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 SF² ]

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 SF² ) 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 - r² 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 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 / δ_st

4) Calculate the dynamic deflection

The Spring Energy (SE) is expressed as

SE = (1/2) kδ_st²

Equate the Spring Energy to the Kinetic Energy.

KE = SE
KE = (1/2) kδ_st²

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/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 (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