1. Welcome to THE NESC GN&C TDT WEBCAST Series
Today’s Moderator: Neil Dennehy (NESC) Today’s Speaker: Scott Starin (GSFC)
Fundamentals of
Spacecraft Attitude Control
Next Webcast: Fundamentals of Aircraft Flight Control”, John Burken, (DFRC),
on 19 September 2012

3. Outline Acknowledgements Introduction to Spacecraft Attitude Control
Attitude Control System Design
Design Process
Requirements Development
Disturbance Torques and Momentum Management
Attitude Determination
Stabilization Methods
Controller Design and Implementation
Errors and Error Budgets
Flexible-Body Dynamics and Jitter
Examples
Hubble
Solar Dynamics Observatory
ST-5
Closing and Q&A
Reference Charts

4. Acknowledgements
Jim O’Donnell, David Ward, David Mangus, and Lou Hallock, all of GSFC, have contributed over the years to a University of Maryland lecture on attitude control, from which I have borrowed extensively
Another source of material is the 4th edition of Larson & Wertz’s Spacecraft Mission Analysis & Design, which in this edition covers much more material and has been titled Spacecraft Mission Engineering: The New SMAD.
Thanks to Alice Liu (GSFC) and Carl Blaurock (Nightsky Systems, Inc.) for providing slides for discussion of the relationship between jitter and attitude control

5. What is Attitude?
Attitude is the three-dimensional orientation of a vehicle with respect to a reference frame
A telescope on a hill can be tilted higher or lower in
Elevation and
Azimuth
The Earth telescope has two degrees of freedom in pointing and is described fully by a
Unit vector
A telescope in space may rotate around its own boresight
Allows imaging hardware to be aligned optimally with the target
Space missions must, in general, consider 3 DOF in their designs

6. Attitude Control Goals and Reference Frames
Most, but not all, spacecraft missions have some need for pointing one or more pieces of hardware at one or more targets.
Solar arrays must be illuminated, at least partially & for some period of time
Focused antennas must point at targets on the ground or at tracking satellites
Scientific instruments must point at their targets to some level of accuracy
Spinning spacecraft often require the spin axis to lie along a desired direction
A typical goal for a spacecraft attitude control system (ACS) is to point a critical instrument at a target and to point another piece of hardware as close to another target as possible
Choosing the proper reference frame for a given mission is important for mission planning and operations
Communications satellites point their antenna arrays at nadir nearly 100% of their mission, often slowly rotating about nadir to allow their solar arrays to track the Sun: Local Vertical/Local Horizontal (LVLH) reference frame
Astrophysics satellites often maintain a constant attitude relative to the stars, periodically performing attitude maneuvers, called slews, to select new targets: inertial reference frame, such as J2000.0

7. Contrasting Spacecraft and Aircraft ControlY X Z
What is needed to control a vehicle in free-fall outside a dense atmosphere, in contrast to propelling a vehicle through a dense fluid?
Disturbance torques on spacecraft are orders of magnitude less than on aircraft
Free-fall is a neutrally stable condition, whereas air flight is not
Air provides a ready sink for mechanical energy (high damping); spacecraft generally have very low damping of flexible motion
In all, space vehicles can achieve very good pointing accuracy with very low-bandwidth (usu. < 0.1 Hz) controllers
The lack of damping and low bandwidth mean spacecraft are susceptible to low-amplitude, high-frequency oscillations (“jitter”) that can blur images

8. Diagram of a Complete ACS
Image credit: Space Mission Engineering, Microcosm, Inc

9. Top-Level Steps in ACS Design
1. Define control modes
Define or derive system-level requirements by control mode
2. Quantify disturbance environment
3. Select type of spacecraft control by attitude control mode
4. Select and size ACS hardware
5. Define determination and control algorithms
6. Iterate and document

10. Typical Attitude Control Mode Requirements
General pointing requirements.
Define what the “target” is, what needs to be pointed there. Some missions require specific scan patterns or spin rates for their instruments, and these should be known early in formulation.
Pointing accuracy
An allocation of the end-to-end payload pointing budget, which also includes mechanical, thermal, and measurement effects.
Pointing stability (also known as jitter)
Typically derived from instrument resolution requirements, since higher frequency spacecraft motion will blur imagery and degrade resolution.
Attitude determination (or knowledge)
Also an allocation of a larger Observatory budget. Knowledge derives from various mission requirements, including requirements to correctly describe the location of an observation in a sky survey, or as an allocation of a higher-level pointing requirement.
Agility/maneuver requirements
Missions often require special calibration maneuvers or the ability to slew between targets at a certain rate. These requirements tend to drive the controller method and actuators selected.
Acquisition requirements
On missions with solar cells for power generation, the ACS typically has a specific time limit to pointing those arrays at the Sun before battery power runs out. It is important to understand the launch vehicle separation conditions to correctly design for all possible initial conditions.
Safing and reliability requirements
What level of reliability/redundancy should be employed for recovery from anomalies, and what should the ACS do in response to an anomaly? What level of stability is necessary for the reliability needed?

