Introduction to Shock & Vibration Response Spectra
By Tom Irvine
SECTION 1
Introduction
Figure 1.1. IRVE 2 Launch & Free-Free Beam Analogy

A rocket vehicle behaves as a free-free beam during flight. The vehicle’s body bending modes can be excited by wind gusts, aerodynamic buffeting, thrust offset, maneuvers, etc. The image shows the IRVE 2 launch from Wallops Island. The vehicle is a Black Brant 9, which has a Terrier Mark 70 first stage. Flight accelerometer data showed that the fundamental bending frequency began at about 8 Hz and then swept up to 13 Hz as propellant mass was expelled. Autopilot guidance and control algorithms need to account for the body bending mode to maintain stability.

Figure 1.2. Tuning Fork, First Mode, A4 Note, Fundamental Frequency 440 Hz

The mode shape from a finite element model is shown exaggerated. The two tines undergo an in-plane bending mode, 180 degrees out-of-phase with one another. The stem also participates in this mode, but its displacement is so relatively small that it is not apparent in the figure.

SECTION 2
Some History

The Ancient Greek philosopher Pythagoras (570–495 BC) studied the vibration of stringed instruments and developed a theory of harmony. He first identified that the pitch of a musical note is in inverse proportion to the length of the string that produces it, and that intervals between harmonious sound frequencies form simple numerical ratios.

The ancient Greeks believed that al objects were composed of some combination of the basic elements: earth, water, air, and fire. Aristotle (384–322 BC) attempted an explanation of earthquakes based on natural phenomena. He postulated that winds within the earth whipped up the occasional shaking of the earth's surface. He noted that earthquakes sometimes caused the water to burst forth in what would now be called a tsunami.

Galileo Galilei (1564-1642) performed numerous experiments with oscillating pendulums during the Renaissance. He discovered that the period of swing of a pendulum was independent of its amplitude. He used his own pulse as a time measurement because there were no watches at that time. Christiaan Huygens (1629-1695) successfully built a pendulum clock based upon Galileo’s work.

Robert Hooke (1635-1703) developed the law of linear spring stiffness. This law has been generalized to the elasticity principle that strain in a body is proportional to the applied stress.

Sir Isaac Newton (1643-1727) derived his laws of motion, which are the foundation of mechanical vibration analysis. He published his findings in a book known by its abbreviated title Principia.

Jacob Bernoulli, Daniel Bernoulli, and Leonhard Euler derived the equation of motion for beam vibration circa 1750.

Lord Rayleigh, John William Strutt, published Theory of Sound in two volumes during 1877-1878. Volume I covered harmonic vibrations, systems with one degree of freedom, vibrating systems in general, transverse vibrations of strings, longitudinal and torsional vibrations of bars, vibrations of membranes and plates, curved shells and plates, and electrical vibrations. Volume II covered aerial vibrations, vibrations in tubes, reflection and refraction of plane waves, general equations, theory of resonators, Laplace’s functions and acoustics, spherical sheets of air, vibration of solid bodies, and facts and theories of audition.

The modern field of mechanical vibration analysis has been built upon the foundation of these authors’ works and has further developed as the result of failures and disasters, as well as the need to design, analyze and test structures and component to withstand dynamic environments. Vibration analysis is also important in other areas including semiconductor manufacturing, human exposure, energy harvesting, machine health monitoring, etc.

SECTION 3
Mechanical Failure
Failure & Damage Photo Gallery
Figure 3.1. Turbine Blade Fatigue Failure

A Qantas Boeing 747-400 aircraft was flying from Sydney to Singapore on May 9, 2011 when the crew noticed abnormalities from the aircrafts No. 4 engine during a climb. The indications included an increase in both the exhaust gas temperature and vibration levels. The plane continued to Singapore for a safe landing and disembarkation of the passengers and crew. The Australian Transport Safety Bureau (ATSB) determined that the cause was a broken turbine blade.

