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Beam Lateral Vibration Frequency

Calculate the damped and undamped beam natural vibration frequency for lateral vibration (simply supported, fixed, and cantilever beams).

Added mass should be included for submerged or wet beams. The added mass coefficient can be calculated in accordance with DNVGL RP F105. The submerged natural frequency is calculated for still water conditions, with no vortex shedding. For beams on a soft foundation such as soil, use the effective length factor to allow for movement at the beam ends. For defined beam ends such as structures, the effective length factor should be set to one.

The mode factor k is dependent on the mode number, and the beam end type. The k factors have been taken from the Shock and Vibration handbook. The damping factor should be set to zero for undamped vibration or set greater than zero and less than or equal to one for damped vibration. For multi layer pipes the bending stiffness can be calculated with the concrete stiffness factor (CSF). The CSF accounts for the additional stiffness provided by the external concrete coating. The concrete stiffness factor is calculated in accordance with DNVGL RP F105. Enter the wall thickness for all layers. Only enter the elastic modulus for layers which affect the pipe stiffness.

Use the Result Table and Result Plot options to display tables and plots. Refer to the figures and help pages for more details about the tools.

References :

Shock And Vibration Handbook, Cyril M Harris, McGraw Hill
Roark's Formulas For Stress And Strain, Warren C Young, McGraw Hill

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CALCULATOR : Beam Lateral Natural Vibration Frequency (Circular Pipe) [PLUS]   ±

Calculate beam damped and undamped lateral natural vibration frequency for single layer pipes (no axial load).

Beam unit mass and EI are calculated for a single layer circular beam with no coatings or internal liners. Beam unit mass can be calculated for full and empty beams. Added mass should be included for submerged or wet beams. The added mass coefficient is calculated in accordance with DNVGL RP F105. The submerged natural frequency is calculated for still water conditions, with no vortex shedding. The effective length factor should be used for beams on a soft foundation such as soil, where the beam ends are poorly defined. For defined beam ends, such as structures, the effective length factor should be set to one (fe = 1). The damping factor = 0 for undamped vibration, and 1 for critically damped vibration.

Select the end type, and vibration mode number (modes 1 to 5). Use the Result Table option to display the natural frequency versus either mode number, end type, or line pipe wall thickness. Use the Result Plot option to display the natural frequency versus beam length and mode number, or beam length and end type. The Fix-Fix and Free-Free modes have the same natural frequencies, but different mode shapes.

Reference : DNVGL RP F105 Free Spanning Pipelines (Download From DNVGL website)

Tool Input

  • schdtype : Schedule Type
  • diamtype : Diameter Type
    • ODu : User Defined Outside Diameter
    • IDu : User Defined Inside Diameter
  • wtntype : Wall Thickness Type
    • tnu : User Defined Wall Thickness
  • modptype : Material Property Type
    • Eu : User Defined Elastic Modulus
    • ρpu : User Defined Density
  • sectype : Section Properties Type
    • EIu : User Defined E x I
  • mmtype : Added Mass Type
    • ρe : External Fluid Density
    • Cmu : User Defined Added Mass Coefficient
    • G : Gap Height
  • mltype : Unit Mass Type
    • mu : User Defined Unit Mass
  • endtype : End Type
  • MN : Vibration Mode Number
  • leftype : Effective Length Type
    • feu : User Defined Effective Length Factor
  • fdtype : Damping Factor Type (0 = Undamped 1 = Critical Damping)
    • fdu : User Defined Damping Factor (0 ≤ fd ≤ 1)
  • ρi : Internal Fluid Density
  • Lo : Nominal Length

Tool Output

  • ρp : Line Pipe Density
  • E : Elastic Modulus
  • EI : E x I
  • ID : Inside Diameter
  • Le : Effective Length
  • OD : Outside Diameter
  • cm : Added Mass Coefficient
  • fd : Damping Factor
  • fn : Natural Frequency
  • k : Natural Frequency K Factor
  • m : Mass Per Length
  • ma : Added Unit Mass
  • mc : Contents Unit Mass
  • md : Displaced Unit Mass
  • mp : Line Pipe Unit Mass
  • tn : Wall Thickness

CALCULATOR : Beam Lateral Natural Vibration Frequency (Multi Layer Pipe) [PLUS]   ±

Calculate beam damped and undamped lateral natural vibration frequency for multi layer pipes (no axial load).

