Reviewers' comments:

Reviewer #1 (Remarks to the Author):

The m

anuscript “Computationally efficient design of directionally compliant metamaterials” by

Shaw et al. reports the design, fabrication and experimental validation of a novel class of

metamaterials which exhibit prescribed deformations, such as torsion, rotation and shear. Their

design strategy hinges on a purely kinematic design approach, which translates target motions

into sets of constraints. In turn, such constraints can be expressed mathematically as a linear

system of equations, that can be solved very rapidly numerically. As a result, the internal

architecture of metamaterials with predefined macroscopic shapes (e.g. 5x5x5 cube) can be

designed for a wide range of mechanical deformations, from motions with unique degrees of

freedom to motions with multiple (up to 3) degrees of freedom. Although their design strategy is

not new (it was already developed for compliant mechanisms), its generalisation in the context of

metamaterials is novel and a very significant addition to the metamaterial toolbox. I believe that

the results reported here will be of strong interest to mechanical engineers, physicists and material

scientists interested in metamaterials. The off

-the

-self computational tool they provide might even

make it readily useable beyond the scientific

community, e.g. by product designers. Therefore I

think that this manuscript is a potential very good fit for Nature Communications. That being said,

I think that as it stands, the paper lacks too much precision and clarity to be fully convincing and

to b

e readable by the broad readership of Nature Communications. For these reasons, I strongly

encourage the authors to address the following comments:

1. The extensive use of acronyms (more than 1 per line on average) makes the paper extremely

difficult to follow. I believe that a significant rephrasing would allow the authors to use less

repetitions and therefore to no rely that much on acronyms.

2. In some instances, the vocabulary is not precise enough, e.g. in the definition of “the yellow

shaded regio

n...” (l 143, p7), what is a “viable topology”? Also, always referring to the labelled

yellow region is extremely confusing because as a reader, we loose track of the underlying

scientific nature of such class of freedom spaces.

3. In all the FEA plots (Fig. 3b, Figs.4g,h,i, figs. 5d,e,f, Figs. 6i,j, Fig. 7i, Figs. 8 b,c,e,f,g), it is

never clearly stated in the caption which numerical protocol has been used to obtain these results.

I understand from the text description of Fig. 4 that modal analysis has be

en used, but what about

for the other figures? In general, the description of the FEA is too scarce to assess their validity

and to reproduce them, e.g. the description of boundary conditions, mesh size, constitutive models

are absent.

4. The description

of the metamaterial geometric feature is incomplete: how are the beams and

blades thickness chosen? How does this choice determine the compliance for target compliant

deformations and for target non

-compliant deformations.

5. In general I feel that despite the generic design approach for the nature (wire, blade, etc..) and

orientation of the slender elements, many design choices remain arbitrary and poorly explained.

For instance, In fig. 2.: what sets the choice of five wires per unit cell? In fig. 4.: what sets the

choice of two blades per unit cell? in fig 4, from cell to cell, the blades are aligned in the vertical

direction but not in the horizontal direction: why such a difference?

6. To convincingly demonstrate that the metamaterials are directionally compliant, it would be

appropriate to provide measurements of the non

-compliant deformation modes, e.g. in figure 9.

7. About the counting of constraints and degrees of freedom in formula (4) of the methods: it is

well known that the effects of multiple constraints can be redundant if they are related by

symmetries, for instance when multiple bars are aligned (See Calladine, IJSS 1978). How do the

authors treat such issue?

8. So far, the design approach has exclusively focussed on kinematics. Howe

ver, I believe that the

approach of the authors could easily be extended using e.g. beam and plate theory to also

determine/design for a target compliance. This is probably outside of the scope of this work, but I

am curious to hear the authors’ take on th

is and I think that this aspect would be worth

mentioning. 

9. A related question concerning fig 9: I understand that the authors designed for the X and Y

direction to be compliant, but they report very similar values of compliances in the X and Y

direc

tions. I find this result striking. Did the authors design for matching the X and Y compliances

or is this just a coincidence? If it has been designed, the authors should explain how.

10. Could the authors comment of the importance of the nonlinear response? Since their

structures comprise many slender elements, I would expect strong geometric nonlinearities to play

a role.

