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Course Outline
1. MATLAB tutorial
2. Motion of systems that can be idealized as particles• Description of motion; Newton’s laws;
• Calculating forces required to induce prescribed motion;
• Deriving and solving equations of motion
3. Conservation laws for systems of particles• Work, power and energy;
• Linear impulse and momentum
• Angular momentum
4. Vibrations• Characteristics of vibrations; vibration of free 1 DOF systems
• Vibration of damped 1 DOF systems
• Forced Vibrations
5. Motion of systems that can be idealized as rigid bodies• Description of rotational motion; Euler’s laws; deriving and solving
equations of motion; motion of machines
Exam topics
Particle Dynamics: Concept Checklist• Understand the concept of an ‘inertial frame’• Be able to idealize an engineering design as a set of particles, and know when this
idealization will give accurate results• Describe the motion of a system of particles (eg components in a fixed coordinate system;
components in a polar coordinate system, etc)• Be able to differentiate position vectors (with proper use of the chain rule!) to determine
velocity and acceleration; and be able to integrate acceleration or velocity to determine position vector.
• Be able to describe motion in normal-tangential and polar coordinates (eg be able to write down vector components of velocity and acceleration in terms of speed, radius of curvature of path, or coordinates in the cylindrical-polar system).
• Be able to convert between Cartesian to normal-tangential or polar coordinate descriptions of motion
• Be able to draw a correct free body diagram showing forces acting on system idealized as particles
• Be able to write down Newton’s laws of motion in rectangular, normal-tangential, and polar coordinate systems
• Be able to obtain an additional moment balance equation for a rigid body moving without rotation or rotating about a fixed axis at constant rate.
• Be able to use Newton’s laws of motion to solve for unknown accelerations or forces in a system of particles
• Use Newton’s laws of motion to derive differential equations governing the motion of a system of particles
• Be able to re-write second order differential equations as a pair of first-order differential equations in a form that MATLAB can solve
Particle Kinematics
r i j k( ) ( ) ( ) ( )t x t y t z t r(t)
r(t+dt)
v(t) dt
i
j
path of particle
kO
( ) ( ) ( ) ( )
( ) ( ) ( )
x y z
x y z
t v t v t v t
d dx dy dzx y z
dt dt dt dtdx dy dz
v t v t v tdt dt dt
v i j k
i j k i j k
• Direction of velocity vector is parallel to path• Magnitude of velocity vector is distance traveled / time
Inertial frame – non accelerating, non rotating reference frameParticle – point mass at some position in space
Position Vector
Velocity Vector
Acceleration Vector
2 2 2
2 2 2
( ) ( ) ( ) ( )
( ) ( ) ( )
yx zx y z x y z
yx zx y z
dvdv dvdt a t a t a t v v v
dt dt dt dt
dvdv dvd x d y d za t a t a t
dt dt dtdt dt dt
a i j k i j k i j k
Particle Kinematics
• Straight line motion with constant acceleration
20 0 0
1
2X V t at V at a
r i v i a i
• Time/velocity/position dependent acceleration – use calculus
0
0
0
( )
0
( )( ) ( )
( )
( )( ) ( )
( )
v t
V
x t t
X
dv g ta f v dv g t dt
dt f v
dx g tv f x dv v t dt
dt f v
0 00 0
( ) ( )t t
X v t dt V a t dt
r i v i
0
( ) ( )
0
( )v t x t
V
vdv a x dx
Particle Kinematics
• General circular motion
R
i
j
Rcos
Rsin
tsin
cos
n
2
22
cos sin
sin cos
( sin cos ) (cos sin )
R
R V
R R
dV VR R
dt R
r i j
v i j t
a i j i j
t n t n
2 2/ / /
/
d dt d dt d dt
s R V ds dt R
• Circular Motion at const speed
22 2
cos sin
sin cos
(cos sin )
R
R V
VR R
R
r i j
v i j t
a i j n n
t s R V R
Particle Kinematics
• Polar Coordinates
r
i
j
ee r
22 2
2 22
r
r
dr dr
dt dt
d r d d dr dr r
dt dt dtdt dt
v e e
a e e
2
V
dV V
dt R
v t
a t n
2 2
2 2
3/222
1
d y dydx d x
d dd d
Rdydx
d d
( ) ( )x y r i j
t
R
t
n
• Arbitrary path
Newton’s laws
• For a particle
• For a rigid body in motion without rotation, or a particle on a massless frame
ij
NA NBTBW
mF a
c M 0
You MUST take momentsabout center of mass
Calculating forces required to cause prescribed motion of a particle
• Idealize system
• Free body diagram
• Kinematics
• F=ma for each particle.
