Dynamic Analysis of a Cantilever Beam with an Offset Mass

Dynamic Analysis of a Cantilever Beam with anOffset Mass

by

Yurui Zhan

Department of Mechanical Engineering and Materials ScienceDuke University

Date:Approved:

Brian P. Mann, Supervisor

Samuel C. Stanton

Earl Dowell

Thesis submitted in partial fulfillment of the requirements for the degree ofMaster of Science in the Department of Mechanical Engineering and Materials

Sciencein the Graduate School of Duke University

2019

Abstract

Dynamic Analysis of a Cantilever Beam with an Offset Mass

by

Yurui Zhan

Department of Mechanical Engineering and Materials ScienceDuke University

Date:Approved:

Brian P. Mann, Supervisor

Samuel C. Stanton

Earl Dowell

An abstract of a thesis submitted in partial fulfillment of the requirements forthe degree of Master of Science in the Department of Mechanical Engineering and

Materials Sciencein the Graduate School of Duke University

2019

Copyright c© 2019 by Yurui ZhanAll rights reserved except the rights granted by the

Creative Commons Attribution-Noncommercial Licence

http://creativecommons.org/licenses/by-nc/3.0/us/

Abstract

This thesis investigates the dynamic characteristics of a cantilever beam with an

offset mass. Starting with a linear system consisting of a cantilever beam with a tip

mass, Hamilton’s principle is utilized to derive the equation of motion for the system,

then similar method is applied to a cantilever beam with an offset mass. The equation

of motion and boundary conditions are nondimensionalized to simplify the situation.

The theoretical trend of natural frequency is also derived to show the effects of mass

ratio, offset ratio and moment of inertia. Experimental results are derived using

a system consisting of a base, a 3D-printed beam and several attachments. After

comparing with theoretical data, several factors including damping ratio, moment

of inertia and Poisson’s ratio are taken into consideration. Both damping ratio and

moment of inertia have very little effect and Poisson’s ratio has opposite influence on

the results. Explanation for the deviation lies on the isotropy of 3D-printed beam,

which also puts forward a question on the qualification of using 3D-printed structures

for dynamical analysis.

iv

Contents

Abstract iv

List of Tables vii

List of Figures viii

List of Abbreviations and Symbols ix

1 Introduction 1

2 Beam Theory 3

2.1 Linear system with a lumped mass . . . . . . . . . . . . . . . . . . . 3

2.2 Offset End Mass System . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 Moment of Intertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3 Experimental Setup 19

3.1 Experimental Setup Design . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Experimental Data and Analysis . . . . . . . . . . . . . . . . . . . . . 22

3.3 Uncertainty Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Analysis and Discussion 27

4.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2 Theoretical Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3 Damping Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.4 Moment of Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.5 Poisson’s Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

v

4.6 Result Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.7 Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

5 Conclusion 38

Bibliography 40

vi

List of Tables

3.1 Beam parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.2 Attachment parameters(Unit of mass: gram/ Unit of length: mm) . . 22

4.1 Experimental Results of natural frequencies with same offset ratio . . 28

4.2 Experimental Results of natural frequencies with same mass ratio . . 28

4.3 Theoretical Results of natural frequencies with same offset ratio . . . 29

4.4 Theoretical Results of natural frequencies with same mass ratio . . . 29

4.5 Damping ratio of natural frequencies with same offset ratio . . . . . . 30

4.6 Damping ratio of natural frequencies with same mass ratio . . . . . . 30

4.7 Comparison of natural frequencies with same offset ratio includingdamping ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.8 Comparison of natural frequencies with same mass ratio includingdamping ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.9 Natural Frequency for same offset ratio with I . . . . . . . . . . . . . 32

4.10 Natural Frequency for same mass ratio with I . . . . . . . . . . . . . 32

4.11 Comparison of natural frequencies with same offset ratio . . . . . . . 34

4.12 Comparison of natural frequencies with same mass ratio . . . . . . . 34

vii

List of Figures

2.1 Schematic of a cantilever beam with a lumped mass . . . . . . . . . . 4

2.2 Schematic of first mode natural frequency with different mass ratios . 8

2.3 Schematic of a cantilever beam with an offset mass . . . . . . . . . . 9

2.4 Effects of offset ratio on natural frequency . . . . . . . . . . . . . . . 17

2.5 Effects of moment inertia on natural frequency . . . . . . . . . . . . . 18

3.1 Experimental setup of a cantilever beam with an offset mass . . . . . 20

3.2 Attachments to the main cantilever beam . . . . . . . . . . . . . . . . 21

3.3 Fillets and rib on attachments . . . . . . . . . . . . . . . . . . . . . . 21

3.4 Schematic of slots for different offset ratio . . . . . . . . . . . . . . . 22

3.5 Time series and FFT results . . . . . . . . . . . . . . . . . . . . . . . 23

4.1 FFT result of the beam with No.1 attachment . . . . . . . . . . . . . 28

4.2 Plot of natural frequency with same offset ratio . . . . . . . . . . . . 34

4.3 Plot of natural frequency with same mass ratio . . . . . . . . . . . . 35

4.4 Plot of natural frequency with same offset ratio after tuning Young’smodulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

viii

List of Abbreviations and Symbols

Symbols

uL Y-direction displacement of end point

u0 Y-direction displacement of clamped end

φ Rotational angle of beam

M Offset mass

m Mass of beam

L Length of beam

EI Flexural rigidity of beam

ρ Density of beam

A Cross-sectional area of beam

I Moment inertia of attachments

µ Ratio of end mass and beam mass

ξ Ratio of offset and beam length

ix

1

Introduction

A cantilever beam having a tip mass is a simplified and basic model for some practical

engineering situations, which has drawn wide interests on its dynamic behavior.

Many researches are conducted from different aspects because this system has many

variations on direction, environment, material, etc. Nonlinearity will also appear in

a cantilever beam system in different conditions, and a large number of literature

deals with these nonlinear terms in different ways.

Many researches are conducted on the analysis of a cantilever beam with tip mass.

To (TO (1982)) derived an method for the exact calculation of a cantilever beam with

tip mass and a base excitation. He used Bernoulli-Euler-type beam to obtain the

natural frequencies and mode shapes. Abramovich and Hamburger (Abramovich and

Hamburger (1991)) studied the theoretical result for a timoshenko cantilever beam

with tip mass and presented plots with different (tip mass)/(beam mass) ratio. Xiao

and Du (Xiao Shifu and Bin (2002)) conducted a research on the dynamical behavior

of a centrifugal cantilever beam with tip mass, and conducted theoretical analysis

and modal test to study the effect on flexible multi-body system.