Figure 3.2. Washington Monument Crack.

A one-inch wide, four-foot long crack formed in the Washington Monument, near the top of the 555-foot obelisk, due to the Mineral, Virginia earthquake on August 23, 2011.

Historical Failures

Railcar axles were failing under repeated “low level” cyclic stress, in the mid nineteenth century. These stresses puzzled engineers because the levels were much lower than the material yield stress. The Versailles rail accident occurred on May 8, 1842 after the leading locomotive broke an axle, and the carriages behind piled into it and caught fire causing many deaths. This prompted German scientist August Wöhler to develop the S-N curves used in fatigue analysis. The term “fatigue” was chosen to describe the “tired metal” in the axles.

The collapse of the Tacoma Narrows Bridge “Galloping Gertie” was captured on film on November 7, 1940. This failure is often referred to as the classic “resonant vibration” failure. But it was more properly a “self-excited” or “flutter” response, as discussed in Section Error! Reference source not found..

Aerospace and Other Industries

Aerospace has myriad examples of potential vibration problems. Helicopters may undergo “ground resonance” prior to takeoff. Launch vehicles may have pogo oscillations in liquid engine propulsion systems. Solid rocket boosters may have thrust oscillations. Both high performance aircraft and launch vehicles must withstand random vibration due to turbulent boundary layers and shock waves as they accelerated through the transonic velocity and encounter the maximum dynamic pressure condition.

Ships, automobiles, machine tools, buildings, nuclear reactors, and other mechanical systems and structures all have their own vibration concerns and failure modes.

Failure Modes

There are many types of potential failure modes including yielding, buckling, ultimate stress, fatigue, fretting, fastener loosening, relay chatter, and loss of sway space. Engineers must understand these hazards so that the components, systems, and structures may be designed and tested accordingly.

Resonant excitation and is often a factor in these failures. Engineers thus have a responsibility to identify equipment natural frequencies through analysis and testing. The natural frequency is the frequency at which the system would oscillate if it were given an initial displacement and then allowed to vibrate freely. Resonance occurs when the excitation frequency is at or near the system’s natural frequency. Damping values, mode shapes, effective modal mass and other dynamic parameters are also needed for this analysis.

SECTION 4
Machine Health Monitoring
Figure 4.1. Vibration Monitoring for Predictive Maintenance.
(https://solutions.borderstates.com/vibration-sensors-minimize-downtime/)

Machines, pumps, and other equipment with reciprocating or rotating parts may experience vibration due to blade passing frequencies, gear mesh frequencies, magnetostriction motor hum, shaft misalignment, rotating imbalance, etc. In addition, fluid handling machines, like fans and pumps will experience broadband turbulence.

Some vibration is normal. But excessive vibration may cause accelerated wear and premature failure. High vibration levels could also indicate a bearing defect. Equipment manufacturers should provide acceptable limits in terms of amplitude and frequency. The amplitude specification may be in terms of displacement, velocity or acceleration.

Vibration monitoring is thus needed to reduce maintenance costs, extend life, and improve safety. Machine vibration can be measured with accelerometers which are permanently mounted on the machine and monitored continuously with a wireless network. Or a technician may use a handheld device with an accelerometer that can be temporarily mounted to the machine via a magnetic base, as a periodic check.

SECTION 5
Vibration Energy Harvesting

Piezoelectric transducers or electromagnetic induction devices can be used to convert ambient vibration energy to electrical current to charge batteries or to power wireless sensor networks. As an example, consider a cooling pump in a factory. The pump’s vibration could be used to power its own health monitoring sensor.

