Understanding product fragility
A reader about motion, shock metrics, modal analysis and survival decisions
Joseph P. McFadden Sr. | November 1, 2026 | R11
Selecting a useful metric begins with how and where a component can fail. We then connect the measured exposure to local response and tested capability. This reader follows the 52-slide presentation How Fragile Are the Parts You Make and can be read on its own. It explains the physical meaning of the metrics, applies them to a lens adhesive, board-to-board solder and an LCD, and introduces the analysis and learning resources in Mechanical Dynamics Workbench, also referred to as MDSP.
How fragile are the parts you make
Joe McFadden here. I would like to present a conversation about how fragile the parts we make really are. Our overall question is simple to ask: will this component survive its environment and still do its job? Answering it takes more than a large acceleration number or a passed test.
The thoughts and teaching approach come from my years of engineering experience. ChatGPT and Claude helped me organize that experience into something we can share and question, and Grok contributed technical review of the explanations and calculations against current references. Gemini also contributed to the review of clarity and flow. We use that review alongside independent calculation checks and engineering judgment. The responsibility for the judgment remains mine. I am not selling anything.
I am addressing what I call Engineering Mind Blindness: the things we overlook when a familiar number stops us from asking another question.
You will meet the Holistic Analyst and Brain Buddy along the way. Their job is to help you picture what happens. We will poke a system, follow the message through it, and listen before we decide what the numbers mean.
A poke might be a tap, a drop, or a vibration. You will hear names such as displacement, velocity, acceleration, jerk, strain, pseudo velocity, and the shock response spectrum, or S R S. You do not need to understand them all right now. We will give each one a physical job, and later a flexible ruler held over the edge of a table will let us watch several of them at once. Supporting references are listed at the end of this reader and included in the PowerPoint.
A route from motion to a decision
You may be an analyst staring at a noisy explicit simulation. You may be an engineer trying to understand an accelerometer record, or a student with a ruler and a curious mind. You are welcome here.
First, we will picture displacement, velocity, acceleration, jerk, and strain. We will give each number a physical job. Then we will build a shock response spectrum and explain pseudo velocity. Finally, we will connect the calculated response to a failure mechanism and the evidence needed for a survival decision.
I want you to feel what the measurement describes before you have to remember its equation. A formula becomes much easier to understand when you can see the object moving in your mind. When we reach the ruler experiment, I will explain the steps and what to look for. You can put this reader down and try it yourself.
A small part carries someone's work
Why spend so much effort on a small bracket or connector? Because people rely on the capability it supports. A scanner that stops reading, a camera that loses alignment, or a connector that opens for an instant can interrupt work even when the housing looks fine.
A small bracket or connection can also change what reaches a critical component. The input we apply at the outside of the product travels through materials, geometry, and joints before it arrives, and the condition of that path can change during testing.
This is why I want us to ask about the entire system. The table, fixture, mounting screws, housing, and internal support all participate. The response at the component may differ greatly from the response at the point where we introduced the loading. Over the next few sections we will give this picture a name.
A rugged scanner is a system of internal listeners. Supports and connections alter the message reaching the scan engine. Illustration for presentation section 3.
A passed test has a scope
Imagine two cameras. One passes a drop test. The other passes the same test, but its bracket almost touches the housing. Both receive a pass, although their remaining clearance may be very different.
Fragility describes susceptibility to a particular failure under specified exposure and conditions. Margin compares capability with demand. They are related ideas, but they are not interchangeable. A pass is useful evidence within the conditions of that test. It does not automatically tell us how much additional loading the product can tolerate.
Even matching units is only a beginning. Two values in millimeters might refer to motion at different locations. Two values in G might describe different directions and pulse durations. Before calculating a margin, match the physical quantity, failure mechanism, location, and operating condition. Then ask how uncertain both the demand and the capability are.
Two shocks ask different questions
Which would you rather apply to your product: fifteen hundred G for two tenths of a millisecond, or fifty G for eleven milliseconds? The first number sounds alarming. The second sounds modest. But let us listen to the whole sentence, including its duration.
Picture a very brief sharp tap and a longer push. The tap can be intense while it lasts. The longer push has more time to change the motion of a slowly responding part. Neither description alone establishes which event will damage your product.
Our teaching inputs are smooth half-sine pulses, starting and ending at zero. They are calculated examples built for this conversation, not measurements from a particular camera; the Workbench can generate the same pulses, and the closing exercise asks you to. We will come back to these two pulses when we build a spectrum. Later, a second pair built from a four-foot drop will carry a worked example. Their different heights and widths give us a useful way to see why a peak number cannot answer every question.
The answer depends on the listener
Your system, whether it is a scan engine or a complete mobile computer, is a mosaic of internal components and subsystems. A lens assembly, circuit board, display, bracket, and connector each has its own natural behavior when poked. That behavior depends on its mass, stiffness, damping, and the way it is supported and connected. A component may have several ways of moving, and its connections allow it to influence its neighbors.
I like to describe this system as a collection of listeners. Our poke, or input, is a message. As that message travels through the housing, fasteners, supports, and joints, it gets edited along the way. Its strength, timing, duration, and frequency content can change. The message that reaches an internal circuit board may therefore be quite different from the one we introduced at the outside of the product.
How each listener responds to its edited message depends on both its natural behavior and its state at that moment. Is it already moving? Are its connections tight, or has something loosened? Has a gap closed, or has earlier loading caused damage? These conditions can change the response and can also change the message passed onward through the system.
We seldom know the natural behavior of every internal component at the start of an investigation. So we imagine a broad family of simple mechanical listeners, each a mass on a spring with damping, from listeners that respond slowly to listeners that respond quickly, and we replay the same input to all of them. That is the idea behind a shock response spectrum, and the plot is one. It compares our two shocks across that family using a quantity called pseudo velocity. Why introduce that metric? Because a large input peak does not tell us how much an internal listener deforms. Pseudo velocity is a useful way to compare that deformation response and, with the right structural connection, the stress demand it creates. Actual stress in a particular component still needs its geometry, mounting, and mode shapes.
I am introducing it now so you know where we are going. Once we have built the spectrum and met pseudo velocity, we will come back to this plot and read it together, including why the curves cross and what that crossing does not tell us about survival. [2]
Calculated half-sine inputs. Five percent damping. The response curves cross near 74 Hz; this compares modeled demand, not product survival.
A mode is a pattern of motion
Place a ruler partly over the edge of a table. Hold the supported end firmly, gently deflect the free tip, and release it. Watch the free end and listen. The ruler bends back and forth in a recognizable pattern. That pattern is a way the structure naturally vibrates: a mode.
A mode shape tells us how different points move relative to one another in that pattern. Near the held end, motion is small. Near the free tip, it is larger. The natural frequency tells us the rhythm, measured in cycles each second. The drawing of a mode shape is usually scaled for visibility. It does not, by itself, tell us the actual amplitude.
A structure has many modes. A small poke may excite several. Which ones matter depends on where and how we apply the loading and where we listen. The first bending mode gives us a simple place to start.
Conceptual illustration: the Holistic Analyst and Brain Buddy explore the motion of a clamped ruler.
