Transient Events: Simulating Impacts, Shocks, and Arrivals

Transient events add short-lived, decaying responses to time series — essential for simulating impacts, seismic arrivals, mechanical shocks, pressure surges, and any phenomenon where a system is suddenly disturbed and returns to equilibrium.

What Are Transient Events?

A transient event models the response of a second-order dynamic system to a sudden input. Unlike oscillations (which repeat indefinitely), transients are localized in time: they start at a specific onset, peak rapidly, and decay back to zero.

Mathematical Model

Phoenix uses the classical second-order system response:

transient(t) = A × h(t - t₀)

Where:
  A   = Amplitude (peak height)
  t₀  = Onset time (seconds)
  h() = Normalized response function (impulse or step)

The response function depends on the damping ratio (ζ) and natural frequency (ωₙ):

ωₙ = 2π × fₙ

Where:
  fₙ = Natural frequency (Hz)
  ζ  = Damping ratio (dimensionless, > 0)

Damping Regimes

The damping ratio determines the character of the response:

Underdamped (ζ < 1) — Oscillating decay - The system overshoots and rings, with each oscillation smaller than the last - Most common in real-world systems (structures, sensors, seismic waves) - Lower ζ = more oscillations before settling

Critically Damped (ζ = 1) — Fastest non-oscillatory return - Returns to equilibrium as fast as possible without oscillating - Theoretical ideal for instrument response

Overdamped (ζ > 1) — Slow exponential return - Returns to equilibrium sluggishly without oscillating - Common in heavily damped or viscous systems

Response Types

Impulse Response — Models instantaneous disturbances:

Underdamped (ζ < 1):
  h(τ) = exp(-ζωₙτ) × sin(ωdτ) / peak_normalization

  Where: ωd = ωₙ√(1 - ζ²)  (damped natural frequency)

Critically Damped (ζ = 1):
  h(τ) = ωₙτ × exp(-ωₙτ) / e⁻¹

Overdamped (ζ > 1):
  h(τ) = exp(-ζωₙτ) × sinh(ω'τ) / max_normalization

  Where: ω' = ωₙ√(ζ² - 1)

The impulse response is normalized so the peak magnitude ≈ 1.0, meaning the amplitude parameter directly controls the peak value.

Step Response — Models sustained disturbances:

Underdamped (ζ < 1):
  h(τ) = 1 - exp(-ζωₙτ) × [cos(ωdτ) + (ζ/√(1-ζ²)) × sin(ωdτ)]

Critically Damped (ζ = 1):
  h(τ) = 1 - (1 + ωₙτ) × exp(-ωₙτ)

Overdamped (ζ > 1):
  h(τ) = 1 - exp(-ζωₙτ) × [cosh(ω'τ) + (ζ/√(ζ²-1)) × sinh(ω'τ)]

The step response settles to 1.0 (scaled by amplitude), with possible overshoot for underdamped systems.

Transient Parameters

Onset Time (t₀) - When the event occurs, in seconds from signal start - Must be ≥ 0 - Signal is zero before onset

Amplitude (A) - Peak height of the response - Units match the signal (mm/s, g, Pa, etc.) - For impulse: directly controls peak magnitude - For step: controls steady-state displacement

Natural Frequency (fₙ) - Characteristic oscillation frequency of the system (Hz) - Higher frequency = faster oscillation within the envelope - Must be > 0

Damping Ratio (ζ) - Controls how quickly the response decays - Must be > 0 - Typical values: 0.01–0.3 for structural systems, 0.5–2.0 for heavily damped systems

When to Use Transient Events

Common Applications

Seismology - P-wave and S-wave arrivals - Aftershock sequences - Surface wave arrivals with different characteristics

Structural Monitoring - Impact events (dropped loads, collisions) - Earthquake response of buildings/bridges - Machine start/stop events

Mechanical Systems - Hammer impacts in modal testing - Sudden load changes - Valve closures (water hammer)

Process Engineering - Pressure surges in pipelines - Flow disturbances - Thermal shocks

Electrical Systems - Switching transients - Lightning-induced surges - Fault response

Configuring Transients in Phoenix

Step 1: Add a Transient Event

Within a channel configuration:

1. Find the "Transient Events" section (below Oscillations)
2. Click "Add Transient"
3. A new transient configuration card appears
4. Configure parameters (see below)
5. Click "Add Transient" again for additional events

Step 2: Set Onset Time

Onset determines when the transient starts within the signal:

Onset = 0     → Event starts at the beginning of the signal
Onset = 5.0   → Event starts 5 seconds into the signal
Onset = 30.0  → Event starts 30 seconds in

For multiple transients (e.g., seismic P-wave then S-wave), space them apart:

Transient 1 (P-wave):  Onset = 5 seconds
Transient 2 (S-wave):  Onset = 15 seconds

Step 3: Set Natural Frequency

The natural frequency controls how fast the transient oscillates:

