1. Overview
Every structure has natural frequencies at which it tends to vibrate. When a machine's forcing frequency coincides with a foundation's natural frequency, resonance occurs, causing vibration amplitudes 5 to 25 times greater than the static response. Natural frequency analysis is the primary tool for preventing resonance in compressor foundation design.
Natural Frequency
f_n = sqrt(K/M) / 2pi
Fundamental frequency of foundation-soil system
Soil Springs
K (lb/ft)
Elastic soil stiffness in each direction
Forcing Frequency
RPM / 60 Hz
Machine operating speed and harmonics
Frequency Ratio
r = f_op / f_n
Must be outside 0.7-1.4 range
Foundation as a Spring-Mass System
A block foundation resting on soil behaves as a mass supported by springs. The soil provides elastic stiffness (springs) and energy dissipation (damping). The foundation block plus machine represents the total mass.
| Component | Physical Analog | Parameter |
|---|---|---|
| Foundation + machine | Mass (M) | Total weight / g (slugs) |
| Soil beneath foundation | Spring (K) | Spring constant (lb/ft or lb/in) |
| Soil damping | Dashpot (C) | Radiation + material damping |
| Machine dynamic force | Applied force (F) | Unbalanced forces at 1x, 2x RPM |
Lumped parameter model: For block foundations, the rigid body assumption is valid when the foundation has no internal flexural modes below 2x the highest forcing frequency. This is typically satisfied for block foundations with depth-to-width ratios greater than 0.6. Table-top (elevated) foundations require structural frequency analysis in addition to soil-structure analysis.
2. Soil Spring Constants
Soil spring constants represent the elastic stiffness of the soil in each direction. They depend on soil properties, foundation geometry, and embedment depth. Accurate spring constants are essential for reliable frequency predictions.
Barkan/Lysmer Spring Constants
Vertical Spring (K_z):
K_z = C_u x A
Where:
C_u = Coefficient of uniform compression (lb/ft3)
A = Foundation base area (ft2)
Horizontal Spring (K_x or K_y):
K_x = C_tau x A
Where:
C_tau = Coefficient of uniform shear (lb/ft3)
Typically: C_tau = 0.5 x C_u
Rocking Spring (K_phi):
K_phi = C_phi x I
Where:
C_phi = Coefficient of non-uniform compression (lb/ft3)
I = Area moment of inertia of base about rocking axis (ft4)
Typically: C_phi = 2 x C_u
Torsional (Yawing) Spring (K_psi):
K_psi = C_psi x I_z
Where:
C_psi = Coefficient of non-uniform shear (lb/ft3)
I_z = Polar moment of inertia of base area (ft4)
Typically: C_psi = 0.75 x C_u
Elastic Half-Space Spring Constants
Based on shear modulus G and Poisson's ratio nu:
Vertical:
K_z = [4 x G x r_0] / (1 - nu)
Horizontal:
K_x = [32 x (1 - nu) x G x r_0] / (7 - 8*nu)
Rocking:
K_phi = [8 x G x r_0^3] / [3 x (1 - nu)]
Torsional:
K_psi = [16 x G x r_0^3] / 3
Where:
G = Shear modulus of soil (lb/ft2 or psi)
nu = Poisson's ratio (0.25-0.45 for most soils)
r_0 = Equivalent radius of foundation base
For rectangular foundation (L x B):
r_0_vertical = sqrt(L x B / pi)
r_0_horizontal = sqrt(L x B / pi)
r_0_rocking = (L x B^3 / (3*pi))^0.25 (about longer axis)
r_0_torsion = ((L x B) x (L^2 + B^2) / (6*pi))^0.25
Typical Soil Spring Values
| Soil Type | C_u (lb/ft3) | G (psi) | nu | Notes |
|---|---|---|---|---|
| Soft clay | 20,000-50,000 | 1,000-5,000 | 0.40-0.45 | Poor for dynamic equipment |
| Stiff clay | 50,000-150,000 | 5,000-15,000 | 0.35-0.40 | Adequate with higher MR |
| Loose sand | 30,000-80,000 | 3,000-8,000 | 0.25-0.30 | May densify under vibration |
| Dense sand/gravel | 100,000-300,000 | 10,000-25,000 | 0.25-0.35 | Good for dynamic foundations |
| Compacted fill | 80,000-200,000 | 8,000-20,000 | 0.25-0.35 | Must be well-compacted |
| Weathered rock | 200,000-500,000 | 20,000-50,000 | 0.20-0.30 | Excellent; stiffest response |
| Intact rock | 500,000+ | 50,000+ | 0.15-0.25 | Best possible site condition |
Embedment Correction
Effect of Foundation Embedment:
Embedment increases effective spring constants due to soil
contact on the sides of the foundation.