Damping changes how motion settles
In the real world, a ruler will not vibrate forever after we bend it and let go. Damping dissipates mechanical energy, so the motion gradually dies away. We will leave a deeper discussion of energy for another day and concentrate on what we can watch.
With underdamping, a released system crosses its resting position and oscillates while the peaks shrink. At critical damping, an ideal linear mass-spring-damper released from displacement with zero initial speed returns without oscillation, as quickly as that nonoscillatory family permits. With more damping, the overdamped return is slower.
Critical damping does not mean an abrupt stop at the resting position. It describes a particular balance between damping, mass, and stiffness in this mathematical model. Our spectrum calculations use five percent of critical damping. That is underdamped. A stated damping value makes the calculation reproducible, but it does not establish the damping of every real component.
Displacement means a change in position
Imagine a camera suspended inside a housing by a flexible bracket. Put one tiny mark on the camera and another on the housing. Watch both from the laboratory floor. The motion of each mark relative to that fixed reference is its absolute displacement.
Now imagine riding along with the housing. What you see is the camera moving relative to the housing. That difference is relative displacement. If both move five millimeters together, each has moved relative to the room, but the distance between them has not changed.
Displacement is a position change, not the accumulated distance traveled. A camera can move away and return to its starting position, leaving zero final displacement after considerable motion. When we care about a gap closing, we need the peak change in separation at that gap, including any contribution from rotation. We cannot choose a convenient point elsewhere and assume it represents the contact location.
z = x − y
Camera and housing displacement share a fixed bench reference. Equal 5 mm displacements give zero relative change. The clear optical window completes the housing illustration for section 9.
Relative travel tells us whether things touch
Picture a board beneath a housing rib. At rest, the gap is one millimeter. During a shock, the board bends toward the rib by eight tenths of a millimeter relative to it. In this simple illustration, two tenths remain.
That is an understandable use of displacement: will these two surfaces meet? No material has to yield for contact to cause a problem. The contact can create a secondary shock, disturb a connector, or damage a component on the board.
The apparent spare gap is not automatically a certified design margin. Manufacturing tolerances, assembly position, temperature, local rotation, and uncertainty in the prediction can consume it. Also, once contact occurs, the linear no-contact model changes. We must account for the new interaction. A good first result is therefore a conditional statement: under these assumptions, this relative travel stays within this specified gap.
Relative board approach reduces the illustrated clearance from 1.0 mm to 0.2 mm. Tolerances, rotation, contact, and uncertainty still matter. Illustration for section 10.
Velocity describes how position changes
Watch the ruler again. At the deepest bend, its tip pauses for an instant before reversing direction. Near its resting position, it moves quickly. The place of greatest displacement is not the place of greatest speed during simple free vibration.
Velocity tells us how quickly position changes and in which direction. A car traveling steadily can have a large velocity while its acceleration is nearly zero. A ruler at its turning point can have zero instantaneous velocity while a restoring force is already accelerating it back.
We must also specify the reference. The camera can have an absolute velocity relative to the room and a different velocity relative to its moving housing. Neither is automatically the pseudo velocity we will calculate later. Keeping these names separate prevents a plausible mental picture from becoming the wrong mathematical quantity.
Velocity change measures the accumulated push
Imagine gently pushing a cart for a while, then giving it a harder push for a much shorter time. Which changes its velocity more? You need to know both how strongly its velocity changes at each instant and how long that change continues.
On an acceleration-versus-time plot, the signed area gives velocity change. Positive and negative portions can cancel. The units show why: meters per second squared multiplied by seconds becomes meters per second.
For a half-sine pulse, the area is two times the peak acceleration times duration, divided by pi. That gives our short pulse about one point eight seven meters per second and our long pulse about three point four three. Those are the two shocks from our opening question. Hold onto the two numbers; they are part of reading the plot we previewed. These are changes in input velocity for the ideal records. They do not tell us the maximum relative speed or stress of every internal part. They are particularly informative when a component responds slowly compared with the pulse duration.
Δv = ∫ a(t) dt Half sine: Δv = 2 A τ / π
A drop connects height with impact speed
Let an object fall from four feet, neglecting air resistance. Just before impact, its speed is about four point eight nine meters per second, or one hundred ninety-three inches per second. Height sets that ideal impact speed through gravity.
Now distinguish arrival speed from the change during contact. If the object stops, the change in velocity has the magnitude of its incoming speed. If it rebounds upward, it has reversed direction, so the change is larger. In a simple one-dimensional model with a rebound coefficient of zero point three, the magnitude of velocity change is about six point three six meters per second. That number is the second pair of pulses I promised: later we will deliver it over one millisecond and over five, and follow one listener to a decision.
That calculation does not establish the pulse duration, the peak acceleration, or local rotation. A cushion may alter rebound as well as duration. An internal camera can move differently from the package center of mass. We use the drop calculation as a physical check, with those assumptions visible.
vimpact = √(2gh) |Δv| = (1 + e) vimpact
Acceleration describes a change in velocity
Imagine sitting in a car as the driver begins to accelerate. Your body needs a force to change its motion. You feel the seat providing it. When the car cruises steadily in a straight line, it can be moving quickly without that same acceleration.
Acceleration describes how velocity changes with time. Slowing down is acceleration in the opposite direction to the motion. Turning is also acceleration because velocity includes direction. A shock record shows this changing quantity over a very short interval.
In this reader, 1 G represents the standard acceleration of 9.80665 m/s². A peak of one thousand G tells us the largest recorded acceleration, but leaves duration, direction, bandwidth, and location unresolved. We need those details to interpret what the component experienced.
Acceleration helps us estimate inertial force
Hold a small mass on a stiff support and accelerate the support slowly enough that the mass follows it almost as a rigid assembly. The net force required is mass times absolute acceleration. For a ten-gram modeled mass at one hundred G, the magnitude is about nine point eight newtons.
Notice what we assumed. The mass follows the support, and we are estimating the net force, not the force in each screw. Geometry and the load distribution still matter. If the support flexes strongly, local mass acceleration can differ from the base acceleration. We must then calculate or measure that response.
Peak acceleration remains useful outside a quasi-static approximation when it is the relevant response acceleration and we use a suitable dynamic model. The problem is not that the metric lies. The problem is asking one peak to answer a question that also depends on time, deformation, and the load path.
0.010 kg × 100 × 9.80665 m/s² ≈ 9.81 N
Jerk describes how acceleration changes
Think about an elevator. It can begin moving with a gradual buildup of acceleration or with an abrupt tug. Jerk describes how rapidly acceleration changes. It is the slope of the acceleration trace.
If acceleration rises from zero to ten G in one millisecond, its average jerk over that interval is ten thousand G per second. The same change spread over ten milliseconds has one tenth of that average jerk. The complete shape of the rise still matters.
Jerk can help us discuss abruptness and motion comfort. Sharp changes also draw attention to higher-frequency content. However, differentiation magnifies high-frequency measurement noise, so a noisy trace can produce a spectacular and misleading jerk peak.