Nonlinearities can arise from the geometry of deformation or from the setup of

1

the system. Large deflections of the continuous beams give rise to geometric nonlin-

earities, and the offset of end mass will also introduce nonlinear terms. Hamid and

Shorya (Moeenfard and Awtar (2014)) utilized Hamilton’s principle, reduced nonlin-

ear partial differential equations to two coupled ordinary differential equations and

then solved the equations analytically. They also compared the analytic results with

numerical ones to validate the accuracy. Shahram and Airton (Shahram Shahlaei-Far

and Balthazar (2016)) analyzed free vibrations of Timoshenko beams using homotopy

analysis method to yield nonlinear natural frequencies and mode shape. Zavodney

and Nayfeh (D. and H. (1989)) conducted research on the nonlinear response of a

slender beam carrying a lumped mass to a principle parametric excitation. Kim

and Bae (Pilkee Kim and Seok (2012)) studied the model of a micro-scale cantilever

beam with tip mass. They used modified Hamilton’s and D’Alembert principle to

mathematically solve the model and applied orthogonality conditions to discretize

the nonlinear equations.

The rest of the thesis is organized as follows: Chapter 2 starts from a linear

system consisting of a cantilever beam with tip mass, and uses Hamilton’s principle

to derive the equations of motion and boundary conditions.Then the system with an

offset mass is analyzed with similar method to derive motion equations. Chapter 3

introduces the design of experimental setup, and the results of experiments. Post-

analysis methods of experimental data are also introduced. Chapter 4 presents all

the theoretical and experimental results, and factors for the deviation are discussed.

2

2

Beam Theory

2.1 Linear system with a lumped mass

Models of vibrating systems can be divided into two classes: discrete and continuous.

In a discrete system, the mass is assumed to be rigid and concentrated, and the

governing equations are ordinary differential equations. Conversely, the mass in a

continuous system is a function of the displacement. Thus the system theoretically

possesses an infinite number of degrees of freedom and the governing equations are

partial differential equations.

To derive governing PDEs, boundary conditions for a partial differential equation

can be divided into two main categories, essential (Dirichlet) and natural (Neumann).

The extended Hamilton principle will be applied for boundary-value problems, and

it is stated asż t2

t1

`

δT ´ δV ` δWnc

˘

dt “ 0 (2.1)

where T is the total kinetic energy, V is the potential energy and δWnc is the virtual

work of non-conservative distributed forces.

We first illustrate Hamilton’s Principle through a simple example. The system

3

with a lumped mass is a good start to carry out analysis because it doesn’t contain

any nonlinear terms. The system consists of a bending cantilever beam with a lumped

mass M at the free end, as shown in Figure 2.1. The displacement of the end is

defined as vpx, tq and the angle is φpx, tq. A small amplitude deflection assumption

is applied in deriving the boundary-value problem, so that sinφ “ φ, cosφ “ φ, and

φpx, tq “ v1px, tq. As a result, the kinetic energy is

T “1

2

ż L

0

ρA

ˆ

Bv

Bt

˙2

dx`1

2M

ˆ

BvLBt

˙2

, (2.2)

and the potential energy is

V “1

2

ż 1

0

EI

ˆ

Bv

Bx

˙2

dx, (2.3)

where EI is the flexural rigidity, ρ is density, A is the cross-sectional area and vL

represents the y-direction displacement at x “ L. For this example, we assume

M “ mL.

Figure 2.1: Schematic of a cantilever beam with a lumped mass

Nondimensionalization is a process that all the dimensional units in every equa-

tion can be factored out to yield a dimensionless equation containing only magnitudes

of quantities. To simplify the analysis in beam theory, we apply nondimensionaliza-

tion to derive equation of motion and boundary conditions with only dimensionless

4

variables and get rid of actual quantities. To start with, dimensional scaling con-

stants are assumed as

x “ Lx

t “ Tcτ

v “ Vcv,

(2.4)

where x, τ, v are the dimensionless variables. After nondimensionalization, the kinetic

and potential energy becomes

T “1

2

ż 1

0

ρA

ˆ

VcTc

˙2ˆBv

Bτ

˙2

d pLxq `1

2M

ˆ

VcTc

˙2ˆBv1Bτ

˙2

“ρAV 2

c

T 2c

«

1

2

ż 1

0

ˆ

Bv

Bτ

˙2

dx`1

2

ˆ

Bv1Bτ

˙2ff

,

V “1

2

ż 1

0

EI

ˆ

VcL2

˙2ˆB2v

Bx2

˙2

d pLxq

“EIV 2

c

L3

1

2

ż 1

0

ˆ

B2v

Bx2

˙2

dx,

(2.5)

According to Buckingham Pi Theorem, all the dimensionless parameters should

be set to 1, so that

ρAV 2c

T 2c

“ 1

EIV 2c

L3“ 1,

(2.6)

thus the scaling constants are

V 2c “

L3

EI

T 2c “

ρAL2

EI.

(2.7)

Because there are no non-conservative forces in this system, δWnc is 0. Applying

5

integration by parts, the variation in the kinetic and potential energy becomes

ż t2

t1

δTdt “ ´

ż t2

t1

„ż 1

0

ˆ

B2v

Bτ 2

˙

δvdx`

ˆ

B2v1Bτ 2

˙

δv1

dt

δV “

ż 1

0

B2v

Bx2δB2v

Bx2dx

“B2v

Bx2δBv

Bx

ˇ

ˇ

ˇ

ˇ

1

0

´B3v

Bx3δv

ˇ

ˇ

ˇ

ˇ

1

0

`

ż 1

0

B4v

Bx4δv

(2.8)

After inserting back into Hamilton principle and collecting terms, the final equation

is obtained as

ż t2

t1

«

´

ż 1

0

ˆ

B2v

Bτ 2

˙

δvdx´

ˆ

B2v1Bτ 2

˙

δv1 ´B2v

Bx2δBv

Bx

ˇ

ˇ

ˇ

ˇ

1

0

`B3v

Bx3δv

ˇ

ˇ

ˇ

ˇ

1

0

´

ż 1

0

B4v

Bx4δvdx

ff

dτ “ 0.

