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Zero-length Spring

Zero-length spring

A zero-length spring has a physical length equal to its stretched length. Its force is proportional to its entire length, not just the stretched length, and its force is therefore constant over the range of flexures in which the spring is elastic (that is, it does not follow Hooke's Law). It was invented in 1932 by Lucien LaCoste, and almost immediately applied to the design of instruments with vertical pendulums, such gravimeters and seismographs. Theoretically, with the correct mass, a pendulum using such a spring as a return can have an infinite natural period. Long-period pendulums enable seismometers to sense the slowest, most penetrating waves of distant earthquakes. Zero-length springs also find use in gravimeters, which need them to have linear sense-pendulums. Some door springs, especially for screen doors, are zero-length springs to reduce the energy of a slammed door. Zero-length springs sometimes smooth auto suspensions. Physically, one common form of a practical zero-length spring is a leaf-spring curled almost in a circle, with the ends mounted to flexible restraints. A convenient form is a helical spring whose wire is twisted while it is being wound (common in screen-door springs). Another common design is a torque-spring or bar. Zero-length springs usually require special compliant mountings, sometimes require precise adjustments to enter zero-length mode, and often have a limited range of motion.

Links

- [http://physics.mercer.edu/earthwaves/zero.html Zero Length Springs in Seismographs]

Hooke's law

In physics, Hooke's law of elasticity is an approximation which states that if a spring is elongated by some distance, x, the restoring force exherted by the spring, F, is proportional to x by a constant factor, k. That is, :

F=-kx.

When this holds, we say that the spring is a linear spring. For many applications, a prismatic rod, with length L and cross sectional area A, can be treated as a linear spring. Its extension (strain) is linearly proportional to its tensile stress, σ by a constant factor, the modulus of elasticity, E. Hence, :

\sigma = E \cdot \varepsilon

or :

\Delta L = \frac \times F \times \frac   = \frac \times L \times \sigma.

It is named after the 17^th century physicist Robert Hooke, who initially published it as the anagram ceiiinosssttuv, which he later revealed to mean ut tensio sic vis, or as the extension, the force. This approximation holds for only some materials under certain loading conditions. Materials for which Hooke's law is a useful approximation are known as linear-elastic or "Hookean" materials. Steel exhibits linear-elastic behavior in most engineering applications; Hooke's law is valid for it throughout its elastic range (i.e., for stresses below the yield strength). For some other materials, such as Aluminum, Hooke's law is only valid for a portion of the elastic range. For these materials a proportional limit stress is defined, below which the errors associated with the linear approximation are negligible. Materials such as rubber, for which Hooke's law is never valid, are known as "non-hookean". The stiffness of rubber is not only stress dependent, but is also very sensitive to temperature and loading rate. The graph below shows a stress-strain curve for low-carbon steel. Hooke's law is only valid for the portion of the curve between the origin and the yield point. stress-strain curve
3. Rupture
4. Strain hardening region
5. Necking region.]] Applications of the law include spring operated weighing machines. Originally the law applied only to stretched springs, but subject to physical constraints it also applies to compression springs.

Spring equation

The most commonly encountered form of Hooke's law is probably the spring equation, which relates the force exerted by a spring to the distance it is stretched by a spring constant, k, measured in force per length. :

F=-kx

The negative sign indicates that the force exerted by the spring is in direct opposition to the direction of displacement. It is called a "restoring force", as it tends to restore the system to equilibrium. The potential energy stored in a spring is given by :

U=kx^2

which comes from adding up the energy it takes to incrementally compress the spring. That is, the integral of work over distance. This potential can be visualized as a parabola on the U-x plane. As the spring is stretched in the positive x-direction, the potential energy increases (the same thing happens as the spring is compressed). The corresponding point on the potential energy curve is higher than that corresponding to the equilibrium position (x=0). The tendency for the spring is to therefore decrease its potential energy by returning to its equilibrium (unstretched) position, just as a ball rolls downhill to decrease its gravitational potential energy. If a mass is attached to the end of such a spring and the system is bumped, it will oscilate with a natural frequency (or resonant angular (circular) frequency) of :

\omega_n = \sqrt.

Generalized Hooke's law

When working a with three-dimensional stress state, a 4^th order tensor (c_ijkl) containing 81 elastic coefficients must be defined to link the stress tensor (σ_ij) and the strain tensor (or Green tensor) (ε_kl). :

\sigma_ = \sum_ c_ \cdot \varepsilon_

Due to the symmetry of the stress and strain tensor, only 36 elastic coefficients are independent. As stress is measured in units of pressure and strain is dimensionless, the entries of c_ijkl are also in units of pressure. Generalization for the case of large deformations is provided by models of neo-Hookean solid and Mooney-Rivlin solid.

