Physical Address
304 North Cardinal St.
Dorchester Center, MA 02124
ASME SI-8-1976 — SI Units in Vibration (First Edition)
🔄 ASME SI-8-1976 is the eighth booklet in the ASME International System (SI) of Units series, dedicated to SI units in vibration, first published in March 1976. A vibration of a machine (or of any body) occurs when the body executes a periodic motion about an equilibrium position; periodic motion is motion that repeats itself at definite intervals of time. The easiest type of vibratory motion to study is simple harmonic motion (SHM), in which the acceleration is negatively proportional to the displacement (usually from the midpoint of the motion).
💡 Core teaching point of the booklet: in SI the length unit is the meter or millimeter and mass is in kilograms, springs are rated in newtons per meter (N/m; a rating in N/mm multiplied by 1000 gives N/m); in the U.S. Customary system, inches, pounds, and seconds are used, springs are rated in pounds per inch, and the acceleration of gravity is taken as 386 in/s² — the same problem is far clearer and more intuitive in SI.
1. Overview and Scope 📌
Vibration is an important subject in machine dynamics. Free vibrations occur when the body is acted upon by forces that depend on the motion of the system (e.g., springs or other elastic media); forced vibrations occur when an external periodic force is additionally applied. The natural frequency is the frequency of free vibration (the number of times the motion repeats itself per unit of time), and the period is the time required to complete one cycle. The booklet guides the reader through the transition from U.S. Customary to SI units in vibration analysis.
The booklet is organized as follows: Section 1, Introduction (SHM, free and forced vibration, natural frequency and period); Section 2, Units and Conversion Factors; Section 3, Solved Problems; Section 4, Practice Problems with Answers. Tables 1–9 cover the principal quantities, derived units, and dedicated conversion tables for length, velocity, acceleration, mass, mass moment of inertia, force, and spring constant.
⚠️ This booklet is a conversion aid, not a vibration textbook; use it together with a formal mechanical vibration course. Its examples cover the basic problem types but provide enough information for an easy transition.
2. Principal Quantities and Unit Systems 🔧
The principal quantities encountered in vibration, in both U.S. Customary and SI units, are tabulated below:
| Quantity | U.S. Customary Unit (Symbol) | SI Unit (Symbol) |
|---|---|---|
| Linear acceleration | ft/s² | m/s² |
| Angular acceleration | rad/s² | rad/s² |
| Force | lbf | newton (N) |
| Mass | slug (lbf·s²/ft) | kilogram (kg) |
| Mass moment of inertia | slug·ft² | kg·m² |
| Energy / work | ft·lbf | joule (J) |
| Power | horsepower (hp) | watt (W) |
| Impulse / momentum | lbf·s | N·s |
| Spring constant (force per unit deformation) | lbf/in, lbf/ft | N/m, N/mm |
| Stress | lbf/in² (psi) | pascal (Pa) |
| Rotational frequency | rev/s, rad/s | s⁻¹, rad/s |
| Linear velocity | ft/s | m/s |
| Angular velocity | rad/s | rad/s |
| Measured temperature | °F | °C |
Derived SI units: energy J = N·m = m²·kg·s⁻²; force N = m·kg·s⁻²; power W = J/s; stress Pa = N/m²; impulse and momentum are both N·s = m·kg·s⁻¹. In the vibration equation m·ẍ + c·ẋ + k·x = F(t), use m in kg, c in N·s/m, k in N/m, and F in N, and every term is dimensionally consistent.
🚨 Spring stiffness is a frequent source of errors: converting lbf/in to N/m requires multiplying by 175.126 (1 lbf/in ≈ 175.13 N/m); if a spring is rated in N/mm, remember 1 N/mm = 1000 N/m. In the natural-frequency formula ωn = √(k/m), k (N/m) and m (kg) must be mutually consistent or the frequency will be off by a factor of 1000.
3. Dedicated Conversion Tables (Length, Velocity, Acceleration, Mass, Moment of Inertia, Force, Spring Constant) ⚖️
Section 2 gives the conversion factors most commonly used in vibration analysis in a row-by-column (multiply) format:
| Quantity | Conversion | Factor |
|---|---|---|
| Length | 1 ft to m | 0.3048 (1 in = 0.0254 m) |
| Velocity | 1 ft/s to m/s | 0.3048 |
| Acceleration | 1 ft/s² to m/s² | 0.3048 |
| Mass | 1 lbm to kg | 0.4536 (0.453592 exact) |
| Mass moment of inertia | 1 slug·ft² to kg·m² | 1.3558 |
| Force | 1 lbf to N | 4.4482 (4.448222 exact) |
| Spring constant | 1 lbf/in to N/m | 1.7513 × 10² |
| Spring constant | 1 lbf/ft to N/m | 1.4594 × 10¹ |
| Spring constant | 1 N/mm to N/m | 1.0 × 10³ |
| Spring constant | 1 lbf/in to N/mm | 1.7513 × 10⁻¹ |
For example, converting 8 ft to meters gives 8 × 0.3048 = 2.4384 m (and converting back, multiply by 3.2808). 1 N/mm equals 5.7102 lbf/in, 1 lbf/in equals 175.13 N/m, and 1 lbf/ft equals 14.594 N/m — the stiffness conversions most used in machine-support and vibration-isolation design.
💡 Quick engineering magnitudes: typical machine-isolation spring stiffnesses range from 10⁴ to 10⁶ N/m with natural frequencies from about 5 to 50 Hz; whether the frequency ratio r = ω/ωn between the rotating excitation and the support natural frequency stays away from 1.0 decides isolation effectiveness and resonance risk.