11. Disturbance Environment
External disturbance torques (four main sources)
Solar radiation pressure
Atmospheric drag, or ram pressure
Magnetic torque due to residual dipole
Gravity gradient
May be secular or cyclical
Internal disturbances
Mechanisms: e.g. solar array drive, moving instruments
Thermal snap
Misalignments and quantization in actuators
Propellant motion (i.e. slosh)
Other minor sources: e.g. astronauts (e.g. ISS), outgassing from polymers, differential IR emission

12. Solar Radiation Pressure
Constant in given heliocentric orbit
Atmospheric Drag
Magnetic Field
Gravity Gradient

13. Effects of External Disturbance Torques
External disturbance torques add momentum to the satellite
Cyclic:
Average to zero over a circular orbit
Analogous to conservative forces
Aerodynamic, gravity gradient, magnetic torques for inertially-pointed
Magnetic, solar pressure, some gravity gradient torques for nadir-pointed
Secular:
Accumulate over the orbit
Analogous to non-conservative forces
Solar pressure torques for inertially-pointed
Aerodynamic, some gravity gradient torques for nadir-pointed

14. Momentum storage and attitude control Actuator commanded momentum
Momentum Management
Disturbance torques add momentum to the satellite
Momentum management
Actuators
Remove (dump) secular momentum change
Magnetic coils
Thrusters
Often a separate control loop run concurrently (especially with magnetics)
Store cyclic momentum change
Reaction wheels must be sized appropriately
Momentum
storage
S/C attitude
Momentum storage and attitude control
Disturbance torque
in inertial frame
Disturbance momentum
buildup
in inertial frame
Momentum
accumulation
Dump
momentum
Actuator commanded momentum

15. Sensor Capability Summary
High
update
rate
Low

16. Actuators Magnetic torquers Wires wound around irons cores
Provide torque by generating a magnetic moment that interacts with Earth’s magnetic field
Only useful in low- to mid-altitude orbits
Reaction wheels and control moment gyroscopes (CMGs)
provide smooth, constant disturbance rejection in any orbit
can be sensitive to very low temperatures, and so are often avoided for deep space missions
Reaction engines (i.e. thrusters)
Useful in practically any flight regime
May double as orbit control actuators (viz. AOCS)
Require propellant, which is expendable and may limit a mission life
May cause vibrations or other rapid motion that is unsuitable for very fine pointing

17. Attitude Determination
The process of developing or deriving an estimate of spacecraft attitude from sensor measurement data
Attitude is expressed as the orientation of the spacecraft reference frame with respect to a standard frame (inertial, orbital, sun-fixed, etc.)
Attitude Knowledge
The level of certainty associated with the attitude determination process
Magnitude of the uncertainty relies on sensor limitations, data quantization, sampling time, and numerical processing
Real-time versus post-processed
On-board systems increasingly rely on powerful computers for filtering
Ground systems generally provide a post-processed “definitive” attitude solution
Single-axis attitude estimation useful for determining orientation of the spin axis of spinning spacecraft
Three-axis attitude estimation is more common
For example, Magnetospheric Multi-Scale (MMS) will be 4 spinners, each with three-axis estimation

18. Error Budgeting
Used as a tool for attitude control (lower frequency), jitter (higher frequency) and attitude determination design to properly allocate errors
Standard practice for combining errors
Accept 3-sigma as a worst case limit
Break budget into three frequency bins
Fixed
Seasonal varying
Short term
Errors in each bin are combined using a Root-Sum-Squared (RSS) operation
Bin errors are combined linearly (added)
Margin goals are typically set up and used to reduce risk through the life-cycle of a mission
Attitude determination errors commonly budgeted
Sensor performance (possibly as updated by estimator), time tag/propagation errors, uncorrected misalignments, seasonal variation (thermal misalignments)
Attitude control/jitter errors commonly budgeted
Reaction wheel disturbances (static/dynamic imbalance, torque noise), torque quantization, attitude determination error, uncorrected low frequency disturbances (sinusoidal disturbance torques), mechanisms, thermal snap