Correction factors (Novak method):
eta_z = 1 + 0.6 x (1 - nu) x (D_embed / r_0)
eta_x = 1 + 0.55 x (2 - nu) x (D_embed / r_0)
eta_phi = 1 + 1.2 x (1 - nu) x (D_embed / r_0) + 0.2 x (2 - nu) x (D_embed / r_0)^3
Where:
D_embed = Embedment depth (ft)
r_0 = Equivalent foundation radius (ft)
Corrected springs:
K_z_embed = K_z_surface x eta_z
K_x_embed = K_x_surface x eta_x
Typical embedment effect:
2 ft embedment: 10-20% increase in spring constants
4 ft embedment: 20-40% increase
6 ft embedment: 30-60% increase
Embedment increases natural frequencies, which can help
move out of the resonance avoidance zone.
Soil variability: Soil spring constants can vary by a factor of 2 or more across a site. ACI 351.3R recommends using both upper-bound and lower-bound soil properties in the analysis to verify that the frequency ratio stays outside the avoidance zone (0.7-1.4) for both cases. If the avoidance zone cannot be avoided with this range, the design must be modified.
3. Frequency Calculations
Natural frequencies are calculated for all six degrees of freedom. Vertical and yawing modes are uncoupled; horizontal translation and rocking are coupled and must be solved simultaneously.
Uncoupled Natural Frequencies
Vertical Natural Frequency:
f_z = (1/2*pi) x sqrt(K_z / M_total)
Where:
K_z = Vertical spring constant (lb/ft)
M_total = (W_machine + W_foundation) / g (slugs)
g = 32.174 ft/s2
Yawing (Torsional) Natural Frequency:
f_psi = (1/2*pi) x sqrt(K_psi / J_z)
Where:
K_psi = Torsional spring constant (ft-lb/rad)
J_z = Mass moment of inertia about vertical axis (slug-ft2)
J_z = M_total x (L^2 + B^2) / 12
For rectangular block with length L and width B.
Coupled Horizontal-Rocking Frequencies
Coupled Mode Equation:
Horizontal translation and rocking about the perpendicular axis
are coupled because the center of gravity is above the base.
The coupled natural frequencies are found from:
f^4 - f^2 x (f_x^2 + f_phi^2) / (1 - alpha) + (f_x^2 x f_phi^2) / (1 - alpha) = 0
Where:
f_x = Uncoupled horizontal frequency = sqrt(K_x / M) / (2*pi)
f_phi = Uncoupled rocking frequency = sqrt(K_phi / J_phi) / (2*pi)
alpha = M x h^2 / J_phi (coupling parameter)
h = Height of CG above base
J_phi = Mass moment of inertia about rocking axis
Solution (quadratic in f^2):
f_1^2, f_2^2 = [(f_x^2 + f_phi^2) +/- sqrt((f_x^2 + f_phi^2)^2 - 4*(1-alpha)*f_x^2*f_phi^2)] / [2*(1-alpha)]
f_1 = lower coupled frequency (combined rocking-sliding)
f_2 = higher coupled frequency (combined sliding-rocking)
Both f_1 and f_2 must be checked against forcing frequencies.
Summary of All Six Frequencies
| Mode | Type | Equation | Typical Range |
|---|---|---|---|
| 1. Vertical (z) | Uncoupled | f = sqrt(K_z/M) / 2pi | 5-25 Hz |
| 2. Lateral (x) | Coupled with #4 | Coupled equation | 3-20 Hz |
| 3. Longitudinal (y) | Coupled with #5 | Coupled equation | 3-20 Hz |
| 4. Rocking (about x) | Coupled with #2 | Coupled equation | 4-25 Hz |
| 5. Rocking (about y) | Coupled with #3 | Coupled equation | 4-25 Hz |
| 6. Yawing (about z) | Uncoupled | f = sqrt(K_psi/J_z) / 2pi | 3-15 Hz |
Practical note: The vertical mode is usually the highest frequency (stiffest) and is most often the mode closest to resonance for slow-speed reciprocating compressors. The rocking modes are typically the lowest and most critical for high-speed centrifugal machines. Always check all modes against all harmonics (1x, 2x, 3x, 4x RPM).
4. Resonance Avoidance
The primary goal of natural frequency analysis is to verify that no natural frequency falls within the resonance avoidance zone for any machine operating frequency or its harmonics.