4. Worked Examples and the SI Approach ✏️
The booklet solves the same single-degree-of-freedom system — a machine with an unbalanced rotating element — in both U.S. Customary and SI units:
- U.S. Customary version: a machine weighing 200 lbf supported by springs of total stiffness k = 4,000 lbf/in, with an unbalanced element producing a disturbing force of 80 lbf at 3,000 r/min and a damping factor ζ = 0.20; the solution gives a natural frequency of 87.86 rad/s (840 r/min), an amplitude of 0.00169 in, a transmissibility ratio T.R. ≈ 0.148, and a transmitted force to the foundation of 11.8 lbf.
- SI version: k ≈ 700,000 N/m, mass m ≈ 90 kg, disturbing force F₀ ≈ 360 N at 3,000 r/min; the solution gives a natural frequency of about 833 r/min, an amplitude of 0.043 mm, a transmissibility of about 0.145, and a transmitted force of 52.4 N.
- Damping-decay example: a massless rigid bar with a 1 oz end mass, a spring k = 10 lbf/in at the midpoint, and a dashpot producing 0.001 lbf at 1 in/s; in SI, k = 1751.27 N/m, with critical damping and decay rate recomputed via the standard formulas.
Notice that the U.S. Customary solution forces the gravitational constant g = 386 in/s² to be written explicitly so that the lbf/in stiffness and the weight work together; the SI solution uses ωn = √(k/m) directly with no g at all — the immediate advantage of the SI system.
⚠️ The g-dependence of U.S. Customary calculations (386 in/s² or 32.174 ft/s²) disappears entirely in SI: use mass in kg and stiffness in N/m and the natural-frequency formula is correct by construction. Before substituting, always verify that k and m are mutual units.
5. Engineering Practice Essentials 🛠️
- 🔍 Vibration isolation design: keep the natural frequency of the equipment–spring support system well below (commonly below 1/√2 times) the dominant excitation frequency, and verify the force transmitted to the foundation with the transmissibility formula; compute stiffness in N/m and mass in kg directly.
- 🔧 Rotor balancing: express unbalance in g·mm (or kg·mm) and compute the disturbance force F = m·e·ω² with m in kg, eccentricity e in m, and ω in rad/s, giving N.
- 📊 Vibration measurement: displacement (m or mm), velocity (mm/s), and acceleration (g or m/s²) sensor calibrations are almost universally SI; keeping analysis inputs and outputs in SI reduces errors.
- 📋 Teaching practice: rewrite U.S. Customary vibration exercises (lbf, lb/in, in/s²) in N, N/m, m/s² and re-solve them, focusing on how g disappears from the natural-frequency formula.
✅ Verification routine: the dimension of ωn = √(k/m) must be √(N/m ÷ kg) = √(kg·m/s² ÷ m ÷ kg) = 1/s → rad/s; if the computed frequency carries other units, k or m is inconsistent — correct the units immediately.
6. Frequently Asked Questions ❓
❓ Q: Why must g = 386 in/s² appear in U.S. Customary vibration calculations?
A: In U.S. engineering units mass and force coexist (lbf and lbm), and when the spring stiffness is rated in lbf/in the mass must be derived as W/g (e.g., 200 lbf ÷ 386 in/s²), so g is unavoidable. In SI, use mass in kg and stiffness in N/m and the natural-frequency formula needs no g.
A: In U.S. engineering units mass and force coexist (lbf and lbm), and when the spring stiffness is rated in lbf/in the mass must be derived as W/g (e.g., 200 lbf ÷ 386 in/s²), so g is unavoidable. In SI, use mass in kg and stiffness in N/m and the natural-frequency formula needs no g.
❓ Q: What is the quick conversion from lbf/in to N/m?
A: 1 lbf/in = 4.448222 N / 0.0254 m ≈ 175.13 N/m (precisely 1.7513 × 10² N/m); conversely 1 N/m ≈ 0.00571 lbf/in. This is the most common stiffness conversion in isolation selection and damper comparisons.
A: 1 lbf/in = 4.448222 N / 0.0254 m ≈ 175.13 N/m (precisely 1.7513 × 10² N/m); conversely 1 N/m ≈ 0.00571 lbf/in. This is the most common stiffness conversion in isolation selection and damper comparisons.
❓ Q: How do I convert between Hz, rad/s, and r/min?
A: ω (rad/s) = 2πf (Hz), so 1 Hz = 6.2832 rad/s; r/min converts to rad/s by multiplying by 2π/60 ≈ 0.10472. For example, 840 r/min ≈ 87.96 rad/s and 3000 r/min ≈ 314.16 rad/s.
A: ω (rad/s) = 2πf (Hz), so 1 Hz = 6.2832 rad/s; r/min converts to rad/s by multiplying by 2π/60 ≈ 0.10472. For example, 840 r/min ≈ 87.96 rad/s and 3000 r/min ≈ 314.16 rad/s.
🔍 Q: How does this booklet coordinate with the rest of the SI series?
A: Start with SI-1 (the orientation guide). Vibration is the specialized application of dynamics (SI-3) and is often used with strength of materials (SI-2, for beam vibration and stress checks) and kinematics (SI-6); rotor dynamics also draws on the mass-moment-of-inertia (kg·m²) conversions.
A: Start with SI-1 (the orientation guide). Vibration is the specialized application of dynamics (SI-3) and is often used with strength of materials (SI-2, for beam vibration and stress checks) and kinematics (SI-6); rotor dynamics also draws on the mass-moment-of-inertia (kg·m²) conversions.