Common Frequencies:

Building sway:          0.5-2 Hz
Bridge vibration:       1-5 Hz
Machine tool:           10-100 Hz
Sensor resonance:       100-1000 Hz
Seismic P-wave:         1-10 Hz
Seismic S-wave:         0.5-5 Hz

Step 4: Set Damping Ratio

Damping controls how quickly the transient decays:

Choosing Damping Ratio:

Very lightly damped:    ζ = 0.01-0.02  (rings for many cycles)
Lightly damped:         ζ = 0.02-0.05  (typical structural)
Moderately damped:      ζ = 0.05-0.15  (typical mechanical)
Heavily damped:         ζ = 0.15-0.30  (damped structures)
Critically damped:      ζ = 1.0        (no oscillation)
Overdamped:             ζ > 1.0        (sluggish return)

Calculating Damping from Desired Decay Time:

ζ = 4 / (t_decay × 2π × fₙ)

Where:
  t_decay = desired time for oscillation to decay (seconds)
  fₙ = natural frequency (Hz)

Example:
  Want decay in 2 seconds, frequency = 5 Hz:
  ζ = 4 / (2 × 2π × 5) = 4 / 62.83 ≈ 0.064

Step 5: Set Amplitude

Amplitude determines the peak magnitude of the transient:

Seismic P-wave:     Amplitude = 50 (mm/s)
Seismic S-wave:     Amplitude = 120 (mm/s, typically larger than P-wave)
Impact test:        Amplitude = 500 (m/s²)
Pressure surge:     Amplitude = 200 (kPa)

Step 6: Choose Response Type

Impulse (default) — For instantaneous disturbances: - Hammer impacts - Seismic wave arrivals - Lightning strikes - Sudden releases

Step — For sustained disturbances: - Valve closures - Sudden load applications - Permanent displacement events - Temperature step changes

Multiple Transient Events

Phoenix supports multiple transients per channel. They combine through superposition (addition).

Why Multiple Transients?

Seismic Recordings - P-wave arrival (faster, lower amplitude) - S-wave arrival (slower, higher amplitude) - Surface wave arrival (slowest, different frequency)

Impact Sequences - Multiple impacts at different times - Machine start + vibration + stop

Modal Testing - Multiple hammer strikes for averaging - Different excitation points

Configuring Multiple Transients

1. Add first transient (e.g., P-wave at 5 seconds)
2. Click "Add Transient" again
3. Configure second transient (e.g., S-wave at 15 seconds)
4. All transients add together in the final signal

Common Transient Patterns

Seismograph Recording (Vertical Component)

Channel Name: "Vertical"
Unit: "mm/s"
Mean: 0
Noise Amplitude: 0.5

Transient 1 (P-wave):
  Onset: 5 seconds
  Amplitude: 50
  Frequency: 8 Hz
  Damping: 0.06
  Type: Impulse

Transient 2 (S-wave):
  Onset: 15 seconds
  Amplitude: 120
  Frequency: 3 Hz
  Damping: 0.04
  Type: Impulse

Duration: 60 seconds
Sampling: 100 Hz

Result: Background noise with a sharp P-wave arrival at 5s, then a larger, lower-frequency S-wave at 15s.

Three-Component Seismograph

Channel 1 ("Vertical"):
  Transient 1: Onset=5s, Amp=50, Freq=8Hz, ζ=0.06 (P-wave)
  Transient 2: Onset=15s, Amp=80, Freq=3Hz, ζ=0.04 (S-wave)

Channel 2 ("North-South"):
  Transient 1: Onset=5s, Amp=20, Freq=8Hz, ζ=0.06 (P-wave)
  Transient 2: Onset=15s, Amp=150, Freq=3Hz, ζ=0.04 (S-wave)

Channel 3 ("East-West"):
  Transient 1: Onset=5s, Amp=15, Freq=8Hz, ζ=0.06 (P-wave)
  Transient 2: Onset=15s, Amp=140, Freq=2.5Hz, ζ=0.04 (S-wave)

Note: P-waves are strongest on the vertical component; S-waves dominate horizontal components.

Impact Test (Modal Analysis)

Channel Name: "Accelerometer"
Unit: "m/s²"
Mean: 0
Noise Amplitude: 0.2

Transient 1 (Impact):
  Onset: 2 seconds
  Amplitude: 500
  Frequency: 45 Hz
  Damping: 0.02
  Type: Impulse

Duration: 5 seconds
Sampling: 500 Hz

Result: Sharp impact peak at 2s followed by long ringing at 45 Hz (typical structural resonance).

Pressure Surge (Water Hammer)

Channel Name: "Pressure"
Unit: "kPa"
Mean: 500
Noise Amplitude: 2

Transient 1 (Valve closure):
  Onset: 10 seconds
  Amplitude: 300
  Frequency: 2 Hz
  Damping: 0.15
  Type: Step

Duration: 30 seconds
Sampling: 50 Hz

Result: Steady pressure at 500 kPa, then a rapid rise to ~800 kPa at 10s with oscillatory overshoot, settling to a new steady state.

