1. Overview
The foundation mass ratio is the ratio of the foundation weight (concrete plus soil above the foundation base) to the total machine weight (compressor, driver, skid, and accessories). This ratio is the primary parameter controlling foundation vibration amplitude for dynamically loaded equipment.
Mass Ratio (MR)
W_foundation / W_machine
Primary design parameter for dynamic foundations
Vibration Amplitude
Inversely ~ MR
Higher mass ratio = lower vibration
Concrete Density
150 lb/ft3
Normal weight concrete for foundations
Reinforcement
0.5-1.0% by Area
Each face, each direction (ACI 351.3R)
Why mass matters: A heavier foundation has more inertia to resist the dynamic forces generated by rotating and reciprocating equipment. Insufficient mass leads to excessive vibration, which causes misalignment, bearing damage, piping fatigue, and structural cracking. The mass ratio is the simplest and most effective tool for controlling vibration.
2. Mass Ratio Criteria
Industry guidelines provide minimum mass ratios based on equipment type. These are starting points; final design must be verified by dynamic analysis.
Recommended Mass Ratios
| Equipment Type | Minimum MR | Typical MR | Preferred MR | Source |
|---|---|---|---|---|
| Reciprocating compressor (slow speed < 600 RPM) | 3.0 | 4.0-5.0 | 5.0+ | ACI 351.3R |
| Reciprocating compressor (high speed > 600 RPM) | 3.0 | 3.0-4.0 | 4.0+ | ACI 351.3R |
| Reciprocating with engine driver | 3.0 | 4.0-6.0 | 5.0+ | Manufacturer |
| Centrifugal compressor (table-top) | 2.0 | 2.5-3.0 | 3.0+ | ACI 351.3R |
| Centrifugal compressor (mat) | 2.0 | 2.0-3.0 | 3.0 | ACI 351.3R |
| Gas turbine | 1.5 | 2.0-3.0 | 2.5+ | ACI 351.3R |
| Electric motor (large) | 1.5 | 2.0-2.5 | 2.5 | ACI 351.3R |
| Pump (API 610) | 2.0 | 2.5-3.0 | 3.0 | API 686 |
| Fan / Blower | 2.0 | 2.5-3.5 | 3.0 | ACI 351.3R |
Machine Weight Components
Total Machine Weight:
W_machine = W_compressor + W_driver + W_skid + W_accessories + W_piping
Where:
W_compressor = Compressor frame, cylinders/casing, crossheads, crankshaft
W_driver = Engine, motor, or turbine (dry weight)
W_skid = Structural steel base frame
W_accessories = Coolers, scrubbers, lube oil system, controls
W_piping = Suction/discharge piping to first support (estimate)
Important:
Use WET weights (include lube oil, cooling water, process gas)
Include all bolted-on components
Get weights from vendor certified drawings, not estimates
Foundation Weight:
W_foundation = V_concrete x gamma_concrete + W_soil_above
Where:
V_concrete = Foundation volume (ft3)
gamma_concrete = 150 lb/ft3 (normal weight)
W_soil_above = Weight of backfill above foundation sides (if applicable)
Mass Ratio:
MR = W_foundation / W_machine
Skid-mounted vs. block-mounted: For skid-mounted reciprocating compressors, the skid weight counts as machine weight, not foundation weight. The foundation must still achieve the required mass ratio relative to the total package weight including the skid. A common error is counting the skid as part of the foundation.
3. Dynamic Forces
Dynamic forces generated by the machine are the loads that the foundation must resist. The magnitude and frequency of these forces determine the required mass ratio and foundation stiffness.
Reciprocating Compressor Forces
Primary Unbalanced Force (1x RPM):
F_primary = M_recip x r x omega^2
Where:
M_recip = Reciprocating mass (piston + crosshead + portion of connecting rod)
r = Crank radius (half of stroke)
omega = Angular velocity (rad/s) = 2 x pi x RPM / 60
Secondary Unbalanced Force (2x RPM):
F_secondary = M_recip x r x omega^2 x (r/L)
Where:
L = Connecting rod length
r/L = Typically 0.20-0.30
Gas Pressure Force (compressor only):
F_gas = A_piston x (P_discharge - P_suction)
This force acts along the cylinder axis at 1x RPM frequency.
Couple (moment about vertical axis):
M_couple = F_unbalanced x d
Where d = distance between cylinder center and foundation CG.