(2.9)

The equation of motion are collected as

v4 ` :v “ 0. (2.10)

Due to a clamped boundary condition at x “ 0, the displacement and slope are

zero at this location. Also, after invoking the arbitrariness of the virtual displacement

δv, the boundary conditions can be obtained as

v0 “ 0

Bv0Bx

“ 0

B2v1Bx2

“ 0

B3v1Bx3

´B2v1Bτ 2

“ 0.

(2.11)

The solution to Equation 2.10 can be separated into variables of displacement

x and time t, and the function of displacement can be assumed as a harmonic mo-

tion expression vpx, tq “ CV pxq cospωt ´ φq. Substituting into Equation 2.10 and

6

boundary conditions, the equation of motion can be written as

V 4 ` ω2V “ 0, (2.12)

and a set of boundary conditions only including the displacement variable x can be

attained as

V p0q “ 0

V 1p0q “ 0

V 2p1q “ 0

V p1q3 ´ ω2V p1q “ 0.

(2.13)

The general solution to Equation 2.12 can be verified to be V “ A sin βx`B cos βx`

C sinh βx `D cosh βx, where β4 “ ω2. Inserting the assumed solution to boundary

equations yields the system

$

’

’

’

’

’

’

’

’

&

’

’

’

’

’

’

’

’

%

B `D “ 0

A` C “ 0

β3 pÁ cos β `B sin β ` C cosh β `D sinh βq

`β4 pA sin β `B cos β ` C sinh β `D cosh βq “ 0

pβ sin β ´ cos βqA` pβ cos β ` sin βqB

`pβ sinh β ` cosh βqC ` pβ cosh β ` sinh βqD “ 0.

(2.14)

A matrix form can be obtained by rewriting the equation set regrading A,B,C and

D

»

—

—

–

0 1 0 11 0 1 0

´ sin β ´ cos β sinh β cosh ββ sin β ´ cos β β cos β ` sin β β sinh β ` cosh β β cosh β ` sinh β

fi

ffi

ffi

fl

»

—

—

–

ABCD

fi

ffi

ffi

fl

“ 0.

From this equation, since the matrix has a non-trivial nullspace, it is not invertible

thus the determinant of this matrix should be zero. After determining the charac-

teristic equation, the first three solutions of β are

β ““

1.2479 4.0311 7.1341‰

, (2.15)

7

then the first three modes of natural frequencies are

ω ““

1.5573 16.2498 50.8954‰

Hz. (2.16)

Figure 2.2 shows the natural frequency trend of first mode corresponding with

mass ratio. For a more general system where the mass ratio µ “ M{m ‰ 1, when

the mass ratio increases, the natural frequency decreases.

Figure 2.2: Schematic of first mode natural frequency with different mass ratios

2.2 Offset End Mass System

Figure 2.3 shows the system with an mass that has an offset. The system consists of

a cantilever beam with length L and an offset mass M with an offset d from the end

of the beam. Small angle assumption is still true for this system, so that sinφ “ φ,

cosφ “ φ, and φpx, tq “ v1px, tq.

In order to derive the kinetic energy and potential energy, the position vector of

the end mass is

rM “ LE1 ` vLE2 ` db2, (2.17)

then the derivative of position vector, which is velocity of the end mass is

9rM “ 9vLE2 ` d 9b2, (2.18)

8

Figure 2.3: Schematic of a cantilever beam with an offset mass

where

9b2 “ 9φˆ b2

“

∣∣∣∣∣∣b1 b2 b3

0 0 9φL0 1 0

∣∣∣∣∣∣“ ´ 9φLb1.

(2.19)

Substituting Equation 2.19 into Equation 2.18, the velocity can be written as

9rM “ 9vLE2 ´ d 9φ2b1

“ 9vLE2 ´ d 9v12Lb1.(2.20)

Accordingly, the kinetic energy for the end mass becomes

TM “1

2M

´

9vL2´ 2d 9v1L 9vL sinφL ` d

2 9vL2¯

“1

2M

`

9v2L ´ 2d 9vL 9vL1v1L ` d

2 9vL2˘

.

(2.21)

Total kinetic energy includes two parts: kinetic energy of the beam and end mass

derived in Equation 2.21. Since nonlinear terms remain after small angle assumption,

they will be maintained to study the effects.

9

Thus, the total kinetic energy will be

T “1

2

ż L

0

ρA

ˆ

Bv

Bt

˙2

dx`1

2M

`

9vL2´ 2d 9vL 9vL

1v1L ` d2 9vL

2˘

, (2.22)

where the first term represents the beam kinetic energy and the second term shows

that of the end mass.

The potential energy is the same because it only comes from the bending of the

beam:

V “

ż L

0

EIv22. (2.23)

For nondimensionalization, dimensional scaling constants are assumed as

v “ Vcv

x “ Lx

t “ Tcτ,

(2.24)

where v, x, τ are the dimensionless variables. After nondimensionalization, the ki-

netic energy becomes

T “1

2ρA

ˆ

VcTc

˙2ˆBv

Bτ

˙2

`

1

2M

«

ˆ

VcTc

˙2ˆBv1Bτ

˙2

´ 2dVcTc

VcTcL

VcL

9v1 9v11v11` d2

VcLTc

´

9v11¯2

ff

“ρAV 2

c

T 2c

¨1

2

ż 1

0

9v2dx`

1

2M

ˆ

VcTc

˙2«

ˆ

Bv1Bτ

˙2

´ 2

ˆ

d

L

˙ˆ

VcL

˙

9v1 9v11v11`

ˆ

d

L

˙2´

9v11¯2

ff

,

(2.25)

10

and the potential energy is

V “1

2

ż 1

0

EI

ˆ

VcL2

˙2

pv2q2d pLxq

“EIV 2

c

L3¨

1

2

ż 1

0

pv2q2dx.

(2.26)

To simplify the equations, all the following equations are in dimensionless forms and

v represents v. Then obtaining and rearranging the Lagrangian as

L “1

2

ż 1

0

ˆ

ρAV 2c L

T 2c

9v2 ÉIV 3

c

L3v22

˙

dx`

1

2M

ˆ

VcTc

˙2«

9v12´ 2

d

L

VcL

9v1 9v11v11 `

ˆ

d

L

˙2

9v112

ff

“ρAV 2

c L

T 2c

1

2

ż 1

0

ˆ

9v2 ÉIT 2

c

ρAL4v22

˙

dx`

1

2M

ˆ

VcTc

˙2«

9v12´ 2

d

L

VcL

9v1 9v11v11 `

ˆ

d

L

˙2

9v112

ff

.

