and rotation are applied respectively to each response. The coupled-equation sets obtained from these two forms of the law are then combined to obtain the single equation of motion. This method will be used to ana- lyze the compound pendulum described next. As noted, we are concerned with small displacements, where the drive acceleration is rarely large enough to gener- ate amplitudes in excess of 1 mrad. Thus, the nonlinear in- fluence of the sin θ term is inconsequential; that is, the instrument is nearly isochronous (sin θ ≈ θ and cos θ ≈ 1 ). Figure 1 shows a pendulum similar to various instru- ments of importance in seismology. It is a true pendulum
in the sense that restoration is due to the gravitational field of the Earth at its surface, little g. Some other instruments common in physics and sometimes labeled pendulums do not employ a restore-to-equilibrium torque based on the Earth’s field. For example, restoration in the Michell– Cavendish balance that is used to measure big G (Newtonian universal gravitational constant) is provided by the elastic twist of a fiber (TEL-Atomic, Inc., 2008). It is sometimes called a torsion pendulum. Many seismic instruments are also called pendulums, even though restoration may be mostly provided by a spring. Two commonly employed spring types are the LaCoste zero-length and the astatic. As discussed in Appendix B, the rotation response at low frequencies of a spring-restored oscillator is significantly dif- ferent from that of a gravity-restored pendulum. While the fiber of the Cavendish balance is secured only at the top, other torsion pendulums use a vertical fiber that is also secured at both ends. The best known example from seismology is the Wood–Anderson seismograph, used by Richter to define the original earthquake magnitude scale. By means of an adjustable period, a similar instrument can be configured to operate with large tilt sensitivity (Peters, 1990). As with any long-period mechanical oscillator, the maximum period (and the maximum sensitivity) of the tilt- meter is regulated by the integrity of its fiber spring. Accept- able stability against spring creep is difficult to achieve when the period is greater than about 30 sec (de Silva, 2007). A common seismology instrument for which the challenge to long-period stability is well known is the garden-gate hori- zontal pendulum. Rodgers (1968) recognized its potential for a variety of measurements. Although the instrument in Figure 1 is idealized in the form of a two-element compound pendulum, it nevertheless is useful for illustrating a variety of important properties. The primary idealization is the assumption of a rigid structure. For reasons of material creep, and the placement of mass M 1 above the axis, real pendulums of this type experience structural deformation. In the absence of integrity sufficient
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Bulletin of the Seismological Society of America, Vol. 99, No. 2B, pp. – , May 2009, doi: 10.1785/
to maintain an essentially rigid structure, the center of mass of M 1 migrates slowly toward ever lower states of potential energy. Even though the rate of this deformation may be low, the rod will, through secondary creep, continue to bend about the axis. Upon initial setup and until primary creep has run its course, the equilibrium position of such an instru- ment is not fixed, not even for the highest Q materials used in fabrication. The defect structures responsible for creep also result in hysteretic internal-friction damping. Ignore for the moment the force F (^) e. As the pendulum oscillates between positive and negative displacements, the three forces R, M 1 g, and M 2 g are responsible for a periodic flexural strain of the pen- dulum arm and pivot. We will not consider here the detailed physics whereby this strain lags behind an associated stress in the rod. Peters (2005a) shows how the Q of the system is naturally proportional to the square of the eigenfrequency. This proportionality results from the flexural interplay be- tween the restoring M 2 g and the destoring M 1 g. This prop- erty also allows one to tune the oscillator to a long period, by increasing M 1 or δ, and thus causing the center of mass to approach the axis. Subsequently, I show that this method can be used to convert the pendulum from an instrument more sensitive to translational acceleration into one that is more sensitive to rotation about O.
