Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ......

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Math Problem a- If A has eigenvalues 1 , 3 2 1 - = = λ λ and corresponding eigenvectors = - = 2 3 , 1 1 2 1 u u Solve the initial value problem ) ( ) ( t Au t u = with = 3 6 ) 0 ( u . b- True or false and why 1. if A is diagonalizable, then A has no multiple eigenvalues. 2. if 2 λ is an eigenvalue of 2 A , then λ is an eigenvalue of A . 3. if A is an n×n real matrix having real eigenvalues and n orthogonal eigenvectors, then A is symmetric. 4. every invertible matrix is diagonalizable. 5. if T xy A = , where , n xy R then y x T = λ is an eigenvalue of A. 6. if T pBp A = with p invertible, then A and B have the same eigenvalues.

Transcript of Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ......

Page 1: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Math Problem

a- If A has eigenvalues 1,3 21 −== λλ and corresponding eigenvectors

=

=2

3,

1

121 uu

Solve the initial value problem )()( tAutu =� with

=3

6)0(u .

b- True or false and why

1. if A is diagonalizable, then A has no multiple eigenvalues.2. if 2λ is an eigenvalue of 2A , then λ is an eigenvalue of A .3. if A is an n×n real matrix having real eigenvalues and n orthogonal eigenvectors, then A

is symmetric.4. every invertible matrix is diagonalizable.

5. if TxyA = , where , nx y R∈ then yxT=λ is an eigenvalue of A.

6. if TpBpA = with p invertible, then A and B have the same eigenvalues.

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Math Problem

Consider the linear general second-order PDE:

a) A x y U B x y U C x y U DU EU FU G x yxx xy yy x y( , ) ( , ) ( , ) ( , ), , , , ,+ + + + + =

Give the three classifications for this PDE, i.e., elliptic, ...

b) What kind of PDE is the following

1) Heat equation CU

x

U

t

∂∂

∂∂

2

2 0− =

2) Wave equation CU

x

U

t2

2

2

2

2 0∂∂

∂∂

− =

3) Laplace equation∂∂

∂∂

2

2

2

2 0U

x

U

y+ =

c) Solve the following equation:

x

x

x

xe

e

t

t1

2

1

2

5

2

2 1

1 22

3

%&'

()*

= ���

���%&'

()*

+%&'

()*

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Math Problem

Let R be a closed bounded region in the x-y plane with the curved boundary C.

R

C

x

y

Let F x y1( , ) and F x y2( , ) be functions that are continuous and have continuous partial derivatives∂ ∂F y1 / and ∂ ∂F x2 / everywhere in some domain containing R. Then, Green's theorem states:

( ) ( )∂∂

∂∂

F

x

F

ydx dy F dx F dy

A C

2 11 2−I = +I . (1)

(a) Use (1) to show that the area of R, A is

A x dy y dxC

= −I1

2( ). (2)

(b) Use (2) to calculate the area of the ellipse

x

a

y

b

2

2

2

2 1+ = . (3)

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Math Problem

a) If A is singular, what can you say about the following:

1) det A2) eigenvalues of A3) relationship between the rows and columns of A

4) the solutions of the inhomogeneous set of linear equations, [ ]{ } { }A x F=

5) the solutions of the homogeneous set of linear equations, [ ]{ } { }0A x =

b) Solve the following:

dx

dtx y= − + 2

dy

dtx y= − −2

Page 5: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Math Problem

If A has eigenvalues 1,3 21 −== λλ and corresponding eigenvectors

=

=2

3,

1

121 uu

a) Solve the initial value problem )()( tAutu =� with

=3

6)0(u .

b) True or false and why:

1. if A is diagonalizable, then A has no multiple eigenvalues.2. if 2λ is an eigenvalue of 2A , then λ is an eigenvalue of A .3. if A is an n×n real matrix having real eigenvalues and n orthogonal eigenvectors, then A

is symmetric.4. every invertible matrix is diagonalizable.

5. if TxyA = , where , nx y R∈ then yxT=λ is an eigenvalue of A.

6. if TpBpA = with p invertible, then A and B have the same eigenvalues.