(2.27)

Similarly, the dimensionless parameters should be set to 1:

EIT 2c

ρAL4“ 1

VcL“ 1,

(2.28)

so the dimensionless constants can be derived as

Vc “ L

Tc “

c

ρAL4

EI.

(2.29)

11

The dimensionless Lagrangian is now

L “ EI

L

«

1

2

ż 1

0

`

9v2 ´ v22˘

dx`1

2

M

ρAL

˜

9v12´ 2

d

L9v1 9v11v

11 `

ˆ

d

L

˙2

9v112

¸ff

“EI

L

„

1

2

ż 1

0

`

9v2 ´ v22˘

dx`1

2µ´

9v12´ 2ζ 9v1 9v11v

11 ` ζ

2 9v112¯

,

(2.30)

where µ “ MρAL

is the ratio of the end mass and the beam, and ζ “ dL

is the offset

ratio.

For the system, variation of L can be written as:

δL “ BLB 9vδ 9v

loomoon

a

`BLBv2

δv2loomoon

b

`BLB 9v1

δ 9v1loomoon

c

`BLB 9v1

1 δ 9v11

looomooon

d

`BLBv11

δv11loomoon

e

. (2.31)

The non-conservative distributed force is also zero in this system, so δWnc “ 0.

Because there are five different variables in the Lagrangian L, δL is divided into

five terms and studied separately. Starting with term a, c, d, these terms come from

kinetic energy and the variables have a connection with time, so it can be written

as:ż t2

t1

δLa “ż t2

t1

ˆż 1

0

9vd

dtδv

˙

dt

“

ż 1

0

˜

9vδv

ˇ

ˇ

ˇ

ˇ

t2

t1

´

ż t2

t1

:vδvdt

¸

dx

“

ż 1

0

˜

9vδv

ˇ

ˇ

ˇ

ˇ

t2

t1

¸

dx´

ż t2

t1

ˆż 1

0

:vδvdx

˙

dt

“ ´

ż t2

t1

ˆż 1

0

:vδvdx

˙

dt.

(2.32)

12

Then, following the same steps in equation.2.32, the integral can be obtained as

ż t2

t1

δLc “BLB 9v1

δv1

ˇ

ˇ

ˇ

ˇ

t2

t1

´

ż t2

t1

ˆ

d

dt

BLB 9v1

δv1

˙

dt

ż t2

t1

δLd “BLB 9v1

1 δv11

ˇ

ˇ

ˇ

ˇ

t2

t1

´

ż t2

t1

ˆ

d

dt

BLB 9v1

1 δv11

˙

dt.

(2.33)

Similarly, variables of terms b, e only have connection with x, so

δLb “ ´δV “ ´ż 1

0

v2δv2dx

“ ´

ż 1

0

v2B

Bxδv1dx

“ ´v2δv1ˇ

ˇ

ˇ

ˇ

1

0

`

ż 1

0

v3δv1dx

“ ´v2δv1ˇ

ˇ

ˇ

ˇ

1

0

` v3δv

ˇ

ˇ

ˇ

ˇ

1

0

´

ż 1

0

v4δvdx,

(2.34)

andż t2

t1

Le “ż t2

t1

ˆ

BLBv11

δv11

˙

dt. (2.35)

Collecting all items and the Lagrangian becomes

ż t2

t1

δLdt “ż t2

t1

ˆż 1

0

´p:v ` v4qδvdx´d

dt

BLB 9v1

δv1 ´d

dt

BLB 9v1

1 δv11 `

BLBv11

δv11

˙

dt

`

ż t2

t1

˜

´v2δv1ˇ

ˇ

ˇ

ˇ

1

0

` v3δv

ˇ

ˇ

ˇ

ˇ

1

0

¸

dt

“

ż t2

t1

„ż 1

0

´p:v ` v4qδvdx`

ˆ

v21 ´d

dt

BLB 9v1

1

˙

δv11

dt

`

ż t2

t1

„

´

ˆ

v31 ´d

dt

BLB 9v1

˙

δv1 ` v20δv

10 ´ v

30 δv0

dt

(2.36)

13

The equation of motion is the same as the one derived from linear system with a

lumped mass:

v4 ` :v “ 0. (2.37)

To derive the boundary value problems, all the boundary equations are collected as:

v20δv10 “ 0 (2.38a)

v30 δv0 “ 0 (2.38b)

ˆ

v21 ´d

dt

BLB 9v1

1

˙

δv11 “ 0 (2.38c)

´

ˆ

v31 ´d

dt

BLB 9v1

˙

δv1 “ 0. (2.38d)

Equation 2.38a can be satisfied in two ways by setting either v20 or δv10 equal to zero.

The first term is the defined as the angular velocity at x “ 0, which cannot be zero

for all times, so that Equation 2.38a is satisfied by setting

v10 “ 0. (2.39)

Similarly, Equation 2.38b is satisfied by setting

v0 “ 0. (2.40)

The process is similar with the linear system with a lumped mass. Substituting

the Lagrangian L and invoking the arbitrariness of the virtual displacement δv, the

displacement and slope cannot be zero for all times at x “ 1, so for Equation 2.38c

and 2.38d, the displacement must satisfy the boundary conditions

v31 ´ µ :v1 “ 0

v21 ` µζ2 :v1

1“ 0,

(2.41)

14

In conclusion, all the boundary conditions become

v0 “ 0

v10 “ 0

v31 ´ µ :v1 “ 0

v21 ` µζ2 :v1

1“ 0.

(2.42)

The solution of length x can still be assumed in the harmonic form of

v “ cV pxq cospωt´ φq, (2.43)

where V pxq are the natural modes and ω is the natural frequency, both derived from

the system eigenvalue problem. After substituting into Equation 2.37 and 2.42, the

equation of motion becomes

V 4 ` ω2V “ 0, (2.44)

and the boundary conditions become

V0 “ 0

V 10 “ 0

V 31 ` µω2V1 “ 0

V 21 ´ µζ2ω2V 11 “ 0.