Beam unit mass and EI are calculated for a circular beam with coatings and or internal liners. Unit mass can be calculated with or without added mass. Added mass is included in the unit mass for submerged beams to account for the fluid which is displaced by the beam. The added mass coefficient is calculated in accordance with DNVGL RP F105. The submerged natural frequency is calculated for still water conditions, with no vortex shedding. The effective length factor should be used for beams on a soft foundation such as soil, where the beam ends are poorly defined. For defined beam ends, such as structures, the effective length factor should be set to one (fe = 1). The damping factor = 0 for undamped vibration, and 1 for critically damped vibration.

The bending stifness can be calculated with the concrete stiffness factor (CSF). The CSF accounts for the additional stiffness provided by the external concrete coating.

Enter the wall thickness for all layers. Only enter the elastic modulus for layers which will contribute to EI.

Select the end type, and vibration mode number (modes 1 to 5). Use the Result Table option to display the natural frequency versus either mode number, end type, or line pipe wall thickness. Use the Result Plot option to display the natural frequency versus beam length and mode number, or beam length and end type. The Fix-Fix and Free-Free modes have the same natural frequencies, but different mode shapes.

Reference : DNVGL RP F105 Free Spanning Pipelines (Download From DNVGL website)

Tool Input

  • schdtype : Line Pipe Schedule Type
  • diamtype : Line Pipe Diameter Type
    • ODu : User Defined Outside Diameter
    • IDu : User Defined Inside Diameter
  • wtntype : Line Pipe Wall Thickness Type
    • tnu : User Defined Wall Thickness
  • eitype : E x I Type
    • Kcu : User Defined Coating Factor
    • CSFu : User Defined Concrete Stiffness Factor
    • EIu : User Defined Pipe E x I
  • mmtype : Mass Type
    • ρe : User Defined External Fluid Density
    • Cmu : User Defined Added Mass Coefficient
    • G : Gap Height
  • mltype : Mass Type
    • mu : User Defined Beam Unit Mass
  • endtype : End Type
  • MN : Mode Type
  • leftype : Effective Length Type
    • feu : User Defined Effective Length Factor
  • fdtype : Damping Factor Type (0 = Undamped 1 = Critical Damping)
    • fdu : User Defined Damping Factor (0 ≤ fd ≤ 1)
  • WTi : Pipe Liner Wall Thickness
  • ρi : Pipe And Liner Density
  • Ei : Pipe And Liner Elastic Modulus
  • WTo : Pipe Coating Wall Thickness
  • ρo : Pipe Coating Density
  • Eo : Pipe Coating Elastic Modulus
  • ρf : Internal Fluid Density
  • Lo : Nominal Length

Tool Output

  • CSF : Concrete Stiffness factor
  • Cm : Added Mass Coefficient
  • EI : Effective E x I
  • EIc : Concrete E x I
  • EIp : Pipe E x I
  • IID : Pipe Inside Diameter Including Liner
  • Kc : Coating Factor
  • Le : Effective Length
  • OD : Line Pipe Diameter
  • OOD : Pipe Outer Diameter Including Coatings
  • fd : Damping Factor
  • fn : Natural Frequency
  • k : Natural Frequency K Factor
  • m : Nominal Mass Per Unit Length
  • md : Displaced Unit Mass
  • mla : Added Unit Mass
  • mlc : Contents Unit Mass
  • mlp : Pipe Unit Mass Including Liner And Coating
  • tn : Line Pipe Thickness

CALCULATOR : Beam Lateral Natural Vibration Frequency (General Beam) [FREE]   ±

Calculate beam damped and undamped lateral natural vibration frequency for general beams (user defined properties - no axial load).