Thermal Shock

Channel Name: "Temperature"
Unit: "°C"
Mean: 25
Noise Amplitude: 0.1

Transient 1 (Thermal shock):
  Onset: 5 seconds
  Amplitude: 50
  Frequency: 0.1 Hz
  Damping: 1.2
  Type: Step

Duration: 120 seconds
Sampling: 10 Hz

Result: Temperature at 25°C, then a slow, overdamped rise to 75°C with no oscillation (overdamped thermal response).

Combining Transients with Other Components

Transient events combine additively with all other signal components:

signal(t) = mean + trend(t) + Σ oscillations(t) + Σ transients(t) + noise(t)

Transient + Background Oscillation

Mean: 0
Noise: 0.1
Oscillation: Frequency=30 Hz, Amplitude=2 (machine vibration)
Transient: Onset=5s, Amplitude=50, Frequency=45 Hz, ζ=0.03 (impact)

Result: Continuous machine vibration with a sharp impact event at 5 seconds

Transient + Linear Trend

Mean: 100
Noise: 1
Trend: Slope=0.5 (gradual drift)
Transient: Onset=20s, Amplitude=30, Frequency=2 Hz, ζ=0.1 (disturbance)

Result: Gradually rising baseline with a transient disturbance at 20 seconds

Using the AI Agent for Transient Events

The AI chat agent can generate transient configurations from natural language descriptions. Example prompts:

Seismograph:

"Simulate a 60-second vertical seismograph recording with a P-wave arrival at 5 seconds and an S-wave arrival at 15 seconds. Use realistic frequencies and damping for a local earthquake."

Impact Test:

"Generate a 5-second accelerometer recording of a hammer impact on a steel beam. The beam has a natural frequency around 45 Hz with very light damping."

Pressure Surge:

"Create a 30-second pressure sensor recording showing a water hammer event at 10 seconds. Initial pressure is 500 kPa."

The agent will set appropriate frequencies, damping ratios, and amplitudes based on domain knowledge.

Tips for Using Transients

Choose Realistic Damping Ratios

Most real systems are underdamped (ζ < 1). Use these as starting points:

Match Amplitude to Physical Scale

Consider the physical phenomenon: - Seismic ground velocity: 0.1–100 mm/s (local events) - Machine vibration: 0.1–50 m/s² - Pressure surges: 10–500% of operating pressure

Set Onset Within Signal Duration

Ensure the onset time allows the transient to develop:

Good:  Duration=60s, Onset=5s   → 55 seconds for decay
Bad:   Duration=10s, Onset=9.5s → Only 0.5 seconds for decay

Consider Sampling Requirements

The natural frequency determines the minimum sampling rate:

Minimum sampling: fₛ ≥ 2.5 × fₙ

Example:
  fₙ = 45 Hz → fₛ ≥ 112.5 Hz (use 250 Hz for good resolution)
  fₙ = 3 Hz  → fₛ ≥ 7.5 Hz (use 50-100 Hz)

Preview After Each Addition

1. Click Preview
2. Verify the transient appears at the correct onset time
3. Check that the decay looks realistic
4. Verify amplitude is appropriate relative to noise

Troubleshooting

Can't see the transient in chart

Possible Causes:

1. Amplitude too small compared to noise Problem: Transient Amp=1, Noise=50 Fix: Increase amplitude or decrease noise

2. Onset time after signal duration Problem: Onset=100s, Duration=60s Fix: Set onset within the signal duration

3. Very high damping (overdamped) Problem: ζ=5, the transient barely rises above zero Fix: Reduce damping ratio to underdamped range (< 1.0)

4. Frequency too high for sampling rate (aliased) Problem: fₙ=50 Hz, Sampling=10 Hz Fix: Increase sampling frequency to ≥ 125 Hz

Transient looks like a spike instead of ringing

Cause: Damping ratio too high (overdamped or near-critical)

Solution: Reduce ζ to underdamped range (0.01–0.3 for visible oscillation).

Transient rings too long

Cause: Damping ratio too low

Solution: Increase ζ. Use the decay time formula:

ζ = 4 / (t_decay × 2π × fₙ)

Multiple transients overlap confusingly

Solution: Space onset times apart to allow each transient to mostly decay before the next begins. Estimate decay time:

t_decay ≈ 4 / (ζ × 2π × fₙ)

Next Steps

• Oscillations — Add periodic behavior alongside transients
• Multi-Channel — Create multi-axis transient recordings
• Channel Correlations — Correlate transient channels
• Data Degradation — Add missing data and outliers
• Technical Reference — Full mathematical specifications

Transient events unlock Phoenix's ability to simulate real-world dynamic phenomena — from seismic recordings to mechanical impacts to process disturbances.