(2.45)

The solution V pxq to Equation 2.44 can be written as V pxq “ A sin βx`B cos βx`

C sinh βx ` D cosh βx, where A,B,C,D are constants to be solved, and β4 “ ω2.

Substituting the assumption solution to Equation 2.45, the boundary condition set

becomes

$

’

’

’

’

’

’

’

’

&

’

’

’

’

’

’

’

’

%

A` C “ 0

B `D “ 0

pÁ cos β `B sin β ` C cosh β `D sinh βq`

µβ pA sin β `B cos β ` C sinh β `D cosh βq “ 0

pÁ sin β ´B cos β ` C sinh β `D cosh βq

´µζ2β3 pA cos β ´B sin β ` C cosh β `D sinh βq “ 0

(2.46)

15

A matrix form can be obtained by rearranging the equation set:

»

—

—

–

0 1 0 11 0 1 0m31 m32 m33 m34

m41 m42 m43 m44

fi

ffi

ffi

fl

»

—

—

–

ABCD

fi

ffi

ffi

fl

“ 0 (2.47)

where

m31 “ ´ cos β ` µβ sin β

m32 “ sin β ` µβ cos β

m33 “ coshµβ ` µβ sinh β

m34 “ sinh β ` µβ cosh β

m41 “ ´ sin β ´ µζ2β3 cos β

m42 “ ´ cos β ` µζ2β3 sin β

m43 “ sinh β ´ µζ2β3 cosh β

m44 “ cosh β ´ µζ2β3 sinh β

(2.48)

By setting the determinant equal to zero, the natural frequency can be obtained.

Figure 2.4 reveals the natural frequency trend with the offset ratio. For the same

mass ratio, the first mode natural frequency decreases along with the increase of

offset ratio. While for the same offset ratio, natural frequency follows the same

tendency when mass ratio increases.

2.3 Moment of Intertia

In the actual experiments, the attachments are a set of thin rectangular plates with

ribs and slots, so that moment of inertia can be included in the system while vi-

brating. When taking the inertia into consideration, the dimensionless Lagrangian

becomes

L “„

1

2

ż 1

0

`

9v2 ´ v22˘

dx`1

2µ´

9v12´ 2ζ 9v1 9v11v

11 ` pI ` ζ

2q 9v11

2¯

, (2.49)

16

Figure 2.4: Effects of offset ratio on natural frequency

so that the matrix form of Equation 2.47 becomes

»

—

—

–

0 1 0 11 0 1 0m31 m32 m33 m34

m41 m42 m43 m44

fi

ffi

ffi

fl

»

—

—

–

ABCD

fi

ffi

ffi

fl

“ 0, (2.50)

where

m31 “ ´ cos β ` µβ sin β

m32 “ sin β ` µβ cos β

m33 “ coshµβ ` µβ sinh β

m34 “ sinh β ` µβ cosh β

m41 “ ´ sin β ´ µpI ` ζ2qβ3 cos β

m42 “ ´ cos β ` µpI ` ζ2qβ3 sin β

m43 “ sinh β ´ µpI ` ζ2qβ3 cosh β

m44 “ cosh β ´ µpI ` ζ2qβ3 sinh β.

(2.51)

The change of natural frequency is shown in Figure. 2.5. When the moment of

17

inertia of the offset increases, the natural frequency shows a steady decrease.

Figure 2.5: Effects of moment inertia on natural frequency

18

3

Experimental Setup

3.1 Experimental Setup Design

An experiment was designed to study the effect of the offset mass. The experi-

mental setup consists of an aluminum base and a cantilever beam coupled with its

attachment, as shown in Figure 3.1. Beam parameters are shown in Table 3.1. The

cantilever beam is clamped on the aluminum base, and two sets of beams are attached

to it according to different conditions, which represents the offset mass.

Table 3.1: Beam parameters

Mass (g) 9.9Length (mm) 250.19Width (mm) 19.8Thickness (mm) 2.18Density (g/cm3) 1.04Young’s Modulus (MPa) 2200

The aluminum base consists of a fixed base and a moving part connected by

screws. The fixed base is mounted on the table using screws so that it cannot have

any movement. After plugging the beam to the predetermined location, the moving

19

Figure 3.1: Experimental setup of a cantilever beam with an offset mass

plate is used to clamp the beam to make sure the actual boundary conditions as

close as possible to theoretical assumptions.

The system is made of ABSplus-P430 by 3D printing, which is a production-

grade thermoplastic that is durable enough to perform virtually the same as produc-

tion parts (Stratasys (2018)). When combined with 3D Printers, ABSplus is ideal

for building 3D models and prototypes. Properties of this light-weight and highly

machinable material can be obtained from data sheet.

The offset mass is realized by attaching different bars to the end of main beam

as shown in Figure 3.2. Each bar has a rib to add rigidity and make its own natural

frequency extremely high compared with the cantilever beam, and a slot is used for

attaching to the beam on the other side. To reduce stress concentration and add

flexibility when plugging the beam, fillets are added to the slot as shown in Figure

3.3.

There are two sets of attachment for this experimental setup. The first set in-

cludes five bars with same length but different widths. These five widths given in

terms of beam widths are respectively 0.25b, 0.5b, 0.75b, b and 1.25b, which pro-

20

Figure 3.2: Attachments to the main cantilever beam

Figure 3.3: Fillets and rib on attachments

vides a set of attachments with different mass but same offset. To illustrate these

attachments more clearly, they are named after No.1 - No.5 attachment with the

increase of width. Parameters for the attachment are shown in Table 3.2. The other

set includes one beam with five slots evenly distributed on it. The gap between slots

is 34mm, and mass for this attachment is 5.5g. When the cantilever beam is plugged

into different slots, the attachments are considered as same mass but different offsets.

To illustrate the experiments more clearly, these five slots are named in Figure 3.4.