Manufacturer data required:
Primary forces and moments (1x)
Secondary forces and moments (2x)
Resultant forces in X, Y, Z directions
Phase angles for multi-cylinder arrangements
Centrifugal Compressor Forces
Residual Unbalance Force:
F_unbalance = m_rotor x e x omega^2
Where:
m_rotor = Rotor mass (lb)
e = Residual eccentricity (in)
omega = Angular velocity (rad/s)
API 617 residual unbalance limit:
U_max = 4 x W / N (oz-in)
Where:
W = Journal static weight (lb)
N = Maximum continuous speed (RPM)
Typical unbalance forces:
Well-balanced rotor: 1-5% of rotor weight
Acceptable: 5-10% of rotor weight
Excessive: > 10% of rotor weight
Centrifugal forces are much smaller than reciprocating forces,
which is why lower mass ratios are acceptable.
| Force Type | Reciprocating | Centrifugal | Frequency |
|---|---|---|---|
| Primary unbalance | 5-30% of machine weight | 1-5% of rotor weight | 1x RPM |
| Secondary unbalance | 1-10% of machine weight | Negligible | 2x RPM |
| Gas forces | Significant | Negligible | 1x, 2x RPM |
| Torque variation | Moderate | Small | 1x, 2x RPM |
| Short circuit (motor) | N/A | 2-10x rated torque | Transient |
Multi-cylinder arrangements: Opposed and V-type cylinder arrangements can partially cancel unbalanced forces. However, they create couples (moments) that must be resisted by the foundation. Always obtain the manufacturer's force and moment summary for the specific cylinder arrangement and throw configuration.
4. Dynamic Amplification
Dynamic amplification occurs when the forcing frequency approaches a natural frequency of the foundation-soil system. The amplification factor determines how much the dynamic response exceeds the static response.
Dynamic Amplification Factor (DAF):
DAF = 1 / sqrt[(1 - r^2)^2 + (2 * D * r)^2]
Where:
r = Frequency ratio = f_forcing / f_natural
D = Damping ratio (typically 0.02-0.10 for soil-foundation systems)
Key values:
r = 0 (static): DAF = 1.0
r = 0.5: DAF = 1.15 (acceptable)
r = 0.7: DAF = 1.6 (marginal)
r = 1.0 (resonance): DAF = 1/(2D) = 5-25 (unacceptable!)
r = 1.4: DAF = 0.6 (super-critical; acceptable)
r = 2.0: DAF = 0.3 (well above resonance)
r > 3.0: DAF approaches zero
Design Criteria (ACI 351.3R):
Frequency ratio should be EITHER:
r < 0.70 (sub-critical: foundation frequency above forcing)
OR
r > 1.40 (super-critical: foundation frequency below forcing)
The range 0.70 < r < 1.40 is the AVOIDANCE ZONE.
Never design a foundation with natural frequency in this range.
Effect of Mass Ratio on Natural Frequency
| Mass Ratio | Effect on f_natural | Typical f_natural (vertical) | Implication |
|---|---|---|---|
| 2.0 | Higher frequency | 15-25 Hz | Lighter; may resonate with 2x of high-speed machines |
| 3.0 | Moderate frequency | 10-20 Hz | Good balance for most applications |
| 4.0 | Lower frequency | 8-15 Hz | Reduces resonance risk at 1x RPM |
| 5.0 | Lowest frequency | 6-12 Hz | Best for slow-speed reciprocating |
Natural Frequency of Foundation-Soil System (Vertical):
f_n = (1/2*pi) x sqrt(K_v / M_total)
Where:
K_v = Vertical soil spring constant (lb/ft)
M_total = Total mass (machine + foundation) / g
g = 32.174 ft/s2
Vertical Spring Constant:
K_v = C_u x A_base
Where:
C_u = Coefficient of uniform compression (lb/ft3)
Typical: 50,000-500,000 lb/ft3 depending on soil
A_base = Foundation base area (ft2)
Higher mass -> lower natural frequency
This moves the frequency ratio further from resonance for
sub-critical design (the most common approach for recip compressors).
Resonance avoidance priority: Avoiding resonance is more important than achieving a specific vibration amplitude. A foundation that operates at r = 0.95 (near resonance) with a high mass ratio will still vibrate excessively. Always check frequency ratio BEFORE increasing mass.
5. Vibration Limits
Foundation vibration must be kept within acceptable limits to prevent damage to the machine, piping, and structure. Multiple criteria apply depending on the application.
