Two points on a rigid body
Constant V Constant A
Superscripts – Frames Subscripts – Points Cosine rule –
hgt (
vgt ) vgh
CIRCULAR MOTION
MASS PROPERTIES ⃗ Provided both points are on the same rigid body Make P a static point ROTATING FRAME Use these only when particle object is moving along rotating rigid body Conversion equation of any free vectors ⃗⃗ – angular velocity of frame B (rotating) in frame A (fixed) Relate rate of change in both frames Use to switch frames Conversion equation for velocity
Tips to solve Make two equations by viewing from different areas for one same value E.g. one from rotating frame, another from rigid body
Conversion equation for acceleration Analyse this by parts 1. acceleration term of Q if it is not moving on the rigid body 2. acceleration of Q with respect to the (rotating) frame B attached to rigid body 3. Coriolis acceleration.
Moment of inertia
COMPLEX SPRING MASS In Series PLANE MOTION – NEWTON’s 2nd LAW Linear momentum ⃗ – absolute velocity measured w.r.t. Newtonian frame Angular momentum – need reference point, normally centre of mass Force ⃗ –absolute acceleration measured w.r.t Newtonian frame
Steps to solve 1. Kinematics analysis 2. Free body diagram 3. Sum up forces 4. Angular momentum 5. Differentiate it Insert into momentum equation
In Parallel (both connected directly to mass) FREE VIBRATION OF RIGID BODIES – Simple harmonic motion Maximum angular velocity
ENERGY METHOD If there are only conservative forces Kinetic energy of a rigid body
Moment Resultant moment Rate of change of angular momentum Only when P is a fixed point ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗
SPRING-MASS SYSTEM
Angular momentum in 3D
(
Parallel axis theorem – initial point must be Centre of Gravity
)
√ or
(
) ( )
FORCED VIBRATION OF A SPRING MASS SYSTEM
BASE EXCITATION
DAMPED VIBRATION Critical value,
√
(
)
(
)
(
)
DAMPED FORCED VIBRATION
ANALYSIS OF MECHANISMS Lower pairs – 1 degree of freedom Revoute pairs – change angles Prismatic pairs – linear displacemtns Higher pairs – >1 degree of freedom
Any structure only 1 frame → treat AFE as one frame Therefore, F only has 2 links attached to it
Degrees of freedom – number of independent relative motions Gruebler’s equation
F=3(n-1) - 2L - h
Where F = total degrees of freedom in the mechanism n = number of links (including the frame) L = number of lower pairs (one degree of freedom) h = number of higher pairs (two degrees of freedom) rotation + sliding = 2 DOFs √
A, J and G are fixed. AJ is the ground link The ground link is also attached to G. Therefore, the count of contribution of L at G is 2.
FOUR BAR LINKAGES
Grashof crank rocker o Shortest link is the input link o Input link has full motion o Output link limited range of motion Grashof crank crank o Shortest link is the ground link o Input and output have full motion Grashof double rocker o Shortest link is the floating link o Input and output have limited motion Grashof rocker crank o Shortest link is the output link o Output link has full motion o Input link has limited range of motion
SLIDER CRANK
Moments of inertia
Description
Figure
Moment(s) of inertia
Comment
A point mass does not have a moment Point mass m at a
of inertia around its own axis, but by
distance r from the axis of
using the parallel axis theorem a
rotation.
moment of inertia around a distant axis of rotation is achieved.
Two point masses, M and m, with reduced mass
—
and
separated by a distance, x.
This expression assumes that the rod is Rod of length L and
an infinitely thin (but rigid) wire. This is
mass m (Axis of rotation at the end
also a special case of the thin rectangular plate with axis of rotation at
[1]
of the rod)
the end of the plate, with h = L and w = 0.
This expression assumes that the rod is an infinitely thin (but rigid) wire. This is a
Rod of length L and mass m
special case of the thin rectangular plate [1]
with axis of rotation at the center of the plate, with w = L and h = 0.
This is a special case of a torus for b = Thin circular hoop of
0. (See below.), as well as of a thick-
radius r and mass m
walled cylindrical tube with open ends, with r1 = r2 and h = 0.
This is a special case of the solid cylinder, with h = 0. Thin, solid disk of radius r and mass m
That
is a
consequence of the Perpendicular axis theorem.
This expression assumes the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube for r1 = r2.
Thin cylindrical shell with [1]
open ends, of radius r and
Also, a point mass (m) at the end of a
mass m
rod of length r has this same moment of inertia and the value r is called the radius of gyration.
This is a special case of the thick-walled [1]
Solid cylinder of radius r,
cylindrical tube, with r1 = 0. (Note: X-Y
height h and mass m
axis should be swapped for a standard right handed frame)
[1][2]
With a density of ρ and the same Thick-walled cylindrical
geometry
tube with open ends, of inner radius r1, outer radius r2, length h and mass m
or when defining the normalized thickness tn = t/r and letting r = r2, then
Tetrahedron of side s and
—
mass m f=ma
Octahedron (hollow) of side s and mass m
—
Octahedron (solid) of
—
side s and mass m
A hollow sphere can be taken to be made up of two stacks of infinitesimally
Sphere (hollow) of
thin, circular hoops, where the radius
radius r and mass m
[1]
differs from 0 tor (or a single stack, , where the radius differs from -r to r).
A sphere can be taken to be made up of two stacks of infinitesimally thin, solid discs, where the radius differs from 0 to r (or a single stack, where the radius Ball (solid) of radius r and
differs from -r to r).
mass m
[1]
Also, it can be taken to be made up of infinitesimally thin, hollow spheres, where the radius differs from 0 to r. When the cavity radius r1 = 0, the object is a solid ball (above). Sphere (shell) of radius r2, When r1 = r2,
with centered spherical cavity of radius r1 and
[1]
mass m
, and the object is a hollow sphere.
[3]
Right circular cone with
—
radius r, height hand mass m
[ 3]
About a
Torus of tube radius a,
diameter: —
cross-sectional radius b and mass m.
About the vertical
axis:
Ellipsoid (solid) of semiaxes a, b, and cwith
—
axis of rotation a and mass m
Thin rectangular plate of height h and of —
width w and mass m (Axis of rotation at the end of the plate)
Thin rectangular plate of —
height h and of [1]
width w and mass m
Solid cuboid of height h,
For a similarly oriented cube with sides
width w, and depthd, and mass m
of length ,
.
Solid cuboid of height D, width W, and length L, and mass m with the longest
For a cube with sides
,
.
diagonal as the axis. Plane polygon with vertices ...,
,
,
,
and This expression assumes that the
mass
uniformly
distributed on its interior, rotating about an axis
polygon is star-shaped. The vectors ,
,
, ...,
are position
vectors of the vertices. perpendicular to the plane and passing through the origin.
Infinite disk with mass normally distributedon two axes around the axis of rotation (i.e.
Where :
—
is the
mass-density as a function of x and y).