21

Table 3.2: Attachment parameters(Unit of mass: gram/ Unit of length: mm)

No. Mass Base length Base Width Base Thickness Rib Length Rib Width

1 0.9 50.09 4.99 2.12 2 2.6

2 1.7 50.02 9.94 2.05 2.92 3.67

3 3.1 49.74 15.25 2 5.01 5.51

4 4.4 49.94 19.94 2.13 5.92 6.72

5 5.6 49.96 24.96 2.09 6.96 7.8

Figure 3.4: Schematic of slots for different offset ratio

The natural frequency was determined by the unforced free oscillations of the

cantilever beam by giving an initial displacement to the system. As a result, an

accelerator is mounted to the vibrating beam to measure the position. It is then

connected to a NI data acquisition (DAQ) device, and the position-time plot will

be shown on a pre-designed MATLAB interface. The data saved can be used in

analyzing other system characteristics. Mass of the accelerator used is 0.7g, which

is 7.1% of the beam mass. It is then considered as a distributed mass to make the

theoretical model closer to actual situation.

3.2 Experimental Data and Analysis

The data obtained from experiments was processed in MATLAB to calculate natural

frequency and damping ratio. Fast Fourier Transform(FFT) rapidly computes the

22

transformation for a signal from time domain to frequency domain. Code using FFT

is generated to analyze the experimental data, and one of the results for the system

is shown in Figure 3.5.

Figure 3.5: Time series and FFT results

The decrease in amplitude from one cycle to the next depends on the extent

of damping in the system. Because the successive peak amplitudes bear a certain

specific relation ship involving the damping of the system, the method of logarithmic

decrement is formed to evaluate the damping ratio of a underdamped system:

δ “ lnx1x2“

2πξa

1´ ξ2, (3.1)

where x1, x2 is the first two displacement. So the damping ratio is

ξ “δ

a

p2πq2 ` δ2. (3.2)

Time response for each experiment gives two displacement as shown in Figure

3.5. Damping ratio can be derived from the method of logarithmic decrement.

3.3 Uncertainty Analysis

The measurements of the variables have uncertainties associated with them, and the

values of the material properties obtained from reference resources also have uncer-

23

tainties (Coleman and Steele (1999)). General uncertainty analysis is an approach to

consider only the uncertainties in each variables, neglecting random errors. If result

r is a function of J variables Xi:

r “ rpX1, X2, ..., XJq,

the general uncertainty is defined as

U2r “ p

Br

BX1

q2U2

X1` p

Br

BX2

q2U2

X2` ...` p

Br

BXJ

q2U2

XJ,

where the UXiare the absolute uncertainties in the variables Xi. So the definition of

relative uncertainties is

ˆ

Urr

˙2

“

ˆ

X1

r

Br

BX1

˙2ÛX1

X1

˙2

`

ˆ

X2

r

Br

BX2

˙2ÛX2

X2

˙2

`...`

ˆ

XJ

r

Br

BXJ

˙2ÛXJ

XJ

˙2

,

(3.3)

where the uncertainty magnification factors (UMFs) are defined as

UMFi “Xi

r

Br

BXi

. (3.4)

In the cantilever beam system, the derivation equation of natural frequency is

ωi “ pβiLq2

c

EI

mL4

“ pβiLq2

d

E 112ct3

ρctL4

“ 0.2887pβiL2qt

L2

d

E

ρ,

(3.5)

where c, t are width and thickness of the beam. The general uncertainty expression

24

becomes

ˆ

Uωω

˙2

“

ˆ

t

ω

Bω

Bt

˙2Ûtt

˙2

`

ˆ

L

ω

Bω

BL

˙2ÛLL

˙2

`

ˆ

E

ω

Bω

BE

˙2ÛEE

˙2

`

ˆ

ρ

ω

Bω

Bρ

˙2Ûρρ

˙2

.

(3.6)

The UMFs are

UMFt “t

ω

Bω

Bt“

t

ω0.2887pβiLq

2 1

L2

d

E

ρ(3.7a)

UMFL “L

ω

Bω

BL“L

ω0.2887pβiLq

2t

d

E

ρp´2q

1

L3(3.7b)

UMFE “E

ω

Bω

BE“E

ω0.2887pβiLq

2 t

L2

c

1

ρ

1

2

1?E

(3.7c)

UMFρ “ρ

ω

Bω

Bρ“ρ

ω0.2887pβiLq

2 t

L2

?E

ˆ

´1

2

˙

ρ´32 . (3.7d)

Substituting Equation 3.7 into Equation 3.6,

pUωωq2“

¨

˝

t

0.2887pβiL2q tL2

b

Eρ

0.2887pβiLq2 1

L2

d

E

ρ

˛

‚

2

pUttq2

` 4

¨

˝

L

0.2887pβiL2q tL2

b

Eρ

0.2887pβiLq2t

d

E

ρ

1

L3

˛

‚

2

pULLq2

`1

4

¨

˝

E

0.2887pβiL2q tL2

b

Eρ

0.2887pβiLq2 t

L2

c

1

ρ

1?E

˛

‚

2

pUEEq2

`1

4

¨

˝

ρ

0.2887pβiL2q tL2

b

Eρ

0.2887pβiLq2 t

L2

?E

ˆ

´1

2

˙

ρ´32

˛

‚

2

pUρρq2

“pUttq2` 4p

ULLq2`

1

4pUEEq2`

1

4pUρρq2.

(3.8)

Relative uncertainties for natural frequency can be obtained from Equation 3.8 using

the relative uncertainties of thickness, length, Young’s modulus and density. From

25

Equation 3.8, the system is most sensitive to beam’s length, while Young’s modulus

and density have same and smallest effect on the results.

26

4

Analysis and Discussion

4.1 Experimental Results

Experiments were carried out using the above-mentioned setup. Experiments of

the beam with each attachments were carried out five times repeatedly to get rid

of random error. When conducting the FFT, the sampling frequency is chosen to

be 1000Hz, and as shown in results, all the FFT results for the one experimental

setup are consistent. After the calculation, as shown in Figure 4.1, the results of

each experimental natural frequency are read directly from the plot and recorded.

Calculating the average of all the results leads to the final experimental natural

frequencies. For beam with same offset but different mass, the offset ratio is

ζ “23

196.92“ 0.117,

and result of FFT for the third trial with attachment 1 is shown in Table 4.1.

For beam with different offset but same mass, results are shown in Table 4.2 after

applying the same approach to experimental data. Mass ratio here is

µ “5.5

9.9“ 0.556.

