Base Excitation at Multiple Supports

What Is It?

Base excitation at multiple supports occurs when a structure is mounted at several points on a vibrating base — for example, equipment bolted at four corners to a vibrating platform. Each support point receives a different input motion, but the motions are correlated because they come from the same base. The structural response depends on the individual support motions and on the correlation between them.

Why It Matters

Multi-support excitation is the most common real-world random vibration configuration. Equipment racks, electronic boxes, satellite payloads and machinery are all mounted at multiple points. The standard single-input base excitation analysis — applying one PSD at all supports — is an approximation that may over-simplify the real condition. Understanding multi-support excitation is essential for accurate equipment response prediction and qualification.

Multi-support excitation is the most common real-world configuration. Single-input base excitation is an approximation. For accurate assessment, the correlation between supports must be considered.

Support Motion

Each support point receives a motion from the base. This motion has its own PSD, but the motions at different supports are correlated — they come from the same base. The correlation depends on the spacing of the supports and the wavelength of the base motion at each frequency. At low frequencies (long wavelengths), the supports move nearly together — high coherence. At high frequencies (short wavelengths), the supports may move independently — low coherence.

Support motion correlation:

At frequency f, wavelength λ = c/f (c = wave speed in base)

If support spacing d << λ:
  Supports move nearly together — coherence ≈ 1
  (rigid-body motion dominates)

If support spacing d ~ λ:
  Supports move with phase difference — coherence < 1
  (flexural motion of base)

If support spacing d >> λ:
  Supports move independently — coherence ≈ 0

Multiple Attachment Points

The number and location of attachment points affect the structural response. More supports generally mean better load distribution and lower response — but only if the supports are modelled correctly. The attachment points define the boundary conditions of the structural model. Incorrect support modelling — wrong stiffness, wrong spacing, wrong correlation — produces incorrect response predictions.

• Number of supports — more supports generally reduce response
• Location — supports at high-stiffness points are more effective
• Spacing — determines correlation characteristics
• Stiffness — support flexibility affects the boundary conditions
• Correlation — must be modelled correctly for multi-support analysis

Correlated Motion

At low frequencies, the base moves as a rigid body — all supports move together. The coherence is near 1 and the inputs are fully correlated. In this regime, the standard single-input base excitation analysis (applying the same PSD to all supports) is a good approximation. At higher frequencies, the base flexes — supports move differently. The coherence drops and the inputs become partially correlated. In this regime, single-input analysis over-simplifies the condition.

Relative Motion

The structural response is driven by the relative motion between supports — not by the absolute motion. If all supports move identically (rigid-body motion), the structure moves with them and there is no relative excitation. The response comes from the differential motion between supports — the bending, twisting and differential displacement that the structure must accommodate. This is why the coherence and phase between supports matter: they determine the differential motion.

The structural response is driven by relative motion between supports, not absolute motion. Rigid-body motion of all supports together produces no structural response. The differential motion — controlled by coherence and phase — drives the response.

Structural Modes

The structural modes determine how the multi-support excitation is converted into response. Modes that involve differential motion between supports are excited by multi-support excitation. Modes that involve rigid-body motion (all supports moving together) are not excited. The mode shapes therefore determine which modes are excited by multi-support excitation and which are not. This is fundamentally different from single-input base excitation, which excites all modes.

• Modes with differential support motion — excited by multi-support excitation
• Modes with rigid-body support motion — not excited by multi-support excitation
• Mode shapes determine which modes are excited
• Different from single-input base excitation, which excites all modes

Equipment Qualification

For equipment qualification, the standard approach is to apply a single PSD at all supports simultaneously. This is conservative if the real environment has partially correlated supports — the single-input case produces more response than the actual partially correlated case. However, it may not be conservative if the real environment has support motion in anti-phase at critical frequencies. The qualification approach should be based on the actual expected support conditions, not on a default assumption.

Single-PSD qualification is conservative for partially correlated supports but may not be conservative for anti-phase support motion. The qualification approach should reflect the expected support conditions.

Analysis Approaches

Several analysis approaches are available for multi-support excitation, ranging from simple to complex. The choice depends on the available data and the required accuracy.

• Single-input — apply same PSD to all supports; simple, conservative for correlated supports
• Independent multi-input — different PSDs at each support, no correlation; use when supports are independent
• Cross-spectral multi-input — full cross-spectral matrix; most accurate, requires cross-spectral data
• Envelope approach — use a conservative envelope PSD; simple but potentially over-conservative

Key Takeaways

• Multi-support excitation is the most common real-world configuration
• Support correlation depends on spacing and wavelength — high at low frequency, low at high frequency
• Structural response is driven by relative motion between supports, not absolute motion
• Single-input base excitation is an approximation — reasonable at low frequency, inadequate at high frequency
• For accurate assessment, use cross-spectral multi-support analysis with the actual support conditions