Newton’s second law requires that the vector sum of the forces external to the pendulum must be equal to the total mass M M 1 M 2 times the acceleration of the center of mass acm. Additionally, the vector sum of external torques acting on the center of mass is equal to the moment of inertia about that point, Icm , times the angular acceleration, d^2 θ=dt^2. The axis at O is stationary in the accelerated frame of reference in which the instrument is located. This axis is ac- celerating in the negative x direction because of the horizon- tal force F (^) e acting through the axis. This external force and the reaction force R are input from the ground via the case that supports the pendulum. The force R balances the weight of the pendulum when it is at equilibrium with F (^) e 0. Be- cause we are concerned with motions for which θ ≪ 1 , we quickly identify the first of several expressions used to de- velop the equation of motion: M 1 M 2 g ≈ R. For various calculations, a convenient reference point is the top of the uniform rod of mass M 2 and length L. Unlike the rod, M 1 that is placed above the axis is dense enough to be approximated as a point mass. For the purpose of torque calculations, the total mass of the pendulum is concentrated at the center of mass, whose position below the reference position at the top of the rod is indicated as d (^) c in Figure 1. Newton’s law for translational acceleration of the center of mass and small rotation about the center of mass yields
F e ≈ Macm; M M 1 M 2 (1)
and
d c dF e Rθ ≈ Icm θ: (2)
An additional relationship required to obtain the equation of motion without damping involves the transformation from the center of mass frame to the axis frame,
acm ≈ dc dθ at; (3)
where at aaxis is the acceleration of the ground respon- sible for the pendulum’s motion. Combining equations (1) to (3) with M 1 M 2 g R one obtains
Icm Mdc d^2 θ dc dMgθ ≈ d (^) c dMat:
Because of the parallel-axis theorem (Becker, 1954), the term in brackets multiplying the angular acceleration is recog- nized to be the moment of inertia I about axis O. Up to this point I have ignored the effects of frictional damping. In the conventional manner a linear damping term is added to equation (4) to obtain the following linear ap- proximation for the equation of motion of this compound pendulum:
Figure 1. A compound pendulum used to illustrate some prop-
erties of interest to seismometry. Force vectors are shown that act on both the pendulum and its axis. c.m stands for center of mass; c.p. stands for center of percussion.
2 R. D. Peters
gθ 0 a 0 ;ground
^
ω^20 ω^20 ω^2 ^2 ω^20 ω^2 =Q^2
p (^) ;
ϕ tan^1
ω 0 ω Qω^20 ω^2
; ω^20
g Leff
The phase term ϕ in equation (12) is plotted in Appendix A for two different values of Q. The pendulum and the drive are in phase when the drive frequency is much lower than the natural frequency. At the high-frequency extreme, the pen- dulum lags behind the drive by π. The magnitude of the transfer function shown in equa- tion (12) is appropriate to an instrument that measures the angular displacement of the pendulum. For an instrument that measures translational displacement of a point on the pendulum at some fixed distance from the axis, the equation is readily modified. For example, placement at the bottom of the rod yields
T (^) f;accel
d L
ω^20 ω^20 ω^2 ^2 ω^20 ω^2 =Q^2
p : (13)
Displacement Transfer Function The displacement transfer function of a compound pendulum is generated from the relationship between accel- eration and displacement, that is, a ω^2 A:
T (^) f;displ
L
Leff
d L
ω^2 ω^20 ω^2 ^2 ω^20 ω^2 =Q^2
p ; (14)
where the displacement implied by equation (14) is the hori- zontal displacement of the bottom end of the compound pendulum relative to the instrument case. The transfer func- tions expressed by equations (13) and (14) are plotted in Fig- ure 4 for the special case of the compound rod pendulum.
Pendulum Measurement of Tilt and Rotation
The response of the compound pendulum to rotation is treated in Appendix B. This appendix provides quantitative support for the claim that the pendulum of Figure 1 can be made sensitive to rotation while simultaneously insensitive to translational acceleration. As seen from the upper right- hand plot of Figure 3, moving the axis closer to the center of mass can readily cause the pendulum’s effective length to become more than 30 times greater than the actual length of
Figure 2. Properties of a meter-stick (rod) simple-compound pendulum. The graphs were generated with equation (10) using M 1 0
and L 1.
4 R. D. Peters
the pendulum. In the process, equation (14) shows that the displacement response derived from translational accelera- tion diminishes. Because the rotational response is indepen- dent of the effective length, the relative response can become
quite large. Figure 5 shows a 50-fold reduction in transla- tional sensitivity for this case in which the center of mass is located 1 mm below the axis.