Page 6: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Math Problem

For the initial/boundary value problem

DE:∂∂

∂∂

u

tk

u

xx L t= ≤ ≤ ≤

2

2 0 0, ,

BC's: u tu L t

xt( , )

( , )0 0 0 0= = ≤and for

∂∂

IC: u x f x x L( , ) ( )0 0= ≤ ≤for

carry out the following steps:

a) Formulate the eigenvalue problem obtained through separation of variablesb) Find the characteristic equation whose roots are the eigenvaluesc) Find the eigenvalues and eigenfunctions and show that the eigenfunctions are orthogonald) Write out the eigenfunction expansion of the solution u x t( , )e) Evaluate the constants in part (d) for the special case f x( ) =1

Page 7: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Math Problem

(a) Given x y z2 2 216 12 9 1/ / /+ + = and z is a dependent variable while x and y areindependent, find ∂ ∂z x/ .

(b) Find the following:

lim( )sin

cos cosx

xe x

x x→

−−0 21

.

(c) Find the point of the plane 2 3 4 25x y z− − = which is the nearest to the point (3,2,1).

Page 8: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Elasticity Problem

The stress tensor in the principal direction is given by

σσ

σσ

ij =�

!

"

$

###

1

2

3

0 0

0 0

0 0

.

Show that the shearing traction S on an infinitesimal area with normal (n1, n2, n3) is given by

S n n n n n n21 2

212

22

1 32

12

32

2 32

22

32= − + − + −( ) ( ) ( )σ σ σ σ σ σ .

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Elasticity Problem

Let I t( ) be a volume integral of a continuously differentiable function A x ti( , ) defined over a spatialdomain V x ti( , ) occupied by a given set of material particles, that is

I t A x t dViV

( ) ( , )= I (1)

(a) Write down the volume expression for the material derivative of I, namely D I t

Dt

( ).

(b) Write in index notation the statement of conservation of angular momentum over the volumeelement V x ti( , ).

(c) Use the results of (a) to show that the conservation of the angular momentum in (b) reduces toshowing that the stress tensor is symmetric, namely, σ σij ji= .

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Elasticity Problem

Let us consider a circular cylinder of length L and radius a oriented so that its axis coincides withthe x1 axis of a Cartesian coordinate system x x x1 2 3. The material is isotropic and there are no bodyforces present. The state of stress is given by the following stress tensor

[ ]2 3 3 2

3

2

0 0

0 0

A x B x C x C x

C x

C x

+ − σ =

.

a) Show that the equilibrium equations are satisfied.b) Show that the compatibility conditions are also satisfied.c) Show that free boundary conditions prevail on the surface of the cylinder.

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Elasticity Problem

2

2 2 2

2 2 2ij

x yz xy yz

xy xy z x yz

yz x yz xyz

σ =

For the stress tensor given above for a body with no acceleration and no couple stresses you are to:

a) At the point (1,2,3), determine principal stresses and the direction of one of the principalstresses.

b) At the same point as above, find the octahedral normal and octahedral shear stresses.

c) For a linearly elastic material, find the strains in terms of (x,y,z), that are associated with thegiven stress tensor.

d) Find the change in the angle between the lines that go through the points (1,2,3) and (2,4,5) andthe points (1,2,3) and (7,4,1) for the above strain state.

e) Show that an infinitesimal displacement duj consists of an infinitesimal strain and rotation

component.

Page 12: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Elasticity Problem

Consider the principal stress tensor

σ

σ

σ

σ

ij =

���

���

1

2

3

0 0

0 0

0 0

a) Show that

T T Tµ µ µ

σ σ σ1

1

2

2

2

2

3

3

2

1�

��

��

+�

��

��

+�

��

��

=

b) Show that

µ σ σ σ σσ σ σ σ1

2 2 32

1 2 1 3= − − +

− −( )( )

( )( )N N S

,

where σ N is the normal component of the stress vector along Gµ and S is the shear stress.