• (for rigid bodies or frames only)
• Solve for unknown forces or accelerationsc M 0
Deriving Equations of Motion for particles
1. Idealize system
2. Introduce variables to describe motion(often x,y coords, but we will see other examples)
3. Write down r, differentiate to get a
4. Draw FBD
5.
6. If necessary, eliminate reaction forces
7. Result will be differential equations for coords defined in (2), e.g.
8. Identify initial conditions, and solve ODE
mF a
2
02sin
d x dxm kx kY t
dtdt
Motion of a projectile
i
j
kX0
V0
0 0 0
0 0 00
x y z
X Y Ztd
V V Vdt
r i j k
ri j k
20 0 0 0 0 0
0 0 0
1
2x y z
x y z
X V t Y V t Z V t gt
V V V gt
g
r i j k
v i j k
a k
Rearranging differential equations for MATLAB
22
22 0n n
d y dyy
dtdt • Example
/v dy dt• Introduce
22 n n
vyd
vdt v y
• Then
( , )yd
f tvdt
ww w• This has form
Conservation laws for particles: Concept Checklist• Know the definitions of power (or rate of work) of a force, and work done by a force• Know the definition of kinetic energy of a particle• Understand power-work-kinetic energy relations for a particle• Be able to use work/power/kinetic energy to solve problems involving particle motion• Be able to distinguish between conservative and non-conservative forces• Be able to calculate the potential energy of a conservative force• Be able to calculate the force associated with a potential energy function• Know the work-energy relation for a system of particles; (energy conservation for a closed
system)• Use energy conservation to analyze motion of conservative systems of particles
• Know the definition of the linear impulse of a force• Know the definition of linear momentum of a particle• Understand the impulse-momentum (and force-momentum) relations for a particle• Understand impulse-momentum relations for a system of particles (momentum
conservation for a closed system)• Be able to use impulse-momentum to analyze motion of particles and systems of particles• Know the definition of restitution coefficient for a collision• Predict changes in velocity of two colliding particles in 2D and 3D using momentum and the
restitution formula
• Know the definition of angular impulse of a force• Know the definition of angular momentum of a particle• Understand the angular impulse-momentum relation• Be able to use angular momentum to solve central force problems
Work and Energy relations for particles
Rate of work done by a force(power developed by force)
ij
k
O
PF
vP F v
Total work done by a force
ij
k
O
PF(t)
r0
r1
1
0
t
W dt F v1
0
W d r
r
F r
2 2 2 21 1
2 2 x y zT m m v v v vKinetic energy
ij
k
O
P
v
r0
r1Work-kinetic energy relation1
0
0W d T T r
r
F r
Power-kinetic energy relationdT
Pdt
Potential energy
ij
k
O
PF
r0
r1
0
( ) constantV d r
r
r F r
Potential energy of aconservative force (pair)
grad( )VF
Type of force Potential energy
Gravity acting on a particle near earths surface
V mgy
Fm
ji
y
Gravitational force exerted on mass m by mass M at the origin
GMmV
r
r
F
r m
Force exerted by a spring with stiffness k and unstretched length
0L
201
2V k r L F
i
j
Force acting between two charged particles
1 2
4
Q QV
r
r
+Q1+Q2
ij
1
F2
Force exerted by one molecule of a noble gas (e.g. He, Ar, etc) on another (Lennard Jones potential). a is the equilibrium spacing between molecules, and E is the energy of the bond.
