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 I_x 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.

Calculation of natural frequency of beam

Natural frequency of beam

INITIAL DATA

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

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

E - Young's modulus;

M - Concentrated mass;

m - Beam mass excluding concentrated mass.

RESULTS DATA

f - First mode frequency.

Metric Imperial

Length (L₁)

Length (L₂)

Moment of inertia (I_x)

Young's modulus (Е)

Concentrated mass (M)

Beam mass (m)

Natural frequency (f)

BASIC FORMULAS

Natural frequency of the mass M, acting alone:

f_M = (1 / 2π)*[3EI_x / (M*L₁²*L₂²)]^0.5;

Natural frequency of the unloaded beam, acting alone:

f_beam = (1.57 / [L₁ + L₂]²)*(EI_xL / m)^0.5;

Total natural frequency of beam of the first mode:

1 / f ² = (1 / f_beam ²) + (1 / f_M²)

INITIAL DATA

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

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

E - Young's modulus;

M - Concentrated mass;

m - Beam mass 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)	-

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