Vibration Amplitude Limits
| Standard / Criterion | Velocity (in/s peak) | Displacement (mils p-p) | Application |
|---|---|---|---|
| ACI 351.3R (general) | 0.10-0.15 | 1.0-2.0 | Foundation top, near machine |
| ACI 351.3R (recip) | 0.10 | 1.0 | Reciprocating compressor foundation |
| Manufacturer (typical) | 0.10-0.20 | 0.5-2.0 | Varies by manufacturer |
| DIN 4150-3 (structural) | 0.20-0.60 | -- | Effect on adjacent structures |
| Human perception | 0.01-0.04 | 0.1-0.5 | Occupied buildings nearby |
| Sensitive equipment | < 0.01 | < 0.1 | Control rooms, instruments |
Vibration Velocity vs. Displacement
Relationship (sinusoidal vibration):
v = 2 * pi * f * d
Where:
v = Velocity (in/s peak)
f = Frequency (Hz)
d = Displacement amplitude (inches, 0-peak)
Converting between units:
mils p-p = 2000 x d (inches 0-peak)
v (in/s peak) = pi * f * d_mils_pp / 1000
Example:
f = 10 Hz, d = 1.0 mil p-p = 0.001 in p-p = 0.0005 in 0-peak
v = 2 * pi * 10 * 0.0005 = 0.031 in/s peak
At this frequency, 1.0 mil p-p corresponds to 0.031 in/s peak velocity.
Transmissibility to Adjacent Structures
| Distance from Foundation | Typical Attenuation | Notes |
|---|---|---|
| 0-10 ft | 0-30% | Minimal attenuation in soil |
| 10-25 ft | 30-60% | Moderate geometric damping |
| 25-50 ft | 60-80% | Significant reduction |
| 50-100 ft | 80-95% | Generally acceptable for structures |
| > 100 ft | > 95% | Negligible vibration transmitted |
Isolation from adjacent foundations: ACI 351.3R recommends a gap of at least 1 inch between the compressor foundation and adjacent floor slabs or structures. This gap should be filled with compressible filler (not grout) to prevent vibration transmission through direct contact.
6. Design Procedure
Foundation mass ratio design follows a systematic procedure starting with rule-of-thumb sizing and progressing through dynamic analysis to verify adequacy.
Step-by-Step Procedure
| Step | Activity | Input Required | Output |
|---|---|---|---|
| 1 | Gather machine data | Vendor drawings, weights, forces | W_machine, F_dynamic |
| 2 | Obtain soil data | Geotechnical report | Bearing capacity, spring constants |
| 3 | Preliminary sizing | Mass ratio table + machine weight | Foundation dimensions |
| 4 | Calculate natural frequencies | Mass, stiffness, soil springs | 6 DOF natural frequencies |
| 5 | Check frequency ratios | Natural freq vs. forcing freq | r values (must avoid 0.7-1.4) |
| 6 | Calculate vibration amplitude | Forces, DAF, mass | Displacement, velocity |
| 7 | Compare to limits | Calculated vs. allowable | Pass/fail |
| 8 | Iterate if needed | Adjust mass, dimensions, soil treatment | Final design |
Six Degrees of Freedom
Foundation Motion Modes:
A rigid foundation on elastic soil has 6 natural frequencies:
1. Vertical (z-translation): Up/down bouncing
2. Horizontal-X (x-translation): Side-to-side sliding
3. Horizontal-Y (y-translation): Front-to-back sliding
4. Rocking about X (pitch): Nodding motion
5. Rocking about Y (roll): Rolling motion
6. Yawing about Z (torsion): Twisting motion
Coupled modes:
Horizontal translation and rocking are typically coupled
(sliding + tilting occur together)
Vertical and yawing are typically uncoupled
(independent motion)
All 6 natural frequencies must be checked
against all forcing frequencies (1x, 2x, and higher harmonics).
Quick Sizing Rules
| Rule | Guideline | Basis |
|---|---|---|
| Plan dimensions | Extend 6-12 inches beyond machine footprint on each side | Load distribution, bolt edge distance |
| Minimum depth | Depth = 0.6 x shortest plan dimension | Prevent rocking instability |
| CG alignment | Foundation CG within 5% of machine CG horizontally | Minimize rocking excitation |
| Eccentricity | CG offset < 5% of base dimension | ACI 351.3R recommendation |
| Soil bearing | Static + dynamic < allowable bearing pressure | Geotechnical report |
| Embedment | Minimum 2 ft below grade | Frost protection, lateral support |
Center of gravity alignment: The horizontal CG of the foundation should coincide with the CG of the machine within 5% of the base dimension. Eccentricity causes rocking vibration that is difficult to control and often exceeds velocity limits at the machine bearings. This is the most common design error in compressor foundations.