19. Active Versus Passive Control
Passive Stabilization
S/C attitude is maintained without using actuators
Gravity gradient (2 axes stable)
Magnetic stabilization (2 axes stable)
Spin stabilized (2 axes stable)
Passive nutation dampers
Stabilization schemes can be used for both rate and position control
Active Control
Computer or analog circuit commands actuators for attitude control
Three-axis control (active control on all axes)
Also supplements passive stabilization
Thruster based control of spin stabilized spacecraft
Momentum wheel or dual-spin bias control
Wheel or magnetic control of yaw for GG stabilized S/C
Preferred
orientation
Earth Center

20. Controller Design & Analysis Summary
Controller Design Requirements
Control system must be stable
In addition to absolute stability, control system must have a reasonable relative stability (gain and phase margins, damping)
The speed of response must be reasonably fast and the control system must be capable of reducing errors to zero or to some small tolerable value
Controller Design Steps
Given spacecraft inertias (the plant), choose appropriate sensor(s) and actuator(s)
Obtain mathematical models of the plant, sensor(s), and actuator(s)
Using the models and analysis, design a controller such that the closed-loop system will satisfy the given specifications
Simulate the model on a computer to test the behavior of the resulting system in response to various signals and disturbances
Construct a prototype physical system
Controller Analysis Methods
Transient response analysis: the determination of the responses of a plant to command inputs and disturbance inputs
Steady-state analysis: the determination of the response after the transient response has disappeared

21. Attitude Control Algorithms
Attitude dynamics are nonlinear, but in most applications easily linearized, so control can often be effected using linear control techniques
Linear control makes analysis of the discrete system simpler
If rotations are about an eigenaxis, only one control loop may be needed
Otherwise, each of the three axes may need a separate control loop, working either in tandem or alternating their commands
Proportional-Integral-Derivative control, these days usually implemented in software, is often used for zero-momentum systems
Because disturbances are small and variable, sometimes sufficient accuracy is achieved without the Integral term.
Nonlinear control methods, such as sliding mode and feedback passivation, are sometimes used
Control of spinning or momentum biased systems
Control of thruster-based systems may be simplified with nonlinear control
As with other control problems, performance must be balanced against stability

22. Effects of Jitter
Sensitive instruments have requirements for attitude stability to avoid image blurring
Jitter, or attitude dynamics at a higher frequency than the controller bandwidth, must be carefully analyzed before launch since there is no way to actively damp it
Onboard mechanisms are the usual source of the energy of jitter, but the frequency content is highly dependent on the flexible modes of the spacecraft structure
The image shown here shows jitter observed in an on-orbit test of SDO
The jitter is driven mainly by the high-gain antenna gimbal steps, so the software prevents gimbal movement (pink) just before images will be taken (blue) to allow the motion to damp

23. Examples Hubble Space Telescope (11 metric tons)
Three-axis stabilized, zero momentum system
Accuracy: 7 milliarcseconds (34 nanoradians)
Sensors: Fine Guidance Sensor (custom interferometric star cameras)
Actuators: Reaction wheels (no thrusters)
Solar Dynamics Observatory (3 metric tons)
Three-axis stabilized, near-zero-momentum
Accuracy: 10 microradians
Sensors: Guide telescopes, star cameras, and digital Sun sensor
Actuators: Reaction wheels, thrusters
ST-5 (3 times 25 kg)
Spin-stabilized
Accuracy: 1.5 degrees (spin-axis accuracy)
Sensors: Magnetometer, spinning Sun sensor
Actuators: Experimental micro-thrusters, nutation damper (passive)

24. Some Useful References
Hughes, Peter C., Spacecraft Attitude Dynamics, 2004
Kaplan, Marshall H., Modern Spacecraft Dynamics and Control, 1976
Thomson, William Tyrrell, Introduction to Space Dynamics, 1986
Wertz, James R (ed.), Spacecraft Attitude Determination and Control, 1986
Wertz, Everett & Puschell (eds.), Space Mission Engineering: The New SMAD, Note that this is the 4th edition of Space Mission Analysis and Design, previous editions of which are also good references.
Wie, Bong, Space Vehicle Dynamics and Control, 2008