Unconventional and Exotic Pendulums
Unconventional or exotic pendulums, such as the ones described in Appendix D, create additional complexities and useful phenomena.
Conclusions
Depending on geometry, compound pendulums can display many different types of responses. The single case treated previously, an idealized compound pendulum, pro- vides a quantitative example. In addition to their diversity, gravity-restored pendulums can be remarkably simple, com- pared to many instruments widely used in seismology. I have spent many years conducting pendulum research. In the course of two decades of intense study, I have come to the following conclusion—the classical pendulum should not be viewed as a relic. With modern digital technology,
Figure 3. Properties of the two-component compound pendulum of Figure 1 for the parameters as indicated. For no value of d=L is the
center of percussion located on the body of the pendulum.
Figure 4. Acceleration and displacement transfer functions
based on equations (13) and (14) using d 0 , M 1 0 , and Q 0 : 7.
Tutorial on Gravitational Pendulum Theory Applied to Seismic Sensing of Translation and Rotation 5
Appendix A
The Simple Pendulum
The simple pendulum (sometimes called a mathematical pendulum) is a convenient starting point for the considera- tion of pendulums in general. A simple pendulum comprises an inextensible, massless string of length L that supports a point mass m on one of its ends (Fig. A1). The other end of this string is the pendulum axis, a fixed stationary point whenever the instrument is located in an inertial coordinate frame. In addition to the only external force of significance (mg in Fig. A1), a tension force T is also shown. It has no component with which to influence the pendulum’s motion; however, it must at all times balance the component of the bob weight along the direction of the string (unit vector r). As the pendulum swings, this tension force varies periodi- cally and would cause a variation in L if the string were ex- tensible. The reaction to this tension force, acting at the top of the string, also would cause periodic sway in the case of any real pendulum. This sway can lengthen the period of os- cillation and also give rise to hysteretic damping. This idealized instrument is not achievable in practice, but the system is pedagogically useful and constitutes a use- ful benchmark against which real systems can be compared and numerically evaluated. First we discuss the homogeneous equation of a sim- ple pendulum’s motion and then what happens to the dy- namics of the pendulum when its axis experiences external accelerations. Newton’s second law is the basis for treating both the simple pendulum and the other instruments considered later. This law is best known to students in the special-case form F ma, where the vector F is the sum of all external forces acting on scalar point mass m experiencing linear accelera- tion vector a as the result of F. The acceleration thus calcu-
lated is valid only if m is constant. Newton actually provided this law in its general form, which says that the time rate of change of the vector momentum of an object is equal to the net external vector force acting on the object. This reduces to F ma for constant m. For systems like a symmetric pendulum that experiences rotation, the angular vector acceleration α is related to the net vector torque τ by way of Newton’s second law of rota- tion in the form τ Iα. The scalar moment of inertia I is specified relative to the axis, which in Figure A1 is the origin O. Because the string is massless, I mL^2. The magnitude of the angular acceleration is the second time derivative of θ or d^2 θ=dt^2. In the equations that follow, Newton’s time de- rivative convention is followed, namely that one dot over the variable designates the first time derivative and two dots, the second time derivative. As with F ma, τ Iα is a special case. The general form of his rotational law says that the time rate of change of vector angular momentum equals the net external applied vector torque. Because it is not an extended body in the usual sense, a simple pendulum is very easily treated using Newton’s sec- ond law in the form of rotation. To begin with, following initialization at θ 0 ≠ 0 , let the pendulum be swinging in free decay in an inertial coordinate frame (Fig. A1). For positive displacement θ corresponding to a specific time (the θ direc- tion is along the z axis), the torque due to gravity at that in- stant is given by
τ L^r × mg Iθ ^z; (A1)
where the caret (hat) over a variable indicates that it is a unit vector. After evaluating the vector cross product, equa- tion (A1) reduces to the following expression in terms of magnitudes:
