1. Overview
Compressor foundations experience periodic dynamic forces from unbalanced masses, gas pressure pulsations, and inertia effects. The foundation must be designed so that vibration amplitudes remain below acceptable limits for equipment integrity, personnel comfort, and adjacent structure protection.
Displacement
mils (peak-to-peak)
Primary criterion for low-speed machines
Velocity
in/s or mm/s RMS
ISO 10816 primary criterion
Acceleration
g's
Important for high-frequency excitation
Resonance
Amplification
Avoid r = 0.7-1.3 range
Vibration Sources in Compressor Systems
| Source | Frequency | Magnitude | Mitigation |
|---|---|---|---|
| Primary unbalance | 1x RPM | High | Counterweights, balancing |
| Secondary unbalance | 2x RPM | Moderate | Opposed cylinders, foundation mass |
| Gas pressure pulsation | 1x-6x RPM | Variable | Pulsation bottles, orifice plates |
| Crosshead forces | 1x, 2x RPM | Moderate | Foundation design, anchoring |
| Torque fluctuation | 0.5x-6x RPM | Low-moderate | Flywheel, coupling selection |
| Piping vibration | Various | Variable | Supports, clamps, snubbers |
Design philosophy: The goal is either (1) a massive foundation with natural frequency well below excitation frequency (under-tuned), or (2) a stiff foundation with natural frequency well above excitation (over-tuned). Never allow the natural frequency to fall within 20-30% of any excitation frequency.
2. Dynamic Response Theory
The foundation-soil system is modeled as a single-degree-of-freedom (SDOF) or multi-degree-of-freedom (MDOF) system. The SDOF model captures the essential dynamics for preliminary design.
SDOF Forced Vibration
Equation of Motion:
m * x'' + c * x' + k * x = F_0 * sin(omega * t)
Where:
m = Total mass (foundation + machine) (lb-s^2/in or slugs)
c = Damping coefficient (lb-s/in)
k = Spring stiffness (lb/in)
F_0 = Unbalanced force amplitude (lb)
omega = Excitation circular frequency (rad/s)
Natural Frequency:
omega_n = sqrt(k / m) (rad/s)
f_n = omega_n / (2 * pi) (Hz)
RPM_n = f_n * 60 (RPM)
Frequency Ratio:
r = omega / omega_n = f_excitation / f_natural
Damping Ratio:
D = c / (2 * sqrt(k * m))
Typical for soil: D = 0.02 - 0.10
Steady-State Amplitude
For constant force excitation (F_0 * sin(omega*t)):
X = (F_0 / k) / sqrt[(1 - r^2)^2 + (2*D*r)^2]
Dynamic Magnification Factor (DMF):
DMF = 1 / sqrt[(1 - r^2)^2 + (2*D*r)^2]
At resonance (r = 1):
DMF = 1 / (2*D)
For D = 0.05: DMF = 10 (dangerous amplification)
For rotating unbalance (F = m_e * e * omega^2):
X = (m_e * e / M) * r^2 / sqrt[(1 - r^2)^2 + (2*D*r)^2]
Where:
m_e = Eccentric mass (slugs)
e = Eccentricity (in)
M = Total system mass (slugs)
Phase Angle
Phase lag between force and response:
phi = arctan[2*D*r / (1 - r^2)]
At r = 1 (resonance): phi = 90 deg
At r << 1: phi approaches 0 deg (in phase)
At r >> 1: phi approaches 180 deg (out of phase)
Vibration Parameters Relationship
| Parameter | Symbol | Relationship | Units |
|---|---|---|---|
| Displacement (peak) | X | X | mils or mm |
| Velocity (peak) | V | V = omega * X | in/s or mm/s |
| Acceleration (peak) | a | a = omega^2 * X | in/s^2 or g |
| RMS from peak | - | RMS = Peak / sqrt(2) | Same units |
| Peak-to-peak | X_pp | X_pp = 2 * X_peak | mils or mm |
3. Vibration Limits & Standards
Multiple industry standards define acceptable vibration levels. The applicable standard depends on machine type, speed, and mounting configuration.
ISO 10816 Vibration Severity Classification
ISO 10816 defines vibration severity zones based on RMS velocity measured on the bearing housing or foundation.
| Zone | Velocity (mm/s RMS) | Velocity (in/s RMS) | Condition |
|---|---|---|---|
| Zone A | < 1.8 | < 0.071 | Newly commissioned; excellent |
| Zone B | 1.8 - 4.5 | 0.071 - 0.177 | Acceptable for long-term operation |
| Zone C | 4.5 - 11.2 | 0.177 - 0.441 | Restricted operation; corrective action |
| Zone D | > 11.2 | > 0.441 | Severe; damage imminent |
Note: Values shown are for Group 2 machines (medium machines, 15-300 kW, rigid foundation). ISO 10816-3 provides specific criteria by machine class.
