Related Resources: vibration
Shock and Vibration Response Equations
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 SF2 ]
Static deflection (in) δst = ( Load per Isolator x Thickness ) / ( Corrected Compressive Modulus x Loaded Area )
Percent Deflection (%δ) = ( δst / Thickness ) x 10
System Natural Frequency Equations
Dynamic Spring Rate (lb/in) Kdyn = Edyn x (1 + 2 x SF2 ) x Loaded Area / Thickness
System Natural Frequency (Hz) fn = ( ( Kdyn x gravity / Load per Isolator )1/2 / ( 2 π )
Transmissibility Vibration Equation
Frequency Ratio (r) = Excitation Frequency (fexc ) / fn
Dynamic Shear Ratio (Grdyn ) = ( Eddyn @ fn ) / ( Eddyn @ fexc )
Transmissibility (T) = ( 1 + (Tan Delta)2 (1 - r2 x Grdyn )2 + (Tan Delta)2 )1/2
Percent Isolation (%) = (1 - T) x 100
Transmissibility at Resonance (Q) = 1 + (Tan Delta @ fexc)2 (Tan Delta @ fexc)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 mV2
For vertical downward free fall drop impacts.
KE (lbf/in) = Wh
3) Calculate the Spring Rate for the part shape
k (lbs/in) = W / δst
4) Calculate the dynamic deflection
The Spring Energy (SE) is expressed as
SE = (1/2) kδst2
Equate the Spring Energy to the Kinetic Energy.
KE = SE
KE = (1/2) kδst2
Arrange terms and solve for dynamic deflection
δdyn = ( ( 2 x KE / k ) )1/2
5) Calculate the dynamic percent deflection
δdyn% = ( δdyn/ t ) x 100
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.
- Weight (W) or Mass (m)
- Velocity (V) or Drop Height (h)
- Acceleration of gravity (g) = 386.1 inches/second2
- 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 (δdyn%) is unitless
- Velocity (V) in inches/second
- Part thickness (t) in line impact
- Static deflection (δst) in inches
- 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.
Harsh Patel, Pune India