25. Attending TODAy’s WeBCAST
Thank you for
Attending TODAy’s WeBCAST
ANY QUESTIONS?

26. Upcoming NESC GN&C TDT Webcasts (as of July 2012)
“Fundamentals of Aircraft Flight Control”, John Burken, (DFRC), on 19 September 2012
“Fundamentals of Adaptive Control”, Irene Gregory (LaRC), on 21 November 2012
“Fundamentals of Spacecraft Attitude Determination”, John Crassidis (Univ. of Buffalo), on 16 January 2013
“Fundamentals of Kalman Filtering and Estimation”, Chris Dsouza (JSC), on 20 March 2013
18 July 2012 Update

27. Additional Reference Charts Reference Frames & ACS Hardware
UMD Space Design Course, ACS Lecture: February 23, 2011

28. Earth Centered Inertial, ECI
X-Axis points toward Vernal Equinox
Y-Axis 90° east in equatorial plane
Z-Axis through the North Pole
Not truly inertial
Equinox moves
Plane of the equator moves
True of Date, TOD
Inertial frame at a fixed epoch
References the true equator and true equinox of date
ECI J2000.0
True of Date frame referenced to the year 2000 epoch (January 1st 12 noon)
X, Y Z

29. Local Vertical Local Horizontal, LVLH
Y-Axis along negative orbit plane normal (orbit plane normal = momentum vector direction)
Z-Axis toward nadir (Earth center)
X-Axis completes the triad
Approximately in direction of motion
Spacecraft centered frame
Not inertial
Moves with spacecraft center of mass
Maintains orientation with respect to the Earth
a.k.a., Orbital Reference Frame

30. Spacecraft-Fixed Fixed to S/C
Typically defined as LVLH for zero attitude error (nadir and zenith pointers)
Two flavors
Origin at center of mass
Might not be geometrically fixed
Propellant expulsion
Articulating components
Used in equations of motion
Geometrically fixed to bus structure
Used as component orientation reference

31. Magnetometer N Converts measured magnetic field into voltages
Provides vector measurement of the B-field
Most common technology used is Fluxgate
Two identical cores with oppositely wound coils
One is excited with alternating current to magnetically saturate the core, which induces saturation on other core
Presence of external field makes the second core saturate at different time than first, and the asymmetry in the alternating pattern can be used to determine B-field
Magnetometer uses
Estimate the S/C attitude in three axes
Compare to geomagnetic model
Feedback attitude error
Accuracy of 0.5º - 10º, limited by accuracy of geomagnetic field model
Estimate S/C rate in two axes
S/C rate is related to B-dot
Feedback B-dot to damp rates
B-field measurement used to determine magnetic torque rod current given controller torque command N

32. Coarse Sun Sensors (CSS)
Photo-diode that converts optical input from the Sun into electrical current
Passive, very reliable
Output proportional to cosine of angle between sensor boresight and sun line
Current highest when Sun is directly along boresight (perpendicular)
Current diminishes as Sun vector moves from boresight
Apertures limit the field-of-view (FOV) of the sensor
FOV is a cone
Half-angle varies widely
Baffles added to limit FOV
Used for sun presence and coarse pointing
~5 accuracy, primarily limited by albedo in LEO
Photo-Diode 
Sun
boresight

33. Fine Sun Sensors Summary
Sun presence and angle outputs
Angle between Sun line and sensor normal when sun is in FOV of sun presence detector
Output signal is an encoded, discrete function of sun angle
Light from the sun illuminates a pattern of slits
Slits are divided into a series of rows with a photocell under each row
Photocell voltage is proportional to the cosine of the Sun angle
First row is used as a threshold setting with respect to using data from the measurement rows
First row slit is half the width of the other slits
Bits are judged to be on if intensity is greater than the threshold bit output
If a bit is at least half illuminated, it is judged to be on
Sign bit defines the side of the sensor the Sun is on
Gray code is used for Sun angle measurement
Converted to binary and then angle
Gray code is equidistant, i.e. only one bit changes for each unit distance
Without fine bits, resolution is 0.5 (Sun angular diameter is 0.53)
Fine bits are interpolated to provide higher resolution (up to 0.05)
Adcole, Ithaco provide Gray code sun sensors, others have developed optical sun sensors

34. ITHACO 2-Axis Digital Sun Sensor
Fine Sun Sensor (FSS)
ITHACO 2-Axis Digital Sun Sensor
(Mask)

35. Gray Code versus Binary Code
For Gray code, if the MSB transition lags,
the readout changes from 3 (i.e., 010) to 4 (i.e., 110)
For binary, if the MSB transition lags,
the readout changes from 3 (i.e., 011) to 0 (i.e., 000) to 4 (i.e, 100)
Example of
Gray code
superiority
for encoders

