ME 563-Fall 2020

Homework No. 1

Due: September 12, 2020 11:59 pm on Gradescope

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ME 563 - Fall 2020 Homework Problem 2.1 A double pendulum consists of two bobs of mass m1 and m2, suspended by inextensible, massless strings of length L1 and L2.

a) Use Lagrange’s equation to find the equations using q1 and q2 as generalized coordinates.

b) The equations of motion are nonlinear, linearize them using a small angle approximation.

M

n

kQ

F X T t yJ 45inO I LCosas82 I try L sin0 thesin02T kicosa tacos9 j

Thevelocity vectors

I 0,4cos0 i 0,4sinOj and Fe 0,4cos0 t02Lcosa it 04sin0 0lasing jThekineticenergy

T Ym ti F t4am 4 2m40 2mLtacos100210,0 t m250

Thepotential energyU mglycoso mglicoso mglzcosa

simplifyingU 2mgL COSIO Mglacos02

Lagrange's Equations

due Yao 2720 2420 0

272g 2m40 Tm 4220510 02782

e Tag 2mi mticacosco 028mL casino 0210 010

27ha ML Lesia Q 02710 020

2420

2m84

sina.ESpojtm2 LzcosC0 0,70 malesina 02052mL Lasin 10 02 0102 2m84sin01 0

dat d 542 20,2

0

2712g m cosCQ 0470 tmLz z

adel gmlkzsinCQ adQ.CO EdML cos10 020 tMead

2420 ML Cassia 0 102082

242g mgLz sin 02

mL cos 10 020 tmLE t 2mL since040,62ML CcSinCIO 02702 tmglzsl.noz

0D2mL7QimLiLzcosCQ 0s7OIz me casino 02052mLLocsin Q 040002 2mgL Sino D

2 m4LzcosCO 020 tmLE t 2mL since 040,62ML CcSinCIO 02702 tmglzsl.noz 0

Angle formulas

cos Qe Oz cos0cosOz sin 0 sin a

Sin Q Qe singcos02 sin02050for small angles

costa 027 cosf.tl s ia sin Ee ltQ 0I Isin Q Qe singcos sided 0 Q

Q Ole

2h0

1 2m48 t m44072 2mgL Sino _O2 MLK240 TmL3 z t Mglzsin012 0

Now took a Newton Euler solution from last HWI 2 I 7

EDmucosa tmc.co z0I mLzsin2IFz

ML sinC0 z tmgs.inOs 0F 272mL To tmLzcos 70402 2M 20

mlzs.in 0 F0dO5t2mgsinQ OO

m L OT T ML a Tmg sin012 0 same as 2 butmultipliedby Lz

2mL t m Dig 12mgsince same as 1 butmultiplied byL

ME 563 - Fall 2020 Homework Problem 2.2 A thin, homogenous bar having a mass of M and length of L is pinned to the ground at point O. A particle P of mass m is free to slide on the smooth surface of the bar. A spring of stiffness k and unstretched length of R0 is attached at point O and particle P. Let r be the radiald distance from O to P and Q be the rotation of the bar from a fixed vertical line.

a) Use Lagrange’s equation to develop the EOM’s for this two-DOF system using generalized coordinates of r and q. Recall that the potential enrgy stored in a spring is related to the square of th stretch in the spring, where the stretch is equal to the difference between its actual length, and the unstretched length. Also, in writing down the velocity of vector of P, you may want to review the polar kinematic expressions for velocity.

b) Usin the equations of motion, determine the equilibrium values for r and q.

A

Datum

toMe

ftp.F

iertroieoFo 4208

F 42mF 42102or

Do Km 12 807 42180Ir Ro orIo BAE

U Agh cosO mgrYakCr Ro

2

Lagrange's Equations

e2 2 24 0

2 mi efdtgr.Jmr.ITgr

mr0 2oy2r mgcos0tkCr Ro

m r t Mr d t k r Ro Mgc 0 0

el 20

t 201

0

2 Mr O Io dge.ca mre8tIo5Ot2mnr0

242,0 0 20 mgrsi.no

Mg4gsin0Cmr24zAtE OIt2mrE0 gcmrTALK sin0 0

ME Mr042 t Kcr Ro Ngos 0 0

most4gA E 2mm tgcmr A4ds.in0 0

04 8 8 0 r req Oleg

KCreq Ro mg cosOleg D

glmreutmyds.in0eq

OQeqnT a D I 2 3 00

cosOleg CDnKlreq Ro MgcD's O

req R t C17mg eq nFIT

n 0 I 2,3 a e D

Linearize 01 05 25 4 0eq Xeq ox

Kxeq F

sinOleg O eq NIT n 0 I 2

Oleg18 X Regt I 0eq 0

plug in and use small angles

1 Atm toner moxa OT mcrTsingtaocoax KI tkoxeqF.ua

same

D Atm Lin 1MCR r II CE tax Dooo

a mCR rlcosqefto.IT CR rtfm tI

TmgCR r si 81 0

MCR r t R rkcmtI.az TmgCR r1 O

Final linearized EON

1 Atm ki MCR r text kx O27 m R mix R r meta mgCR r 8 0

In matrix form

Atm MCR r a

math crown 1 jm EMIT

it nl3CT CIT KIT

ME 563 - Fall 2020 Homework Problem 3.1 A double pendulum consists of two bobs of mass m1 and m2, suspended by inextensible, massless strings of length L1 and L2.