This is where a good signal tool earns its keep. Mechanical Dynamics Workbench lets you inspect a trace in time and frequency, investigate suspected noise, try an appropriate filter, and compare the processed result with the original record, so you can see what an edit changed and whether the evidence supports it. Keep the original record and document the settings. We will come back to that laboratory near the end.
For structural damage, ask what response that abrupt input creates. If a material is sensitive to deformation rate, its local strain rate is a more direct quantity than input jerk.
j(t) = da(t) / dt
Stretching and compression produce strain
Stretching and compression produce strain: the resulting relative deformation of the material. We can measure or calculate that strain. For a material line in this axial example, engineering normal strain is the change in length divided by its original length. This definition applies regardless of the overall size of the part. Imagine drawing two marks ten millimeters apart on a strip of material. If their separation increases by one hundredth of a millimeter, the engineering strain between them is one thousandth: zero point one percent, or one thousand microstrain.
Now bend a ruler. Material on one side stretches while material on the other side compresses, producing tensile and compressive strain. The ruler's tip displacement is easy to see, while strain is the local relative deformation within that moving shape. Two structures can have the same tip movement and very different local strains because their geometry and curvature differ. Shear deformation also produces strain; this illustration focuses on stretching and compression.
This distinction matters for a circuit board, solder joint, ceramic component, adhesive, or glass surface. Board strain may be a useful measured indicator, but the relationship to local component failure needs evidence. Location, orientation, package geometry, defects, and the failure mechanism determine what strain value means. A single universal strain limit does not describe all of them.
ε = ΔL / L₀ 0.01 mm / 10 mm = 0.001 = 1,000 με
Stretching and compression produce strain. The displayed length change gives 0.001 engineering normal strain, or 1,000 microstrain. Extension is exaggerated. Illustration for section 17.
Stress and strain answer different questions
Picture a strip of material being pulled. Stress is the internal force carried per area; F/A is the average axial stress for the illustrated strip. Strain is the resulting change in length relative to the original length. They are different physical quantities connected by the material's behavior.
For a simple linear elastic uniaxial model, stress equals Young's modulus times strain. That relation is helpful when its assumptions fit. Plastics, adhesives, solder, and other materials can behave differently as temperature, deformation rate, and prior loading change. Yielding or fracture requires a suitable criterion, not merely a familiar equation.
Strain rate asks how fast strain changes. Reaching one thousand microstrain in one millisecond corresponds to an average strain rate of one per second. Reaching the same strain in one second gives one thousandth per second. The same final deformation can therefore arrive through very different material responses. Temperature and loading rate belong in the capability evidence.
Linear uniaxial elastic model: σ = E ε
Stress and strain are different physical quantities. The average axial-stress expression and the linear elastic uniaxial relationship have the stated scope. Illustration for section 18.
Each metric answers a physical question
Let us pause before adding spectra. Displacement tells us where something moves. Relative displacement tells us how its separation from something else changes. Velocity tells us how quickly position changes. Velocity change summarizes the signed acceleration area.
Acceleration tells us how velocity changes, and jerk tells us how acceleration changes. Strain is the local relative deformation resulting from stretching, compression, or shear. Stress describes internal force intensity. None is a universal damage meter.
Try saying the failure question before choosing the metric. Will the board hit the rib? That leads us toward relative travel. Will a heavy attachment overload its support? That leads us toward inertial loads and the resulting local stress. Will a solder connection open after repeated events? We need a model and test evidence appropriate to that mechanism and history. This habit keeps a sophisticated plot attached to the practical decision.
Three failures will come back to test this habit: a lens adhesive that cracks, a board-to-board solder joint that opens over time, and a display that fails in tumble testing. Each one starts with the failure evidence, not with an input number.
These three examples begin with failure evidence. A lens assembly with a brittle adhesive suggests a bondline failure question, but microscopy and the failure surface should establish whether the adhesive, adherend or interface failed. For board-to-board solder, distinguish mechanical bending fatigue from thermal expansion, creep or manufacturing defects. For a liquid crystal display (LCD), delamination, glass fracture and functional loss can have different causes. There is no single input metric that answers all three. Use input spectra to describe exposure, then map that exposure to the failure location and compare a tested capacity. The examples here describe engineering workflows.
Each product still needs its own measurements and tested capacity. [5] [6] [7]
Local response quantities for the three examples
Example
Local quantity to evaluate
Modal clue
Lens adhesive
Bond-edge peel or shear, or a calibrated fracture criterion
Lens rocking and opening at the bond edge
Board-to-board solder
Local cyclic stress or strain with a suitable fatigue model
Differential board bending and connector rotation
LCD during tumble
Interface demand, glass strain or contact load, selected by the failed layer
Display bending, edge prying and support contact
Input metrics describe the exposure. A local response model and measured capacity are needed to judge the component.
A shock spectrum uses imagined listeners
Now let us build the imagined family of listeners more explicitly. Return to the camera on a flexible mount. We may not yet know the frequencies of all the internal parts, so we represent a range of possible behavior with many simple mass-spring-damper systems. Each has a different natural frequency. For a chosen damping value, we give every listener the same base acceleration history and calculate its response.
This bank of listeners explores the demands of that particular input record. If we use a record measured on the housing, it describes the message there. If we use a record measured at an internal support, it describes the message after the real assembly has edited it along that path. Choosing the input location is therefore part of choosing the engineering question.
The calculation uses a simple equation of motion. In relative coordinates, inertia, damping, and spring stiffness balance the effect of base acceleration. You do not need to solve that equation by hand to understand the procedure: replay the same event for each listener, keep a chosen peak, and plot those peaks against listener frequency.
That is a response spectrum. Each imagined listener represents one possible response behavior. A real component can contribute several modes, and modes can involve motion of several connected components. The bank provides a common comparison; identifying the actual listeners requires information about the product. A real mode also has a shape and a participation in the input. Multiple modes and nonlinear joints can require more detailed analysis. The simple bank helps us see where an input is demanding. [5]
z̈ + 2ζωₙ ż + ωₙ² z = −abase(t)
An acceleration SRS keeps response peaks
For an absolute-acceleration shock response spectrum, we keep the largest absolute acceleration reached by each modeled mass. We include response during the pulse and the residual motion afterward. The horizontal coordinate is the listener's natural frequency, not a timestamp. In the presentation's acceleration spectrum, the two curves are our two shocks again, seen through listeners that report acceleration.
Imagine a shelf of bells, each with its own tone. The same tap excites them differently. A spectrum is like recording how strongly each bell responded, although our actual listeners are numerical oscillators and the chosen measurement here is acceleration.
An SRS is not a Fourier spectrum of the input. It also does not say that all these peaks happened simultaneously. Each oscillator can reach its maximum at a different time. State whether the reported spectrum is absolute acceleration, relative displacement, or another response, and whether it includes positive, negative, or absolute maxima. The word spectrum alone is not enough to reproduce the result.
Why pseudo velocity is useful
Here is the reason for pseudo velocity in our Holistic approach. We want to follow the message all the way to the part that must keep working. Peak input acceleration describes one feature of that message. It does not, by itself, tell us how far an internal listener bends. That bending can open an adhesive edge, strain a joint, or bring a display into contact with its support.