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Elasticity Problem

Show for plane-strain problems that the general equations of linear isotropic elasticity lead to thebiharmonic equation

∇ + −−

∇ =4 21 2

10Φ ν

νV ,

where Φ is the Airy stress potential defined by τ ∂∂xx y

=2

2

Φ, τ ∂

∂yy x=

2

2

Φ, and τ ∂

∂ ∂xy x y= −

2Φ, and

V is the body force potential such that FV

xx = − ∂∂

and FV

yy = − ∂∂

.

Page 14: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Elasticity Problem

Consider a continuum region having the volume V and surface area S.

(a) If I t A x t dViV

( ) ( , )= I

Show details of the derivation leading to

D I

Dt

D A

DtA

v

xV

dVi

i= +

���

���I

∂∂

,

where vi is the velocity vector and D I

Dt is the total time derivative of the function I t( ).

(b) Give the statement of conversation of linear momentum and show, using the results of (a), that

ρ σDv

DtXi

ij j i= +, ,

where Xi is the body force vector and σij are the stress tensor components.

Page 15: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Elasticity Problem

Consider an elastic isotropic solid. Use the equations of equilibrium and the constitutive andcompatibility relations to show that

σν

σ νν

δij kk kk ij i j j i k k ijF F F, , , , ,( )++

+ + +−

=1

1 10

Page 16: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Elasticity Problem

Consider a solid body subjected to the stress σij .

(a) Using the index notation, determine the expression for the shear stress τ on the cut whosenormal is n .

(b) Specialize this shear stress to the case where the components of the shear stress σij vanish and

show that τ2 reduces to

τ σ σ σ σ σ σ212

22

11 222

12

32

11 332

22

32

22 332= − + − + −n n n n n n( ) ( ) ( )

(c) Use the results of (b) to calculate the octahedral shear stress.

Page 17: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Elasticity Problem

An unbounded elastic plate contains a circular inclusion of a different elastic material. The plate isloaded in tension as shown in the figure. Determine the stress distribution in both the plate and theinclusion.

a2

λ , µ1 1 λ , µ2 2

σo σo

Page 18: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Elasticity Problem

Use the equations of Equilibrium

σij j iF, + = 0

with the constitutive relation

σ λ δ µij kk ij ije e= + 2

and compatibility

e e e eij kl kl ij ik jl jl ik, , , ,+ − −

to show that

σν

σ νν

δij kk kk ij i j j i k k ijF F F, , , , ,( )++

+ + +−

=1

1 10.

If needed, use the relation

ν λλ µ

=+2( )

.

Page 19: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Elasticity Problem

Use the Field Equations

σij j iF, + = 0 (1)

σ λ δ µij kk ij ije e= + 2 (2)

e e e eij kl kl ij ik jl jl ik, , , ,+ − − = 0 (3)

and show that

σ λ µλ µ

σ λλ µ

δij kk kk ij i j j i k k ijF F F, , , , ,( )+ +

++ + +

+=2

3 2 20. (4)

Hints:

(i) Start by specializing equation (3) to k l= and call it equation (5).(ii) Also specialize equation (3) to i k= and j l= and call it equation (6).(iii) Use equations (1), (2), (5) and (6) to show (4).

Page 20: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Strength of Materials Problem

Let us consider a rotating flat disk of inner and outer radii b and a, respectively. In terms of theradial displacement u, the radial equilibrium equation can be written as follows:

d

dr r

d

drr u

Er

1 1 22( )�

! "$#

= − − ν ρω .

Solve this differential equation and derive equations for the radial σr r( ) and hoop σθ( )r stressdistributions by assuming plane stress conditions. What is the ratio of the maximum hoop stress to themaximum radial stress with no boundary loading? What should be the ξ = a b/ ratio to assure thatσ σθ max max/ r = 4 if the Poisson ratio is ν = 1 3/ .

Page 21: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Strength of Materials Problem

A very long, prismatic elastic shaft of elliptical cross section is embedded in an elastic medium. Theshear modulus G of the shaft and the two semi-axes a and b of the elliptical cross section areknown. When a length dx of the shaft rotates an amount θ, the medium applies a retaining torquedT k dx= θ to the length dx. Let us assume that a torque To is applied to the end of the shaft atx = 0. Obtain an expression for the rotation angle θ( )x .