11. What determines the optimal choice for the number of unit cells?

Reviewer #2 (Remarks to the Author):

A new computationa

l tool was created to design directionally compliant metamaterials more

efficiently, since design of metamaterials is complex due to the number and locations of flexible

elements. The new software tool takes orders of magnitude less time (tens of seconds v

s tens of

hours) than current computational tools for this application.

This computational model was based on the Freedom and Constraint Topologies (FACT)

methodology with some simplified assumptions in order to more efficiently compute designs. The

simp

lified assumptions are that only geometry and orientation are considered in designing and the

beams are infinitely stiff along the constraint-

force lines but compliant in all other directions.

Basic rules for use of the FACT methodology and mathematical definitions to model the

methodology within the tool were presented to explain how the tool determines the design. Case

studies were performed on single degree-

of-freedom (DOF), multi

-DOF, multi-

DOF with CS outside

the yellow shaded region, and bulk shape systems to show the design works for many different

cases.

The discussion on the FACT methodology used in each case studies along with the accompanying

figures clearly shows the process by which the computational tool efficiently designs these

compliant metamaterials. This helped me to better understand how the methodology was applied

to design the computational tool.

There is an impressive variety of case studies presented. This showed that t

he tool has been

tested and can be used for many different designs. The supplementary information added even

more case studies to show that this tool works for many different cases.

CHANGES THAT SHOULD BE MADE BEFORE PUBLICATION

At Line 435, “existing” should be “exiting”.

CHANGES THAT SHOULD BE CONSIDERED TO IMPROVE THE PAPER

The flow and readability of Figure 2 could be improved by placing 2d, 2e prior to 2c. This would

give the figure a more linear feel as to how the process works.

Suggest placin

g the comments about scale bars and color definitions in the corresponding

subfigure description in Figures 3, 6, 7, and 9. For example, for Figure 3 –

Place “(scale bar in a,

50 μm)” at the end of the description of a. As another example, for Figure 6 –

Place “colors in i

and j are defined in Fig. 3b” at the end of the description of i, j.

Lines 165 –

170 discuss that layer extensions may be added to the design. Is there any time

where these extensions could interact with a different layer and would thus

affect the desired

compliance of the design? In looking at this in the context of the rest of the paper I don’t think

there would be interactions but discussing that the layer extensions won’t affect the desired

compliance may be helpful.

1

Reviewer #1:

The manuscript “Computationally efficient design

of directionally compliant metamaterials” by

Shaw et al. reports the design, fabrication and

experimental validation of a novel class of

metamaterials which exhibit prescribed deformati

ons, such as torsion, rotation and shear. Their

design strategy hinges on a purely

kinematic design approach, whic

h translates target motions

into sets of constraints. In turn, such constr

aints can be expressed mathematically as a linear

system of equations, that can be solved very

rapidly numerically. As a result, the internal

architecture of metamaterials

with predefined m

acroscopic shapes (e.g. 5x5x5 cube) can be

designed for a wide range of mechanical deform

ations, from motions with unique degrees of

freedom to motions with multiple (up to 3) degrees of freedom.

Minor clarification: Our appro

ach can design for any combina

tion of DOFs from none up to

six since six is the maximum number of indepe

ndent twists that any 3D system can possess.

Hence the reason for the 7 columns of the FACT

library. Note that we provide a 4 DOF example

(Fig. 7i) and a 6 DOF exampl

e (Supplementary Fig. 4i).

Although their design stra

tegy is not new (it was already de

veloped for compliant mechanisms),

its generalization in the

context of metamaterials is novel a

nd a very significant addition to the

metamaterial toolbox. I believe that

the results reported here will

be of strong interest to

mechanical engineers, physicists and material scie

ntists interested in metamaterials. The off-the-

self computational tool they provide might ev

en make it readily useab

le beyond the scientific

community, e.g. by product designers. Therefore, I th

ink that this manuscrip

t is a potential very

good fit for Nature Communications. That being said,

I think that as it sta

nds, the paper lacks too

much precision and clarity to be

fully convincing and to

be readable by the

broad readership of

Nature Communications. For thes

e reasons, I strongly encourag

e the authors to address the

following comments:

1. The extensive use of acronyms (more than 1 pe

r line on average) make

s the paper extremely

difficult to follow. I believe that a significant

rephrasing would allow the authors to use less

repetitions and therefore to

no rely that much on acronyms.