Pull the ruler tip down and let go. At the deepest bend, it pauses. As it springs toward its resting position, it moves faster. Now imagine two simple vibrating systems reaching the same relative displacement. One moves through its natural cycle slowly. The other moves through it much faster. Pseudo velocity brings the peak relative deformation together with that natural rhythm by multiplying the displacement by angular natural frequency.
Why use that combination? It expresses the displacement response on a velocity scale that is useful for comparing shocks across a family of listeners. For a particular mode, a model can connect that deformation to local stress or strain. It gives us a way to screen where the message deserves closer attention and compare changes to the exposure.
At a specified frequency and damping, pseudo velocity contains the same peak-displacement information in different units. Its value is in how we interpret and use that information. It is not automatically the speed of a material point. A mode shape is a pattern; actual point speeds depend on location, amplitude, and time. We will make that distinction precise next. [2]
Pseudo velocity combines peak relative deformation with natural frequency. The ruler illustration directs attention to Gaberson and Tom Irvine. Illustration for section 22.
Why the word pseudo matters
For an ideal undamped oscillator vibrating freely in one mode, the peak relative speed equals angular natural frequency times peak relative displacement. That is the physical connection behind the name. It gives our ruler picture a mathematical anchor.
During a general forced shock, with damping, the actual maximum relative speed can differ. The event can still be pushing while the system moves, and damping changes the exchange of motion and deformation. We keep the exact definition of pseudo velocity rather than quietly substituting an actual velocity.
Our calculated four-foot-drop example makes this visible. For the five-millisecond pulse and a four-hundred-hertz listener, pseudo velocity is about zero point nine six meters per second. The computed peak relative speed is only about zero point three four. Both are correct, because they describe different quantities. The word pseudo is doing useful work here. It reminds us which number we calculated.
PV = ωₙ Sd Sd = max |z(t)|
A small calculation makes PV tangible
Suppose our listener reaches one tenth of a millimeter of relative displacement and has a natural frequency of five hundred hertz. Convert the displacement to meters: zero point zero zero zero one. Multiply the frequency by two pi to obtain angular frequency, then multiply by displacement. The result is about zero point three one four meters per second, or twelve point four inches per second.
Now imagine the same displacement at fifty hertz. The pseudo velocity is one tenth as large, because the natural rhythm is ten times slower. That does not establish that one real component is ten times safer. Its shape, coupling, material, and failure mechanism still matter.
The calculation gives you a way to read the metric without mystique. It is the peak relative deformation response expressed on a velocity scale using that listener's natural frequency. You can work backward from it to the same displacement.
For the same listener, cutting pseudo velocity in half also cuts its predicted peak relative travel in half. That is useful when comparing two inputs or isolation choices. It does not tell us that a different listener, with a different shape and connection, has half the stress. [2]
PV = 2π × 500 × 0.0001 ≈ 0.314 m/s
Pseudo acceleration is a separate quantity
Starting with the same peak relative displacement, we can multiply by angular frequency twice. That gives pseudo acceleration. Pseudo displacement scaling, pseudo velocity, and pseudo acceleration are exact transformations of the same displacement peak.
Absolute acceleration SRS is calculated from the actual acceleration response of the oscillator. With damping, that response contains both the spring and damping contributions. Their peaks need not occur together. Therefore pseudo acceleration is not generally identical to absolute-acceleration SRS, even when the curves are close.
This matters when converting a supplied spectrum. Dividing pseudo acceleration by angular frequency gives pseudo velocity exactly. Dividing an absolute-acceleration SRS by angular frequency is an approximation that needs justification. For our earlier five-hundred-hertz example, pseudo acceleration is about one hundred one G. That number follows from the displacement definition, not from an independent measurement of absolute acceleration.
PA = ωₙ² Sd aabsolute = −2ζωₙ ż − ωₙ² z
One chart can show three linked quantities
A four-coordinate plot can look crowded at first. Read it in layers. Find the natural frequency along the bottom. Move up to the pseudo-velocity curve. That gives the velocity-scale value for that listener.
The sloping guide lines let you read the equivalent peak relative displacement and pseudo acceleration without redrawing the spectrum. Constant displacement rises with frequency because pseudo velocity equals angular frequency times displacement. Constant pseudo acceleration falls because pseudo velocity equals pseudo acceleration divided by angular frequency.
All three readings describe the same calculated displacement spectrum. The diagonal acceleration values are pseudo acceleration. They are not a separate absolute-acceleration SRS. Begin with a single point, such as our five-hundred-hertz example, and check the arithmetic. Once that point feels familiar, the rest of the chart becomes a map you can navigate rather than a page of mysterious diagonals. [1]
Diagonal guides show relative displacement and pseudo acceleration derived from the displacement spectrum.
The stored spring energy has a useful connection
We promised not to open every doorway into energy today. Here is only the piece that supports this metric. For an ideal linear spring, the elastic energy stored at maximum deformation is one half stiffness times displacement squared. The oscillator relation between stiffness, mass, and natural frequency lets us write that as one half mass times pseudo velocity squared.
This is an exact algebraic connection for that linear oscillator. It explains one reason the scale is useful: for this oscillator, pseudo velocity squared represents twice the maximum spring energy per unit mass. It lets us express elastic deformation demand in a compact form. It does not say that the real part is moving at that speed at the instant of maximum deformation. In undamped free motion, the speed is zero at maximum displacement.
It also does not measure all the energy put into a whole product, the energy dissipated by its joints, or the work required to fracture a component. Keep the statement tied to the spring and model we actually defined. [2]
½ k Sd² = ½ m PV²
The two shocks, read as listeners
Let us go back to the plot from the start of our conversation. Now you can read it. Along the bottom are the natural frequencies of our imagined listeners, slow on the left and fast on the right. Up the side is pseudo velocity: the peak relative deformation of each listener, scaled by its own natural rhythm.
Below a crossing near seventy-four hertz, the long fifty G pulse produces the larger value. Above it, the very short fifteen hundred G pulse does. The long pulse changes the input velocity by about one hundred thirty-five inches per second, the short one by about seventy-four, and a slow listener feels mostly that accumulated push. A fast listener can follow the sharp pulse and is driven harder by it.
So the question we began with, which shock would you rather apply, has no single answer. It depends on which listener you are asking about.
And remember what this plot is. It compares calculated response demand in a specified linear model with five percent damping. It does not say that every part below seventy-four hertz fails under the longer pulse. We still need the real component's connection to that motion and its capability. In our Holistic approach, this is a map of where to investigate. Locate the modes that bend the vulnerable feature, compare the input at the appropriate support, then recover the local response. A change to cushioning, support stiffness, or damping can help one listener while changing what reaches another. Check the assembly again after a design change. The next two sections take one listener all the way to a decision. [2]
The pseudo-velocity comparison from the opening section. Five percent damping. The curves cross near 74 Hz; this compares modeled demand, not product survival.
Cushioning changes the time available to respond
Imagine catching a moving object by allowing your hands to move with it. You extend the stopping distance and time. Packaging can use deformation to change the pulse delivered to the product in a related way.