Zero-length springs

Hooke's law does not apply in some special physical conditions. In 1932 Lucien LaCoste invented the zero-length spring. A zero-length spring has a physical length equal to its stretched length. Its force is proportional to its entire length, not just the stretched length, and its force is therefore constant over the range of flexures in which the spring is elastic (that is, it does not follow Hooke's Law). Theoretically, with the correct mass, a pendulum using such a spring as a return can have an infinite natural period. Long-period pendulums enable seismometers to sense the slowest, most penetrating waves of distant earthquakes. Zero-length springs also find use in gravimeters, which need them to have linear sense-pendulums. Some door springs, especially for screen doors, are zero-length springs to reduce the energy of a slammed door. Zero-length springs sometimes smooth auto suspensions. Physically, one common form of a practical zero-length spring is a leaf-spring curled almost in a circle, with the ends mounted to flexible restraints. A convenient form is a helical spring whose wire is twisted while it is being wound (common in screen-door springs). Another common design is a torque-spring or bar. Zero-length springs usually require special compliant mountings, sometimes require precise adjustments to enter zero-length mode, and often have a limited range of motion.

Links

- [http://www.mssu.edu/seg-vm/bio_lucien_lacoste.html A Biography of Lucien LaCoste, inventor of the zero-length spring]
- [http://physics.mercer.edu/earthwaves/zero.html Zero Length Springs in Seismometers] Category:Continuum mechanics Category:Eponymous laws ko:훅의 법칙 ja:振動運動#フックの法則

Lucien LaCoste

Lucien LaCoste 1908-1995 was a prominent physicist, metrologist. He was coinventor of the modern gravimeter, invented the zero-length spring, and vehicle-mounted gravimeters. He was also co-founder of a prominent company selling gravimetric instruments. LaCoste discovered the zero-length spring in 1932 while performing an assignment in Arnold Romberg's undergraduate physics course. A zero-length spring is a spring supported in such a way that its exerted force is proportional to its length, rather than the distance it is compressed. That is, over at least part of its travel, it does not conform to Hooke's Law of spring compression. The zero-length spring is extremely important to seismometers and gravimeters because it permits the design of vertical pendulums with (theoretically) infinite periods. In practice, periods of a thousand seconds are possible, a hundred-fold increase from other forms of pendulum. Over a short period starting in 1932, the design of these instruments was revolutionized, obsoleting all previous designs. During this period, LaCoste and his physics teacher Arnold Romberg invented the first modern seismographs and gravimeters, using steel and quartz (respectively) zero-length springs. While a graduate student, LaCoste decided to go into business together with Romberg, selling advanced gravimeters to oil-exploration companies. LaCoste's most famous invention is the ship, and aircraft-mounted gravimeter. These revolutionized exploration for minerals by allowing wide-ranging geological surveys. The chief problem that Lacoste defeated was to distinguish the accelerations of the vehicles from the accelerations due to gravity, and measure the minute changes in gravity. Since the accelerations from the vehicle typically are hundreds to thousands of times more forceful than the measured changes, this invention was considered impossible until LaCoste demonstrated it. These inventions give no flavor for LaCoste's fun-loving, often puckish character. These anecdotes were related by one of his many friends, C.R. Dawson. #As a young man, Dr. LaCoste once caused a near-riot in an Austin speakeasy (a prohibition-era illegal bar) by looking up from a book and ordering a glass of milk. #Dr. LaCoste was a fine tennis player. One summer, while he was studying at his family home near Fort Sam Houston in San Antonio, he received a phone call from some friends saying that another friend (later to become one of San Antonio's top surgeons) had just been given a rude and unsporting drubbing in a tennis match at the San Antonio Country Club. LaCoste interrupted his reading, ran the few miles from his house to the country club, handily defeated the offender in straight sets, and ran back home to resume his studies.

Links

- [http://www.agu.org/sci_soc/lacoste.html AGU's LaCoste page]

References

- December 1984 issue of The Leading Edge.
- Eos, December 12, 1995, p. 516. LaCoste, Lucien LaCoste, Lucien LaCoste, Lucien LaCoste, Lucien Category:gravimetry

Seismograph

Seismometer (in Greek seismos = earthquake and metero = measure) are used by seismologists to measure and record seismic waves. By studying seismic waves, geologists can map the interior of the Earth, and measure and locate earthquakes and other ground motions. The term seismograph is usually interchangeable, but seismometer seems to be a more common usage. The seismometer was first invented by Zhang Heng in China in 132AD. Later John Milne invented the horizontal pendulum seismograph at the Imperial College of Engineering in Japan in 1880. This marked the beginning of modern seismology. 1880. This model is a K2 made by Kinemetrics and part of the Pacific Northwest Seismograph Network.]]

