Vibration isolation for machine foundations - Guidelines

IS 13301: Key Formulas, Tables & Specifications (Scope Related)

1. Rubber Springs (Clause 5.2)

• Axial Compression Stiffness:

[ k_o = \frac{E + B}{t} \times A \times (1 + 2A + x) ]

Where:

• (k_o) = vertical stiffness (N/mm)
• (t) = rubber pad thickness (mm)
• (A) = bearing area (mm²)
• (E, B, x) = constants from Table 2 (properties of natural rubber)

• Horizontal stiffness:

[ k_n = \frac{G \times A}{t} ]

• Damping ratio: Recommended 5% for preliminary design (range 2%-10%).

2. Metal Springs (Clause 7.1)

• Shear Stress under Axial Load:

[ \tau_v = \frac{8 P D}{\pi d^3} \times C_v ]

Where:

[ C_v =

IS 13301 Key Formulas, Tables & Specifications for Vibration Isolation

1. Rubber Springs (Clause 5.2)

• Vertical stiffness under axial compression:

[ k_o = \frac{E \cdot A}{t} \left(1 + 2A + B \right) ]

• Horizontal stiffness:

[ k_n = \frac{G \cdot A}{t} ]

• Parameters:

• Damping ratio: Recommended 5% for preliminary design (range 2%-10%).

2. Natural Frequency and Isolation (Clause 4)

• Natural frequency of single DOF system:

[ f_n = \frac{1}{2\pi} \sqrt{\frac{g}{\delta}} ]

where
(g) = acceleration due to gravity,
(\delta) = static deflection.

• Effective vibration isolation:
  (f_n < 0.4 f_m) (where (f_m\

IS 13301 Key Definitions & Formulas

1. Natural Frequency (fn) of Single Degree Freedom System

[ f_n = \frac{1}{2\pi} \sqrt{\frac{k}{m}} = \frac{1}{2\pi} \sqrt{\frac{g}{\delta}} ]

• (k) = stiffness
• (m) = mass
• (g) = acceleration due to gravity
• (\delta) = static deflection under load

Effective vibration isolation requires:
[ f_n < 0.4 f_m ] where (f_m) = machine operating frequency.

2. Rubber Springs Stiffness (Clause 5.2)

• Vertical stiffness under axial compression: [ k_o = \frac{E \cdot A}{t} \left(1 + 2A + B \right)^x ]

• (k_o) = vertical stiffness

• (t) = thickness of rubber pad

• (A) = bearing area

• (A) (area ratio) = force-free surface area / bearing area

• Constants (E, B, x) from Table 2 below.

• Horizontal stiffness: [ k_n = \frac{G \cdot A}{t} ]

• (k_n) = horizontal stiffness

• (G) = shear modulus

3. Table 2: Properties of Natural Rubber Compounds

IS 13301: Key Formulas & Tables for Types of Vibration Isolators

1. Natural Frequency Formula (Single DOF system)

[ f_n = \frac{1}{2\pi} \sqrt{\frac{g}{\delta}} ]

• (f_n) = natural frequency (Hz)
• (g) = acceleration due to gravity (≈ 9.81 m/s²)
• (\delta) = static deflection under supported weight (m)

2. Effective Vibration Isolation Criterion

• For effective isolation:
  [ f_n < 0.4 f_m ]
  where (f_m) = machine operating frequency (Hz)

3. Table 1: Natural Frequency Ranges of Vibration Isolators

4. Rubber Spring Design Guidelines (Clause 7.2)

• Allowable Bearing Pressure: 0.8 to 1.6 N/mm² (Shore hardness 40-70)
• Allowable Shear Stress: 0.3 to 0.5 N/mm² (Shore hardness 40-70)
• Thickness Limit: Thickness ≤ 1/5th of pad width
• Dynamic Behavior: Non-linear stiffness & damping; lab resonance tests recommended
• Installation: Ensure free, unrestrained sides for pad isolators

Summary Diagram: Vibration Isolation Concept

graph LR
A[Machine Frequency \(f_m\)] --> B{Isolator Natural Frequency \(f_n\)}
B -->|\(f_n < 0.4 f_m\)| C[Effective Isolation]
B -->|\(f_n \geq 0.4 f_m\)| D[Poor Isolation]

**Use IS 13301 guidelines to select isolators with natural frequencies

Key Formulas and Tables from IS 13301 on Dynamic Properties for Vibration Isolation

1. Natural Frequency of Vibration Isolator (Single DOF System)

[ f_n = \frac{1}{2\pi} \sqrt{\frac{g}{\delta}} ]

• ( f_n ) = natural frequency (Hz)
• ( g ) = acceleration due to gravity (m/s²)
• ( \delta ) = static deflection under load (m)

Effective isolation:
[ f_n < 0.4 f_m ] where ( f_m ) = machine operating frequency.

2. Effective Frequency Ranges for Vibration Isolators

3. Rubber Spring Stiffness

• Vertical (Axial) Stiffness: [ k_0 = \frac{E \cdot A}{t} \left[ (1 + 2x) + B \right] ] where

• ( k_0 ) = vertical stiffness (N/mm)

• ( E ) = Young’s modulus (N/mm²)

• ( A ) = bearing area (mm²)

• ( t ) = thickness (mm)

• ( B ), ( x ) = constants from Table 2 below

• Horizontal Stiffness: [ k_n = \frac{G \cdot A}{t} ] where

• ( k_n ) = horizontal stiffness (N/mm)

• ( G ) = shear modulus (N/mm²)

4. Rubber Properties (Natural Rubber Compounds)

Key Formulas and Specifications for Design of Vibration Isolators (IS 13301)

1. Natural Frequency of Isolator (Single Degree Freedom System)

[ f_n = \frac{1}{2\pi} \sqrt{\frac{g}{\delta}} ]

• (f_n) = Natural frequency (Hz)
• (g) = Acceleration due to gravity (9.81 m/s²)
• (\delta) = Static deflection under supported weight (m)

Design Criterion:
[ f_n < 0.4 f_m ] where (f_m) = machine operating frequency for effective isolation.