Avoidance Zone Definition
ACI 351.3R Frequency Ratio Limits:
For each natural frequency f_n and each forcing frequency f_f:
r = f_f / f_n
Acceptable ranges:
r < 0.70 (sub-critical: well below resonance)
r > 1.40 (super-critical: well above resonance)
Avoidance zone:
0.70 < r < 1.40 (NEVER design in this range)
Forcing frequencies to check:
1x RPM = Operating speed
2x RPM = Second harmonic (major for reciprocating)
3x RPM = Third harmonic (significant for some machines)
4x RPM = Fourth harmonic (check if applicable)
Blade passing = RPM x Number_of_blades (centrifugal)
Electrical: 2x line frequency = 120 Hz for 60 Hz systems
With soil variability (upper/lower bound):
f_n varies by factor of sqrt(K_upper/K_lower)
Typical soil variability: K varies by factor of 2
Therefore f_n varies by factor of sqrt(2) = 1.41
Must clear avoidance zone for BOTH upper and lower f_n values
Resonance Check Matrix
| Mode | f_n (Hz) | 1x (Hz) | r_1x | 2x (Hz) | r_2x | Status |
|---|---|---|---|---|---|---|
| Vertical | 12.5 | 15.0 | 1.20 | 30.0 | 2.40 | FAIL (1x in zone) |
| Lateral-1 | 6.2 | 15.0 | 2.42 | 30.0 | 4.84 | PASS |
| Lateral-2 | 8.1 | 15.0 | 1.85 | 30.0 | 3.70 | PASS |
| Rock-1 | 5.8 | 15.0 | 2.59 | 30.0 | 5.17 | PASS |
| Rock-2 | 9.5 | 15.0 | 1.58 | 30.0 | 3.16 | PASS |
| Yawing | 7.0 | 15.0 | 2.14 | 30.0 | 4.29 | PASS |
Common resonance scenario: A 900 RPM (15 Hz) reciprocating compressor on medium-stiff soil often has a vertical natural frequency near 12-18 Hz, placing the 1x forcing in or near the avoidance zone. Solutions include increasing foundation mass (lowers f_n) or soil improvement (raises f_n) to move r outside 0.7-1.4.
5. Design Strategies
When initial analysis shows a natural frequency in the avoidance zone, several strategies can shift the frequency ratio to an acceptable range.
Sub-Critical Design (r < 0.7)
Goal: Make f_n HIGHER than f_forcing / 0.7
f_n_required > f_forcing / 0.7
Methods to increase f_n:
1. Increase soil stiffness (K):
Soil improvement: compaction, grouting, piling
Pile foundations: dramatically increase K
Effect: f_n ~ sqrt(K), so doubling K increases f_n by 41%
2. Decrease total mass (M):
Reduce foundation size (contradicts mass ratio need!)
Conflict: Lower mass raises f_n but increases vibration amplitude
Only feasible if mass ratio is already well above minimum
3. Increase base area (A):
Larger base increases K proportionally
Also increases moment of inertia for rocking resistance
Most effective single change for vertical mode
Sub-critical is the preferred approach for most compressor
foundations because it also provides the lowest vibration amplitude.
Super-Critical Design (r > 1.4)
Goal: Make f_n LOWER than f_forcing / 1.4
f_n_required < f_forcing / 1.4
Methods to decrease f_n:
1. Increase total mass (M):
Deeper/larger foundation block
Effect: f_n ~ 1/sqrt(M), so doubling M decreases f_n by 29%
Also improves mass ratio (double benefit)
2. Decrease soil stiffness (K):
Add resilient layer (cork, rubber) under foundation
Spring-mounted foundation (vibration isolation)
Effect: f_n ~ sqrt(K), so halving K decreases f_n by 29%
3. Pile-supported with isolation:
Piles to transfer load; flexible isolation layer on top
Provides bearing capacity without high stiffness
Super-critical design accepts passage through resonance
during startup and shutdown. Adequate damping is required to limit
transient vibration during speed transitions. This is acceptable
for machines that start/stop infrequently.
Strategy Comparison
| Strategy | Pros | Cons | Best For |
|---|---|---|---|
| Sub-critical | Lowest vibration; no resonance passage | Requires stiff soil or piles | Reciprocating; fixed speed |
| Super-critical | High mass ratio; good damping | Resonance during startup/shutdown | High-speed centrifugal; variable speed |
| Spring isolation | Very low transmitted vibration | Complex; maintenance-intensive | Sensitive environments; retrofit |
| Pile foundation | High stiffness; bearing capacity | Expensive; difficult to modify | Soft soil sites; heavy machines |
Sensitivity Analysis
Parameter Sensitivity on Natural Frequency:
f_n = (1/2*pi) x sqrt(K/M)
Effect of 10% change in each parameter:
K increases 10%: f_n increases 4.9%
K decreases 10%: f_n decreases 5.1%
M increases 10%: f_n decreases 4.7%
M decreases 10%: f_n increases 5.4%
A (base area) increases 10%: f_n increases 4.9% (via K)
A decreases 10%: f_n decreases 5.1%
Most sensitive parameter:
Soil stiffness (K) has the highest uncertainty (can vary by 50-100%)
and has the most influence on f_n. This is why soil investigation
quality is the most important factor in foundation design.