7. Worked Examples
Example 1: Reciprocating Compressor Foundation
Given:
Reciprocating compressor: 2,000 HP, 4-throw, 900 RPM
Compressor weight: 45,000 lb
Engine driver weight: 28,000 lb
Skid weight: 12,000 lb
Accessories (scrubber, cooler): 8,000 lb
W_machine = 45,000 + 28,000 + 12,000 + 8,000 = 93,000 lb
Target mass ratio: 4.0 (high-speed reciprocating)
Step 1: Required foundation weight
W_foundation = MR x W_machine = 4.0 x 93,000 = 372,000 lb
Step 2: Required concrete volume
V = W / gamma = 372,000 / 150 = 2,480 ft3
Step 3: Foundation dimensions
Machine footprint: 22 ft x 8 ft
Foundation plan: 24 ft x 10 ft (1 ft extension each side)
Required depth: 2,480 / (24 x 10) = 10.3 ft
Check depth/width ratio: 10.3 / 10 = 1.03 (acceptable; > 0.6)
Step 4: Verify mass ratio
W_actual = 24 x 10 x 10.3 x 150 = 371,000 lb
MR = 371,000 / 93,000 = 3.99 (rounds to 4.0) -- OK
Step 5: Check soil bearing
Total weight = 93,000 + 371,000 = 464,000 lb
Static bearing = 464,000 / (24 x 10) = 1,933 psf
Requires soil bearing capacity > 3,000 psf (with dynamic factor)
Example 2: Centrifugal Compressor Foundation
Given:
Centrifugal compressor: 8,000 HP, 11,000 RPM
Compressor casing weight: 18,000 lb
Gas turbine driver: 22,000 lb
Gearbox: 8,000 lb
Lube oil console: 5,000 lb
W_machine = 18,000 + 22,000 + 8,000 + 5,000 = 53,000 lb
Target mass ratio: 2.5 (centrifugal compressor)
Step 1: Required foundation weight
W_foundation = 2.5 x 53,000 = 132,500 lb
Step 2: Foundation volume
V = 132,500 / 150 = 883 ft3
Step 3: Table-top foundation sizing
Table-top (elevated) foundation:
Top deck: 18 ft x 10 ft x 2.5 ft = 450 ft3 (67,500 lb)
Columns (4): 2 ft x 2 ft x 6 ft each = 96 ft3 (14,400 lb)
Base mat: 20 ft x 12 ft x 2 ft = 480 ft3 (72,000 lb)
Total concrete: 450 + 96 + 480 = 1,026 ft3 = 153,900 lb
MR = 153,900 / 53,000 = 2.90 (exceeds 2.5 minimum) -- OK
Step 4: Frequency check
f_operating = 11,000 / 60 = 183 Hz (1x)
f_natural_vertical ~ 12-18 Hz (estimated)
r = 12 / 183 = 0.066 (well below 0.70; sub-critical design)
No resonance concern at the primary forcing frequency.
Check 1x compressor speed with table-top deck frequencies.
Example 3: Effect of Mass Ratio on Vibration
Given:
Unbalanced force: F_0 = 5,000 lb at 15 Hz (900 RPM)
Soil vertical spring: K_v = 2,000,000 lb/ft
Damping ratio: D = 0.05
Compare MR = 3.0 vs MR = 5.0 for 93,000 lb machine:
MR = 3.0: W_fdn = 279,000 lb
M_total = (93,000 + 279,000) / 32.174 = 11,562 slugs
f_n = (1/2*pi) * sqrt(2,000,000/11,562) = 2.09 Hz
r = 15 / 2.09 = 7.18
DAF = 1 / sqrt[(1-51.6)^2 + (2*0.05*7.18)^2] = 0.020
d_static = F_0 / K_v = 5,000/2,000,000 = 0.0025 ft
d_dynamic = 0.020 x 0.0025 = 0.00005 ft = 0.60 mils p-p
MR = 5.0: W_fdn = 465,000 lb
M_total = (93,000 + 465,000) / 32.174 = 17,343 slugs
f_n = (1/2*pi) * sqrt(2,000,000/17,343) = 1.71 Hz
r = 15 / 1.71 = 8.77
DAF = 0.013
d_dynamic = 0.013 x 0.0025 = 0.0000325 ft = 0.39 mils p-p
Result: Increasing MR from 3.0 to 5.0 reduces vibration by 35%.
Both are below 1.0 mil limit, but MR = 5.0 provides more margin.
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