27

Figure 4.1: FFT result of the beam with No.1 attachment

Table 4.1: Experimental Results of natural frequencies with same offset ratio

Set Mass Ratio µ Frequency(Hz)Beam 0 12.2Beam with No.1 Attachment 0.091 10Beam with No.2 Attachment 0.172 8.714Beam with No.3 Attachment 0.313 7.286Beam with No.4 Attachment 0.444 6.5Beam with No.5 Attachment 0.566 5.9

Table 4.2: Experimental Results of natural frequencies with same mass ratio

Set Offset Ratio ζ Frequency(Hz)Beam with No.1 Slot -0.361 5.5Beam with No.2 Slot -0.254 5.7Beam with No.3 Slot 0 6.1Beam with No.4 Slot 0.254 5.8Beam with No.5 Slot 0.361 5.4

28

4.2 Theoretical Result

Following the steps discussed in Chapter 2.2 , MATLAB code is generated to derive

the natural frequency theoretically and the results are shown in Table 4.3 and Table

4.4. Because all these theoretical results come from dimensionless equations, they

don’t have any unit here.

Table 4.3: Theoretical Results of natural frequencies with same offset ratio

Set Natural FrequencyBeam 3.516Beam with No.1 Attachment 3.0007Beam with No.2 Attachment 2.69Beam with No.3 Attachment 2.3129Beam with No.4 Attachment 2.07Beam with No.5 Attachment 1.91

Table 4.4: Theoretical Results of natural frequencies with same mass ratio

Set Natural FrequencyBeam with No.1 Slot 1.76Beam with No.2 Slot 1.8994Beam with No.3 Slot 1.9446Beam with No.4 Slot 1.8994Beam with No.5 Slot 1.76

From Table 4.3, natural frequency decreases when offset is constant and mass

increases. Similarly, it also shows a downtrend when offset increases. It is reasonable

from a physical point of view because the beam will obviously have lower natural

frequency when the mass becomes larger.

4.3 Damping Ratio

Damping ratio is considered as an influencing factor for the system. Damping ratio

of each system is calculated using the method of logarithmic decrement as discussed

29

in Chapter 3.2 for each time response, and then taking the average of results for

five trials leads to the final damping ratios as shown in Table 4.5 and Table 4.6.

Table 4.5: Damping ratio of natural frequencies with same offset ratio

Set x1 x2 lnpx1{x2q Damping ratio ξBeam 0.0727 0.0646 0.1158 0.0184Beam with No.1 Attachment 0.0748 0.0674 0.1046 0.0166Beam with No.2 Attachment 0.0709 0.0669 0.0581 0.0093Beam with No.3 Attachment 0.0729 0.0672 0.0809 0.0129Beam with No.4 Attachment 0.0745 0.0676 0.0965 0.0154Beam with No.5 Attachment 0.0758 0.0685 0.0995 0.0158

Table 4.6: Damping ratio of natural frequencies with same mass ratio

Set x1 x2 lnpx1{x2q Damping ratio ξBeam 0.0727 0.0646 0.1158 0.0184Beam with No.1 Slot 0.0658 0.0600 0.0913 0.0145Beam with No.2 Slot 0.0608 0.0561 0.0822 0.0131Beam with No.3 Slot 0.0595 0.0548 0.0824 0.0131Beam with No.4 Slot 0.0501 0.0463 0.0787 0.0125Beam with No.5 Slot 0.0488 0.0435 0.1143 0.0182

Undamped natural frequency, which is more similar to the condition of theoretical

assumption, can be obtained using

ωn “ωd

a

1´ ξ2, (4.1)

where ωn is undamped natural frequency, and ωd is the damped natural frequency

obtained from the experiments.

Taking damping ratio into consideration gives to the undamped natural frequency

as shown in Table 4.7 and 4.8. From the tables, the undamped frequencies have very

little difference with damped frequencies, so we can conclude that damping ratio in

this system is small in this system and has little effect on the system.

30

Table 4.7: Comparison of natural frequencies with same offset ratio including damp-ing ratio

µ ωed(Experimental) ωed ωenrad/s

Beam 0 76.655 3.555 3.556

Beam with No.1 Attachment 0.091 62.832 2.913 2.914





Table 4.8: Comparison of natural frequencies with same mass ratio including damp-ing ratio

ξ ωed(Experimental) ωed ωenrad/s

Beam with No.1 Slot -0.361 35.673 1.608 1.608

Beam with No.2 Slot -0.277 36.186 1.678 1.678

Beam with No.3 Slot 0 37.442 1.736 1.737

Beam with No.4 Slot 0.277 36.814 1.707 1.707

Beam with No.5 Slot 0.361 34.301 1.590 1.591

4.4 Moment of Inertia

When calculating the moment of inertia for the attachments, the slot will be ne-

glected, so that they are considered as standard plates with ribs. Non-dimensionalizing

the moment of inertia gives to Equation 4.2.

I “IMML2

“

112mp4h2 ` w2q ` 1

12m2p4h

22 ` w

22q `m2d

2

ML2,

(4.2)

where m, h and w are the mass, length and width of the plate, m2, h2 and w2 are

the parameters of the rib, and d shows the offset between the plate and rib.

Including moment of inertia of offset bar gives to the results as shown in Table

4.9 and Table 4.10. The tables indicate that moment of inertia, similar to damping

31

ratio, doesn’t have significant influence on results.

Table 4.9: Natural Frequency for same offset ratio with I

Set µ ωtn ωtn with inertia

Beam 0 3.516 3.516

Beam with No.1 Attachment 0.091 3.001 2.979





Table 4.10: Natural Frequency for same mass ratio with I

Set ξ ωtn ωtn with inertia

Beam with No.1 Slot -0.361 1.760 1.743

Beam with No.2 Slot -0.277 1.890 1.810

Beam with No.3 Slot 0 1.945 1.914

Beam with No.4 Slot 0.277 1.890 1.810

Beam with No.5 Slot 0.361 1.760 1.743

4.5 Poisson’s Ratio

Poisson’s ratio is the ratio of transverse contraction strain to longitudinal extension

strain in the direction of stretching force. It has an effect on the Young’s modulus

according to the analysis of Arafat (Arafat (1999)). As shown in his dissertation,

Lame constants are introduced as

µ “E

2p1` νq

λ “Eν

1´ ν ´ 2ν2,

where ν is Poisson’s ratio for material. For a simple beam,

Q1111 “ λ` 2µ

“ Epν

1´ ν ´ 2ν2`

1

p1` νqq.

32

Poisson’s ratio for ABS plastic is about 0.35, which makes Young’s modulus E bigger.

