Calculate the damped and undamped beam natural vibration frequency for general beams (simply supported, fixed, and cantilever beams). For other beam types (eg circular beams) refer to the module links below.
The lateral natural vibration frequency beam end conditions are: pinned ends (simply supported beams), fixed ends, free fixed ends (cantilever beams), pinned fixed ends, and for beams with no load, pinned free ends, and free ends (unsupported 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 buckling load can be calculated using either the Euler equation (suitable for long beams), or the Johnson equation (suitable for short beams). The buckling load is dependent on the end type, and is used for mode 1 vibration only. Buckling normally occurs on the axis with lowest stiffness (I1 or I2). The bending stiffness for vibration and buckling can be defined separately for cases where vibration and buckling are not parallel.
`fn = ca.cd k / (2 π) √((EI) / (m . Le^4)) `
`ca =1 / (1 + F / (Fb))) `
`cd = √(1 - fd^2) `
where :
fn = lateral natural frequency [Hz]
ca = axial load coefficient
cd = damping coefficient
fd = damping factor (0 = undamped 1 = critical damping)
k = mode factor
L = effective beam length
EI = beam E I (bending modulus)
m = beam unit mass or mass per length
F = axial load (+ve in tension and -ve in compression)
Fb = buckling load
The longitudinal natural vibration frequency end conditions are: free fixed ends (cantilever), fixed ends, and free ends (unsupported). The fixed ends and free ends modes have the same natural frequencies, but different mode shapes. The longitudinal natural frequency is independent of cross section, and depends on the beam elastic modulus and density.
`fn = cd k / (2 π L) √(E / ρ) `
where :
fn = natural frequency [Hz]
cd = damping coefficient
k = mode factor
L = beam length
E = beam elastic modulus
ρ = beam density
The torsional natural vibration frequency end conditions are: free fixed ends (cantilever), fixed ends, and free ends (unsupported). The fixed ends and free ends modes have the same natural frequencies, but different mode shapes. The torsional natural frequency is independent of cross section, and depends on the beam shear modulus and density.
`fn = cd k / (2 π L) √(G / ρ) `
where :
fn = natural frequency [Hz]
cd = damping coefficient
k = mode factor
L = beam length
G = beam shear modulus
ρ = beam density
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.
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. Refer to the links below for other beam options.
References :
Shock And Vibration Handbook, Cyril M Harris, McGraw Hill
Roark's Formulas For Stress And Strain, Warren C Young, McGraw Hill
Change Module :