This is a great point. We eliminated the three

most confusing and frequently used acronyms

throughout the manuscript (i.e., freedom space

(FS), intermediate freedom space (IFS), and

constraint space (CS)) and more clearly phrased

sentences as suggested. These changes were too

numerous to highlight but you’ll see much fe

wer acronyms and clearer

sentences where these

words are used.

2. In some instances, the vocabulary is not precis

e enough, e.g. in the definition of “the yellow

shaded region...” (l 143, p7), what is a “viable t

opology”? Also, always refe

rring to the labelled

yellow region is extremely confusing because as

a reader, we lose tr

ack of the underlying

scientific nature of such

class of freedom spaces.

We clarified what we mean by “viable topology

” in the text (pp. 7)

, and gave the spaces

within the yellow region a technical name (i.e., ce

ll spaces) to clarify their scientific significance.

All instances referring to the yellow shaded re

gion were altered accord

ingly (pp. 7, 8, 12, 14, 15,

16, 18, 20, 24, and 30).

2

3. In all the FEA plots (Fig. 3b, Fi

gs.4g,h,i, figs. 5d,e,f, Figs. 6i,j, Fi

g. 7i, Figs. 8 b,c,e,f,g), it is

never clearly stated in the caption which numerical

protocol has been used

to obtain these results.

I understand from the text descrip

tion of Fig. 4 that modal analysis

has been used, but what about

for the other figures? In general,

the description of the FEA is too scarce to assess their validity

and to reproduce them, e.g. the description of

boundary conditions, mesh size, constitutive

models are absent.

Another very important point! Thanks for catch

ing this. We added details about all the FEA

in the paper in two new Methods sections (

pp. 10, 11, 18-19, 30 and 34-35, 38). We also added a

clause in the data availability

section about the CAD files necessary

to replicate the results (pp.

35).

4. The description of the metamaterial geometri

c feature is incomplete: how are the beams and

blades thickness chosen? How does this choice

determine the compliance for target compliant

deformations and for target non-compliant deformations.

The power of our approach is that it genera

tes topologies (i.e., the

kind, number, location,

and orientation of flexible elements within

a DCM) without the comp

lexities of geometry

considerations (e.g., lengths, wi

dths, and thicknesses of the elem

ents) or constituent material

properties. This simplification is

the secret sauce th

at enables rapid desi

gn generation. Topology

is such a dominant factor in determining directions of compliance that regardless of constituent

material properties or the geometry specified (a

s long as wires still look

like wires (i.e., they are

much longer than their diameter) and blades still

look like blades (i.e., they are much thinner

than they are long or wide)) the resulting DOFs remain largely unaffected. However, that being

said, for our automated tool to draw our desi

gns and conduct the modal analysis, the tool does

need to be assigned material properties and the

diameter of the wires.

You’ll notice these are

asked for in the GUI during the design process. Th

e reason that it’s not discussed in the paper

though, is that the topology (i.e., how many wires a

nd their locations and orientations within the

serially stacked layers) is calculated without

material properties or wire diameters being

specified. This is at the heart of why our comput

ational approach is so disruptively fast at

generating designs.

The consequence of this rapid design capability

has trade offs though. We can’t, for instance,

initially set actual target stiffness values in th

e various directions. We can only guarantee that we

will generate a topology that will polarize th

e compliant directions specified from the

constrained unwanted directions so

that there is as large a difference in stiffness as possible

between them once material properties and geom

etries are assigned to the topology. Once a

polarized topology has been generated though, an op

timizer could be applied to determine the

optimal diameter and material property of each wi

re to achieve target values of stiffness and

compliance. But that’s a task for a differen

t and much less impactfu

l optimization paper.

See pp. 6-7 for a discussion

of this in the paper.

5. In general I feel that despite th

e generic design approach for the

nature (wire, blade, etc..) and

orientation of the slender elements, many design

choices remain arbitrary and poorly explained.

For instance, In fig. 2.: what sets the choice of fi

ve wires per unit cell? In fig. 4.: what sets the

3

choice of two blades per unit cell? in

fig 4, from cell to cell, the bl

ades are aligned in the vertical

direction but not in the horizontal

direction: why such

a difference?

