**Related Resources: vibration**

### Shock and Vibration Response Equations

**Machine Design and Engineering **

**Mechanical Shock and Vibration Response Equations **

*Shape Factor Equations *

Shape Factor SF = Loaded Area / Unloaded Area

Rectangular Prism SF = Length x Width / 2 x Thickness x (Length + Width)

Square Prism SF = Length / 4 x Thickness

Disk SF = Diameter / 4x Thickness

Ring SF = ( Outside Diameter / 4 x Thickness ) - ( Inside Diameter \ 4 x Thickness )

Spherical Cap SF = ( 2 x Radius - Thickness ) / 2 x Radius

*Static Deflection Equations with Vibration Isolator *

Compressive Modulus (psi) = Stress (Compression) / ( Assumed Percent Deflection / 100 )

Corrected Compressive Modulus (psi) = (Compressive Modulus) x [ 1 + 2 x SF^{2} ]

Static deflection (in) δ_{st} = ( Load per Isolator x Thickness ) / ( Corrected Compressive Modulus x Loaded Area )

Percent Deflection (%δ) = ( δ_{st} / Thickness ) x 10

System Natural Frequency Equations

Dynamic Spring Rate (lb/in) K_{dyn} = E_{dyn} x (1 + 2 x SF^{2} ) x Loaded Area / Thickness

System Natural Frequency (Hz) f_{n} = ( ( K_{dyn} x gravity / Load per Isolator )^{1/2} / ( 2 π )

**Transmissibility Vibration Equation**

Frequency Ratio (r) = Excitation Frequency (f_{exc} ) / f_{n}

Dynamic Shear Ratio (Gr_{dyn} ) = ( Ed_{dyn} @ f_{n} ) / ( Ed_{dyn} @ f_{exc} )

Transmissibility (T) = ( 1 + (Tan Delta)^{2} (1 - r^{2} x Grdyn )^{2} + (Tan Delta)^{2} )^{1/2}

Percent Isolation (%) = (1 - T) x 100

Transmissibility at Resonance (Q) = 1 + (Tan Delta @ f_{exc})^{2} (Tan Delta @ f_{exc})^{2} )^{1/2}

*Shock Response Equations*

1) Convert Weight in pounds-force to Mass:

m (slugs) = W / g

2) Calculate the Kinetic Energy (KE) for the impact:

For horizontal impacts only the mass is considered.

KE (lbf/in) = 1/2 mV^{2}

For vertical downward free fall drop impacts.

KE (lbf/in) = Wh

3) Calculate the Spring Rate for the part shape

k (lbs/in) = W / δ_{st}

4) Calculate the dynamic deflection

The Spring Energy (SE) is expressed as

SE = (1/2) kδ_{st}^{2}

Equate the Spring Energy to the Kinetic Energy.

KE = SE

KE = (1/2) kδ_{st}^{2}

*Arrange terms and solve for dynamic deflection *

δ_{dyn} = ( ( 2 x KE / k ) )^{1/2}

5) Calculate the dynamic percent deflection

δ_{dyn}% = ( δ_{dyn}/ t ) x 100

Note:

For Shape Factors less than 1.2 and percent dynamic deflections less than 40% the expected fatigue life is considered to be in excess of one million cycle (indefinite).

For Shape Factors less than 1.2 and percent dynamic deflections between 40% and 60% the expected fatigue life is considered to be in excess of 1,000 cycles.

If the results achieved fail to achieve the desired performance then revise shape and/or durometer and repeat calculations.

The percent static deflection (continuous load without impact) must not exceed 20%.

There is no accepted methodology for higher shape factors or higher percent dynamic deflections.

Where:

- Weight (W) or Mass (m)
- Velocity (V) or Drop Height (h)
- Acceleration of gravity (g) = 386.1 inches/second
^{2} - Free fall drop height (h) in inches
- Dynamic deflection (δ) in inches
- Force (F) in pounds-force
- Kinetic Energy (KE) in pounds-force-inch
- Mass (m) in slugs
- Nominal spring rate* (k) in Pounds-force/inch
- Percent deflection (δ
_{dyn}%) is unitless - Velocity (V) in inches/second
- Part thickness (t) in line impact
- Static deflection (δ
_{st}) in inches

Terms:

- Vibration: A periodic motion around a position of equilibrium.
- Random Vibration: Vibration whose magnitude is not specified for any given instant of time.
- Shape Factor: The ratio of the loaded area to unloaded area.
- Static Deflection: The distance that a given mass compresses.
- Percent Deflection: The fraction of static deflection to uncompressed thickness.
- Frequency: The number of times the motion repeats itself per unit of time measured in Hertz (Hz).
- Natural Frequency: The frequency of free vibration of a system.
- Resonant Frequency: A frequency at which resonance exists.
- Resonance: The frequency match between the natural frequency of the system and the external forced vibration frequency.
- Damping: The dissipation of energy in an oscillating system.
- Transmissibility: The ratio of the response amplitude of a system in steady state forced vibration to the excitation amplitude.
- Isolation: A reduction in the capacity of a system to respond to an excitation.

Contribution:

Harsh Patel, Pune India

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