Soil investigation recommendation: For compressor foundations, perform site-specific dynamic soil testing (SASW, crosshole, or downhole) rather than relying on standard geotechnical borings. Static soil properties (SPT, bearing capacity) do not accurately predict dynamic stiffness. Dynamic shear modulus from field testing is 2-5x more accurate than correlations from SPT blow counts.
6. Soil Investigation
The quality of natural frequency predictions depends directly on the quality of soil data. Dynamic soil properties differ significantly from static properties used in conventional geotechnical engineering.
Required Soil Parameters
| Parameter | Symbol | Test Method | Accuracy |
|---|---|---|---|
| Shear modulus | G (psi) | SASW, crosshole, resonant column | Best: +/- 15% |
| Poisson's ratio | nu | Crosshole (Vp and Vs) | +/- 20% |
| Unit weight | gamma (pcf) | Undisturbed sampling | +/- 5% |
| Damping ratio | D | Resonant column, free vibration | +/- 30% |
| Shear wave velocity | Vs (ft/s) | SASW, crosshole, downhole | +/- 10-15% |
| Compression wave velocity | Vp (ft/s) | Crosshole, seismic refraction | +/- 10-15% |
Converting Vs to Shear Modulus
Shear Modulus from Shear Wave Velocity:
G = rho x Vs^2
Where:
G = Shear modulus (lb/ft2)
rho = Mass density = gamma / g (slugs/ft3)
Vs = Shear wave velocity (ft/s)
Poisson's Ratio from Wave Velocities:
nu = (Vp^2 - 2*Vs^2) / (2*(Vp^2 - Vs^2))
Strain Dependence:
G decreases with increasing strain level.
Laboratory G_max (resonant column): at strains < 0.001%
Field G (dynamic foundation): at strains of 0.001-0.01%
Reduction factor: G_design = 0.5 to 0.8 x G_max
Always use strain-compatible G values.
G_max from low-strain tests overestimates stiffness at
operational strain levels, leading to non-conservative
(too high) frequency predictions.
Test Methods Comparison
| Method | Depth Range | Cost | Accuracy | Notes |
|---|---|---|---|---|
| SASW (surface waves) | 0-100 ft | Low-moderate | Good | Non-invasive; no boreholes needed |
| Crosshole | Any (between boreholes) | High | Best | Requires 2-3 cased boreholes |
| Downhole | Any (single borehole) | Moderate | Good | One cased borehole required |
| Resonant column (lab) | Sample depth | Moderate | Excellent (G_max) | Disturbed sample issue for sands |
| SPT correlation | Any | Lowest | Poor (factor of 2-3) | Last resort only; not recommended |
Design recommendation: For compressor foundations exceeding 1,000 HP, always perform field dynamic soil testing (SASW minimum, crosshole preferred). The cost of dynamic soil testing ($5K-$15K) is small compared to the cost of a foundation redesign or chronic vibration problems ($50K-$500K). SPT-based correlations should only be used for preliminary estimates.
7. Worked Examples
Example 1: Vertical Natural Frequency
Given:
Foundation: 24 ft x 10 ft x 8 ft deep
Concrete density: 150 lb/ft3
Machine weight: 93,000 lb (total package)
Soil C_u = 150,000 lb/ft3
Embedment: 4 ft
Step 1: Foundation weight
W_fdn = 24 x 10 x 8 x 150 = 288,000 lb
Step 2: Total mass
M_total = (93,000 + 288,000) / 32.174 = 11,837 slugs
Step 3: Base area
A = 24 x 10 = 240 ft2
Step 4: Vertical spring constant (surface)
K_z = C_u x A = 150,000 x 240 = 36,000,000 lb/ft
Step 5: Embedment correction
r_0 = sqrt(240 / pi) = 8.74 ft
eta_z = 1 + 0.6 x (1 - 0.30) x (4 / 8.74) = 1 + 0.192 = 1.192
K_z_embed = 36,000,000 x 1.192 = 42,912,000 lb/ft
Step 6: Vertical natural frequency
f_z = (1/2*pi) x sqrt(42,912,000 / 11,837)
f_z = (1/6.283) x sqrt(3,625) = 0.1592 x 60.2 = 9.58 Hz
Step 7: Check against forcing (900 RPM = 15 Hz)
r_1x = 15 / 9.58 = 1.57 (> 1.40 -- PASS, super-critical)
r_2x = 30 / 9.58 = 3.13 (> 1.40 -- PASS)
Vertical mode is in the super-critical range. Acceptable.