2. Types of Vibration Isolators & Frequency Ranges (Table 1)

3. Rubber Spring Design Limits (Clause 7.2)

• Allowable Bearing Pressure: 0.8 to 1.6 N/mm² (for Shore hardness 40–70)
• Allowable Shear Stress: 0.3 to 0.5 N/mm² (for Shore hardness 40–70)
• Thickness Limit: Thickness ≤ Width / 5 (for stability)

4. Damping Ratio (Clause 3.5)

• Defined as the ratio of actual damping to critical damping.
• Important for vibration transmissibility and isolator performance.

5. Recommendations

• Perform steady-state resonance tests on rubber isolators under expected static and dynamic loads to determine true dynamic stiffness and damping.
• Ensure unrestrained free sides for pad-type isolators to avoid constraint effects.

graph LR
A[Machine Frequency \(f_m\)] --> B[Isolator Natural Frequency \(f_n\)]
B --> C{Is \(f_n < 0.4 f_m\)?}
C -- Yes --> D[Effective Isolation]
C -- No --> E[Redesign Isolator]

For detailed design,

Key Formulas & Tables from IS 13301 - Other Design Considerations

1. Rubber Springs (Clause 5.2)

• Vertical stiffness under axial compression:

[ k_v = \frac{E}{1 + 2A + B} \times \frac{A}{t} ]

Where:

• ( k_v ) = vertical stiffness

• ( E, B, x ) = constants from Table 2

• ( A ) = bearing area

• ( t ) = thickness of rubber pad

• ( A ) (area ratio) = ratio of force-free surface area to bearing area

• Horizontal stiffness:

[ k_h = \frac{G A}{t} ]

• Damping ratio: Typically 2% to 10%, design value recommended at 5%.

Table 2: Properties of Natural Rubber Compounds

2. Metal Springs (Clause 7.1)

• Shear stress under axial loading:

\

Trench Isolation (IS 13301 - Clause 8)

Trench isolation is an effective vibration isolation technique used in industrial environments.

Key Specifications:

• Trench Depth (d):
  [ d \geq 0.6 \times L ] where L = length of Rayleigh wave (≈ length of shear wave (L_g)).

• Rayleigh Wave Length (L):
  [ L = L_g = \frac{\sqrt{G/\rho}}{f} ]

  • (G) = shear modulus of soil (Pa)
  • (\rho) = mass density of soil (kg/m³)
  • (f) = frequency of incoming wave (Hz)

• Determination of (L):
  Obtained from in-situ wave propagation tests as per IS 5249:1991.

Types of Isolation (Fig. 4):

• (a) Active Isolation: Uses trench depth ≥ 0.6L to block wave propagation.
• (b) Passive Isolation: Uses other methods like cork pads, springs, etc.

Additional Notes on Cork Pads (Clause 7.3):

• Allowable bearing pressure: 1 to 4 kg/cm² (manufacturer's test-based).
• Cork pads must be enclosed in steel frames to prevent lateral expansion.
• Treat cork pads against oil/water to maintain efficiency.
• Dynamic properties influenced by thickness, static stress, vibration amplitude, and creep.

flowchart LR
    A[Incoming Vibration Wave] --> B[Soil with Shear Modulus G]
    B --> C[Trench Isolation]
    C --> D{Depth ≥ 0.6L?}
    D -- Yes --> E[Effective Isolation]
    D -- No --> F[Partial/No Isolation]

Summary:
For trench isolation, design trench depth ≥ 0.6 times the Rayleigh wave length, calculated using soil properties and vibration frequency, ensuring effective vibration attenuation.

IS 13301: Testing and Evaluation - Key Formulas & Tables

1. Frequency Ratio & Transmissibility

• Transmissibility ( T ) varies with frequency ratio ( r = \frac{\text{excitation frequency}}{\text{natural frequency}} ).
• Refer Fig. 2 & 3 for ( T ) vs. ( r ) curves.
• Notations:
  • ( P_r ) = transmitted force
  • ( P ) = peak force
  • ( \tau ) = pulse duration
  • ( T_p ) = natural period

2. Rubber Springs (Clause 5.2)

• Vertical stiffness under axial compression:

[ k_o = \frac{E}{t} \left[(1 + 2A) + B\right] ]

• Variables:
  • ( k_o ) = vertical stiffness
  • ( t ) = thickness
  • ( A ) = bearing area
  • ( A ) (area ratio) = force-free surface area / bearing area
  • ( E, B, x ) = constants (see Table 2)

IS 13301 Key Formulas, Tables & Specifications

1. Rubber Springs (Clause 5.2)

• Vertical stiffness under axial compression:

[ k_o = \frac{E \cdot A}{t} \left[ (1 + 2A) + B \right]^{x} ]

Where:

• (k_o) = vertical stiffness (N/mm)

• (E, B, x) = constants from Table 2

• (t) = thickness of rubber pad (mm)

• (A) = bearing area (mm²)

• (A) (area ratio) = force-free surface area / bearing area

• Horizontal stiffness:

[ k_n = \frac{G \cdot A}{t} ]

Where:

• (k_n) = horizontal stiffness (N/mm)

• (G) = shear modulus from Table 2

• Damping ratio: Recommended 5% for preliminary design (range 2%-10%).

2. Table 2: Properties of Natural Rubber Compounds