Figure 5.1. Cantilever Piezoelectric Energy Harvesting Kit
(https://www.mide.com/products/vlt-9001-piezoelectric-energy-harvesting-kit)

Piezoelectric crystals are asymmetric. The asymmetry in the unit cell of the material sets up the mechanism whereby deforming the crystal leads to a small potential difference. The piezoelectric harvesters are typically cantilever beam structures. A mass may be added to the free end of the beam to tune the device to the source’s dominant vibration frequency. Note that steady vibration with a dominant frequency is best suited for harvesting. Outputs of 10 to 50 milliwatts are possible, depending on the vibration amplitude and frequency.

Another potential benefit of an energy harvesting device is that it removes energy from a system, thus providing some damping which can extend the system’s life.

SECTION 6
Human Vibration

There are also biomedical concerns for the case where humans are exposed to vibration. Motion sickness and seasickness may result from vibration exposure in the 0.1 Hz to 0.5 Hz domain. This may occur because the brain has difficulty processing apparently conflicting data from the eyes, ear canals, and other somatosensory organs.

Furthermore, operators of farm equipment, busses, and trains may suffer spinal damage due to long-term exposure. The human spinal column natural frequency is 10 to 12 Hz. Each organ and bodily part has its own natural frequency. Whole body vibration is addressed in ISO 2631 [1]. Hand-arm vibration is another concern for operators of power tools.

SECTION 7
Vibration Excitation Sources
Source Characterization

Engineers must also determine the characteristics of the shock or vibration excitation so that they can properly analyze and test the affected components or structures. Design modifications can then be made to avoid dynamic coupling between the excitation frequencies and the structure’s natural frequencies. The excitation sources can be grouped into four types.

Initial Displacement or Velocity

A common example is a guitar string which is given an initial displacement and then suddenly released. Another is the Pegasus drop transient which is discussed in Section Error! Reference source not found.. An initial velocity example is a drop shock test machine.

Applied Force

Wind, acoustic pressure, and ocean currents impacting structures are all cases of applied force. So is footfall excitation on a pedestrian bridge. Shafts with rotating imbalance also impart an applied force within a machine. Another example is the force from a hammer striking an object. Impulse hammers are used to purposely excite a structure in modal testing to identify the structure’s natural frequencies, damping ratios and mode shapes. The force may also be applied by a small shaker attached to the structure via a stinger rod.

Base Excitation

Seismic excitation is the classic example of base excitation. Another is an automobile traveling over a speed bump or down a washboard road. Aircraft and launch vehicle avionics components are subjected to base excitation from their mounting structures during flight. These components are tested on large shaker tables to verify that they can withstand flight vibration prior to installation into the vehicle.

Self-Excited Vibration

Self-excited vibration is a special case where the alternating force that sustains the motion is created or controlled by the motion itself. This source type is also referred to as negative damping. Airfoil and bridge flutter are two examples in this category. See the Tacoma Narrows Bridge failure in Section Error! Reference source not found.. The pogo oscillation in a launch vehicle with liquid engines is another case.

Source Photo Gallery, Base Excitation
Figure 7.1. Washboard Road, Base Excitation
(https://www.trailerblocks.com/blogs/trailer-blocks-blog/tagged/trailer-tech)

Ripples can form on gravel and dirt roads with dry, granular road materials. This pattern creates an uncomfortable ride for the occupants of traversing vehicles and hazardous driving conditions for vehicles that travel too fast to maintain traction and control. The resulting vibration may also damage suspension components.

Figure 7.2. Loma Prieta Earthquake 1989, Base Excitation

The earthquake caused the Cypress Viaduct to collapse, resulting in 42 deaths. The Viaduct was a raised freeway which was part of the Nimitz freeway in Oakland, which is Interstate 880. The Viaduct had two traffic decks. Resonant vibration caused 50 of the 124 spans of the Viaduct to collapse. The reinforced concrete frames of those spans were mounted on weak soil. As a result, the natural frequency of those spans coincided with the excitation frequency of the earthquake ground motion.

The Viaduct structure thus amplified the ground motion. The spans suffered increasing vertical motion. Cracks formed in the support frames. Finally, the upper roadway collapsed, slamming down on the lower road. The remaining spans which were mounted on firm soil withstood the earthquake.