For our comparison, we hold the assumed velocity change at six point three six meters per second and use half-sine pulses lasting one and five milliseconds. The longer pulse has a lower peak acceleration. At a four-hundred-hertz listener, the calculated pseudo velocity falls from about one hundred ninety-nine to about thirty-eight inches per second.
This is a controlled comparison, not proof that a particular cushion achieves those pulses. Real cushioning may change rebound, bottom out, or introduce its own resonances. If a component remains in the impulsive limit, merely reducing peak acceleration may do little. Extending duration enough can change that regime. Available travel and the resulting interface pulse must be verified.
Calculated PVSS for equal assumed input velocity change and different pulse durations. Actual peak relative speed is a separate quantity.
The worked example becomes a clearance decision
Let us finish that example with a failure quantity we can actually interpret. Suppose the modeled motion is the relevant relative travel at a possible contact, and the verified minimum available gap is one millimeter. These are explicit teaching assumptions.
With the one-millisecond pulse, the no-contact model predicts about two point zero one millimeters of travel. That exceeds the gap and flags possible contact. The prediction beyond contact is no longer a complete physical solution because the new contact force was omitted.
With the five-millisecond pulse, predicted travel is about zero point three eight millimeters. Under the same linear model, that leaves about zero point six two millimeters of nominal gap. We have answered a clearance question within the assumptions. We have not proved the product survives, established a steel yield threshold, or shown that changing material automatically doubles the margin. Functional checks and other failure modes remain part of the decision.
Stress needs a physical connection to motion
Imagine an entire small block moving rigidly across a table. It can move quickly without stretching internally. Now imagine a longitudinal elastic wave traveling down a slender rod. Nearby material points change their separation as the wave passes, creating strain and stress.
For a one-dimensional linear elastic traveling wave, the magnitude of stress relates to particle velocity through density times wave speed. That is a useful physical relationship under those assumptions. It does not mean that every velocity in a product, or every input pseudo-velocity value, directly gives its local stress.
A vibrating structure has boundaries, reflected waves, mode shapes, and locations with different motion and stress. Research continues to examine the theoretical basis for pseudo velocity as a severity measure. We can value the metric without turning a conditional stress relationship into a universal product survival rule. [2]
Traveling rod wave: |σ| = ρ c |vparticle|
A mode connects input to local response
Think of pushing a playground swing. The direction and location of your push determine how effectively you excite its motion. A structural mode also couples to input in a particular way. Modal participation describes that coupling within a chosen mathematical normalization.
After calculating the modal response, the mode shape tells us how that coordinate produces displacement at a location. A modal stress field relates the same coordinate to local stress. A finite-element model can provide these relationships, and testing helps establish whether the model represents the actual structure.
For a single linear mode, a stress estimate can therefore use the input displacement spectrum together with the participation and modal stress coefficient. Multiple modes may require time histories or an appropriate modal combination method. Effective modal mass is useful, but multiplying a pseudo-velocity curve by a mass percentage is not a general substitute for this calculation. We need the mapping, not just the mode's frequency.
This is the practical why of pseudo velocity for stress. For one unchanged linear mode, direction, and damping, its local stress scale is proportional to the input pseudo velocity, with a coefficient supplied by the structure. That coefficient includes participation, the local stress mapping, and natural frequency. Reducing pseudo velocity then reduces that mode's calculated stress demand in proportion. If you change the structure or its state, update the modes and the mapping before making that comparison. [1] [5] [2]
Single-mode estimate: |σpeak| ≈ |S Γ| Sd
Modal analysis of the mounted assembly
Picture the lens, the board, or the display not on a bench by itself, but installed: screwed down, cabled, pressed against its gasket. Installation changes frequencies and load paths, so that is the assembly to study.
Use an impact-hammer or shaker test with a measured input and responses at enough locations and directions to identify the relevant frequencies, damping, and mode shapes of that mounted assembly. Correlate those results with the model before trusting the model's details.
Then look for the modes that deform the failure site. For the lens, watch for rocking and opening at the bond edge. For the boards, watch for coupled bending and rotation at the connector. For the display, watch for curvature at the edges and contact with the supports.
A mode with little effective mass can still produce a large local strain, so completeness of the global modes does not prove that the stress at a small feature has converged. Check that the local failure-related response changes little as you add modes, refine the mesh, and widen the bandwidth, using a quantity appropriate to the feature.
One limit to keep in view: an intact linear modal model cannot follow progressive debonding or contact that switches on and off. When that begins, the model has to change. [5]
Modal response and the failure location
Here is how a spectrum connects to a real location. For one mode excited through its base, the input spectrum gives the response of a reference listener.
To reach the real structure, multiply that peak relative displacement by the mode's participation, and then by the mapping from the modal coordinate to displacement, strain, or stress at the failure site. Those factors come from the same normalization and must be kept consistent.
At a fixed natural frequency and damping, peak relative displacement, pseudo velocity, and pseudo acceleration carry the same information in different units. Choose the one that connects most directly to the validated failure quantity. Absolute acceleration is the exception, because it includes the damping force.
When several modes matter, remember that their peaks were calculated independently and generally occur at different times. Adding them directly does not give a predicted time history. Adding their magnitudes can be labeled a bound under the right assumptions, and a statistical combination needs its own justification. Where phase matters, reconstruct signed time histories.
Lens assembly with a brittle adhesive
Start at the failure surface. Did the adhesive itself crack, did it separate from the glass or the barrel, or did the alignment shift without a visible crack? Microscopy answers that, and the answer decides what to measure. Choose a repeatable endpoint: crack initiation, bond separation, or an optical alignment shift.
For an intact bond in its linear range, compare local stress or strain from a calibrated modal response with an appropriate initiation criterion. Be careful with a finite-element corner: the stress at a mathematically sharp bond edge can grow as the mesh is refined. Such a singular peak is not a usable material allowable.
Use an averaged, process-zone, cohesive, or fracture-mechanics approach where that is justified. Once a crack grows or surfaces touch, the stiffness changes and the intact linear model is no longer enough.
Design changes worth trying: bring the lens center of mass closer to its support, limit prying at the bond edge, and evaluate bond thickness or a tougher adhesive at the relevant rate and temperature. For an unchanged mounted mode, lower input pseudo velocity means less calculated modal deformation. Use the local mapping to check whether that reduces the bond's peel, shear, or fracture demand. If the support or adhesive changes, reassess the modes and the mapping. Jerk is a diagnostic here, not a brittle-adhesive capacity. [6]
Mother to daughter board solder fatigue
Picture a daughter board standing on a connector above its mother board. During a shock the two boards bend, and they do not have to bend together. The solder sits where their motions differ.
Before we blame vibration, ask what else could have opened that joint: thermal cycling, creep, a process defect, corrosion, or one overload. Failed over time does not by itself mean mechanical fatigue.
If it is mechanical, the quantity to reduce is the differential motion at the joint, not the peak G of one board. Measure strain near the connection, keep both boards' motions and their timing, and connect that to the joint through its geometry and a suitable model. Then use rainflow counting to identify the cycles in that local history and a fatigue relation that fits solder, which often needs an inelastic strain or energy model rather than a simple elastic exponent.