API 684 Criteria for Rotating Equipment
| Criterion | Limit | Application |
|---|---|---|
| Foundation amplitude | < 1.0 mil p-p at operating speed | Block foundations |
| Foundation velocity | < 0.12 in/s peak | All machinery foundations |
| Bearing housing vibration | < sqrt(12,000/RPM) mils p-p | API 617, 618 machines |
| Amplification at resonance | < 5x static deflection | Design limit for damping |
ACI 351.3R Amplitude Limits
Allowable vibration displacement (ACI 351.3R):
For reciprocating machines:
X_allow = 150 / f (mils peak, where f = frequency in CPM)
Example: At 600 RPM: X_allow = 150/600 = 0.25 mils peak
For rotating machines:
X_allow = 80 / f (mils peak)
Velocity-based criterion:
V_allow < 0.10 in/s peak (general)
V_allow < 0.06 in/s peak (sensitive equipment nearby)
V_allow < 0.20 in/s peak (robust industrial)
Frequency Ratio Design Criteria
| Frequency Ratio (r) | Design Region | DMF (D=0.05) | Assessment |
|---|---|---|---|
| < 0.5 | Under-tuned (safe) | < 1.33 | Massive foundation approach |
| 0.5 - 0.7 | Caution zone | 1.33 - 2.13 | Verify amplitude acceptable |
| 0.7 - 1.3 | Resonance zone | 2.13 - 10.0+ | UNACCEPTABLE - redesign |
| 1.3 - 1.5 | Caution zone | 2.13 - 1.33 | Verify amplitude acceptable |
| > 1.5 | Over-tuned (safe) | < 1.33 | Stiff foundation approach |
Critical requirement: The foundation natural frequency must not coincide with 1x RPM, 2x RPM, or any significant harmonic of the operating speed. ACI 351.3R requires checking all harmonics up to the 6th order for reciprocating compressors.
4. Transmissibility & Isolation
Transmissibility describes the fraction of dynamic force transmitted from the machine through the foundation to the soil (or conversely, from the soil to the machine).
Force Transmissibility:
TR = F_transmitted / F_applied
TR = sqrt[1 + (2*D*r)^2] / sqrt[(1 - r^2)^2 + (2*D*r)^2]
For isolation (TR < 1.0):
Required: r > sqrt(2) = 1.414 (regardless of damping)
Isolation efficiency:
IE = (1 - TR) * 100%
At r = 3.0, D = 0.05: TR = 0.125 (87.5% isolation)
At r = 5.0, D = 0.05: TR = 0.042 (95.8% isolation)
Isolation System Selection
| Isolator Type | Frequency Range | Isolation % | Application |
|---|---|---|---|
| Steel springs | 2-8 Hz | 85-98% | Large reciprocating compressors |
| Rubber pads (neoprene) | 8-25 Hz | 70-90% | Small to medium machines |
| Air springs | 1-5 Hz | 90-99% | Precision, sensitive equipment |
| Inertia block (no springs) | N/A | 30-60% | Mass addition for amplitude reduction |
| Cork/fiberglass pads | 15-40 Hz | 50-80% | High-frequency isolation |
Important tradeoff: Adding damping reduces peak amplitude at resonance but decreases isolation effectiveness at high frequency ratios. For compressor foundations, moderate damping (D = 0.03-0.08) provides a practical balance between resonance protection and force isolation.
5. Soil-Structure Interaction
The soil provides both stiffness (spring) and energy dissipation (damping) for the foundation system. Soil dynamic properties directly affect natural frequencies and vibration amplitudes.
Equivalent Soil Springs (Rigid Circular Foundation)
Lysmer analog for vertical vibration:
k_v = 4 * G * r_0 / (1 - nu)
Horizontal (sliding):
k_h = 32 * (1 - nu) * G * r_0 / (7 - 8*nu)
Rocking:
k_r = 8 * G * r_0^3 / [3 * (1 - nu)]
Torsion:
k_t = 16 * G * r_0^3 / 3
Where:
G = Dynamic shear modulus (lb/in^2)
r_0 = Equivalent radius of foundation (ft)
r_0 = sqrt(A / pi) for vertical and sliding
r_0 = (4*I / pi)^(1/4) for rocking
nu = Poisson's ratio
A = Foundation base area (ft^2)
I = Moment of inertia of base area (ft^4)
Radiation Damping
Geometric (radiation) damping ratios:
Vertical: D_v = 0.425 / sqrt(B_z)
B_z = (1 - nu) * m / (4 * rho * r_0^3)
Horizontal: D_h = 0.288 / sqrt(B_x)
B_x = (7 - 8*nu) * m / (32 * (1-nu) * rho * r_0^3)
Rocking: D_r = 0.15 / [(1 + B_theta) * sqrt(B_theta)]
B_theta = 3*(1-nu)*I_theta / (8 * rho * r_0^5)
Where:
rho = Soil mass density (slugs/ft^3)
I_theta = Mass moment of inertia about rocking axis
B_z, B_x, B_theta = Dimensionless mass ratios
Note: Radiation damping is typically 3-15% for vertical,
2-8% for horizontal, and 1-5% for rocking.