Basic principles

Seismometers have: #A frame securely affixed to the earth. The foundation is critical, and often the most expensive part of a seismic station. #An inertial mass suspended in the frame by some method, using springs or gravity to establish a steady-state reference position. #A damper system to prevent long term oscillations in response to an event. #A means of recording the motion of the mass relative to the frame. Early seismometers used optics, or motion-amplifying mechanical linkages, while modern instruments use electronic amplification of signals generated by position or motion sensors. Passing seismic waves move the frame, while the mass tends to stay in a fixed position due to its inertia. The seismometer measures the relative motion between the frame and the suspended mass. Professional seismic observatories usually have instruments measuring three axes, north-south, east-west, and up-down. Seismologists generally prefer a vertical seismograph if only one instrument is available. A professional station is often mounted on bedrock with an uncracked connection to a continental plate. The best mountings may be in deep boreholes, which avoid thermal effects, ground noise and tilting from weather and tides. Amateur, or less exotic instruments are often mounted in insulated enclosures on small buried piers of unreinforced concrete. Reinforcing rods and aggregates would distort the pier as the temperature changes. A site should always be surveyed for ground noise with a temporary installation before pouring the pier and laying conduit.

An early example

The principle can be shown by an early special purpose seismometer. This consisted of a large stationary pendulum, with a stylus on the bottom. As the earth starts to move, the heavy mass of the pendulum has the inertia to stay still in the non-earth frame of reference. The result is that the stylus scratches a pattern corresponding with the earth's movement. This type of strong motion seismometer recorded upon a smoked glass (glass with carbon soot). While not sensitive enough to detect distant earthquakes, this instrument could indicate the direction of the initial pressure waves and thus help find the epicenter of a local earthquake — such instruments were useful in the analysis of the 1906 San Francisco earthquake. Further re-analysis was performed in the 1980s using these early recordings.

Early Designs

After 1880, most seismometers were descended from those developed by the team of John Milne, James Alfred Ewing and Thomas Gray, who worked together in Japan from 1880-1895. These seismometers used damped horizontal pendulums. Later, after World War II, these were adapted into the widely-used Press-Ewing seismometer. Later, professional suites of instruments for the world-wide standard seismographic network had one set of instruments tuned to oscillate at fifteen seconds, and the other at ninety seconds, each set measuring in three directions. Amateurs or observatories with limited means tuned their smaller, less sensitive instruments to ten seconds. The basic damped horizontal pendulum seismometer swings like the gate of a fence. A heavy weight is mounted on the point of a long (from 10 cm to several meters) triangle, hinged at its vertical edge. As the ground moves, the weight stays unmoving, swinging the "gate" on the hinge. The advantage of a horizontal pendulum is that it achieves very low frequencies of oscillation in a compact instrument. The "gate" is slightly tilted, so the weight tends to slowly return to a central position. The pendulum is adjusted (before the damping is installed) to oscillate once per three seconds, or once per thirty seconds. The general-purpose instruments of small stations or amateurs usually oscillate once per ten seconds. A pan of oil is placed under the arm, and a small sheet of metal mounted on the underside of the arm drags in the oil to damp oscillations. The level of oil, position on the arm, and angle and size of sheet is adjusted until the damping is "critical," that is, almost having oscillation. The hinge is very low friction, often torsion wires, so the only friction is the internal friction of the wire. Small seismographs with low proof masses are placed in a vacuum to reduce disturbances from air currents. Zollner described torsionally-suspended horizontal pendulums as early as 1869, but developed them for gravimetry rather than seismometry. Early seismometers had an arrangement of levers on jeweled bearings, to scratch smoked glass or paper. Later, mirrors reflected a light beam to a direct-recording plate or roll of photographic paper. Briefly, some designs returned to mechanical movements to save money. In mid-twentieth-century systems, the light was reflected to a pair of differential electronic photosensors. The recording device in most such machines was paper on a slowly-turning drum.