The answers to your questions are addressed in

the Methods section, which are referenced in

the main text for each example. The number of wire

s or blades that should be used to synthesize

each cell such that resulting ce

lls achieve their intended DOFs onl

y and are as close to exactly

constrained as possible is disc

ussed on pp. 12, 27-30. The purpose of Supplementary Fig. 7 is to

provide instructions for selecting elements

from constraint spaces and the purpose of

Supplementary Fig. 8 is to give examples of how

to choose the kind (e.g., wires, blades, circular

hyperboloid elements, etc.), numbe

r, location, and orientation of

elements within constraint

spaces to generate correct cells. See also pp. 25-26.

As for the vertical vs, horizontal question, bot

h designs will achieve

the same DOF if their

elements are selected from the correct constrai

nt spaces using the rules described in Methods,

which they both are. Thus, from a FACT sta

nd point, they achieve the same DOF which is

infinitely compliant compared to the infinitely stiff constrained directions. In practice they of

course differ with actual finite

stiffness values, but generating t

opologies rapidly that are in the

general stiffness and compliant

ball park without these consid

erations is a huge benefit.

6. To convincingly demonstrate that the metamate

rials are directionally

compliant, it would be

appropriate to provide measuremen

ts of the non-compliant deform

ation modes, e.g. in figure 9.

Although in principle I believe this would ha

ve been a good idea, there are an infinite

number of other directions both

translational and rotational in na

ture, which were designed to be

constrained. The question is which of these sh

ould we have measured to compare against the

compliant directions and why would we choose

any of those over any others to compare?

There’s no good answer to these questions and we can

’t measure all the infinite options. This is

why we chose to use modal analysis to verify

directions of complian

ce throughout the majority

of the paper’s examples because the lowest natura

l frequency mode shapes are associate with the

most compliant directions (i

.e., DOFs). By providing lowest

frequency mode shapes we

demonstrate that all other infinite directions ar

e effectively stiffer than those, whatever those

stiffness values may be.

If we were to measure the stiffness in other

directions on the DCM of

Fig. 9 we would have

needed to take those measurements when the

other directions were measured because the

material that is 3D printed changes over time as

it is exposed to natural light. So, we’d have to

re-print and retest the entire

design, which would be very costly, time-consuming, and we

believe would be unnecessary given all the other

modal analysis verification in the paper.

7. About the counting of constraint

s and degrees of freedom in form

ula (4) of the methods: it is

well known that the effects of multiple constrai

nts can be redundant if they are related by

symmetries, for instance when multiple bars ar

e aligned (See Calladine, IJSS 1978). How do the

authors treat such issue?

We only shy away from over-constraint when

we are synthesizing each cell to reduce

computational time and effort because exactly

constrained cells require the placement of less

elements than over-constrained cells by definitio

n. We do not shy away from over-constraint in

general though because every layer of cells is

intended to massively over-constrains the final

4

DCM. This is why DCMs can be shaped in any

way desired to still achieve the target DOFs

because each unit cell redundantly constrains th

e system to achieve those DOFs. Thus, we

leverage over-constraint to enable

shape versatility. A detailed disc

ussion about how to deal with

exact and over constrai

nt is provided on pp. 27-30.

8. So far, the design approach has exclusively fo

cused on kinematics. However, I believe that the

approach of the authors could easily be extend

ed using e.g. beam and

plate theory to also

determine/design for a target compliance. This is probably outside of the scope of this work, but

I am curious to hear the authors’ take on this

and I think that this

aspect would be worth

mentioning.

This paper focuses on kinematics because kinema

tics requires much less computation to deal

with than kinematics combined with elastomechanic

s like most other approaches must deal with.

But yes, your idea is fantastic and would provid

e a worthwhile extension to our theory for a

future paper. Although we haven’t in

corporated plate theory (the pa

rt that excites me), we have

in a way begun to incorporate beam theory when we

construct our stiffness matrix in our custom

modal analysis approach to verify our DOFs as

discussed in the new se

ction we added (pp. 34-

35). Note that we can currently use our stiffness matrix to calculate specific stiffness values in

any desired direction.

9. A related question concerning fig 9: I understa

nd that the authors designed for the X and Y

direction to be compliant, but they report very

similar values of compliances in the X and Y

directions. I find this result

striking. Did the authors de

sign for matching the X and Y

compliances or is this just a coincidence? If

it has been designed, th

e authors should explain

how.