Coupled Rocking-Sliding Mode
Horizontal and rocking motions are coupled because horizontal forces applied above the foundation base create moments. A coupled analysis is required for accurate results.
| Mode | DOF | Typical Natural Freq | Primary Concern |
|---|---|---|---|
| Vertical | 1 | 15-40 Hz | Vertical displacement at base |
| Horizontal | 1 | 10-30 Hz | Sliding at CG elevation |
| Rocking | 1 | 8-25 Hz | Angular rotation, top-of-block amplitude |
| Coupled rock-slide | 2 | Two frequencies | Most critical for tall foundations |
| Torsion | 1 | 20-60 Hz | Uneven cylinder firing, moment couples |
6. Worked Examples
Example 1: Vertical Vibration Amplitude
Given:
Reciprocating compressor: 900 RPM, primary unbalanced force = 5,000 lb
Foundation + machine mass: 80,000 lb (m = 80,000/386.4 = 207 slugs)
Soil: Dense sand, G = 3,000 ksf, nu = 0.30
Foundation base: 10 ft x 14 ft
Step 1: Equivalent radius
A = 10 * 14 = 140 ft^2
r_0 = sqrt(140 / pi) = 6.68 ft
Step 2: Vertical spring stiffness
k_v = 4 * G * r_0 / (1 - nu)
k_v = 4 * 3,000 * 1,000 * 6.68 / (1 - 0.30) * (1/144)
k_v = 4 * 3,000,000 * 6.68 / 0.70 = 114.5 x 10^6 lb/ft
k_v = 114.5 x 10^6 / 12 = 9.54 x 10^6 lb/in
Step 3: Natural frequency
omega_n = sqrt(k_v / m) = sqrt(9.54 x 10^6 / 207)
omega_n = 214.7 rad/s
f_n = 214.7 / (2 * pi) = 34.2 Hz = 2,050 RPM
Step 4: Frequency ratio
r = 900 / 2,050 = 0.439 (< 0.5, under-tuned -- GOOD)
Step 5: Damping (assume D = 0.05 total)
DMF = 1 / sqrt[(1 - 0.439^2)^2 + (2*0.05*0.439)^2]
DMF = 1 / sqrt[(1 - 0.193)^2 + (0.0439)^2]
DMF = 1 / sqrt[0.651 + 0.00193] = 1 / 0.808 = 1.24
Step 6: Vibration amplitude
X_static = F_0 / k_v = 5,000 / 9.54 x 10^6 = 0.000524 in
X_dynamic = X_static * DMF = 0.000524 * 1.24 = 0.00065 in = 0.65 mils peak
Step 7: Velocity
omega = 900 * 2*pi / 60 = 94.2 rad/s
V = omega * X = 94.2 * 0.00065 = 0.061 in/s peak
V_rms = 0.061 / sqrt(2) = 0.043 in/s = 1.1 mm/s (Zone A -- OK)
Example 2: Checking Multiple Harmonics
Given:
Same compressor at 900 RPM, f_n = 34.2 Hz
Check harmonics:
1x: 900 RPM = 15.0 Hz, r = 15.0/34.2 = 0.44 -- OK
2x: 1,800 RPM = 30.0 Hz, r = 30.0/34.2 = 0.88 -- RESONANCE ZONE
3x: 2,700 RPM = 45.0 Hz, r = 45.0/34.2 = 1.32 -- Caution
4x: 3,600 RPM = 60.0 Hz, r = 60.0/34.2 = 1.75 -- OK
Problem: 2x harmonic (secondary unbalance) at r = 0.88
falls in resonance zone (0.7-1.3).
Solutions:
1. Increase foundation mass to lower f_n below 21.4 Hz
(so 2x at 30 Hz gives r > 1.4)
2. Stiffen soil (compaction) to raise f_n above 42.9 Hz
(so 2x at 30 Hz gives r < 0.7)
3. Reduce 2x unbalanced force through opposed-pair balancing
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