The Workbench can produce a fatigue damage spectrum from the acceleration record. Treat it as a way to compare two exposures, not as a life prediction, until the local response and the fatigue law have been calibrated. And a board's travel divided by a solder-ball diameter is not a joint strain.
Pseudo velocity can help compare excitation near the coupled board modes. To reduce failures over time, carry that comparison through to the joint's local cycle amplitudes and counts. The two boards' separate spectral peaks cannot supply their relative motion or its phase. [7]
Why the phase between boards matters
Here is a deliberately simple illustration, not a solder-strain calculation. Suppose the mother board at the connector moves two tenths of a millimeter each way at one frequency. If the daughter board moves with the same amplitude and in step, the relative motion between them is zero. If it moves exactly out of step, the relative amplitude is four tenths of a millimeter. In between, you get something in between.
Rotation and the local attachment geometry matter too in a real connector.
Now notice what a spectrum stores: a peak amplitude for each board, with the timing thrown away. Subtracting two spectral peaks does not reconstruct their differential history. To keep the phase, use synchronized measurements or signed modal superposition.
That is the point of this illustration. When the goal is fewer joint failures, the best metric is local joint demand, and that needs the relative motion, not two separate peaks. [7]
LCD failure during tumble testing
Picture a handheld unit tumbling in a drum. Each fall hits a different corner, in a different direction, and the damage can accumulate. Start by identifying the failed layer: adhesive delamination, glass cracking, housing contact, flex-cable damage, or an electrical or optical failure. These have different causes and different measurements.
Record the orientation and sequence of impacts, and inspect function or interface condition between defined blocks of tumbles, so the moment of failure is bracketed.
For an intact mounted display, modal analysis can locate the bending and prying patterns and help you choose measurement locations. Pseudo velocity at those modes can help compare the deformation demand from different impact orientations, while the local model or measurements tell us what happens at the failing layer. Once gaps close, an adhesive slips, or delamination starts to propagate, you need a nonlinear transient, contact, or cohesive model, or a direct local measurement.
Check design changes for their trade-offs: reducing one contact load can increase travel or strain somewhere else. Confirm an improvement with representative repeated tumble sequences and the actual failure endpoint. Input jerk alone neither proves nor rules out delamination. [6]
Material choice changes the whole system
It is tempting to give every material a velocity budget, then compare the spectrum with that number. A simplified rod or beam calculation can illustrate a relationship, but it is not a universal allowable for a finished product.
Replace a steel bracket with magnesium and several things may change together: stiffness, density, natural frequency, local geometry, manufacturing details, damping, and joint behavior. Even if a simplified strength-to-impedance ratio looks favorable, the new bracket does not automatically inherit a known survival margin.
For glass, surface flaws and edge condition can govern fracture. For polymers and adhesives, temperature and loading rate can strongly influence behavior. For a joint, preload and slip may dominate. Use material properties within an appropriate model, then compare the predicted failure-related response with relevant capability evidence. The useful question is what changes in this design, rather than which material wins a generic ranking.
A screening number needs its context
You may encounter historical pseudo-velocity screening levels such as fifty or one hundred inches per second. Treat them as context-dependent engineering heuristics, not as a boundary between harmless and damaging shocks for every product.
A camera can lose alignment, an electrical connection can open, or a brittle part can crack under conditions that a rugged metal assembly survives. Conversely, a large spectrum value does not prove failure in an insensitive structure. The relevant construction and failure mechanism matter.
If a program uses a screening criterion, identify its source, purpose, assumptions, applicable hardware, and required follow-up. Do not turn a convenient line into a promise of survival. A particularly useful outcome of screening is deciding where to investigate next: which modes, connections, or measurements deserve attention. That is a valuable role without claiming that one number closes the entire engineering decision.
The measurement location is part of the story
In my shock-testing work with scan engines, the question was why engines were failing when they were rated for two thousand G and a two-tenths-of-a-millisecond pulse. The table accelerometer indicated two thousand G. To investigate, I added another accelerometer near the plate holding the engines.
The engine-side measurement showed over eight thousand G, and the pulses differed. Examination of the rig found evidence of relative movement and joint slip as fasteners loosened. The fixture had changed during the testing. Our original message was being edited by the condition of the load path.
Those observations led us to investigate the mounting system instead of treating the table reading as the whole exposure. They do not establish a universal fourfold transmissibility curve or prove that every engine point experienced an identical trace. They show why a second listening point and inspection of the physical rig can change our understanding.
Processing should preserve the evidence
Before trusting a peak or spectrum, ask where the sensor sat, which direction it measured, whether it saturated, and whether its mounting or mass affected the measurement. Confirm sampling, anti-alias filtering, bandwidth, and timing. For simulation data, identify the coordinate system, units, history output, and time increments.
Then inspect what processing changes. A cutoff sweep can help compare sensitivity to filtering, but a smoother curve is not automatically more physically correct. Compare peak, pulse shape, velocity change, and spectra over the frequency band relevant to the decision. Preserve the raw record and record the settings.
Integration provides another check, but a mismatch with an ideal drop velocity is a reason to investigate, not automatic proof of bad data. Sensor orientation, gravity treatment, rotation, local motion, rebound, and the integration interval all matter. For a spectrum, state damping, frequency range, response definition, and the treatment of residual response.
A damage boundary comes from tests
Imagine gradually exploring which combinations of shock amplitude and duration cause a specified product failure. A damage boundary summarizes the tested relationship between exposure and failure under the chosen setup.
Mechanical-shock fragility testing with shock machines can distinguish between critical velocity change and critical acceleration. Those ideas connect with our slow- and fast-response intuition, but an empirical product damage boundary is not simply a pseudo-velocity spectrum drawn on different axes.
The boundary depends on the product, mounting, direction, failure definition, pulse family, and test history. A curve cannot promise no damage for every possible waveform outside a shaded corner. If you plan a standards-based test, use the applicable complete procedure and its requirements. This reader explains the physical purpose; it does not replace the complete test procedure. [3]
A component fragility test plan
Suppose you are planning the test yourself. Decide first what you will observe as failure: bond initiation, alignment loss, loss of solder continuity, a crack size, a delaminated area, or display function. Use conditioned, representative units with controlled supports and assembly conditions.
Vary both the input amplitude and the pulse duration, so the test explores the component's relevant modes instead of leaning on a single peak-G limit. Record the input and a local response channel or strain that can validate the mapping. Replicate, because scatter is part of the answer.
Keep the survivors in the analysis beside the failures, as censored observations. A specimen that has survived several escalating tests is not equivalent to a fresh unit at the final level. For progressive tumble damage, keep the event order and history. Validate the fitted relation on independent units before using it to rank design changes.
This is a planning framework. It does not claim compliance with a particular standard. [3] [4]
Scatter changes what a median tells us
Suppose two hypothetical product populations both have a median failure level of eighty-five inches per second under an agreed test and exposure definition. In each population, half the units fail by the median. Does that tell us the risk at sixty inches per second? No. We need the shape of the distribution as well.