mL^2 θ mgL sin θ 0 : (A2)
The equation of motion of a simple pendulum is in general nonlinear because of the sine term in equation (A2). Unlike archtypical chaotic pendulum motion (Peters, 2007), in seis- mology applications θ 0 ≪ 1 rad is nearly always an accept- able approximation. It should be noted that the pendulum of Figure A1 cannot exhibit chaos because the string supports only tension; this simplification disallows winding modes (θ > π) that are part of chaotic motion. We have assumed that the restoring force is due to a uni- form Earth gravitational field of magnitude g ≈ 9 : 8 m=sec^2 , and that Coriolis acceleration due to the rotating Earth is neg- ligible. If energy is supplied to a simple pendulum to offset its damping, and if the pendulum is located somewhere other than the equator, then the plane of its oscillation is steadily altered because of the Earth’s rotation. Aczel (2004) provides a fascinating account of Leon Foucault’s invention of this famous pendulum. For pendulums of a type common to seis- mology, physical constraints against axis rotation make the
Figure A1. Geometry of the simple pendulum. Coriolis influence inconsequential for most purposes. For a
Tutorial on Gravitational Pendulum Theory Applied to Seismic Sensing of Translation and Rotation 7
pendulum whose effective length is an appreciable fraction of the Earth radius, such as the Schuler pendulum (Aki and Richards, 2002), the uniform-field approximation is not valid. Although not shown in Figure A1, further assume that the only other external force acting on the pendulum is one of frictional retardation, viscous damping. With all these approximations, the equation of motion for the nondriven simple pendulum takes on the simple linear form
θ ωo Q
θ_ ω^2 oθ 0 ; ω^2 o g L
: (A3)
This simple-harmonic-oscillator-with-viscous-damping equation is the heart of linear-approximate models of the pendulum and many other mechanical oscillators. It is ap- plicable to the pendulum only to the extent that sin θ ≈ θ is acceptable and also that its loss of oscillatory energy derives from damping friction that is proportional to the first power of the velocity. With damping included in the equation of motion
τ Lc_θ mgL sin θ mL^2 θ; (A4)
where cθ_ is the viscous friction force, assumed proportional to the angular velocity through the coefficient c, acting on the bob of mass m at a distance L from the axis. It should be noted that c is not always constant (Peters, 2004). The adjective “mathematical” is appropriate to describe a simple pendulum because real instruments are never as simple as the assumptions made concerning its structure. In addition, a damping term proportional to the velocity does not fully describe the behavior of real oscillators. The first person to introduce viscous damping to the simple harmonic oscillator may have been physicist Hendrik Anton Lorentz (1853–1928). Neither Lorentz nor George Gabriel Stokes (1819–1903) treated the viscous model as loosely as has been common in recent years (Peters, 2005a). The quality factor is defined by Q 2 πE=ΔE, where E is the energy of oscillation and ΔE is the energy lost per cycle due to the damping. One can easily estimate Q to a few percent from an exponential free decay as follows. After ini- tializing the motion at a given amplitude, count the number of oscillations required for the amplitude to decay to 1 =e ≈ 0 : 368 of the initial value. Q is then obtained by multiplying this number by π. For linear viscous damping, a simple relationship exists between Q and the damping (decay) coefficient β, used with the exponential to describe the turning points of the motion through expβt:
Q
ωo 2 β
ωo
mL c
: (A5)
It should be noted that if β were a constant, then Q would be proportional to the natural frequency f 0 through ω 0 2 πf 0. As demonstrated by Streckeisen, circa 1960, (E. Wielandt,
personal comm., 2000) with a LaCoste-spring vertical seis- mometer, the Q of practical instruments is not proportional to f 0. At least for instruments configured to operate with a long-natural period, the proportionality is one involving f^20. For these systems, the damping derives from internal friction in spring and pivot materials, and the best simple model is nonlinear (Peters, 2005a). Although it involves the velocity only by way of algebraic sign, this form of damping is non- linear, even so resulting in exponential free decay. Although the coefficient β may be reasonably called a damping coeffi- cient, it is not proper to call it a damping constant, because it is not constant but varies with frequency. Swinging in a fluid such as air, a real pendulum experi- ences two drag