Beam unit mass and EI are user defined. Select the end type, and vibration mode number (modes 1 to 5). The effective length factor should be used for beams on a soft foundation such as soil, where the beam ends are poorly defined. For defined beam ends, such as structures, the effective length factor should be set to one (fe = 1). The damping factor = 0 for undamped vibration, and 1 for critically damped vibration.

Use the Result Table option to display the natural frequency versus either mode number, or end type. Use the Result Plot option to display the natural frequency versus beam length and mode number, or beam length and end type. The Fix-Fix and Free-Free modes have the same natural frequencies, but different mode shapes.

Tool Input

  • endtype : End Type
  • MN : Vibration Mode Number
  • leftype : Effective Length Type
    • feu : User Defined Effective Length Factor
  • fdtype : Damping Factor Type (0 = Undamped 1 = Critical Damping)
    • fdu : User Defined Damping Factor (0 ≤ fd ≤ 1)
  • EI : Bending Stiffness Modulus (E x I)
  • m : Unit Mass
  • Lo : Nominal Length

Tool Output

  • Le : Effective Length
  • fd : Damping Factor
  • fn : Natural Frequency
  • k : Natural Frequency K Factor
CALCULATOR : Beam Added Mass Coefficient (General Beam) [FREE]   ±

Calculate general beam added mass coefficient and added mass from gap height and characteristic length.

Added mass is included in the unit mass for submerged beams to account for the fluid which is displaced by the beam. The added mass coefficient is calculated in accordance with DNVGL RP F105. The equation is suitable for undamped vibration of circular pipes in still fluid. For circular pipes the diameter should be used as the characteristic length. For other profile shapes the width can be used as the characteristic length. The method may not be valid for other profile shapes (engineering judgement is required). Refer to the help pages for more details.

Reference : DNVGL RP F105 Free Spanning Pipelines (Download From DNVGL website)

Tool Input

  • cmtype : Added Mass Coefficient Type
    • Cmu : User Defined Added Mass Coefficient
  • mb : Beam Mass Per Unit Length
  • ρe : External Fluid Density
  • W : Beam Characteristic Length
  • G : Gap Height
  • AX : Beam Cross Section Area

Tool Output

  • Cm : Added Mass Coefficient
  • G/W : Gap Over Characteristic Length Ratio
  • m : Total Mass Per Unit Length
  • ma : Added Mass Per Unit Length
CALCULATOR : Beam Vibration Line Pipe Schedule [FREE]   ±

Calculate line pipe schedule outside diameter inside diameter and wall thickness.

Select the pipe schedule (NPS or ISO etc), pipe diameter and wall thickness, or use the user defined option. Use the Result Table option to display the pipe schedule for the selected diameter.

Tool Input

  • schdtype : Line Pipe Schedule Type
  • diamtype : Line Pipe Diameter Type
    • ODu : User Defined Outside Diameter
    • IDu : User Defined Inside Diameter
  • wtntype : Wall Thickness Type
    • tnu : User Defined Wall Thickness

Tool Output

  • ID : Nominal Inside Diameter
  • OD : Nominal Outside Diameter
  • OD/tn : Diameter Over Wall Thickness Ratio
  • tn : Nominal Wall Thickness
CALCULATOR : Beam Vibration Material Property [FREE]   ±

Calculate beam elastic modulus, shear modulus, bulk modulus, density, and thermal expansion coefficient.

The table values of Poisson ratio and bulk modulus are calculated from the elastic modulus and shear modulus. Use the Result Table option to display a table of properties versus material type.

Tool Input

  • modptype : Material Type
    • Eu : User Defined Elastic Modulus
    • Gu : User Defined Shear Modulus
    • Ku : User Defined Bulk Modulus
    • νu : User Defined Poisson Ratio
    • ρu : User Defined Density
    • αu : User Defined Thermal Expansion Coefficient

Tool Output

  • α : Thermal Expansion Coefficient
  • ν : Poisson Ratio
  • ρ : Density
  • E : Elastic Modulus
  • G : Shear Modulus
  • K : Bulk Modulus
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