CIVE40005 Mechanics Flashcards
Newtons 1st Law & explanation?
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).
Newton’s 2nd Law & explanation ?
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
Newton’s 2nd Law when time not constant ?
F = v* dm/dt + m* dv/dt = vdm/dt + ma
(F = ma)
Newton’s 3rd Law & explaination ?
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
work = ?
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
define kinetic energy
related to the body’s mass and velocity
K = 1/2 * m * v^1/2
potential energy ?
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
gravitational energy ?
- Gravitational potential energy
Increase in gravitational potential energy by lifting a body of mass m by height h is:
U = mgh
elastic potential energy ?
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
what is a conservative force ?
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.
what is kinematics ?
Kinematics
describes motion of bodies without reference to forces
what are degrees of freedom ?
- 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:
what is right hand coordinate system ?
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
what are the 3 fundemental types of motion ?
Rectilinear (Cartesian coordinates)
Plane curvilinear (Plane-polar coordinates)
General curvilinear (Cylindrical polar coordinates, Spherical polar coordinates)
what is difference between cylindrical, plane & spherical polar coordinate system ?
describe vectors in cartesian coordinates
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
expression for instantaneous velocity
v and acceleration a in Rectilinear Motion ?
v = ds /dt
a = dv / dt = d^2 s / dt^2 = v dv / ds (eliminating t)
relationships for constant acceleration ?
v = v0 + a(t-t0)
integrating to give
s = v0t + 1/2at^2
elminating t
v^2 = v0^2 + 2as
SUVAT
describe projectile motion
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
what is the number of DOFs in plane polar coordinates ?
In plane polar coordinates, the radius of motion
r is fixed
and only the angle
θ varies
this reduces the problem to 1-DOF.
unit vectors for plane polar coordinates ?
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)
relationship between cartesian & polar plane unit vectors ?
er = cos θi + sin θj
eθ = − sin θi + cos θj.
why is circular motion a special case?
r (radial component) is constant
what are the unit vectors for cylindrical polar coordinates ?
er - in radial direction
eθ - tangential direction
k- in vertical direction
in what cases does displacement occur ?
To allow any displacement in this system without releasing the
assumption of element rigidity:
one of the supports needs to allow a translation:
how can we transform mechanisms to statiic systems (structures)?
introduce deformable elements (e.g. longitudinal & rotational springs)
DOFs define how much:
- force is carried or
- strain energy is stored
within those deformable elements
how can we transform mechanisms to statiic systems (structures)?
introduce deformable elements (e.g. longitudinal & rotational springs)
DOFs define how much:
- force is carried or
- strain energy is stored
within those deformable elements
what is kinetics
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
acceleration occurs when ?
forces are unbalenced
define amplitude
harmonic motion
Maximum displacement of the body from
equilibrium, in the above example it has the symbol
a.
define period
harmonic motion
Time taken to complete one cycle of the oscillation, it
has the symbol
T, where:
T = 2π/ω
frequency definition
harmonic motion
Number of cycles that are completed in one
second, usually has the symbol
f, where:
f = 1/T = ω / 2π
define phase
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
what is the damp ratio?
r / 2mw where w is the natural circular frequency ( sqrt(k/m) )
describe an overdamped system
(ξ > 1)
2 real, distinct and negative eigenvalues
No oscillations
As t → ∞ then x→0
decaying temporal amplitude.
describe a critically damped system
(ξ = 1)
2 repeated, real and negative eigenvalues λ1=λ2=−ω
As t → ∞ then x→0
decaying temporal amplitude
describe an underdamped system
(ξ < 1)
2 complex conjugate eigenvalues
λ1=−ξω + iωd
λ2=−ξω−iωd
Most common case in structural dynamics
what is stable equilibrium
after any small perturbation the system returns to (or
oscillates about) its original configuration
what is unstable equilibrium
completely different equilibrium
configuration after a pertubation
(or possibly (but improbably) NEUTRALLY STABLE)
what is conservative static loading
- Load magnitude or direction is invariant during
deformation - Define stability of a conservative static system in terms of
the equilibrium path (load–deflection diagram
what is the Equilibrium path
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
what is Buckling
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
what is the energy axiom 1
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
definition for total potential energy
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)