forces, one acting on the bob and the other acting on the string. This problem was first treated analyti- cally by Stokes (1850), originator of the drag-force law f 6 πηRv for a small sphere of radius R falling in a liquid at terminal velocity v in a fluid of viscosity η. However, this equation does not in general allow an accurate theoretical estimate of Q based simply on the fluid’s viscosity. Stokes’ law can be used only when working with very small parti- cles. In particular, for a macroscopic pendulum,the Reynolds number is generally too large to allow its use. In most cases the air influence must include a quadratic velocity term as well as the linear term assumed for equation (A3) (Nelson and Olsson, 1986). In other words, even air damping is not necessarily linear. In the case of extended rigid bodies undergoing periodic flexure during oscillation, several damping mechanisms gen- erally are present. Internal friction in pivot and structure usu- ally contributes significantly, sometimes dominantly (Peters, 2004). The net quality factor describing the decay is given by
QNet
Q 1
Q 2
; (A6)
where it is seen that the damping mechanism with the low- est quality factor dominates. Only for those mechanisms that give rise to exponential decay is Q independent of the pendulum’s amplitude. With quadratic-in-velocity fluid damping, the amplitude trend of Q is opposite to that of Coulomb friction. Unlike hysteretic and viscous damping, neither of these nonlinear mechanisms yields an exponential free decay. The subscript zero used with ω in equation (A3) is a natural consequence of the mathematical solution to the dif- ferential equation. A subscript corresponding to the eigen- frequency of the pendulum in the absence of damping (Q → ∞) is used to distinguish this value from the red- shifted frequency when there is damping; that is, ω ω^20 β^2 ^1 =^2. This redshift is negligible except, perhaps where the pendulum damping is near critical (Q ≈ 0 : 5 ). When internal-friction hysteretic damping is the dominant source of damping, the redshift has no meaning because there is no mechanism to cause it (Peters, 2005a).
8 R. D. Peters
that tilt influence on a pendulum (due, for example, to a sur- face seismic wave) is negligible compared to the influence of the wave’s acceleration components unless the frequency of the wave is very low. For a surface Rayleigh wave having amplitude A, such that
yx; t A sinkx ωt; k
2 π λ
; ω
2 π T
(B1)
then the maximum of the spatial gradient in a linear elastic solid is given by
∂y ∂x
max
kA 2 πA λ
jθjtilt; (B2)
the amplitude of the pendulum response to the harmonic tilt- ing that occurs when the wave passes. For ω less than the pendulum’s natural frequency ωo, one finds from equation (5) in the body of this article that the acceleration response of the pendulum is
jθaccelj
ω^2 A g
: (B3)
Thus, the ratio of tilt-to-acceleration response is
θtilt θaccel
^
gT 2 πv
; (B4)
where v is the phase speed of the surface wave. From equa- tion (B4) it is seen that the period of the wave must exceed 2 πv=g for the tilt influence to become greater than the accel- eration influence. The crossover period is actually about 50% smaller than the value calculated because the vertical ampli- tude of particle motion in a Rayleigh wave is roughly 50% greater than the horizontal amplitude. For a wave speed of 2500 m=sec, the crossover period is about 800 sec, beyond which tilt dominates instrument response. It should be noted that significant deviations in the di- rection of the Earth’s gravitational field occur at frequencies below about 1 mHz as the result of eigenmode oscillations (normal modes). Although the magnitude variations in g are exceedingly small, the direction changes in g are readily measured with a pendulum acting as a tiltmeter and using a displacement sensor. As will be seen in the discussion of sensor choice, a velocity sensor is not well suited to such measurements. Although the next section is concerned with pendulum measurement of rotation, it should be understood that tilt from Earth normal modes is a special case of rotation in which the direction of the Earth’s field changes for an in- strument located along a line of nodes. This matter is impor- tant because of the need to better understand the mechanisms of Earth hum, which is comprised of such modes. Along with my student M. H. Kwon, around 1990 I accidentally ob- served persistent oscillation components corresponding to the lowest eigenmode frequencies of the Earth. These ob-
servations were made with a pendulum similar to that of Peters (1990), which was designed for surface-physics re- search. The results were documented by Kwon (Kwon and Peters, 1995).