The plotted examples use Weibull curves with the same median and different shape parameters. At sixty, one predicts roughly twenty-two percent failure and the other roughly two percent. These are illustrations of the model, not measured glass or metal populations.
Real inference requires the observations, sample count, exposure history, failure definition, and uncertainty. Include units that did not fail rather than discarding them. If a unit passed one level and failed at the next, its threshold may be interval-censored under an appropriate threshold model. A median alone cannot support a claimed one-in-ten risk at another level. [4]
Hypothetical Weibull curves with the same median and different shape parameters. These are not measured material or product populations.
The sequence of events can change the part
A product may change while we test it. Fasteners can loosen. A crack can grow. Contact surfaces can wear. Some events may produce no lasting change, while others can alter the response to the next event.
A peak-response spectrum does not retain the complete order, signs, cycle count, or duration of the response. Different histories can share similar spectrum peaks and produce different cumulative damage. For fatigue, we need suitable local stress or strain histories and a validated life model or relevant tests.
Increasing the severity repeatedly on the same unit can be useful, but the resulting failure is conditioned by everything that happened earlier. It is not automatically a fresh-unit threshold. Keep the sequence and inspect the unit's state. Use a statistical model appropriate to the experiment, with censored observations and uncertainty. The lesson is very close to our fixture story: history can change the system that receives the next message. [4]
Single impact and accumulated exposure
Two records can have nearly the same peak spectrum and do very different amounts of damage, because one of them happens a thousand times. A shock response spectrum keeps the peak response of each listener and nothing about how many cycles it rang through.
A fatigue damage spectrum is the tool for repetition. It counts the response cycles of each listener and sums their contributions with a chosen fatigue exponent.
For an isolated shock, a listener may keep ringing after the input ends. The analysis must include enough of that post-event response that extending it no longer materially changes the fatigue index. Its importance depends on the pulse, natural frequency, and damping. For continuous vibration, supply one long continuous history rather than restarting the analysis on each segment.
Treat the result as a comparison between exposures under the same definitions: the same damping, the same response quantity, the same cycle-counting convention, a stated exposure duration or common normalization, and consistent boundary treatment. Mean stress, plasticity, temperature, and the order of events can all invalidate a simple linear damage sum. Until local response and a calibrated fatigue law give it physical meaning, the index ranks exposures. It does not predict life. [8]
The Workbench as a laboratory
Before the exercise, a word about the tools. I developed Mechanical Dynamics Workbench in partnership with ChatGPT and Claude to make these ideas visible, and it has grown over months of continuous improvement. It is free, and you can find it and its walkthrough guides on my blog.
At a high level, it lets you generate example signals or work with measured and simulated data, examine motion in time and in frequency, explore filtering and sampling, calculate shock and vibration response, and compare the effects of different exposures. You do not need field measurements to begin.
I build Learning Centers into the tools because instructions can tell you which button to press, while a Learning Center should help you understand why you would press it, what assumptions you are making, and what the result can and cannot tell you. Think of it as a hands-on laboratory. Change the damping. Change the frequency. Change the duration of the poke. Predict what will happen, then look.
Two habits the Workbench is built to support. Name the response quantity you are using, absolute acceleration, pseudo acceleration, pseudo velocity, or relative displacement, so the two acceleration quantities are never silently interchanged. And treat capacity not established as a real result: it tells you which test comes next. Detailed walkthroughs are in the Learning Centers, in written guides, and in video presentations.
The purpose is to follow the whole chain: the message we apply, the message that arrives at a support, the listeners it excites, the local demand, and the capability evidence. Use each plot to ask a better question about that chain. That is how the tools support Holistic engineering judgment.
For practice, use the accompanying Mechanical Dynamics Workbench Guided Walkthroughs R11 and Filtering and Aliasing Walkthroughs R11. The first carries the input through modal response, local failure demand, and capability evidence, including the lens, solder, and LCD examples. The second retains the fourteen guided filtering and aliasing lessons, with predictions, observations, technical explanations, and a record of the editing decisions.
In the reviewed Workbench 7.6.2-FR4 release:
The Workbench's Gaberson dashboard includes a four-coordinate pseudo-velocity plot. Constant displacement diagonals rise with frequency, and constant pseudo-acceleration diagonals fall. A separate panel shows true absolute acceleration and pseudo-acceleration so the two quantities cannot be silently interchanged. The dashboard does not assign generic printed circuit board frequency bands or universal glass or solder pass/fail thresholds. Jerk reports its derivative method and smoothing time window. FDS reports its response quantity, range/amplitude convention and residual-tail treatment. Built-in component examples do not establish tested capacities. Use the learning center's lens, board and LCD workflows to decide the local measurement and calibration needed before making a design acceptance decision.
Older saved calculations require their recorded conventions when replayed or compared.
A useful result names the next decision
Start with the performance requirement. Is the concern image blur during an exposure, permanent optical misalignment, contact, fracture, electrical interruption, or fatigue? Each leads to a different response quantity and timescale.
Follow the message through the load path. Identify the loading at the appropriate interface and the current state of the component. Calculate or measure the response needed for that failure question. Then compare it with capability evidence under compatible conditions. Document uncertainty and any limits in the model.
The outcome may be supported survival within scope, a concern requiring a design change, or insufficient evidence. All three can be useful engineering conclusions. If evidence is insufficient, choose the next measurement or test that reduces the uncertainty that matters. Do not process the same data indefinitely merely because the software offers another plot. The purpose of the analysis is to improve a decision.
A learning exercise you can repeat
Here is a learning exercise for the Workbench or another suitable analysis tool. Generate our two half-sine pulses, preserving their units and durations. Before looking at any spectrum, predict which has the larger peak acceleration and which has the larger velocity change. Then calculate both and explain the difference.
Next, compare displacement and pseudo-velocity response spectra at the same damping. Pick one low frequency and one high frequency. Describe the listener's motion in words before quoting its number. Change damping and notice which results change. At one fixed frequency, convert pseudo velocity back to peak relative displacement. Explain why those two plots carry the same displacement information, and why a peak-G comparison alone could miss it.
Finally, choose a hypothetical clearance and compare it with the corresponding relative travel, stating the model assumptions. You do not need a proprietary model or an expensive test to learn this reasoning. The important habit is to predict, calculate, inspect, and explain. When a result surprises you, it gives you a better question to investigate.
A common input excites reference listeners with different natural behavior. The illustrated rig is an analogy for the independent oscillator calculations in the spectrum. Illustration for section 50.
AI is a partner in the investigation
The same habits apply when we work with AI. It can help us organize explanations, explore calculations, and produce useful teaching material. It can also repeat a relationship beyond the conditions where that relationship applies.
That is why I want us to remain drivers of our path. Ask what the quantity actually represents. Check the units, the physical model, and the evidence behind a conclusion. Compare an important calculation with an independent check. When an explanation sounds elegant, ask whether it still works in a simple example you can picture.
My work with ChatGPT and Claude helps package years of engineering experience into something presentable and accessible. Grok has contributed technical review, and Gemini has contributed review of clarity and flow. The responsibility to think remains with us. Used thoughtfully, these tools can strengthen curiosity and judgment. The fragility we should avoid is allowing a confident answer to replace our own examination of the system.