Poisson’s ratio will lead to an increase in the Young’s modulus, which is not

consistent with the trend of experimental results. Therefore, it will not be taken into

consideration as an influencing factor for the deviation.

4.6 Result Discussion

The mass of beam becomes 9.9` 0.7 “ 10.6g after including the mass of accelerator.

As a result, the density has a 7% increase to 1.25 g/cm3. Based on other parameters

in the experiment, the scaling constants Tc is

Tc “

c

ρAL4

EI

“

c

1.25ˆ 103 ˆ 0.000043164ˆ 0.196924

2.2ˆ 109 ˆ 1.70944ˆ 10´11

“ 0.0463 1{s

Applying the scaling constants for time and damping ratio calculated to experimental

results leads to the final comparison of experimental and theoretical results. For

experiments of beam with different masses where ζ “ 0.117, results are shown in

Table 4.11 and Table 4.12.

ωed(Experimental), ωed, ωen and ωtn represents damped natural frequencies from

experiments, damped nondimensional natural frequencies, undamped nondimensional

natural frequencies and theoretical natural frequencies.

Results for each natural frequency are shown in Figure 4.2 and Figure 4.3.

4.7 Uncertainty

Uncertainty can be derived from Equation 3.8 as discussed in Chapter 3.3. Taking

results from same offset ratio as an example, the uncertainty comes from many

33

Table 4.11: Comparison of natural frequencies with same offset ratio

µ ωen ωtnBeam 0 3.556 3.516






Table 4.12: Comparison of natural frequencies with same mass ratio

ξ ωen ωtnBeam with No.1 Slot -0.361 1.608 1.743

Beam with No.2 Slot -0.277 1.678 1.810

Beam with No.3 Slot 0 1.736 1.914

Beam with No.4 Slot 0.277 1.707 1.810

Beam with No.5 Slot 0.361 1.591 1.743

Figure 4.2: Plot of natural frequency with same offset ratio

34

Figure 4.3: Plot of natural frequency with same mass ratio

aspects, including the material properties and geometric features of the beam. From

the equation of scaling time constants, uncertainties of other parameters can be

minimized using methods like repeated measurement except Young’s modulus. Thus

all weight of uncertainty is put on Young’s modulus. The theoretical property is

calculated by tuning the Young’s modulus to match the last experimental result

with the theoretical one, which has the largest deviation of 9.47%. Matching the last

point in the experiment to the theoretical data

ωe5 “ ωt5 “ 1.8823, (4.3)

so the nondimensionalized damped natural frequency is

ωe5 “ 1.8823â

1´ ζ2

“ 1.8823ˆ?

1´ 0.01582

“ 1.8821,

(4.4)

35

then the corresponding time scaling constant is

Tc “ωepnondimensionalq

ωe

“1.8821

37.071

“ 0.05076,

(4.5)

so the tuned Young’s modulus is

E “ρAL4

T 2c I

“1.25ˆ 103 ˆ 4.3ˆ 10´5ˆ 0.19692

0.050762 ˆ 1.709ˆ 10´11

“ 1.84ˆ 109Pa.

(4.6)

From Equation 3.8, the uncertainty for the Young’s modulus is

UE “ Een ´ Etn

“ 2.2ˆ 109´ 1.84ˆ 109

“ 3.64ˆ 108

UEE“

3.64ˆ 108

2.2ˆ 109

“ 16.5%,

(4.7)

so the uncertainty for natural frequency is

Uω1

ω1

“

c

1

4pUEEq2

“

c

1

4ˆ 0.1652

“ 8.25%.

(4.8)

The plot of natural frequency with tuned Young’s modulus is shown in Figure. 4.4.

36

Figure 4.4: Plot of natural frequency with same offset ratio after tuning Young’smodulus

37

5

Conclusion

This thesis investigates the natural frequency of an asymmetric system consisting of

a cantilever beam and different offset masses.

The beam theory of a cantilever beam is first generated using Hamilton’s princi-

ple. After deriving the kinetic and potential energy, equation of motion is written for

the system. Nondimensionalization is applied to simplify the analysis. Essential and

natural boundary conditions are defined to obtain the harmonic motion expression.

The system including the cantilever beam with different offset masses is then ana-

lyzed using the same method. Analysis including moment of inertia is also performed

to get closer to the experimental environment. An experiment is then designed to

verify the theoretical calculation. After repeating with different offset masses, a set

of data is obtained and used to derive other parameters in the system.

The theoretical analysis shows the decreasing trend of first-mode natural fre-

quency when the attachment mass or the offset of attachment increases. From the

plots, the experimental results have a reasonable trend showing the decrease but have

big deviation with the theoretical ones. We take several influencing factors into con-

sideration: damping ratio in the system, moment of inertia of attachments, Poisson’s

38

ratio of the material and beam properties. The method of logarithmic decrement is

applied to derive damping ratio, and it is proved to have little effect on the result.

Moment of inertia for the attachments is also calculated but doesn’t have significant

difference in this system. Poisson’s ratio will make Young’s modulus higher, which

results in opposite tendency of the natural frequency. We conclude that the biggest

influencing factor lays on the beam properties.

ABS plastic is the material chosen in this experiment instead of commonly used

metal setup. All the beam and attachments are printed out by 3D printers, which

increases unknown uncertainty to the process. The material properties of ABS plastic

comes from the data sheet provided by the manufacturer.The beam and attachments

are printed out using a 3D printer whose smallest resolution is 0.5mm. Printing

orientation is calculated by the corresponding software for the printer and defined

by the chosen position on the printing platform. Filling level is also a factor for

the printed beam. 3D printing provides a new way to build models for structural

analysis. It has much flexibility and can be used to build complex system, but it also

has many limitations learned from the analysis. The precision of printer determines

the quality of the printed model, which is important for a system to be consistent

in property for all parts. Making sure that the printed structure is isotropic is also

essential to perform all beam theory analysis, but 3D printing increases uncertainty

from this aspect. As a result, using 3D printing to complete dynamic analysis is

convenient but needs further consideration.

39

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Shahram Shahlaei-Far, A. N. and Balthazar, J. M. (2016), “Nonlinear Vibrations ofCantilever Timoshenko Beams: A Homotopy Analysis,” Latin American Journalof Solids and Structures, 13, 1866–1877.

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Dynamic Analysis of a Cantilever Beam with an Offset Mass

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