CIVE40005 Mechanics

Question 1

Newtons 1st Law & explanation?

Answer 1

A body remains at rest or continues to move at a
uniform velocity if there is no external force acting on it.

Law 1 introduces the concept of inertia:
a body’s reluctance to change its current motion,
a measure of a body’s inertia is given by its mass
(kg).

Question 2

Newton’s 2nd Law & explanation ?

Answer 2

The rate of change of momentum of a body is directly
proportional to the external force acting on a body and takes
place in the direction of the force.

F ∝ d / dt (mv)

F = kF * d / dt (mv)
where kF is constant of proportionality

Question 3

Newton’s 2nd Law when time not constant ?

Answer 3

F = v* dm/dt + m* dv/dt = vdm/dt + ma
(F = ma)

Question 4

Newton’s 3rd Law & explaination ?

Answer 4

If body A exerts a force on body B then body B exerts a force equal in magnitude and opposite in direction on body A

Question 5

work = ?

Answer 5

W = Force × Distance
where the “distance” is in the distance the body moves in the
direction of the force.

Vectors:
- incremental work done dW is a scalar product (·) of small increments of force F and displacement r:

dW = F . dr

result is scalar

Graphically : area under force - displacement graph

Question 6

define kinetic energy

Answer 6

related to the body’s mass and velocity
K = 1/2 * m * v^1/2

Question 7

potential energy ?

Answer 7

potential energy U :
gained if a body is moved in the opposite direction to an
internally applied force,
there is no absolute value of
U, it is relative to any point
defined as zero — known as a datum

Question 8

gravitational energy ?

Answer 8

1. Gravitational potential energy
   Increase in gravitational potential energy by lifting a body of mass m by height h is:
   U = mgh

Question 9

elastic potential energy ?

Answer 9

Elastic potential energy (usually called strain energy)
Stored in a linearly elastic longitudinal spring of
stiffness k compressed or stretched by x
U = 1/2 * k * x ^ 2’

Stored in a linearly elastic rotational spring of
stiffness c and rotated by θ:

U = 1/2 * c * θ ^2

Question 10

what is a conservative force ?

Answer 10

If the work done by the force depends only on the body’s net
change in position and not on the path followed, the
corresponding is known as a Conservative Force.
Gravitational force is an example;
certain fluid forces such as jet forces or wind loads are
non-conservative forces.

Question 11

what is kinematics ?

Answer 11

Kinematics
describes motion of bodies without reference to forces

Question 12

what are degrees of freedom ?

Answer 12

• define the current position and orientation
• More constraints to the rigid body can reduce the number
  of DOFs.

eg in 2-dimensional analysis : 3 DOFs 3 independent displacement components (x, y and θ)
in 3D :
6 degrees of freedom:
- 3 translation components (x, y and z)
- 3 rotational components (θx, θy and θz)
which in a “right-hand” coordinate system are
defined:

Question 13

what is right hand coordinate system ?

Answer 13

Cartesian directions:
* Thumb:
x
* Index finger:
y
* Middle finger:
z

Rotations: use “right-hand screw”:
* Thumb: describes positive Cartesian direction
about which rotation occurs
* Fingers: describe positive rotation sense

Question 14

what are the 3 fundemental types of motion ?

Answer 14

Rectilinear (Cartesian coordinates)

Plane curvilinear (Plane-polar coordinates)

General curvilinear (Cylindrical polar coordinates, Spherical polar coordinates)

Question 15

what is difference between cylindrical, plane & spherical polar coordinate system ?

Question 16

describe vectors in cartesian coordinates

Answer 16

unit vectors ( i, j, k) do not vary with time t
Velocity and acceleration vectors can be written thus :

v = dr/dt = dx/dt* i + dy/dt * j + dz/dt * k

a = d^2r/dt^2 = d^2x/dt^2 * i + d^2y/ dt^2 * j + d^2z/dt^2 * k

Question 17

expression for instantaneous velocity
v and acceleration a in Rectilinear Motion ?

Answer 17

v = ds /dt

a = dv / dt = d^2 s / dt^2 = v dv / ds (eliminating t)

Question 18

relationships for constant acceleration ?

Answer 18

v = v0 + a(t-t0)
integrating to give
s = v0t + 1/2at^2
elminating t
v^2 = v0^2 + 2as

SUVAT

Question 19

describe projectile motion

Answer 19

Motion of projectiles is an example where constant
acceleration is assumed. Assumptions are:
no aerodynamic drag
projectile altitude is sufficiently small
we can say the following about the acceleration vector a:

a = axi + ayj

ax = 0 = dvx / dt

ay = -g = dvy / dt

Integrating to give :

dvx / dt = 0 => vx = vx0

dx/dt - vxo => x = vx0*t + x0
where vx0 and x0 are the initial horizontal velocity and
displacement components respectively.

dvy / dt = −g ⇒ vy = −gt + vy0
and:
dy / dt = −gt + vy0 ⇒ y = −1/2gt^2 + vy0t + y0,
where vy0 and y0 are the corresponding initial vertical velocity
and displacement components respectively.
where vy0 and y0 are the corresponding initial vertical velocity
and displacement components respectively

Question 20

what is the number of DOFs in plane polar coordinates ?