Pendulum Sensing of Rotation
There is a significant difference in the rotational equa- tion of motion for a pendulum that is restored by a spring, as opposed to a pendulum that is restored by the Earth’s gravi- tational field. Although spring-restored instruments could be used in some cases for measurements in all three axes needed to completely specify the Earth's motion, there is a signifi- cant advantage at low frequencies to using gravity-restored instruments for the horizontal axes; measuring rotation around the local vertical axis requires a spring-restored pen- dulum. The equations developed by Graizer (2006) are re- stricted in applicability to a spring-restored instrument. Here we consider a gravity-restored pendulum having an effective length Leff. The homogeneous part (left-hand side) of the equation of motion is identical to equation (5) in the body of this article. Excitation of the pendulum relative to an inertial coordinate frame can only arise from work done by the damping force. In the absence of damping, the pendulum would remain stationary in the inertial frame, while the case holding the instrument oscillated around a horizontal axis. Responding to relative motion between the pendulum and case, the output from the sensor would be the negative of the case motion θ in that inertial frame. With damping, there is motion of both the case and pendulum, and their differ- ence ϕ is what is measured by the sensor. This motion is governed by
ϕ^ ωo Q
ϕ_ ω^2 oϕ α ω^2 oα; ω^2 o g Leff
; (B5)
where the Earth rotation variable α is the time varying ori- entation of the case relative to a horizontal axis in the inertial frame. For a pendulum that is spring restored, the right- most term of the right-hand side of equation (B5) is missing. It should also be noted (Graizer, 2006), that triaxial instru- ments of this type generally show cross coupling between orthogonal axes. As before, we can readily predict from equation (B5) the response for both the low- and high- frequency limits. Unlike equation (5) in the body of this article, these limiting cases prove to be identical:
ϕ α; for ω ≪ ωo and also for ω ≫ ωo : (B6)
The full transfer function, obtained as before using the method of Steinmetz phasors, is given by
ϕo αo
^
ω^2 o ω^2 ω^2 o ω^2 ^2 ω^2 oω^2 =Q^2
p ;
phase tan^1
ωoω Qω^2 o ω^2
(B7)
10 R. D. Peters
The place where the transfer function differs from ϕ α is in the vicinity of the pendulum’s resonance frequency, where a decline in sensor output occurs. For high Q, the near-resonance steady-state decline of the response is nar- row. As Q decreases, the width of the region of declination gets larger, as illustrated in Figure B1.
Response, Rotation Compared to Acceleration
The pendulum of Figure 1 in the body of this article can be made sensitive to rotation and simultaneously insensitive to acceleration. As seen from the upper right-hand plot of Figure 3 in the body of this article, moving the axis close to the center of mass causes the pendulum’s effective length to be significantly greater than the actual length of the pen- dulum. In the process, it is seen from equation (14) in the body of this article that the displacement response derived from linear acceleration drops. Because the rotational re- sponse is independent of the pendulum’s effective length, its sensitivity remains unchanged while the acceleration re- sponse is dramatically reduced.
Choice of Sensor
A truly broadband sensor for measuring all Earth mo- tion is a practical impossibility. Force-balance seismographs, such as those built by Gunar Streckeisen, come as close to this ideal as any. Because these instruments employ velocity sensing, there is a commensurate loss in very low-frequency sensitivity, as can be understood from Figure B2. As can be seen in that figure, if all other things are equal a velocity sen- sor will outperform a position sensor when detecting Earth motions with characteristic frequencies higher than the natu- ral frequency of the seismic instrument. A position sensor is superior for detecting Earth motions with frequencies lower than the natural frequency of the instrument.