12 6
2a a
Er r
ri
j
1
F2
m1
m2
m3
m4
R21
R12
R13 R31
R23
R32
F2ext
F3ext
F1ext
Energy relations forconservative systems subjected to external forces
ijRInternal Forces: (forces exerted by one part of the system on another)
External Forces: (any other forces) extiF
0
0
( ) ( )t t
extext i
forces t
W t t dt
F vWork done by external forces
0 0extW T V T V
System is conservative if all internal forces are conservative forces (or constraint forces)
Energy relation for a conservative system 0 0t t t t
Kinetic and potential energy at time 0t 0 0T V
Kinetic and potential energy at time t T V
Linear Impulse of a force
ij
k
O
F(t) v
m
1
0
( )t
t
t dtF
Linear momentum of a particle mp v
Impulse-momentum relations d
dt
pF 1 0 p p
Impulse-momentum relations
m1
m2
m3
m4
R21
R12
R13 R31
R23
R32
F2ext
F3ext
F1ext
0
0
( )t t
exttot i
particles t
t dt
FTotal external impulse
tot i iparticles
m p vTotal linear momentum
( )ext toti
particles
dt
dt p
FImpulse-momentum tot tot p
Impulse-momentum for a system of particles
Impulse-momentum for a single particle
0 0tot tot pSpecial case (closed system) (momentum conserved)
Collisions
*
A B
A B
vxB1vx
A1
vxB0
vxA0
1 1 0 0A B A BA x B x A x B xm v m v m v m v
1 1 0 0B A B Av v e v v
1 0 0 0
1 0 0 0
(1 )
(1 )
B B B AA
A B
A A B AB
A B
mv v e v v
m m
mv v e v v
m m
AB
A B
B0vA0v
v A1 B1v
n
1 1 0 0 0 0(1 )B A B A B Ae v v v v v v n n
1 1 0 0B A B AB A B Am m m m v v v v
1 0 0 0(1 )B B B AA
B A
me
m m
v v v v n n
1 0 0 0(1 )A A B AB
B A
me
m m
v v v v n n
Restitution formula
Momentum
Momentum
Restitution formula
Angular Impulse-Momentum Equations for a Particle
i
j
k
x
y
zO
F(t)
r(t)0
0
( ) ( )t t
t
t t dt
r F
m h r p r v
d
dt
hMImpulse-Momentum relations
1 0 h h
Useful for central force problems
Angular Momentum
Angular Impulse
Special Case 1 0 0 h h Angular momentum conserved
Free Vibrations – concept checklistYou should be able to:1.Understand simple harmonic motion (amplitude, period, frequency, phase)2.Identify # DOF (and hence # vibration modes) for a system3.Understand (qualitatively) meaning of ‘natural frequency’ and ‘Vibration mode’ of a system4.Calculate natural frequency of a 1DOF system (linear and nonlinear)
5.Write the EOM for simple spring-mass-damper systems by inspection
6.Understand natural frequency, damped natural frequency, and ‘Damping factor’ for a dissipative 1DOF vibrating system7.Know formulas for nat freq, damped nat freq and ‘damping factor’ for spring-mass system in terms of k,m,c8.Understand underdamped, critically damped, and overdamped motion of a dissipative 1DOF vibrating system9.Be able to determine damping factor and natural frequency from a measured free vibration response10.Be able to predict motion of a freely vibrating 1DOF system given its initial velocity and position, and apply this to design-type problems
Vibrations and simple harmonic motion
Period, T
Peak to PeakAmplitude A