Improved designs

In 1894, Milne invented a basic, undamped horizontal-pendulum seismometer with a continuous photographic record. He succesfully advocated a system of seismic stations, and the British adopted his seismograph for them. In 1895, von Rebeur Paschwitz in Germany used a tiny, 42 g horizontal pendulum with optic recording to record the first-ever confirmed Japanese earthquake to be recorded in Germany. The expense and fuzziness of photographic seismographs reduced their utility. In 1904 Wiechert of Gottingen, Germany put a 1000 kg mass atop a vertical pendulum and held it upright with weak springs. This gave excellent sensitivity, and permitted a mechanical seismograph with jeweled bearings and conventional paper records to receive distant earthquakes. The inverted pendulum significantly reduces the pendulum length required for a suitably low frequency. This reduces the overall size of the instrument. In 1906, Galitizine produced the first electromagnetic seismograph. A pendulum with a magnet induced current in a coil which then drove a galvanometer. The Omori seismograph used Zollner's suspension on Milne's horizontal pendulum (Omori was a pupil and colleague of Milne in Japan). It was the prototype of the Bosch-Omori seismograph used worldwide in the early 20th century. It uses two torsion wires or (for the vertical seismometer) a pair of springs for its hinge. Basically, one wire pulls down on the side away from the mass, while another pulls up on the side toward the mass. Bosch added damping that Omori omitted. In 1932 Lucien LaCoste invented the zero-length spring. A zero-length spring has a physical length equal to its stretched length. Its force is proportional to its entire length, not just the stretched length, and is therefore constant over a range of flexures (that is, it does not follow Hooke's Law). Theoretically, a pendulum using such a spring can have an infinite natural period. Long-period pendulums enable seismometers to sense the slowest, most penetrating waves of distant earthquakes. WIthin two years, zero-length spring versions of many seismometers were available, and the resonant period of the lowest-frequency seismometers went from 90 seconds to more than 900 seconds. The Wood-Anderson torsion seismometer is one of the most elegant horizontal damped pendulums that was adapted to use zero length springs. A 2 cm pendulum is attached like a flag to the middle of a long, vertical steel torsion wire. A mirror on the pendulum reflects a light beam. A magnet wraps around the pendulum to damp motion by inducing eddy currents in the pendulum. The pendulum and wire are sometimes mounted in an evacuated aluminum pipe with a window to pass the light. This compact, lightweight seismometer is sometimes used with electronic photocells and amplification. A practical amateur design was commissioned by Scientific American for their "Amateur Scientist" feature. Basically, the design is a classic small horizontal pendulum (similar to von Rebeur's). The weight is a large sense coil, moving in the magnetic field of a magnetron magnet (cheaply available from microwave oven repair shops). The damper is a one-megaohm variable resistance across the sense coil. The hinges are very thin sheets of brass, held in clamps. The frame is square aluminum tubing. The device senses velocity rather than position, but requires very little care, is very sensitive with modern electronic amplifiers, and it is easy to construct and tune. A special feature is that the pendulum's frequency and damping can be tested remotely by running a pulse of current through the coil. The strain seismometer by E. Oddone measures the distance between two piers, which changes when a ground-wave passes the instrument. Oddone specifically wanted to check seismic theory with a seismometer that did not use pendulums. The greatest single improvement was the long term drum recorder. A large cylinder is wrapped with paper. The cylinder is rotated by clockwork (or a synchronous electric motor) and, turning on a spiral screw, advances along the axis of rotation. A recording stylus is linked to the proof mass by a series of levers (or uses an electric galvanometer movement), to amplify small relative motions of the mass to drive the stylus. This apparatus collects a recording for an extended period of time (usually a week). Clockwork displaces the recording stylus once per minute to allow time comparisons between charts recorded at different locations. On modern seismometers, two such recorders are coupled to the mass to determine motions in each of two axes.