Thank you for reading
Thank you for spending this time with me. We began with a practical question: will the component survive its environment and still do its job? We pictured motion, gave each metric a physical meaning, and explored how a spectrum helps us compare the response of different mechanical listeners.
The next step is to connect that response to your component, its present state, and its actual failure mechanism. The Mechanical Dynamics Workbench walkthroughs give you opportunities to practice with generated signals, measured records, simulation histories, filtering, spectra, and response comparisons. You can return to this reader at your own pace.
I welcome your questions, experiences, and feedback. You can find my work at McFaddenCAE.com, or reach me at mcfadden@snet.net. Keep asking what the system is telling you, and what evidence would help you understand it better.
Have a wonderful and thoughtful day.
Calculation details and definitions
All numerical shock examples use an initially resting linear base-excited oscillator. The relative coordinate is z = x − y. The damping ratio is 0.05, equivalent to Q = 1/(2ζ) = 10. The input is A sin(πt/τ) over 0 ≤ t ≤ τ, followed by zero acceleration. Peaks include the forcing interval and residual response. These ideal records are teaching inputs, not complete measured drop histories.
Quantity
Definition
Meaning
Sd
max |z(t)|
Peak relative displacement
PV
ωₙ Sd
Pseudo velocity
PA
ωₙ² Sd
Pseudo acceleration
Sa
max |−2ζωₙ ż − ωₙ² z|
Peak absolute acceleration
Sv
max |ż(t)|
Peak actual relative speed
The continuous sinusoidal forcing was integrated numerically, followed by the analytical underdamped free response. The residual search covers three natural periods, which includes the first and largest residual extrema for this unforced decaying model. Repeating selected cases at 10, 75, 400, 2,000 and 10,000 Hz with tighter integration tolerances and doubled peak sampling changed the reported response metrics by less than 0.005%. This is numerical convergence evidence, not validation of a particular physical product.
400 Hz response
1 ms pulse
5 ms pulse
Input peak acceleration
1,018.3 G
203.7 G
Input velocity change
6.357 m/s
6.357 m/s
Sd
2.0121 mm
0.3829 mm
PV
5.0569 m/s
0.9624 m/s
PV
199.09 in/s
37.89 in/s
Peak actual relative speed
4.6861 m/s
0.3378 m/s
PA
1,295.98 G
246.64 G
Sa
1,302.49 G
246.90 G
The input velocity change assumes a four-foot vertical free fall, gravitational acceleration 9.80665 m/s², rebound coefficient 0.3 and one-dimensional impact. A specified pulse is then imposed as the oscillator base input. No cushion material, nonlinear contact, or real bracket geometry was simulated.
Modal mapping
For one linear mode under uniform base excitation, q̈j + 2ζjωj q̇j + ωj² qj = −Γj abase. If Sj is the local modal stress coefficient for that same normalization, σj(t) = Sj qj(t). The single-mode peak stress magnitude is therefore |SjΓj| Sd at the mode frequency and damping. Do not substitute a modal mass percentage for Γj or combine several modal peaks as though they necessarily occur at the same time. Nonlinear contacts, changing joints and yielding require an appropriate extension of the model.
The illustrative statistical curves
For exposure level u and median 85 in/s, the plotted hypothetical probability is F(u) = 1 − exp[−ln(2)(u/85)^β]. The two shape parameters are β = 3 and β = 10. At 60 in/s, the resulting probabilities are about 21.6% and 2.1%. No product-specific probability follows from these curves. Actual inference requires observations, an applicable model, uncertainty estimates and the test history.
Using the fatigue comparison reproducibly
For a joint-life investigation, rainflow-count the relevant local stress or strain history and use a fatigue relation calibrated for the failure mechanism, units, cycle convention, temperature and loading conditions. A comparative oscillator index does not establish solder failure at D = 1 or life as 1/D. The mapping from board response to the joint must represent the actual geometry, attachment and participating modes. [7] [8]
The previously reviewed MDSP 7.6.2-FR4 implementation distinguishes isolated_event, which appends and checks a converged post-event oscillator response tail, from record_only, which analyzes the supplied record from its stated zero initial condition. The tail belongs to the modeled oscillator response. For continuous vibration, retain a sufficiently long continuous history. Processing segments consistently would require both oscillator-state and rainflow-residue carryover.
In that reviewed implementation the default cycle measure is peak-to-peak range. An amplitude convention uses range/2 and needs a matching fatigue coefficient. The documented defaults b = 4 and C = 1 are comparative settings, not tested component properties. With b = 4, halving the cycle measure while leaving the coefficient unchanged reduces the computed index by a factor of 16. Record the software version, response quantity, units, convention, exponent, coefficient and boundary treatment when comparing results. [8]
For equal-duration comparisons, keep exposure duration consistent. To compare complete service histories of different durations, retain their actual duration and repetition counts; alternatively state a common normalization. Mean stress, plasticity, temperature and load sequence can invalidate a simple linear damage accumulation model. More processing cannot replace local calibration and representative testing.
References and further reading
1 Gaberson
Howard A. Gaberson. Pseudo Velocity Shock Spectrum Rules for Analysis of Mechanical Shock. Background on displacement-based spectra and four-coordinate plotting. Conditional stress relationships require the appropriate structural model.
2 Hou and Li 2025
Desen Hou and Q. M. Li. About the theoretical basis of using pseudo-velocity as shock severity metric. Mechanical Systems and Signal Processing 236, 113024. The paper examines limitations of a modal-velocity justification and develops a lower-damage-bound basis.
3 ASTM D3332
ASTM D3332-99(2023), Standard Test Methods for Mechanical-Shock Fragility of Products, Using Shock Machines. The public scope distinguishes critical velocity change and critical acceleration. This reader explains the concepts and does not replace the complete test procedure.
4 NIST SEMATECH
Engineering Statistics Handbook, section 8.1.3.1, Censoring. Background on retaining incomplete failure observations in reliability analysis. Threshold and sequential-shock applications also need a statistical model suited to the experiment.
5 Irvine 2012
Tom Irvine. An Introduction to the Shock Response Spectrum, Revision S, July 9, 2012. A tutorial on oscillator response, damping and shock-spectrum calculations.
6 Mall and Johnson 1985
S. Mall and W. S. Johnson. Characterization of Mode I and Mixed-Mode Failure of Adhesive Bonds Between Composite Adherends. NASA Technical Memorandum 86355. Background on debond growth and fracture measures in the tested bonded systems. It supplies no allowable for the lens or LCD examples.
7 Zhou 2009
Yuxun Zhou. Harmonic and Random Vibration Durability Investigation for SAC305 (Sn3.0Ag0.5Cu) Solder Joint. University of Maryland dissertation. Background on relating board vibration to local solder response and fatigue calibration. Properties from a specific assembly require validation before transfer to another.
8 Irvine fatigue damage spectrum
Tom Irvine. Direct Fatigue Damage Spectrum Calculation for a Shock Response Spectrum, June 25, 2014. Background on response rainflow cycles, comparative fatigue indices, and the explicit peak-to-valley amplitude convention.