Figure 7.3. Shaker Table Testing Lateral Base Excitation

A small satellite is mounted to a slip table which is driven by a large electromagnetic shaker. The purpose of the test is to verify that the satellite can withstand the flight vibration that will be imparted by the launch vehicle.

Figure 7.4. Shaker Table Testing, Vertical Base Excitation (courtesy Unholtz-Dickie)

An equipment rack is mounted via wire rope isolators to an expander head which is mounted in turn on an electromagnetic shaker.

Source Photo Gallery, Applied Force
Figure 7.5. Automobile Modal Test, Applied Force
(Image courtesy of the Modal Shop)

A small shaker applies a force excitation to an automobile fender via a stinger rod. The applied force and resulting acceleration at the input point are measured by an impedance head transducer. Response accelerometers may be mounted at various locations on the vehicle.

Figure 7.6. Thumper Truck, Applied Force

A thumper truck is a vehicle-mounted ground impact system which can be used to provide a seismic source to perform both reflection and refraction seismic surveys for oil, natural gas and mineral exploration. A heavy weight is raised by a hoist at the back of the truck and dropped about three meters, to impact the ground. The resulting ground waveforms are measured with geophones. Some thumpers use a technology called "Accelerated Weight Drop" (AWD), where high pressure gas is used to accelerate a heavy weight Hammer (5,000 kg) to hit a base plate coupled to the ground.

Figure 7.7. Space Shuttle Orbiter Ku-Band Antenna Dithering, Applied Force to Orbiter

The Ku-band antenna is the disk below the lower Atlantis decal. Dither is a vibration employed in some mechanical systems to avoid stiction and to ensure smooth motion. Stiction is short for static friction. The antenna was dithered via a command signal at a frequency of 17 Hz to maintain its ability to smoothly search for Tracking and Data Relay Satellite System (TDRSS) satellites. There was a concern that the antenna’s resulting vibration would interfere with sensitive microgravity experiments such as crystal growth.

Figure 7.8. Space Shuttle & Ares I Solid Rocket Boosters

Solid rocket boosters have elongated internal combustion cavities which act as organ pipes. Vortex shedding within the hot exhaust gases drives standing pressure waves inside these cavities. These pressure waves have a fundamental thrust oscillation frequency with integer harmonics. The cavities can be modeled as closed-closed pipes because the nozzle throat diameter is relatively small compared to the cavity diameter. The Space Shuttle boosters’ thrust oscillation frequency was 15 Hz. The Ares I vehicle was projected to have a 12 Hz oscillation, but this vehicle was cancelled prior to what would have been its first flight. Note that the speed of sound in the internal hot exhaust gas is about 3500 ft/sec.

Source Photo Gallery, Self-Excited Vibration
Figure 7.9. Titan II Gemini Vehicle, Pogo Oscillation

Rocket vehicles with liquid engines, such as the Titan II, may experience combustion instability, which causes excessive vibration forces. This is a potential source of self-excited vibration, whereby the elastic vehicle structure and the propulsion system form a closed-loop feedback system.

There are several types of combustion instability vibration effects. The most common effect is “Pogo,” which is similar to Pogo stick motion. In this case, a low frequency oscillation in the combustion chamber, or propellant feed system, excites the longitudinal vibration mode of the entire rocket vehicle, or some other structural mode. This may create a cyclical energy exchange between the vibration mode and the propulsion system oscillation. The problem may also be initiated when a wind gust or some other perturbation excites the vibration mode. This vibration in turn causes an oscillation in the propulsion system, which further excites the vibration.

Placing accumulators in the fuel and oxidant lines to damp out the pressure fluctuations solves this Pogo problem. The accumulator contains a volume of gas that acts like a soft spring to reduce the propellant frequency to well below that of critical structural frequencies. The accumulator volume must be carefully selected to meet this goal.

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