# Derivation of the coupled equations of motion for a beam subjected to three translational accelerations and three rotational accelerations

## Abstract

Equations of motion are derived to describe the coupled (axial, torsional, and lateral) dynamic response of a uniform Bernoulli--Euler beam, of constant circular cross-section, subjected to three specified translational accelerations and three specified rotational accelerations. The boundary condition equations are those for a cantilever beam with a symmetrical rigid mass attached to the free end of the beam. The resulting equations of motion (and boundary conditions) are non linear. However, for many problems, these equations can be reduced and simplified to a form more amenable to solution. An illustrative example is given where the equations are reduced and simplified to those for the coupled quasi-static response of a cantilever beam with a rigid symmetrical mass attached to its free end. Thus, the quasi-static problem corresponds to the solution of a two point, linear boundary value problem. A numerical example is also provided. The results of a computer solution for the two point, linear boundary value problem are compared with the experimental results obtained from three spinning beam experiments. The agreement between the measured and predicted beam bending strains versus angular speed is excellent. Consequently, it is concluded the mathematical model for the spinning beam and the equations associated with themore »

- Authors:

- Publication Date:

- Research Org.:
- Sandia Labs., Livermore, CA (USA)

- OSTI Identifier:
- 6267061

- Report Number(s):
- SAND-79-8202

- DOE Contract Number:
- EY-76-C-04-0789

- Resource Type:
- Technical Report

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 42 ENGINEERING; PROJECTILES; STRUCTURAL BEAMS; STRAINS; ACCELERATION; DIAGRAMS; DYNAMIC LOADS; EQUATIONS OF MOTION; EXPERIMENTAL DATA; ROTATION; STRESSES; VARIATIONS; DATA; DIFFERENTIAL EQUATIONS; EQUATIONS; INFORMATION; MOTION; NUMERICAL DATA; 420200* - Engineering- Facilities, Equipment, & Techniques

### Citation Formats

```
Benedetti, G.A.
```*Derivation of the coupled equations of motion for a beam subjected to three translational accelerations and three rotational accelerations*. United States: N. p., 1979.
Web.

```
Benedetti, G.A.
```*Derivation of the coupled equations of motion for a beam subjected to three translational accelerations and three rotational accelerations*. United States.

```
Benedetti, G.A. Thu .
"Derivation of the coupled equations of motion for a beam subjected to three translational accelerations and three rotational accelerations". United States.
```

```
@article{osti_6267061,
```

title = {Derivation of the coupled equations of motion for a beam subjected to three translational accelerations and three rotational accelerations},

author = {Benedetti, G.A.},

abstractNote = {Equations of motion are derived to describe the coupled (axial, torsional, and lateral) dynamic response of a uniform Bernoulli--Euler beam, of constant circular cross-section, subjected to three specified translational accelerations and three specified rotational accelerations. The boundary condition equations are those for a cantilever beam with a symmetrical rigid mass attached to the free end of the beam. The resulting equations of motion (and boundary conditions) are non linear. However, for many problems, these equations can be reduced and simplified to a form more amenable to solution. An illustrative example is given where the equations are reduced and simplified to those for the coupled quasi-static response of a cantilever beam with a rigid symmetrical mass attached to its free end. Thus, the quasi-static problem corresponds to the solution of a two point, linear boundary value problem. A numerical example is also provided. The results of a computer solution for the two point, linear boundary value problem are compared with the experimental results obtained from three spinning beam experiments. The agreement between the measured and predicted beam bending strains versus angular speed is excellent. Consequently, it is concluded the mathematical model for the spinning beam and the equations associated with the model are accurate.},

doi = {},

journal = {},

number = ,

volume = ,

place = {United States},

year = {1979},

month = {2}

}