Advances in Structural Engineering pp 215-220 | Cite as

# Vibration of Multi-span Thin Walled Beam Due to Torque and Bending Moment

## Abstract

In this paper, the flexural torsional vibration of a thin walled beam due to combined action of bending moment and torque has been analyzed to calculate the natural frequencies and corresponding eigenvectors under different combination of bending moment and torque. The aim is to account the effect of intermediate support on the dynamic behavior of beam compared to single span beam. Also, the interaction formula has been derived for axial force, bending moment and torque for single span and multiple span beams. In the case of lateral buckling of beams, the critical bending moment is inversely proportional to length, which is the condition of zero natural frequency. Therefore, the lowering of natural frequency due to increased length is expected. However, the effect of length on the ratio of torsional and flexural component of vibration is not so obvious in spite of the fact that a phenomenon of flexural torsional vibration is important in long span beam only. The calculation shows that as the length increases, given the bending moment, the torsional component increases as the length increases. This fact is important in case of long span bridges to show that in-plane flexural vibration will trigger the flexural torsional vibration. The problem of multi-span beam is also important in hydropower plants where due to large length of turbo generator shaft, to control the lateral vibration of shaft several intermediate guide bearings are provided to latterly support the shaft. This is a case of multi-span shaft with lateral constraint but torsionally unrestrained at intermediate support. The impact of intermediate support on the on the torsional component of flexural torsional vibration may warrant the consideration of flexural torsional vibration in the design of intermediate support.

## Keywords

Differential equation Thin walled beam Torque Vibration## Notations

- A
_{m}, B_{m}, C_{m} Amplitude of mth harmonic of displacement

- A
_{r} Cross sectional area

- E
Young’s modulus

- G
Modulus of rigidity

- I
_{0} Polar moment of inertia

- I
_{b} Moment of inertia about binormal

- I
_{n} Moment of inertia about principal normal

- I
_{ω} Sectorial moment of inertia

- K
_{t} St. Venent torsional constant of section

- L
Length of the beam

- m, m1
Rate of change of phase angle per unit length

- M
_{n}, M_{b} Bending moment along unit vectors

**n**and**b**- M
_{t} Twisting moment

- m
_{t} Distributed torque

- Q
_{n}, Q_{b} Shear force along unit vectors

**n**and**b**- Q
_{t} Axial force

- Q
_{y}, Q_{x} Euler’s buckling load

- Q
_{φ} Torsional buckling load

- q
_{t}, q_{n}, q_{b} Uniformally distributed load along unit vector

**t**,**n**and**b**- s
Distance measured along the arc of the curve from the fiducial point

**t**,**n**,**b**Unit vectors parallel to tangent, normal and binormal

- u
_{t}, u_{n}, u_{b} Displacement along unit vectors

**t**,**n**,**b**- x, y, z
Coordinate axes along

**b**,**n**,**t**- ΔM
_{n} Second order term of normal bending moment

- ΔM
_{b} Second order term of binormal bending moment

- ΔM
_{t} Second order term of twisting moment

- θ
_{t}, θ_{n}, θ_{b} Rotation along unit vectors

**t**,**n**,**b**- ρ
_{n}, ρ_{b} Change in curvature along unit vectors

**n**,**b**- μ
Mass per unit length

- ω
Vibration frequency in rad/s

## References

- 1.Verma VK (2009) Buckling of beams under combined action of bending moment and torque. In: 54th congress of Indian Society of theoretical and applied mechanics (ISTAM), Netaji Subhas Institute of Technology (NSIT), New Delhi, IndiaGoogle Scholar
- 2.Verma VK (1997) Geometrically nonlinear analysis of curved beams. In: 42nd congress of Indian Society of theoretical and applied mechanics (ISTAM), South Gujrat University, Surat, Gujarat (GJ), IndiaGoogle Scholar
- 3.Verma VK (2001) Vibration of beams under combined action of bending moment and torque. In: 46th congress of Indian Society of theoretical and applied mechanics (ISTAM), Regional Engineering College (now, National Institute of Technology), Hamirpur, Himachal Pradesh (HP), IndiaGoogle Scholar