Modern instruments

Modern instruments use electronic sensors, amplifiers, and recording instruments. Most are broadband, operating on a wide range of frequencies. Some commercially-available research seismometers receive frequencies from 30 Hz (0.03 seconds per cycle) to 1/850 Hz (850 seconds per cycle). Seismometers unavoidably introduce some distortion into the signals they measure, but professionally-designed systems have carefully-characterized frequency transforms. Sensitivities come in three broad ranges: geophones, 50 to 750 V/m; local geologic seismographs, about 1,500 V/m; and teleseismographs, used for world survey, about 20,000 V/m. Instruments come in three main varieties: short period, long period and broad-band. The short and long period measure velocity and are very sensitive, however they 'clip' or go off-scale for ground motion that is strong enough to be felt by people. A 24-bit analog-to-digital conversion channel is commonplace. Practical devices are linear to roughly a part per million. V Delivered seismographs come with two styles of output: analog and digital. Analog seismographs require analog recording equipment, possibly including an analog-to-digital converter. Digital seismographs simply plug in to computers. They present the data in standard digital forms (often "SE2" over ethernet). The modern broad-band seismometer (so called because of the capacity to record a very broad range of frequencies) consists of a small 'proof mass', confined by electrical forces, driven by sophisticated electronics. As the earth moves, the electronics attempt to hold the mass steady through a feedback circuit. The amount of force necessary to achieve this is then recorded. Another type of seismometer is a digital strong-motion seismometer, or accelerograph. This data is essential to understand how an earthquake affects human structures. A strong-motion seismometer measures acceleration. This can be mathematically integrated later to give velocity and position. Strong-motion seismometers are not as sensitive to ground motions as teleseismic instruments but they stay on scale during the strongest seismic shaking. Accelerographs and geophones are often heavy cylindrical magnets with a spring-mounted coil inside. As case moves, the coil tends to stay stationary, so the magnetic field cuts the wires, inducing current in the output wires. They receive frequencies from several hundred hertz down to 4.5 Hz (cheap) to as low as 1 Hz (pretty expensive). Some have electronic damping, a low-budget way to get some of the performance of the closed-loop wide-band geologic seismographs. Strain-beam accelerometers constructed as integrated circuits are too insensitive for geologic seismographs (2002), but are widely used in geophones. Some other sensitive designs measure the current generated by the flow of a non-corrosive ionic fluid through an electret sponge or a conductive fluid through a magnetic field. Today, the most common recorder is a computer with an analog-to-digital converter, a disk drive and an internet connection. Many observatories now use computers. For amateurs, a PC with a sound card and software is adequate, and saves a lot of paper. An algorithm often used to eliminate insignificant observations uses a short-term average and a long term average. When the short term average is statistically significant compared to the long term average, the event is worth recording.

Interconnected seismometers

Seismometers spaced in an array can also be used to precisely locate, in three dimensions, the source of an earthquake, using the time it takes for seismic waves to propagate away from the hypocenter, the initiating point of fault rupture (See also Earthquake location). Interconnected seismometers are also used to detect underground nuclear test explosions. In seismography, an array of seismometers images sub-surface features. The data are reduced to images using algorithms similar to tomography. The data reduction methods resemble those of computer-aided tomographic medical imaging X-ray machines (CAT-scans), or imaging sonars. A world-wide array of seismometers can actually image the interior of the Earth in wave-speed and transmissivity. This type of system uses events such as earthquakes, impact events or nuclear explosions as wave sources. The first efforts at this method used manual data reduction from paper seismograph charts. Modern digital seismograph records are better adapted to direct computer use. With inexpensive seismometer designs and internet access, amateurs and small institutions have even formed a "public seimograph network." (See references). Seismographic systems used for petroleum or other mineral exploration historically used an explosive and a wireline of geophones unrolled behind a truck. Now most short-range systems use "thumpers" that hit the ground, and some small commercial systems have such good digital signal processing that a few sledgehammer strikes provide enough signal for short-distance refractive surveys. Exotic cross or two-dimensional arrays of geophones are sometimes used to perform three-dimensional reflective imaging of subsurface features. Basic linear refractive geomapping software (once a black art) is available off-the-shelf, running on laptop computers, using strings as small as three geophones. Some systems now come in an 18" (0.5 m) plastic field case with a computer, display and printer in the cover! Small, inexpensive seismic imaging is now sufficiently inexpensive that it is used by civil engineers to survey foundation sites, locate bedrock, and find subsurface water.

External links

- [http://www.geophys.uni-stuttgart.de/seismometry/man_html/man2001.html Erhard Wielandt's 'Seismic Sensors and their Calibration']- Current (2002) reference by a widely-consulted expert.
- [http://neic.usgs.gov/neis/seismology/history_seis.html The history of early seismometers]
- [http://www.ifg.tu-clausthal.de/java/seis/sdem_app-e.html A Java code applet demonstrating the operation of a damped-mass seismometer]
- http://neic.usgs.gov/neis/seismology/keeping_track.html
- [http://physics.mercer.edu/earthwaves/zero.html Zero Length Springs in Seismographs]
- [http://www.geonet.org.nz/mrz-drum.html Link to live Seismic Drum at Geonet's Mangatainoka River station in New Zealand]
- [http://psn.quake.net Public seismograph network- many resources for amateurs and underfunded institutions]
- [http://www.eas.slu.edu/People/STMorrissey/index.html A USGS standard seismometer designed by a seismic instrumentation engineer for home construction and use.]
- [http://psn.quake.net/lehman.txt The Lehman amateur seismograph, from Scientific American]- not designed for calibration.
- [http://geohazards.cr.usgs.gov/staffweb/mcnamara/Pubs/Reports/sts2OFR2.0.pdf USGS evaluation of Streckheisen STS-2 Seismometer models]- Streckheisen is a common make of research seismometers Category:Seismology Category:Measuring instruments ja:地震計

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