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Added mass, beam

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ExampleReferenceRecordsLargest deviation
Weightless simply supported beam, point mass at the mid-span nodeClosed form10%
Weightless simply supported beam, point mass at the middle of a single elementClosed form10%
Weightless simply supported beam, point mass inside the element at L/3Closed form10%
Weightless simply supported beam, point mass inside the element on a fine meshClosed form10%

Problem. Equipment or a point load placed on a beam is taken into the modal analysis as a point mass where it stands. When the mass sits inside an element rather than at a node, no new node is added. The mass is distributed to the translations and rotations of the element's two end nodes with the consistent mass approach. The vertical frequency of a 5 t mass on a simply supported beam whose own weight is neglected is tested.

Reference document. The frequency is computed in closed form from the deflection that a unit force at the point mass produces at that point. When the mass is at a node this deflection comes exactly from beam theory and the reference is the true frequency. When the mass lies between two nodes, the cubic element sees the deflection under the mass as slightly smaller. That share is taken off the reference, so the reference is the frequency the engine should give on that mesh. The share vanishes as the mesh is refined.

The flexibility at the point mass and the angular frequency are computed from these relations. When the mass is at a node the second term is zero.

δ=a2b23EIL−a′3b′33EI l3,ω2=1M δ\delta = \dfrac{a^2 b^2}{3EIL} - \dfrac{a'^3 b'^3}{3EI\,l^3},\qquad \omega^2 = \dfrac{1}{M\,\delta}

Numerical model. The mass is tested in four positions, at a node, at the middle of a single element, at a third of the span on a four-element mesh and inside an element on a sixteen-element mesh.

Weightless simply supported beam, point mass at the mid-span node

Parameters. L = 6 · element count = 2 · a = 3 · M = 5 · mode = 2

Closed form. The flexibility at an in-span point mass and the angular frequency that follows

δ=a2b23EIL−a′3b′33EI l3,ω2=1Mδ\delta = \dfrac{a^2 b^2}{3EIL} - \dfrac{a'^3 b'^3}{3EI\,l^3},\qquad \omega^2 = \dfrac{1}{M\delta}
QuantityReferenceComputedDeviation
Vertical mode frequency [Hz]7.117637.117630%

Weightless simply supported beam, point mass at the middle of a single element

Parameters. L = 6 · element count = 1 · a = 3 · M = 5 · mode = 2

Closed form. The flexibility at an in-span point mass and the angular frequency that follows

δ=a2b23EIL−a′3b′33EI l3,ω2=1Mδ\delta = \dfrac{a^2 b^2}{3EIL} - \dfrac{a'^3 b'^3}{3EI\,l^3},\qquad \omega^2 = \dfrac{1}{M\delta}
QuantityReferenceComputedDeviation
Vertical mode frequency [Hz]8.218738.218730%

Weightless simply supported beam, point mass inside the element at L/3

Parameters. L = 6 · element count = 4 · a = 2 · M = 5 · mode = 2

Closed form. The flexibility at an in-span point mass and the angular frequency that follows

δ=a2b23EIL−a′3b′33EI l3,ω2=1Mδ\delta = \dfrac{a^2 b^2}{3EIL} - \dfrac{a'^3 b'^3}{3EI\,l^3},\qquad \omega^2 = \dfrac{1}{M\delta}
QuantityReferenceComputedDeviation
Vertical mode frequency [Hz]8.021278.021270%

Weightless simply supported beam, point mass inside the element on a fine mesh

Parameters. L = 6 · element count = 16 · a = 2.22 · M = 5 · mode = 2

Closed form. The flexibility at an in-span point mass and the angular frequency that follows

δ=a2b23EIL−a′3b′33EI l3,ω2=1Mδ\delta = \dfrac{a^2 b^2}{3EIL} - \dfrac{a'^3 b'^3}{3EI\,l^3},\qquad \omega^2 = \dfrac{1}{M\delta}
QuantityReferenceComputedDeviation
Vertical mode frequency [Hz]7.633677.633670%