time
Displacementor Acceleration
y(t)
Typical vibration response
• Period, frequency, angular frequencyamplitude
Simple Harmonic Motion 0( ) sin
( ) cos( )
( ) sin )
x t X X t
v t V t
a t A t
V X A V
Vibration of 1DOF conservative systems
Harmonic Oscillator
2
02
m d ss L
k dt Derive EOM (F=ma)
Compare with ‘standard’ differential equation
Solution
2 2 20 0 0 0( ) ( ) / sin( )n ns t L s L v t
0 0 0
2
1
n
x s C L x s
m
k
nk
m Natural Frequency
k
m mk k
x1 x2
Vibration modes and natural frequencies
•Vibration modes: special initial deflections that cause entire system to vibrate harmonically•Natural Frequencies are the corresponding vibration frequencies
Number of DOF (and vibration modes)
In 2D: # DOF = 2*# particles + 3*# rigid bodies - # constraintsIn 3D: # DOF = 3*# particles + 6*# rigid bodies - # constraints
Expected # vibration modes = # DOF - # rigid body modes
A ‘rigid body mode’ is steady rotation or translation of the entire system at constant speed. The maximum number of ‘rigid body’ modes (in 3D) is 6; in 2D it is 3. Usually only things like a vehicle or a molecule, which can move around freely, have rigid body modes.
k
m mk k
x1 x2
Calculating nat freqs for 1DOF systems – the basics
EOM for small vibration of any 1DOF undamped system has form
m
k,L0y
2
2 2
1
n
d yy C
dt
1. Get EOM (F=ma or energy)2. Linearize (sometimes)3. Arrange in standard form4. Read off nat freq.
n is the natural frequency
Tricks for calculating natural frequencies of 1DOF undamped systems
k,L0
m
L0+
n
g
• Nat freq is related to static deflection
• Using energy conservation to find EOM
s
k, dm
22
0
2
02
2
02
1 1( )
2 2
( ) ( ) 0
dsKE PE m k s L const
dt
d ds d s dsKE PE m k s L
dt dt dt dt
d sm ks kL
dt
Linearizing EOM
2
2( )
d yf y C
dt Sometimes EOM has form
We cant solve this in general… Instead, assume y is small
2
20
2
20
(0) ...
(0)
y
y
d y dfm f y C
dt dy
d y dfm y C f
dt dy
There are short-cuts to doing the Taylor expansion
Writing down EOM for spring-mass-damper systems
s=L0+xk, L0
mc
2
2
22
2
0
2 02
n n n
d x c dx km x
m dt mdt
d x dx k cx
dt m kmdt
F a
k1
k2
Commit this to memory! (or be able to derive it…)
x(t) is the ‘dynamic variable’ (deflection from static equilibrium)
Parallel: stiffness 1 2k k k
c2
c1
Parallel: coefficient 1 2c c c
k1 k2
Series: stiffness 1 2
1 1 1
k k k
c2c1
Parallel: coefficient1 2
1 1 1
c c c
s=L0+xk, L0
mc
2
02
d s dsm c ks kL
dtdt EOM
2
2 2
1 2
nn
d x dxx C
dtdt
Standard Form
02
nk c
C Lm km
21d n
Overdamped 1
Critically Damped 1
Underdamped 1
Canonical damped vibration problem
Overdamped 0 0 0 0( )( ) ( )( )( ) exp( ) exp( ) exp( )
2 2n d n d
n d dd d
v x C v x Cx t C t t t
1
Critically Damped 1 0 0 0( ) ( ) ( ) exp( )n nx t C x C v x C t t
0 00
( )( ) exp( ) ( )cos sinn
n d dd
v x Cx t C t x C t t
Underdamped 1
0 0 0dx
x x v tdt
with 0 0 0ds
s s v tdt
0 0x ss x