No, we didn’t design for it specifically and we were

also surprised to see how similar they ended

up being. After some thought though we convinced

ourselves that the reason why is that the

relative dimensions between the parent design and

the child cell design are the same (i.e., if a

single scale factor were multiplied to all the dime

nsions in the small child cell, it would look

identical to the homogenous parent

design shown in Fig. 9a. Thus,

since a smaller version of the

design was rotated 90 degrees inside

of a larger version of itself,

I suppose we shouldn’t be too

surprised that they have

the same stiffness although

this interests me a lot. It may be a scientific

principle that we write a future

paper about regarding hierarchical

metamaterials if we can prove

it for general scenarios. We didn’t want to discuss it

in this paper, since it’s not the focus of the

paper, the paper is already very lengthy, and

we aren’t yet comfortable making any definitive

statements about the underlying prin

ciple or reason for the similarity.

10. Could the authors comment of the importa

nce of the nonlinear response? Since their

structures comprise many slender elements, I

would expect strong geom

etric nonlinearities to

play a role.

Nonlinearities should only be

considered once a DCM actually

deforms a finite amount.

Directions of compliance exist w

ithout anything ever needing to

deform at all (just like a spring

can be considered stiff whether or not it is lo

aded). Since we are designing DCMs that achieve

desired directions of compliance

before anything is ever deform

ed, we don’t need to consider

5

finite deformation nonlinearities,

especially during the design pr

ocess. Such nonlinearities would

substantially complicate the unde

rlying mathematics and would ag

ain make it impossibly slow

and impractical to manage the design of such

materials. The advance of this paper came by

stripping out all unnecessary c

onsiderations and focusing on

topology and linear mathematics

only.

11. What determines the optimal c

hoice for the number of unit cells?

It depends on the desired shape of the final DC

M. The more cells in the desired volume, the

more bulk shapes you’ll be able to access that

simultaneously achieve the desired DOFs, but the

more computational time and effort is necessary.

A discussion of this is provided on pp. 9, 12,

14, and 23.

Thank you for all the wonderful suggestions. We f

eel that the paper is now much stronger and

we hope we’ve cleared up your concerns.

Reviewer #2:

A new computational tool was cr

eated to design directionally compliant metamaterials more

efficiently, since design of metama

terials is complex due to the num

ber and locations of flexible

elements. The new software tool takes orders of

magnitude less time (tens of seconds vs tens of

hours) than current computationa

l tools for this

application.

This computational model was based on the Freedom and Constraint Topologies (FACT)

methodology with some simplified assumptions in

order to more efficiently compute designs.

The simplified assumptions are that only geometry

and orientation are c

onsidered in designing

and the beams are infinitely stiff along the cons

traint-force lines but

compliant in all other

directions.

Basic rules for use of the FACT methodology a

nd mathematical definitions to model the

methodology within the tool were

presented to explain how the to

ol determines the design. Case

studies were performed on single degree-of-

freedom (DOF), multi-DOF, multi-DOF with CS

outside the yellow shaded regi

on, and bulk shape systems to s

how the design works for many

different cases.

The discussion on the FACT methodology used

in each case studies along with the

accompanying figures clearly shows the process by which the computational tool efficiently

designs these compliant metamaterials. This helped me to better understand how the

methodology was applied to desi

gn the computational tool.

There is an impressive variety of case studies

presented. This showed th

at the tool has been

tested and can be used for many different desi

gns. The supplementary information added even

more case studies to show that this

tool works for many different cases.

CHANGES THAT SHOULD BE MADE BEFORE PUBLICATION

At Line 435, “existing” should be “exiting”.

Thank you for catching this. We made the change (pp. 25).

CHANGES THAT SHOULD BE CONSIDERED TO IMPROVE THE PAPER

6

The flow and readability of Fi

gure 2 could be improved by placing 2d, 2e prior to 2c. This would

give the figure a more linear feel

as to how the process works.

We see how this could be confusing if each part

of the figure were intended to represent each

step of the process. Since this is the first exam

ple of the paper, however, we wanted to provide a

high-level picture of the approach, which is actua

lly represented by parts a through c only (a is

the empty template volume that will be filled with

elements (input), b is the approach that fills

the space with elements, and c is the final result (output)). Parts d and e are simply describing the

parts of the final design a

nd we feel that it’s important to sh

ow the final design first before we

begin to describe what it’s made of. We decide

d to add arrows between a and b and b and c to

clarify this concept in Fig. 2.

