The objective of this work is to analytically study the nonlinear dynamics of beam flexures with a tip mass undergoing large deflections. Hamilton's principle is utilized to derive the equations governing the nonlinear vibrations of the cantilever beam and the associated boundary conditions. Then, using a single mode approximation, these nonlinear partial differential equations are reduced to two coupled nonlinear ordinary differential equations. These equations are solved analytically using the multiple time scales perturbation technique. Parametric analytical expressions are presented for the time domain response of the beam around and far from its internal resonance state. These analytical results are compared with numerical ones to validate the accuracy of the proposed analytical model. Compared with numerical solution methods, the proposed analytical technique shortens the computational time, offers design insights, and provides a broader framework for modeling more complex flexure mechanisms. The qualitative and quantitative knowledge resulting from this effort is expected to enable the analysis, optimization, and synthesis of flexure mechanisms for improved dynamic performance.

References

References
1.
Howell
,
L. L.
,
2001
,
Compliant Mechanisms
,
Wiley
,
New York
.
2.
Smith
,
S. T.
,
2000
,
Flexures: Elements of Elastic Mechanisms
,
Gordon & Breach
,
Amsterdam, Netherlands
.
3.
Awtar
,
S.
,
Slocum
,
A.
, and
Sevincer
,
E.
,
2007
, “
Characteristics of Beam-Based Flexure Modules
,”
ASME J. Mech. Des.
,
129
(6), pp.
625
639
.10.1115/1.2717231
4.
Su
,
W.
, and
Cesnik
,
C. E. S.
,
2011
, “
Strain-Based Geometrically Nonlinear Beam Formulation for Modeling Very Flexible Aircraft
,”
Int. J. Solids Struct.
,
48
, pp.
2349
2360
.10.1016/j.ijsolstr.2011.04.012
5.
Awtar
,
S.
,
2004
, “
Synthesis and Analysis of Parallel Kinematic XY Flexure Mechanisms
,” Sc.D., Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA.
6.
Awtar
,
S.
, and
Parmar
,
G.
,
2013
, “
Design of a Large Range XY Nanopositioning System
,”
ASME J. Mech. Rob.
,
5
(2), p.
021008
.10.1115/1.4023874
7.
Fowler
,
A. G.
,
Laskovski
,
A. N.
,
Hammond
,
A. C.
, and
Moheimani
,
S. O. R.
,
2012
, “
A 2-DOF Electrostatically Actuated MEMS Nanopositioner for On-Chip AFM
,”
J. Microelectromech. Syst.
,
21
, pp.
771
773
.10.1109/JMEMS.2012.2191940
8.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
,
1979
,
Nonlinear Oscillations
,
Wiley
,
New York
.
9.
Rao
,
S. S.
,
2007
,
Vibration of Continuous Systems
,
Wiley
,
Hoboken, NJ
.
10.
Crespo Da Silva
,
M. R. M.
,
1988
, “
Non-Linear Flexural-Flexural-Torsional-Extensional Dynamics of Beams-I. Formulation
,”
Int. J. Solids Struct.
,
24
, pp.
1225
1234
.10.1016/0020-7683(88)90087-X
11.
Crespo Da Silva
,
M. R. M.
,
1998
, “
A Reduced-Order Analytical Model for the Nonlinear Dynamics of a Class of Flexible Multi-Beam Structures
,”
Int. J. Solids Struct.
,
35
, pp.
3299
3315
.10.1016/S0020-7683(98)00017-1
12.
Nayfeh
,
A. H.
,
1973
, “
Nonlinear Transverse Vibrations of Beams With Properties That Vary Along the Length
,”
J. Acoust. Soc. Am.
,
53
, pp.
766
770
.10.1121/1.1913389
13.
Zaretzky
,
C. L.
, and
Crespo da Silva
,
M. R. M.
,
1994
, “
Experimental Investigation of Non-Linear Modal Coupling in the Response of Cantilever Beams
,”
J. Sound Vib.
,
174
, pp.
145
167
.10.1006/jsvi.1994.1268
14.
Hijmissen
,
J. W.
, and
van Horssen
,
W. T.
,
2008
, “
On the Weakly Damped Vibrations of a Vertical Beam With a Tip-Mass
,”
J. Sound Vib.
,
310
, pp.
740
754
.10.1016/j.jsv.2007.06.014
15.
Zavodney
,
L. D.
, and
Nayfeh
,
A. H.
,
1989
, “
The Non-Linear Response of a Slender Beam Carrying a Lumped Mass to a Principal Parametric Excitation: Theory and Experiment
,”
Int. J. Non-Linear Mech.
,
24
, pp.
105
125
.10.1016/0020-7462(89)90003-6
16.
Yu
,
Y. Q.
,
Howell
,
L. L.
,
Lusk
,
C.
,
Yue
,
Y.
, and
He
,
M. G.
,
2005
, “
Dynamic Modeling of Compliant Mechanisms Based on the Pseudo-Rigid-Body Model
,”
ASME J. Mech. Des.
,
127
(4), pp.
760
765
.10.1115/1.1900750
17.
Awtar
,
S.
, and
Sen
,
S.
,
2010
, “
A Generalized Constraint Model for Two-Dimensional Beam Flexures: Nonlinear Strain Energy Formulation
,”
ASME J. Mech. Des.
,
132
(8), p.
081009
.10.1115/1.4002006
18.
MATLAB Product Help, R2010a, ode45 command.
You do not currently have access to this content.