a) Determine the expression for potential energy U and the generalized forces corresponding to F for the generalized coordinates. Use these results to determine the angles θ1 and θ2 corresponding to static equilibrium. Leave these angles in terms of the ratio F/mg. b) Write down an expression for the kinetic energy T in terms of the generalized coordinates θ1 and θ2 and their time derivatives. From this expression, identify the elements mij , where:

c) Determine the mass matrix [M] and the stiffness matrix [K] corresponding small oscillations about the equilibrium state for F/mg = 0.

d) Determine the mass matrix [M] and the stiffness matrix [K] corresponding small oscillations

about the equilibrium state for F/mg = 2. Compare these with those found in part c).

1

ed

8TO V

M M meme

rMa F

a U mglcos0 mallcos0 tacos02

2m84cos0 mytacos02

E Lsing Lasin a i Lcosa Lacosadj

85 LCosa80 Lacosaz802 i t 4sin0,80 22514028021

SWING F SIFi Lcosa80 tacos0280127 it 4sin0,80 2251402802 j

FLcos0 80 Flaco02802

Q FL cos0 and Q FLacosaz

Determineequilibriumpoints

34 Q 2mgLosing F4cos0

2mgsinO Fcoso2tanO Flag TanO Fang

Qew tan Fang

and

342 012 mylosing Flacos02

mgsin a FoosOztan02 FlingGreat tan Fmgb write kinetic EnergyF X T t yJ 45in I LCosas82 I try L sin0 thesin 2 Y Licoso tacos02 j

Thevelocity vectors

I 0,4cos0 i 0,4sing and Fe 0,4cos0 02Lcosa it 04sind 0lasing jThekineticenergy

T YMF ti t4am 4 2m40 2mLtacos10020,0 t m250

mi 2m42Miz Ma ML L cos10OzMaz my

stiffness calculation

ki 34q2mglocosOrea 142 ka 1gal O

22 24 mgtacosOrea

c consider Fling 0 and Flamg D

new Drew tan O 0cosOien cosOzew 1

Thelinearmassandstiffness matrices

EMI 2m42

myok ma 24

m L L

M422

O

d d

Qew tan 1 TileOzew tan 2 1,0715

cosOien 1 2

cosOzew 0,447

Thelinearmassandstiffness matrices

RAI e 2m42 0.949mL la O

myi ok M8 1,4144

0,949mLL2 O 0,4474

ME 563 - Fall 2020 Homework Problem 3.2 Consider the system below, whose motion is described by the absolute coordinates shown.

a) Write down the potential energy function U for this four-DOF system and use the following results from lecture to develop the stiffness matrix for the system:

b) Use the method of influence coefficients to develop the flexibility matrix [A] =[K]-1.

c) Check your results in a) and b) above by verifying that [A][K]=[I] where [I] is the identity matrix.

A

a The potentialenergy can be writtenas

U 42KNE 42131445 422k NoNot YKHcNa

Equilibriumposition NaNa XB Nc 0

242m KYA 2426 31146 2KHoXB 5146 2KXB

22 2kW no t ka Nc 3Kxp 2kx Knc

2 Kae NaNi Noaa Nogo req

ki 34 K Kis ka 34m10 143 1231 Iffy 0 144 ELEO

Kaz 23,44 5k 1423 152225 2k 424 142 34 0

133 24 3K 134 143 LIYE K

1444 312 k

KI K O O O

O

g

k k o

2K 3K K

O K k

b Apply a unit load a A

gnkqgÉ rant o arm

92 93 941 0 sincenotconnected

Byreciprocity 912 921 0 a3 931 0 ay 914 0

Apply a unit load a B

É Itm 4m 922 923 924 zero strainto right

122 083 Egg Fox 3kazzt O

K 1124 922 13k923 134

924 134

By reciprocity 942 924 113k 923 932 134

Apply a unit load a C

intake in

ayyy yayKeg 43K t 42k

key 13,44342 IkAzz Haze

2Fox key933 0 933 Ikea 516k

a34 943 516k

d

rj tma.tn Kew 3k t 42K Kew 34142 Gkf

F VicensView t k k t k44 near i s

Kew 9,4 k k

EFox Kena44 1 0

944 Year 6k

a K O O O

O BK 43K 43k0 113k 96k SlotO BK 96K 16k

a K K O O 0 Hk O O O

O SK 2k O O 43K 43K KskO 2k 3K K O 113k 46k k

O O K K O BK 516k 16K

I O O GO l O 0O O I 0O O O I