36. Earth Sensors
Used to determine the satellite orientation relative to the Earth
In LEO, the Earth covers up to 40% of the sky
Hard edge of horizon is difficult to detect
Radiated energy from the Earth decreases gradually
Presence of atmosphere (~70 km)
Variations between illuminated and shaded regions
Limits sensor accuracy to 0.05º º range
Sensors measure the horizon in the infrared range
Energy emitted in the IR spectrum from all parts of the Earth is more homogeneous
Defines the horizon more sharply
Distinguishes horizon regardless of viewpoint (day or night)
m (CO2) spectrum is the most suitable for attitude determination
Can be coupled directly into fine attitude control for Earth-Pointers
Horizon crossing sensors
Mounted on a spinning S/C or spinning component on the S/C to sweep detector field of view (FOV) across the horizon
Detect times radiance detector output crosses a threshold
Pitch error determined by comparing “center” time to expected time
Roll error determined by comparing chord length to expected chord length at spacecraft altitude
Static Earth sensors
Earth horizon is maintained in detector FOV
Measures of horizon location in detector FOV are used to determine attitude
Pitch and roll errors are determined by differential measurements of opposing detectors

37. Body Mounted Horizon Crossing Sensor
Used on a spinning S/C
Derive Earth width from crossing times
Use Earth width to derive nadir angle
Use nadir angle, Sun angle as measured by a Sun sensor, and ephemeris data (Sun and Earth) to determine direction of spin axis
S/C Spin Axis
Transition from space to Earth
AOS
Transition from Earth to space
LOS
S/C spin axis along
the negative orbit plane normal
(i.e., along Y-axis of LVLH)

38. Scanning Horizon-Crossing Sensors
Used on non-spinning S/C
Sensor FOV is deflected so its bore-sight rotates over a cone
When the detector crosses into and out of the Earth the temperature difference between Earth and space is sensed
Optical encoder measures the phase angle between conical Earth crossings and a reference
Use measurements to derive pitch and roll attitude information
Another sensor is added (e.g., sun sensor) if yaw is to be measured
If a single scanner is used, altitude must be known
If two scanners are used, attitude determination is independent of altitude

39. Infrared Static Earth Sensor
Static determination of the Earth’s horizon inside the instrument’s FOV
Optical system projects the Earth horizon onto radiance detectors
Filter limits the observed spectral band to the CO2 range
Lens focuses the Earth image onto thermopile surfaces
Difference in the output between opposite thermopiles provides a measurement of attitude error (differential mode)
Reliable since there are typically no moving parts
Thermopile FOV
Tuned to a narrow range of orbits and altitudes
As altitude increases, the diameter of the Earth as seen by the sensor decreases
Thermopiles will no longer cover the Earth image
Sensor can no longer detect the Earth
As altitude decreases, the diameter of the Earth as seen by the sensor increases
All thermopiles will cover the Earth image
Sensor loses ability to measure attitude

40. Star Trackers
Used to measure spacecraft attitude with respect to the celestial sphere
Often uses a charge-coupled device (CCD) to image the starfield
Position and magnitude of bright objects on the CCD are referenced to a star catalog
If a match is made the star is identified and tracked
Typical Outputs
Attitude quaternion with respect to inertial space
Star identification performed by star tracker
Individual star horizontal and vertical components and magnitude
Star ID must be done using the spacecraft processor
Requires use of an on-board star catalog
Sometimes rate information
Sensor Components
Sun shade
Optical system
Sensing element or detector (CCD)
Electronic signal processing unit
Thermoelectric or Peltier CCD Cooler

41. Star Trackers continued
Basic Types
Gimbaled star tracker searches for stars via mechanical motion of gimbal
Fixed Head Star Tracker scans electrically until it locates a star and then tracks it
Accuracy ranges from < 1 arcsecond to ~45 arcseconds
Limiting factors vary for different units, but include:
Optical distortion (low spatial frequency noise)
Centroiding accuracy (high spatial frequency noise)
Accuracy of star catalog (particularly in high accuracy sensors)
Spacecraft velocity relative to sensed stars causes apparent position shift in those stars (aka velocity aberration)
Velocity aberration must be accounted for by spacecraft processing, since star trackers do not include a data interface to describe spacecraft position/velocity
Potential errors if velocity aberration left unaccounted for:
Spacecraft velocity about earth < 5 arc-sec
Earth velocity about sun < 20 arc-sec
Solar System wrt Galaxy < 10 arc-sec per century (star catalog driven)
Fine Error Sensors a special, high accuracy class of star tracker
Shares optics with instrument to eliminate misalignment errors
Often uses a small FOV to maximize accuracy across the FOV