Calculating natural frequency and damping factor from a measured vibration response
Displacement
time
t0 t1 t2 t3
T
x(t0) x(t1) x(t2) x(t3)
t4
0( )1log
( )n
x t
n x t
Measure log decrement:
2 2
2 2
4
4n T
Measure period: T
Then
Forced Vibrations – concept checklist
You should be able to:1.Be able to derive equations of motion for spring-mass systems subjected to external forcing (several types) and solve EOM using complex vars,or by comparing to solution tables2.Understand (qualitatively) meaning of ‘transient’ and ‘steady-state’ response of a forced vibration system (see Java simulation on web)3.Understand the meaning of ‘Amplitude’ and ‘phase’ of steady-state response of a forced vibration system4.Understand amplitude-v-frequency formulas (or graphs), resonance, high and low frequency response for 3 systems 5.Determine the amplitude of steady-state vibration of forced spring-mass systems.6.Deduce damping coefficient and natural frequency from measured forced response of a vibrating system 7.Use forced vibration concepts to design engineering systems
x(t)k, L0
m
x(t)
m
m
F(t)=F0 sin t
External forcing
Base Excitation
x(t)
Rotor Excitation
y(t)=Y0 sint
m0
y(t)=Y0sint
L0
k, L0
L0
k, L0
L0
2
02 2
1 2sin
nn
d x dxx KF t
dtdt
1, ,
2n
kK
m kkm
EOM for forced vibrating systems
2
2 2
1 2 2
n nn
d x dx dyx K y
dt dtdt
, , 12
nk
Km km
22 20
2 2 2 2 2
1 2sin
nn n n
Yd x dx K d yx K t
dtdt dt
00
2n
mkK M m m
M MkM
2
02 2
1 2sin( )
nn
d x dxx C KF t
dtdt
0 0 0dx
x x v tdt
( ) ( ) ( )h px t C x t x t
Equation Initial Conditions
Full Solution
0( ) sinpx t X t Steady state part (particular integral)
100 1/2 2 22 22 2
2 /tan
1 /1 / 2 /
n
nn n
KFX
Transient part (complementary integral)
Overdamped0 0 0 0( ) ( )
( ) exp( ) exp( ) exp( )2 2
h h h hn d n d
h n d dd d
v x v xx t C t t t
1
Critically Damped 1 0 0 0( ) exp( )h h hh n nx t C x v x t t
0 00( ) exp( ) cos sin
h hh n
h n d dd
v xx t C t x t t
Underdamped 1
0 0 0 0 0 0 0 00
(0) sin cosph h
pt
dxx x C x x C X v v v X
dt
21d n
Steady-state and Transient solution to EOM
0( ) sinpx t X t
10 0 1/2 2 22 22 2
2 /1( / , ) tan
1 /1 / 2 /
nn
nn n
X KF M M
2
02 2
1 2sin( )
nn
d x dxx C KF t
dtdt
Steady state solution to 2 / T
s=L0+xk, L0
mc
0( ) sinF t F t
Canonical externally forced system (steady state solution)
1
2n
k cK
m kkm
0 0.5 1 1.5 2 2.5 310
-1
100
101
Frequency Ratio /n
Mag
nific
atio
n X
0/KY
0
= 0.01
= 0.05 = 0.1 = 0.2 = 0.4
= 1.0 = 0.6
2
2 2
1 2 2
n nn
d x dx dyx C K y
dt dtdt
0( ) sinpx t X t
1/223 3
10 0 1/2 2 2 22 22 2
1 2 / 2 /( , , ) tan
1 (1 4 ) /1 / 2 /
nn
nn
n n
X KY M M
Steady state solution to
Canonical base excited system (steady state solution)
12
nk c
Km km
0( ) sinpx t X t
0 0 ( , , )nX KY M
Steady state solution to2 2
2 2 2 2
1 2
nn n
d x dx K d yx C
dtdt dt
0 siny Y t
2 21
1/2 2 22 22 2
/ 2 /tan
1 /1 / 2 /
n n
nn n
M
c
0
0 002 ( )n
mk cK
m m m mk m m
Canonical rotor excited system (steady state solution)