In this work, the classic theory of Timoshenko beams is revisited using screw theory. The theory of screws is familiar from robotics and the theory of mechanisms. A key feature of the screw theory is that translations and rotations are treated on an equal footing and here this means that bending, torsion, and extensions can all be considered together in a particularly simple manner. By combining forces and torques into a six-dimensional vector called a wrench, Hooke’s law for the Timoshenko beam can be written in a very simple form. From here simple expressions can be found for the kinetic and potential energy densities of the beam. Hence equations of motion for small vibrations of the beam can be easily derived. The screw theory also leads to a new understanding of the boundary conditions for beams. It is demonstrated that simple boundary conditions are closely related to mechanical joints. In order to set up the boundary conditions for a beam attached to a joint, a system of wrenches dual to the screws representing the freedoms of the joint must be found. Finally, a screw version of the Rayleigh–Ritz numerical method is introduced. An example is investigated in which the boundary conditions on the beam lead to vibrational modes of the beam involving bending, torsion, and extension at the same time.

1.
Ball
,
R. S.
, 1900,
The Theory of Screws
,
Cambridge University Press
,
Cambridge
.
2.
Selig
,
J. M.
, 2005,
Geometrical Fundamentals of Robotics
, 2nd ed.,
Springer
,
New York
.
3.
Selig
,
J. M.
, and
Ding
,
X.
, 2001, “
A Screw Theory of Static Beams
,”
Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems
, Maui, HI, pp.
2544
2550
.
4.
Timoshenko
,
S.
, 1955,
Strength of Materials: Part 1
, 3rd ed.,
van Nostrand
,
New York
.
5.
von Mises
,
R.
, 1924, “
Motorrechnung, ein neues Hilfsmittel in der Mechanik
,”
ZAMM
0044-2267,
4
(
2
), pp.
155
181
(English translation by E. J. Baker and K. Wohlhart, Motor Calculus: A New Theoretical Device for Mechanics, Institute for Mechanics, University of Technology Graz, Austria, 1996).
6.
Goldstein
,
H.
,
Poole
,
C.
, and
Safko
,
J.
, 2002,
Classical Mechanics
, 3rd ed.,
,
New York
.
7.
Pipes
,
L. A.
, and
Harvill
,
L. R.
, 1970,
Applied Mathematics for Engineers and Physicists
, 3rd ed.,
McGraw-Hill
,
Tokyo
.
8.
Meirovitch
,
L.
, 1967,
Analytical Methods in Vibrations
,
Macmillan
,
New York
.
9.
Aalami
,
B.
, and
Atzori
,
B.
, 1974, “
Flexural Vibrations and Timoshenko’s Beam Theory
,”
AIAA J.
0001-1452,
12
(
5
), pp.
679
685
.
10.
Oguamanam
,
D. C. D.
, and
Heppler
,
G. R.
, 1996, “
Effect of Rotating Speed on the Flexural Vibration of a Timoshenko Beam
,”
Proceedings of IEEE International Conference on Robotics and Automation
, Minneapolis, MN, pp.
2438
2443
.
11.
Wang
,
S.
, 1997, “
Unified Timoshenko Beam B-spline Rayleigh–Ritz Method for Vibration and Buckling Analysis of Thick and Thin Beams and Plates
,”
Int. J. Numer. Methods Eng.
,
40
(
3
), pp.
473
491
. 0029-5981
12.
Bercin
,
A. N.
, and
Tanaka
,
M.
, 1997, “
Coupled Flexural-Torsional Vibrations of Timoshenko Beams
,”
J. Sound Vib.
0022-460X,
207
(
1
), pp.
47
59
.
13.
Bishop
,
R. E. D.
, and
Price
,
W. G.
, 1977, “
Coupled Bending and Twisting of a Timoshenko Beam
,”
J. Sound Vib.
0022-460X,
50
(
4
), pp.
469
477
.
14.
Antman
,
S. S.
, 2005,
Nonlinear Problems of Elasticity
, 2nd ed.,
Springer
,
New York
.
15.
Rubin
,
M. B.
, 2000,
Cosserat Theories: Shells, Rods and Points
,
Kluwer
,
Dordrecht
.