Answer 20

In plane polar coordinates, the radius of motion
r is fixed
and only the angle
θ varies
this reduces the problem to 1-DOF.

Question 21

unit vectors for plane polar coordinates ?

Answer 21

For plane polar coordinates where
r >= 0 and θ = [0, 2π]:
er: unit vector in the radial direction,
eθ: unit vector in the tangential direction.

( UNIT VECTORS ARE NOW TIME VARIANT )
position of the particle changes with time:
- r and θ also vary with t relative to the origin
- unit vectors in polar coordinates are functions of time t
(this is unlike the Cartesian unit vectors i, j and k)

Question 22

relationship between cartesian & polar plane unit vectors ?

Answer 22

er = cos θi + sin θj

eθ = − sin θi + cos θj.

Question 23

why is circular motion a special case?

Answer 23

r (radial component) is constant

Question 24

what are the unit vectors for cylindrical polar coordinates ?

Answer 24

er - in radial direction
eθ - tangential direction
k- in vertical direction

Question 25

in what cases does displacement occur ?

Answer 25

To allow any displacement in this system without releasing the
assumption of element rigidity:
one of the supports needs to allow a translation:

Question 26

how can we transform mechanisms to statiic systems (structures)?

Answer 26

introduce deformable elements (e.g. longitudinal & rotational springs)
DOFs define how much:
- force is carried or
- strain energy is stored
within those deformable elements

Question 27

how can we transform mechanisms to statiic systems (structures)?

Answer 27

introduce deformable elements (e.g. longitudinal & rotational springs)
DOFs define how much:
- force is carried or
- strain energy is stored
within those deformable elements

Question 28

what is kinetics

Answer 28

Branch of dynamics concerning relationships between:
unbalanced forces and resulting changes in motion
Newton’s 2nd law of motion becomes vitally important
For constant mass, Newton’s 2nd law of motion gives:
F = ma

Question 29

acceleration occurs when ?

Answer 29

forces are unbalenced

Question 30

define amplitude

harmonic motion

Answer 30

Maximum displacement of the body from
equilibrium, in the above example it has the symbol
a.

Question 31

define period

harmonic motion

Answer 31

Time taken to complete one cycle of the oscillation, it
has the symbol
T, where:
T = 2π/ω

Question 32

frequency definition

harmonic motion

Answer 32

Number of cycles that are completed in one
second, usually has the symbol
f, where:

f = 1/T = ω / 2π

Question 33

define phase

Answer 33

The difference in motion as described by the following:
if we have an additional oscillatory motion (s2)
described by the following expression:
s=a sin(ωt + α) (≡s2)
in phase if phase angles are equal

Question 34

what is the damp ratio?

Answer 34

r / 2mw where w is the natural circular frequency ( sqrt(k/m) )

Question 35

describe an overdamped system

Answer 35

(ξ > 1)
2 real, distinct and negative eigenvalues
No oscillations
As t → ∞ then x→0
decaying temporal amplitude.

Question 36

describe a critically damped system

Answer 36

(ξ = 1)
2 repeated, real and negative eigenvalues λ1=λ2=−ω
As t → ∞ then x→0
decaying temporal amplitude

Question 37

describe an underdamped system

Answer 37

(ξ < 1)
2 complex conjugate eigenvalues
λ1=−ξω + iωd
λ2=−ξω−iωd

Most common case in structural dynamics

Question 38

what is stable equilibrium

Answer 38

after any small perturbation the system returns to (or
oscillates about) its original configuration

Question 39

what is unstable equilibrium

Answer 39

completely different equilibrium
configuration after a pertubation
(or possibly (but improbably) NEUTRALLY STABLE)

Question 40

what is conservative static loading

Answer 40

• Load magnitude or direction is invariant during
  deformation
• Define stability of a conservative static system in terms of
  the equilibrium path (load–deflection diagram

Question 41

what is the Equilibrium path

Answer 41

all displacement-velocity combinations where equilibrium is present (on displacement-velocity graph)

(At points off the path, no equilibrium exists:
relationship describe by an equation of motion)

In general, there will be n displacement components:
for this the equilibrium path will be a curve (or curves)
in an (n + 1) dimensional space
* so-called configuration space

Question 42

what is Buckling

Answer 42

dynamic process which occurs when an equilibrium path loses stability or becomes unstable
After buckling occurs:
- structure can be said to have buckled;
- behaviour is called the post-buckling response

There are different types of buckling

Question 43

what is the energy axiom 1

Answer 43

A stationary value of the total potential
energy V with respect to the generalized coordinates Qi is
necessary and sufficient for the equilibrium of the system

Question 44

definition for total potential energy

Answer 44

V = U + Φ
U: gain in potential energy
Φ: work done by the load

Φ = − loadP × distance load moves ∆ in loading direction. V is commonly written thus:
V = U − P∆. (70)
U is given by the distance the mass m rises:
U = mgRθ
P∆ is given by the vertical distance the load P moves:
P ∆ = P*R(1 − cos(θ)

Hence, total potential energy
V is:
V = mgRθ − P*R(1 − cos(θ)