42. Rate Sensors Rate gyros Used to measure spacecraft angular rates
Integrate output to determine the spacecraft attitude
Typically dominates the “high bandwidth” control loop required for thruster and acquisition operations
“Fills” in between position reference sensors
Rate integrating gyros
Measure angular displacements for a fixed time interval
Postprocessing required to determine angular rate
Types of rate sensors
Mechanical gyro
Hemispherical Resonator Gyro (HRG)
Optical gyros
Interferometric Fiber-Optic Gyro (IFOG)
Ring Laser Gyro (RLG)

43. Mechanical Gyro
When the spacecraft rotates about the gyro input axis, the gimbal supporting the gyro attempts to precess relative to the gyro about the output axis
Gyro noise comes from ball bearings and electronics
Gyro gimbal motion is inhibited or nulled
Viscous damping and spring restraint
Angular excursion is proportional to the spacecraft rate
Electromagnetic torque rebalance
Nulling current is proportional to the spacecraft rate
Dynamically tuned gyro (DTG) is a unique family of mechanical gyros
Two DOF, electrically rebalanced RIG
Rotor suspension is dynamically tuned to frequency of spin motor
Result is greatly attenuated noise from motor and bearings
Allows designer to use more robust motor/bearing system, eliminates exotic damping fluids, and increases reliability without harming performance
DTG’s used on SWAS, TRACE, TRMM, XTE, WMAP

44. Hemispherical Resonator Gyro (HRG)
Quartz crystal shell oscillated at specific amplitude and frequency
Angular rate sensor
Determines changes in angular orientation of the spacecraft by measuring force to rebalance standing wave pattern on quartz hemispherical resonator
Rate integrating sensor
Angle change measured by nodal displacement on rim
Standing wave established on shell and shell is rotated about its axis
Oscillating mass elements experience Coriolis forces that cause standing wave to precess w.r.t the shell
Precession angle is a constant fraction of the shell rotation angle

45. Interferometric Fiber-Optic Gyro (IFOG)
Two beams, one CW and one CCW traveling around a circle of radius (R) enclosed in a fiber of length (L), will arrive back at a moving reference point moving at an angular rate () with a phase difference due to rotation
Send two light beams (CW and CCW) into a coil of fiber and collect signals on a detector
With no rotation of the coil, the two signals add in phase and produce maximum signal power at the detector
As the coil rotates, the signals interfere and the detector power decreases
The detector power is a function of the phase difference between the two beams and related to the rotation rate

46. Ring Laser Gyro (RLG)
Two beams, one CW and one CCW traveling around optical cavity
Standing wave pattern is inertially fixed
Optical detector senses the intensity of the light at a point in the standing wave
Interpret changes in the measured output intensity as rotation of the case relative to the standing wave
Wave can “stick” to cavity due to imperfections: dithering corrects

47. Accelerometers
Correlate a change in acceleration to a change in voltage
When case is accelerated the mass tends to remain stationary
Results in a relative movement of the mass with respect to the case
Springs cause the mass to stop moving with respect to the case
Displacement of the mass with respect to the case is proportional to the acceleration of the case
F = ma = kx
a = acceleration m = mass k = spring constant x = spring displacement x m
k/2
k/2

48. Pendulous Accelerometer
Consists of a mass suspended from a calibrated leaf spring in a manner similar to a pendulum.
The mass of the accelerometer is enclosed within a case that is filled with a damping liquid, which helps keep the pendulum from oscillating.
The accelerometer is mounted so that acceleration in only the desired geometrical plane is detected.
The amplitude of the swing is proportional to the amplitude of the acceleration
The output of the device indicates acceleration direction and amplitude
Output is within the limits of the equipment and is limited by physical stops

49. Piezoelectric Accelerometers
Relies on piezoelectric effect of crystal to generate an electrical output proportional to the applied acceleration
Piezoelectric effect produces an opposed accumulation of charged particles on the crystal
Charge is proportional to applied force or stress
Stress on crystal occurs as a result of the seismic mass imposing a force on the crystal
Accumulated charge is proportional to pressure or acceleration
Over a specified frequency range the structure approximately obeys Newton’s Law, F = ma