UNIVERSITY OF TORONTO

Faculty of Applied Science and Engineering

Final Examination

Third Year— Program 5a, 5r

AER3O1 F 12 y11 ardc

Examiner: G M T D'Eleuterio

18 December 2018

Student Name: Last Name First Names

Student Number:

Instructions:

Attempt all questions.

The value of each question is indicated in the table opposite.

Write the final answers only in the boxed space provided for each question.

This is a Type B examination: An aid sheet is provided.

No calculator is permitted.

There are 17 pages and 6 problems in this examination paper.

For Examiner Only

Problem Value Mark

A

1 10

B

2 10

C

3 15

D

4 20

5 20

6 25

Total 100

Part A

Provide very short answers.

1(a). Define the tangential force as seen in a rotating frame.

/2

Give two equivalent definitions of a conservative force.

/2

State principle of work and kinetic energy.

/2

Define power.

/2

1 (e). Define the Hamiltonian.

/2

University of Toronto - ENGINEERING SCIENCE - Programs 5a, 5r

AER301F Aid Sheet Dynamics

(uTw)v - (uTv)w [vu - (uTv)1]w (u><v)'<Ux V x - vX uX _uvT + vuT

1 0 0 cos 8 0 - sin 9 cos 8 sin 9 0 C1 0 cos 8 sin 9 , C2 0 1 0 C3 - sin 0 cos 0 0

0 - sin 9 cos 8 sin 0 0 cos 0 0 0 1

C = C(03)C(02)C(O1) W = C3(03)C2 (92 )1191 + C3(83)1202 + 1303

C = cos q1+ (1 - cos q5)aaT - sin cbaX

L L" _T r12

Wb ba CbaCab 0 0

[cube]

'AA 'me2

A

£

A _

h 0 + w xh 0 +v0 xp=i 0

b - - Cma rnb

La Lb r 3

- r62

" r9+2O

0

- 1+ Cos 8

[rod]

'AA = 12 12

I 2eh2

P [t

2 (2 1 V =iI — — —

a

T= 2nrl-

A

[sphere]

'AA =

d(3L\ 0L \ 3qk) - Oqk

= Q,A + =jkqk +jt = 0

[cube]

'AA = 'rn2

A

[sphere]

'AA = mr2

[rod]

'AA = 12

University of Toronto - ENGINEERING SCIENCE - Programs 5a, 5r

AER301F Aid Sheet Dynamics

(uTw)v - (uTv)w [vuT - (uTv)1]w (u < v)'< u'< v>< - v'<u'< _uvT + vuT

1 0 0 cos 8 0 - sin cos 8 sin 0 0 C1 = 0 cos 0 sinG C2 0 1 0 C3 - sin 0 cos 8 0

0 - sin 0 cos 0 sin 0 0 cos 8 0 0 1

C = C(83 )C(82 )C(81 ) = C3(03)C2(82)1101 + C3(83)1292 + 1303

C = cos 01 + (1 - cos c5)aaT - sin a><

V ) = pT

, TT r'v2

ba X - p 0 C baCab 0

h+w xh0+v0 xp=r0

b - a - Gmarnb La Lb

'F - r2

" = 0 r6+20

0

A h 2 A 2eh FI+--

It t t2

V 2 =t ( (2

-

1 - -

a

£

1 -1- E cos 8

df'3L'\ 3L kqk +jt -- 0

Part B

Determine if the following statements are true or false and indicate by "T" (for true) and

"F" (for false) in the box beside the question. The value of each question is 2 marks.

A golf ball struck due south in the northern hemisphere will appear to veer to the left.

A spacecraft in low-Earth orbit cannot change its orbital angular mo-mentum by firing its thrusters toward the center of the Earth.

If a velocity-dependent potential V = V(qk, k; k = 1 . . . n) satisfies Lagrange's equations then the corresponding generalized force due to Vis

d (3V'\ 3V

Qk= — --- J ---- dt \dqkj aqk

Euler's equations of rotational motion for a rigid body apply only when written for the body's center of mass.

A continuum model of a cantilevered beam has theoretically an infinite number of modes.

3

Part C

Provide just the answers.

3(a). What is the eccentric anomaly?

/3

3(b). How many parameters are required to describe fully an orbit around the Earth, including being able to determine the position of the satellite in time?

/3

3(c). If L = L(j) satisfying Lagrange's equation is not explicitly dependent on q or time t, what can be said about ç?

/3

11

3(d). Briefly describe Poinsot's geometric interpretation of rigid-body rotation.

/3

3(e). Give two advantages of the Lagrangian formulation for dynamics over the Newton-Euler formulation.

13

Part D

Full solutions are required.

4. Consider two vectors 'a and v. As expressed in frames Ta and T5, these vectors have referential forms Ua, Va and Ub, Vb. Define matrices

Da = UaV, Db = UbV

If Tb rotates with respect to Ta with angular velocity w = .F w, show that

Cab(Db+W < DDW >< )Cba

where Cab is the rotation matrix from Tb to Ta.

No

120

5. A massless string is wound around a uniform spool of radius a, mass m and second moment of mass I = mr 2 about its center. One end of the string passes around a massless pulley and is tied to a particle of mass rn0 . The spool rolls without slipping. Solve for the horizontal position x of the spool's center given the initial conditions x(0) (0) = 0.

5. .

/20

6. A crane is modeled by two uniform beams, each of mass rn and length L Consider

the crane in a position where the beams are at right angles as shown below. The

beams are pinned, where there are rotational springs of stiffness 4k at the ground and

k = mge at the connection point. In the following, consider only planar motion,

which is ultimately to be linearized.

P

m

How many degrees of freedom describe this system? Clearly identify your

choice of coordinates for the remainder of this problem on the diagram

above.

Derive an expression for the kinetic energy.

Derive an expression for the potential energy.

What are the equations of motion in matrix form?

What is the static equilibrium state of the crane?

Determine the natural frequencies of vibration.

Determine and sketch the mode shapes of the system.

If the crane were placed on a mobile base rolling frictionlessly on a hor-

izontal surface, what would the mode shapes look like? (Sketch only, do

not solve.)

6(a). How many degrees of freedom describe this system? Clearly identify your

choice of coordinates for the remainder of this problem on the diagram above.

/2

6(b). Derive an expression for the kinetic energy.

/4

6(c). Derive an expression for the potential energy.

12

/4

6(d). What are the equations of motion in matrix form?

6(e). What is the static equilibrium state of the crane?

/2

/2

13

6(f). Determine the natural frequencies of vibration.

Ii'

/4

6(g). Determine and sketch the mode shapes of the system.

15

16

6(h). If the crane were placed on a mobile base rolling frictionlessly on a horizontal surface, what would the mode shapes look like? (Sketch only, do not solve.)

17

/3