Suggest placing the comments a

bout scale bars and color definitions in the corresponding

subfigure description in Figures

3, 6, 7, and 9. For example, for Figure 3 – Place “(scale bar in a,

50 μm)” at the end of the description of a. As

another example, for Figure 6 – Place “colors in i

and j are defined in Fig. 3b” at the

end of the description of i, j.

We put the scale bar and color definition at th

e end of the figure cap

tions to conform with

Nature

Communications

formatting guidelines. At least all ot

her paper examples we saw that are

published in

Nature Communications

puts these at the end

of the figure caption.

Lines 165 – 170 discuss that layer extensions ma

y be added to the design. Is there any time

where these extensions could interact with a di

fferent layer and would t

hus affect the desired

compliance of the design? In looking at this in

the context of the rest of the paper I don’t think

there would be interactions but discussing that

the layer extensions won’t affect the desired

compliance may be helpful.

This is a very insightful comment and an inte

resting question. The extensions would definitely

have an effect on the stiffness

of various DCMs, but th

ey aren’t a dominant

factor in polarizing

directions of compliance and stiffness. The

topology alone dominates

in determining which

directions will be compliant and which will be st

iff. All other considerations (e.g., geometry of

elements, constituent material properties, and geometry of stage extensions) would determine the

specific stiffness values and how different th

ey are from each other. We leverage this

observation and only consider the topology when

designing DCMs so we

can quickly generate

topologies that are guaranteed to have the DOFs be

the most compliant directions. Then all other

considerations could be optimized to tune thei

r actual values. Since the paper is already very

lengthy, we don’t feel that a discus

sion of this rises to the level

of a necessary discussion in the

paper.

Thank you for taking the time to review

this paper to help

make it better.

REVIEWERS' COMMENTS:

Reviewer #1 (Remarks to the Author):

Dear Editor,

the authors have successfully addressed most of my comments. The paper is now clearer and

more precise.

A minor point: I want to follow up on my question 4:

I understand that the authors do not want to

mention beam/blade geometry for the design approach. However, I could not find the values used

by the authors for their real 3D printed samples. I believe that mentioning these values could be

very useful to th

e interested reader: 3D printing such large aspect

-ratio structural details is likely

to be the main fabrication bottleneck.

Reviewer #2 (Remarks to the Author):

The authors have done a nice job responding to the reviews. Here is one minor note to con

sider

before publication:

Pg 31 Line 583, says "perform the linear modal analyses for the case studies of Figs. 4g-

i, 5d

-e",

but looks to me like it should be "5d

-f"

Response to Reviewers

Reviewer #1 (Remarks to the Author):

Dear Editor,

the authors have successfully

addressed most of my comments

. The paper is now clearer and

more precise.

A minor point: I want to follow up

on my question 4: I understand th

at the authors do not want to

mention beam/blade geometry for the design approach. However, I could not find the values

used by the authors for their real 3D printed samples. I believe that mentioning these values

could be very useful to the intere

sted reader: 3D printing such larg

e aspect-ratio structural details

is likely to be the main fabrication bottleneck.

Thank you for your helpful feedback in making the

paper clearer and more precise. As to your

point, we set the diameter

s of the wire elements and the thic

knesses of the blade elements in all

of our real 3D printed samples to be the sma

llest feature size our

printer was comfortable

printing. The blade widths were made as wide as

possible within the space they had available

within their respective cells so they would behave

as much like an ideal blade as possible. All

element lengths were determined by the distance between the DCM layers and their angles. We

decided that because there were so many geometri

c parameters that define

d each cell within each

3D printed design that instead

of significantly lengthening the

paper by providing all of these

parameters for every design that we printed (and

possibly derailing the focus of the paper since

FACT operates independent of these parameters),

we would provide CAD or .stl files of the

printed structures upon the reader’s

request and declare that in our da

ta availability statement. Of

course all the actual geometric parameters used

are embedded in those files if any reader did

want to exam them. Such information would be

more important to provide for case studies

directly within a follow-on paper that optimizes

the geometry of the topologies that FACT

generates.

Reviewer #2 (Remarks to the Author):

The authors have done a nice job responding to the

reviews. Here is one

minor note to consider

before publication:

Pg 31 Line 583, says "perform the

linear modal analyses for the ca

se studies of Fi

gs. 4g-i, 5d-e",

but looks to me like it should be "5d-f"

Good catch! We made the change in track

changes. Thank you again for your help.