 Frequency of Simply Supported Beam | Online Calculator

Natural Frequencies Calculators

# Vibration of a Beam with Simply Supported Ends and Concentrated Mass

In this calculation, a beam of length L1 + L2 with a moment of inertia of the cross-section Ix and own mass m is considered. The ends of the beam are simply supported. At the distance L1 from the left end of the beam, a concentrated mass M is located. For the calculation, the elastic modulus E of the beam should be specified. As a result of calculations, the natural vibration frequency of the beam f is determined for the first vibration mode.  ### INITIAL DATA

L1, L2 - Distances from the beam ends to concentrated mass;

Ix - Moment of inertia of the beam in the direction of vibration;

E - Young's modulus;

M - Concentrated mass;

m - Beam weight excluding concentrated mass.

### RESULTS DATA

f - First mode frequency.

Length (L1)

Length (L2)

Moment of inertia (Ix)

Young's modulus (Е)

Concentrated mass (M)

Beam weight (m)

Natural frequency (f)

### BASIC FORMULAS

Natural frequency of the mass M, acting alone:

fM = (1 / 2π)*[3EIx / (M*L12*L22)]0.5;

Natural frequency of the unloaded beam, acting alone:

fbeam = (1.57 / [L1 + L2]2)*(EIx / m)0.5;

Total natural frequency of beam of the first mode:

1 / f 2 = (1 / fbeam 2) + (1 / fM2)

### INITIAL DATA

L1, L2 - Distances from the beam ends to concentrated mass;

Ix - Moment of inertia of the beam in the direction of vibration;

E - Young's modulus;

M - Concentrated mass;

m - Beam weight excluding concentrated mass.

### RESULTS DATA

f - First mode frequency.

### MATERIALS PROPERTIES

 Material Young’s modulus Pa (psi) Poisson’s ratio Steel 1.86÷2.1×1011 (2.7÷3.05×107) 0.25÷0.33 Cast iron 0.78÷1.47×1011 (1.1÷2.1×107) 0.23÷0.27 Copper 1.0÷1.3×1011 (1.45÷1.9×107) 0.34 Tin bronze 0.74÷1.22×1011 (1.1÷1.8×107) 0.32÷0.35 Brass 0.98÷1.08×1011 (1.4÷1.6×107) 0.32÷0.34 Aluminum alloy 0.7×1011 (1.0×107) 0.33 Magnesium alloy 0.4÷0.44×1011 (5.8÷6.4×106) 0.34 Nickel 2.5×1011 (3.6×107) 0.33 Titanium 1.16×1011 (1.7×107) 0.32 Lead 0.15÷0.2×1011 (2.2÷2.9×106) 0.42 Zinc 0.78×1011 (1.1×107) 0.27 Glass 4.9÷5.9×1010 (7.1÷8.5×106) 0.24÷0.27 Concrete 1.48÷2.25×1010 (2.1÷3.3×106) 0.16÷0.18 Wood (along the grain) 8.8÷15.7×1010 (12.8÷22.8×106) - Wood (across the grain) 3.9÷9.8×1010 (5.7÷14.2×106) - Nylon 1.03×1010 (1.5×106) -