Page 22: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Strength of Materials Problem

Let us consider the torsional deformation of a prismatic bar of elliptical cross section. Calculate theratio between the maximum strain energy density U0 max and the average strain energy density

U avg0 as a function of the aspect ratio c a b= / of the cross section (a and b are the longer and

shorter semi-axes of the cross section, respectively). How does this strain energy density ratioU U avg0 0max/ compare to the same ratio for a prismatic bar of narrow rectangular cross section?

Page 23: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Strength of Materials Problem

Let us consider a homogeneous, isotropic, and linearly elastic disk under the influence of an axi-symmetric temperature distribution T T r= ( ). Assume that Young's modulus E, Poisson's ratio ν ,and the thermal expansion coefficient α are known. Derive an equation for the distribution of theradial stress σr r( ). How can you specialize this result to the case of a disk whose inside and outsideboundaries at r b= and r a= , respectively, are free of tractions?

Page 24: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Strength of Materials Problem

Consider a thick-walled cylinder with outer and inner radii of a and b , respectively, subjected to aninternal pressure of p. Poisson's ratio ν and Young's modulus E are known. Calculate thedifference in the deformed wall thickness between plane-stress and plane-strain conditions.

Page 25: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Strength of Materials Problem

Consider a bicycle wheel of radius a rotating at angular velocity ω . Assume that all spokes areradial, lie in the same plane, and are of the same material as the rim. Let Ar and As be cross-sectional areas of the rim and a single spoke, respectively. Also assume that n, the number of spokes,is large and the stresses are zero when ω = 0. Derive equations for the hoop stress σθ in the rim andthe maximum radial stress σrm in the spokes. What is the asymptotic value of the σ σθrm / ratio forvery small and very large n A As r/ cross-sectional area ratios between the spokes and the rim?

Page 26: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Strength of Materials Problem

y

z

a

Apply the Saint-Venant torsion theory to a solid bar whose cross section is an equilateral triangle.Determine the Prandtl stress function φ (τ ∂φ ∂xy z= / and τ ∂φ ∂xz y= − / ) by satisfying the dφ = 0

boundary condition at all three sides. Assure that for a given rate of twist β and shear modulus G

the solution satisfies the ∇ = −2 2φ βG compatibility equation. Calculate the torsional rigidityC T G= / ( )β , where T dA= II2 φ is the torque.

Page 27: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Strength of Materials Problem

Consider a thick-walled cylinder with outer and inner radii of a and b , respectively, under theinfluence of outer and inner pressures po and pi , respectively. Assume that plane stress conditionsprevail, i.e., σz = 0 and that Poisson's ratio ν and Young's modulus E are known. Calculate theradial displacement u as a function of the radial coordinate r . What is the radial displacement u r( )in the simple case of hydrostatic pressure, i.e., when p pi o= ?

Page 28: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Strength of Materials Problem

Consider the torsional stiffness k T= / β of a prismatic bar, where T is the applied torque and βis the resulting rate of twist. According to Saint-Venant's approximate formula for prismatic bars ofarbitrary cross section, k G A J≈ 4 24/ ( )π , where G, A, and J denote the shear modulus, crosssectional area, and polar moment, respectively. Determine how accurate this approximation is for aprismatic bar of elliptical cross section with an aspect ratio of 3.

Page 29: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Structural Mechanics Problem

For the beam configuration and loading condition shown below, you are to find all reactions and thevertical deflection at point A (located at the mid-point of the left side of the beam). Also draw theshear and bending moment diagrams, with all the important points clearly labeled and with their valuesshown. Assume that Young's modulus E and the moment of inertia I are known for the beam.

L L

k E IL

= 33

f0

A

Page 30: Math Problem 1 3 A 2 = 2 - University of Cincinnati · Consider the linear general second-order ... Math Problem Let R be a closed bounded region in the ... Math Problem (a) Given

Structural Mechanics Problem

For the triply redundant frame shown below:

(a) Find all reactions.

(b) Draw the shear and bending moment diagrams for each member of the frame.

(c) Find the deflection at point A located at the center of the horizontal bar.

PA

A A /2

A2

A E I3

E I2

E I