Appendix C
Some Pendulum Characteristics
Kater Pendulum
The rod-only pendulum is useful for recognizing a his- torically significant property important to precision measure- ments of g, Earth gravitational acceleration, according to the method of Kater (Peters, 1999). When restricted to measure- ments involving a single axis of rotation, the accuracy with which the effective length of a pendulum can be measured is limited. This limitation is removed by using a pendulum whose motion is measured for two different, conjugate axes. The pendulum oscillates with the same period when a given first axis is replaced by a second parallel axis passing through the center of percussion calculated from the position of the first axis. The system is equivalent to a simple pendulum of length equal to the distance between the two axes, which can be measured very accurately. The method was used (taking an average of six Kater pendulums) to measure the absolute reference for the Earth's field in Potsdam, Germany, in 1906 (National Geodetic Survey, 1984).
Figure B1. Rotational pendulum transfer function magnitude
(upper graph) and phase angle in radians (lower graph).
Figure B2. Instrument-self-noise PSD plots of importance to
the choice of a sensor. The curves were generated using equa- tion (12) in the body of this article.
Tutorial on Gravitational Pendulum Theory Applied to Seismic Sensing of Translation and Rotation 11
quency of the pendulum, the intersection of this line-pair moves downward. The term percussion implies a short lived impulse; as the driving period lengthens, the percussion point is no longer a meaningful reference for the inertial center of rotational mo- tion. The center of oscillation remains meaningful but is no longer located at the center of percussion. As the drive fre- quency goes toward zero, the center of oscillation moves to- ward infinity. A (false) assumption that some part of the inertial mass of a seismometer remains stationary in space as the instrument case moves is acceptable when the instru- ment is functioning as a vibrometer (i.e., drive frequencies above the natural frequency), but it is not true for the low- frequency extreme of the pendulum’s response.
Appendix D
Unconventional and Exotic Pendulums
Unconventional Pendulums
Rotation Sensor Two very different, unconventional gravitational com- pound pendulums are described in this appendix. Illustrated in Figure D1 is a rotation sensor capable of operating over a broad frequency range. Whereas the pendulum of Figure 1 in the body of this article is a rigid vertical-at-equilibrium structure that oscillates about a horizontal axis, the rigid beam of the pendulum illustrated in Figure D1 is horizontal- at-equilibrium. I believe there are three advantages to this system, although they have not all been experimentally veri- fied. First, the influence of creep is expected to be less for the horizontal configuration as compared to the vertical one. Creep in the members of the long-period vertical pendulum alters the equilibrium position, whereas creep of the boom in the horizontal pendulum alters the period. Secular change in the equilibrium position decreases the maximum possible sensitivity of an instrument’s detector, unless force feedback is employed. Period change is inconsequential except as it increases responsiveness to translational acceleration. Be- cause the instrument is designed to minimize this response, the creep influence is of secondary rather than primary im- portance as in the case of the vertical pendulum. The second advantage involves air currents. A thermal gradient within the container that holds the instrument can
cause convective flows, and the resulting circulation is ex- pected to have greater influence on the vertical pendulum than on the horizontal pendulum. The final advantage results from the simple means by which one mounts a pair of displacement detectors on oppo- site ends of the beam. Operating differentially in phase op- position, they yield a better signal-to-noise ratio ( SNR) than is possible from a single detector. The greater the length of the beam, the greater will be the sensitivity of the instrument.
Some Other Inertial Rotation Sensors Parts from a pair of STS-1 horizontal seismometers were used to build a rotation sensor with mechanical properties similar to the rotation pendulums I have described. The in- strument tested by Hutt et al. (2004) differs, however, by containing springs; the lower flexures are placed under ten- sion by means of a large brass counterweight. Morrissey (2000) also built a beam-balance broadband tiltmeter with similar mechanical properties. His instrument used a pair of lead masses mounted on opposite ends of a horizontal aluminum bar suspended at the center with a flexural axis. It used force-feedback balancing and he claimed a sensitivity of 120 mV=μrad, with a resolution of better than 0.1 nrad, using linear variable differential trans- formers ( LVDT ). Wielandt (2002) notes that a capacitive sen- sor is superior to an LVDT for the reason of the granular nature of ferromagnetism of the latter. Compared to a capac- itive sensor of (singly) differential type which is customary, there is an SNR advantage to using a pair of (doubly fully) differential capacitive sensors with the pendulum of Fig- ure D1, one such fully-differential detector for each end, with the pair operating in phase opposition.
