ME 563-Fall 2020

Homework No. 3

Due: September 19, 2020 11:59 pm on Gradescope

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).

ProbleMD

48

Q v

M

yOz

M m m

ya

Mz F

a U mg cos mgkcos cos

2mgL cosQ mg cos012

EE Lsing sin e 9 CLcosQ t cos j8rJ 4051080 12205102802 i t Lsin0,8012251N2810dgSWING F 882

FT 4050810 12205102802 i t Lsin0,801225in02810dg

FLcos01 80 F cos02802Q FL cos0 and Qz FLzcos zDetermineequilibriumpoints

22 Q 2mg sin F4cosQ

2mgs.inQFcosQ2tanQFmg tanQi Fgmg

Qew tan Fangand

22 Q2 mglzsinoz

FLzcosQzmgs.in0zFcos0ztan0zFmgGear tan 1 Fmgb write kinetic Energy5 4,9 t gJ 45in 9 LcosQj

I try L sin0 1 gin 2 T Chaos cos023J

Thevelocity vectors

I OiLcos it 10,4sinya and FE 04cos0 tIQLecos it 014sin it alasina jThekineticenergy

T ymTt I YgmVIVI Yg2m42 2mL cosCQOdioiz t mEziom 2m42mis ma ML cos10 012Mor mL

Stiffness calculation

Oki 22geq2mg4cosl0iea 142 141 30,1

22 222 1g mgtacos eenc consider Mmg O and Flgmg 0

Qew zeq tan O OcosQew coscozen

Thelinearmassandstiffness matrices

IAI 2m42 mLLz If my 24 Or

m L L2 m O L

d consider Hmg L and Flgmg p

Qew tan 1 Fly2eq tan 2 1,0715

cosQew 11oz

cos023 0,447Thelinearmassandstiffness matrices

AAI 2m42 0,949mLLz 4 MY 1,4144 Os0,949mLL2 Mhz 0 0,44742

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 The potentialenergy can be writtenas

a 42km7 42131445 t4212koxoAGT 42kExcNB

Equilibriumposition maNoAB Nc O

ManaKHA 2 3K4g 2KHoXD 5KX 2KXB

2 2KExpooxo Ktx xD 3koxo 2Koxo kik

2qKKc Xaoh na ar Nogo q

Ki 21,1 K Kia ka 23 0 143 431 221 0 144 212 2502min24s

Kee_222 5k Kas132 2245 2K k24 442 222,125 0

Kss 221 3K Kz4 K z K22ND doxodale

K44 221 L24C

K K O O OO SK 2k OO 2k 3K K

O O K K

b Apply a unit load ANoforce here

m_ff Efx ka 0 a c00attail 921 931 941 0 sincenotconnected

Byreciprocity 9,2 921 0 a3 931 0 ay 9,4 0

Apply a unit load Oa B3k f 2x k 922 923 924 zero strainrun In mn to right

take e e EFX 3Kazz 1 O00_00122 Hazy 922 134 924 134

923 134Bg reciprocity 942 924 1 34 923 932 1310

Apply a unit load a C

3k 2K krvvy ke.mn mn Ykeg 43k 42k

f key 13k2K0 OF c E ktazz Haze

2Fox keyAzz tf _0 933 Ykeq ka34 943 564

Apply a unit load D

y kimoe.me keg 43kt42k Keat 133454 63kf

00 YKewEYkew.tlk shek t Yu44 keys keg K Gsk K

Keg 1k 61gk t K

Keqq K2 Gf kGK

Fox Key 44 1 0

944 Ykcqz 1 4

a Hk O O O

O 43K 43K 43K0 113k 516k 46k0 113k 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 k k

O O K K O BK 516k 16K

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

ME 563 - Fall 2020 Homework Problem 3.3 A disk of mass m and radius r rolls without slip on a cart of mass 3m. A force F is applied to the cart. The disk has an angular displacement q. Let x, describe the absolute motion of the cart. The springs is unstretched when xA=0. Assume the surface between the cart and ground is smooth.

a) Use a Newton Euler formulation to find the equations of motion. Your final equations should not include any forces of reaction. The final differential equation are in terms of xA and q and should be in matrix form.

b) Use Lagrange’s Equations to find the equations of motion. The generalize coordinates are xA and q. The final differential equation are in terms of xA and q and should be in matrix form.

c) Compare the form of the mass matrix, stiffness matrix and forcing vector of the Newtonian formulation with the Lagrangian formulation. In particular, are they the same? If not, in which formulation are the matrices symmetric ? Do the forcing vectors differ? Explain any similarities or differences.

2KWMV of

m 46 4A T ROKM A 3M

ooo

XA

ta NAT atra dataTE Ha t r079 DE XA T rd i

Fa KAT to

xatR0JtTYqC3mJta.ratYgmFo.Fo 121602

T 4213M aint 42M Kiat2 2riario 142 02

U Yz k Xt t Yz 2K XA tr2

dw at fi.de 8doxa t rFd0Lagrange's Equations

at Fania 252 21

Qala

Iae Mag 20

20 Qa

2172nF 3M Eat mnin tmr0 smixatmr.IO

dat Mania 4maiamro.LTgqa Oi2U2qy KNAt2ktxn tr0 and QxA A

4mix'At mr t 3KXA T2Kr a f

2Tgq mm I I mrna

da mr O I O

mriiin.LTgo

O 2Ugo

2kloxn trdr and r f

tmr 2 tmornin T 2KrXa 12km r f

f I L3mr

2KMMA offrmKM A 3M

ooo

4A

mg IG2K AA100

V f Incintroia IFRat

2km C N 3mgfree

V 3m a

fatI EFX frzrtf 2kkxatr01 mcniatr.IO D9 Sfg ng t N O 2A EMG Farr IDT 3Is EFA KHA face 3M 41The wheel FOM can be found by combing 1 and3

Iq tf MWA To 12k Exa Trio

Minin f Iq Tmr c 2kfxAtrQ f

Combine and 4

Zinnia 1 Ige'd c KHA O

t.mn mny nz l.Y.ok BLory