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MEEN 617:. Appendix D. Note on assumed mode method for a continuous system 2008 1

Appendix D. Technical Note on Assumed Modes

The fundamental vibrating mode

of a cantilever beam and itsassociated natural frequency can bemodeled as a single degree offreedom lumped mass on a spring.

The beam equivalent stiffness and mass can be determined byequating the beam strain energy (V)and kinetic energy (T)of thevibrating beam to the strain and kinetic energy of the lumpedspring and mass, respectively. The equivalent displacementcoordinate should be equal for both energies.

The beam has continuously distributed mass and elasticproperties. Let: beam material density, E:beam elastic modulus,A: beam cross section area, L:beam length,

I:area moment of inertia.

y(x,t) is the displacement of a beam material point, i.e. a function

of its location (x)and time (t).yL(t)denotes the beam dynamicdisplacement at x=L. The beam potential (strain) and kineticenergies, Vand T, are defined as:

22

2

2

0

1 12 2

L

eq L

yV E I dx K y

x = =

;

(1)22

2

2

0

1 1

2 2

L

eq L

yT A dx M y

t

= =

L

xyL

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MEEN 617:. Appendix D. Note on assumed mode method for a continuous system 2008 2

In practice, an assumed shape of vibration(x)is used to estimatethe equivalent stiffness (Keq) and mass (Meq).

Let y(x,t)=(x)yL(t) (2)

The mode shape(x)must be twice differentiable and consistentwith the essential boundary conditions of the cantilever beam,i.e. no displacement or slope at the fixed end. That is, from

y(0,t)=0

(x=0) =0(y/x)x=0=0 d/dx|x=0=0for all times t>0.

Substitution of y(x,t)=(x)yL(t)into Eq. (1) gives:

22

2

0

L

eq

dK E I dx

d x

=

; ( )

2

0

L

eqM A dx = (3)

The fundamental natural frequency of the vibrating beam is thengiven by

eq

eqn M

K= (4)

Using ( )2

/Lx= , then 31

; 4 ;5

eq eq

E IM AL K

L= =

And

1/ 2

2

120n

E I

L A

(5)

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MEEN 617:. Appendix D. Note on assumed mode method for a continuous system 2008 3

The equivalent mass of the beam, Meq, is a fraction of the totalmass (~1/5) since the material points composing the beamparticipate differently in the vibratory motion.

Keq=3EI/L3 (more exact value) follows if the static deflection

curvefor the beam with a point load at its free end is used as theassumed mode shape, i.e.

2 31

( ) 32

x xx

L L

=

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