Example 2: Soil Variability Check
Given (from Example 1):
f_z = 9.58 Hz at C_u = 150,000 lb/ft3
Soil variability: +/- 50% on C_u
Lower bound: C_u = 75,000 lb/ft3
K_z_lower = 75,000 x 240 x 1.192 = 21,456,000 lb/ft
f_z_lower = (1/2*pi) x sqrt(21,456,000 / 11,837) = 6.78 Hz
r_1x = 15 / 6.78 = 2.21 (PASS -- super-critical)
Upper bound: C_u = 225,000 lb/ft3
K_z_upper = 225,000 x 240 x 1.192 = 64,368,000 lb/ft
f_z_upper = (1/2*pi) x sqrt(64,368,000 / 11,837) = 11.74 Hz
r_1x = 15 / 11.74 = 1.28 (FAIL -- in avoidance zone 0.7-1.4!)
Problem: At upper-bound soil stiffness, the vertical
frequency falls in the avoidance zone for 1x forcing.
Solutions:
a) Increase foundation mass to lower f_z_upper below 15/1.4 = 10.7 Hz
Need f_z_upper < 10.7 Hz
Need K/M < (2*pi*10.7)^2 = 4,516
Need M > 64,368,000 / 4,516 = 14,252 slugs = 458,700 lb total
Need W_fdn > 458,700 - 93,000 = 365,700 lb -> MR = 3.93
Increase depth from 8 ft to 10.2 ft
b) Reduce base area to lower K_upper
Less effective because it also reduces mass
Recommended: Option (a) - increase foundation depth to 10.5 ft
Example 3: Coupled Rocking-Sliding Check
Given (from Example 1 with 10.5 ft depth):
Foundation: 24 x 10 x 10.5 ft
W_total = 93,000 + (24 x 10 x 10.5 x 150) = 93,000 + 378,000 = 471,000 lb
M_total = 471,000 / 32.174 = 14,636 slugs
CG height above base: h = 10.5/2 + 2.0 = 7.25 ft
(assuming machine CG is 2 ft above top of foundation)
Step 1: Uncoupled horizontal frequency (x-direction)
K_x = C_tau x A = 75,000 x 240 = 18,000,000 lb/ft (using lower bound)
f_x = sqrt(18,000,000 / 14,636) / (2*pi) = sqrt(1,230) / 6.283 = 5.58 Hz
Step 2: Uncoupled rocking frequency (about y-axis)
I_y = (10 x 24^3) / 12 = 11,520 ft4
K_phi = C_phi x I_y = 300,000 x 11,520 = 3,456,000,000 ft-lb/rad
J_phi = M_total x [(10^2 + 10.5^2)/12 + h^2]
J_phi = 14,636 x [17.52 + 52.56] = 14,636 x 70.08 = 1,025,431 slug-ft2
f_phi = sqrt(3,456,000,000 / 1,025,431) / (2*pi) = sqrt(3,370) / 6.283 = 9.24 Hz
Step 3: Coupling parameter
alpha = M_total x h^2 / J_phi = 14,636 x 52.56 / 1,025,431 = 0.750
Step 4: Coupled frequencies
a = (f_x^2 + f_phi^2) / (1-alpha) = (31.1 + 85.4) / 0.25 = 466.0
b = f_x^2 x f_phi^2 / (1-alpha) = 31.1 x 85.4 / 0.25 = 10,624
f^2 = [466.0 +/- sqrt(466^2 - 4 x 10,624)] / 2
f^2 = [466.0 +/- sqrt(217,156 - 42,496)] / 2
f^2 = [466.0 +/- 417.9] / 2
f_1^2 = (466.0 - 417.9) / 2 = 24.05 -> f_1 = 4.90 Hz
f_2^2 = (466.0 + 417.9) / 2 = 441.95 -> f_2 = 21.02 Hz
Step 5: Check ratios at 15 Hz (1x)
r_1 = 15 / 4.90 = 3.06 (PASS -- super-critical)
r_2 = 15 / 21.02 = 0.71 (MARGINAL -- barely outside 0.70)
The second coupled mode is very close to the avoidance zone boundary.
Consider increasing foundation mass or width to improve this margin.
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