Microseism Detector Although conventional seismographs always operate with damping near 0.7 or 0.8, an undamped vertical pendu- lum with merit is next described. Valuable information con- cerning hurricanes (via microseisms) could be gleaned from a large network of inexpensive pendulums operating with a reasonably high Q. Lengthening the period by moving the center of mass close to the axis has the following advantage. The sensitivity of the pendulum to frequencies other than resonance is significantly decreased as shown in Figure D2. This is especially important for high-frequency noises that derive from localized, cultural disturbances. This would al- low the SNR of the electronics employed to be relaxed with- out a significant loss of microseism detectability (Fig. D2). The transient response of this high-Q pendulum would disallow meaningful analysis of time domain data; however, analyses in the frequency domain, using power spectral den- sity ( PSD) plots or cumulative spectral power ( CSP ) plots would not be similarly limited (Peters, 2008a,b). Knowledge of the Q, used to correct the spectra in generating the PSD, allows for the generation of a wealth of useful information.
Figure D1. A horizontally oriented gravitational pendulum that
is sensitive to rotation but insensitive to translation. The axis of ro- tation is perpendicular to the plane of the figure, and the arrows show directions of the end-arm motion.
Tutorial on Gravitational Pendulum Theory Applied to Seismic Sensing of Translation and Rotation 13
Exotic Pendulums
Appreciation for the physics of pendulum dynamics is improved by consideration of unconventional instru- ments such as the Schuler pendulum and a gravity-gradient- stabilized satellite that behaves like a pendulum
Schuler Pendulum Tuned to a period of 84 min, the Schuler pendulum is important to navigation. That period matches near-earth sat- ellite periods and is described in detail elsewhere (Aki and Richards, 2002).
Gravity-Gradient-Stabilized-Satellite Pendulum Because of the inverse-square dependence of the Earth’s field, a cigar-shaped satellite behaves as a very long-period pendulum. The radial gradient of the field is responsible for a separation of the center of mass and the center of gravity, so that a torque exists when the long axis deviates from the ver-
tical. The method selected by De Moraes and Da Silva (1990) to treat this problem is based on Hamiltonian dy- namics. The following analysis uses Newton’s second law and his expression for the gravitational field. Assume a body in the shape of the rod-only pendulum discussed in Appendix C, having mass m and length L. The center of mass is located at the center of the rod, and the center of gravity is calculated using Z (^) L
0
GM (^) e dm R y^2
mGM (^) e R yg^2
; dm
m L
dy; (D1)
where G 6 : 67 × 10 ^11 N m^2 =kg^2 is the Newtonian con- stant, M (^) e 6 × 1024 kg is the mass of the Earth, R 6 : 4 × 106 m is the radius of the Earth, y is measured from the bottom of the rod, and y (^) g is the center of gravity. The integral of equation (D1) yields to first order the expression
yg ≈
L
L^2
2 R
: (D2)
Application of Newton’s second law yields
I (^) c θ mg
L^2
2 R
θ ≈ 0 ; I (^) c
mL^2 12
; (D3)
from which it is seen that ω^20 6 g=R. With g ≈ 9 : 2 m=sec^2 at an altitude of 200 km, the pendulum’s effective length is R= 6 , yielding a period of about 36 min. Oscillation in the absence of imposed damping would prevent the satellite from accomplishing mission objectives. A method that has been used to attenuate oscillation employs hysteretic damp- ing derived from electrical currents that are powered by solar cells.
Physics Department Mercer University 1400 Coleman Avenue Macon, Georgia 31207
Manuscript received 1 July 2008
Figure D2. The displacement response of a compound pendu-
lum tuned to resonate with a Q of 40 at a period of 4 s. The response of a near-critically damped simple pendulum of comparable physi- cal length is shown for comparison (dashed curve).
14 R. D. Peters
