DYNAMICS

72 ^DYNAMICS

73 ^DYNAMICS where ω and α are the absolute angular velocity and absolute

angular acceleration of the relative position vector r_A/B of constant length, respectively.

Rotating Axis Y

X x

y A

r_A/B

r_B B r_A

r_A = r_B + r_A/B

v_A = v_B + X × r_A/B + (v_A/B)_xyz

a_A = a_B + Xo × rA/B + X × (X × rA/B) + 2X × (vA/B)_xyz + (a_A/B)_xyz where X and Xo are, respectively, the total angular velocity and acceleration of the relative position vector r_A/B.

Plane Circular Motion

A special case of radial and transverse components is for constant radius rotation about the origin, or plane circular motion.

r

e_r s

x e_θ

θ y

Here the vector quantities are defined as r

r , where

the radius of the circle the angle from the axis to

r e

v e

a e e

r x

r

r r

2 r

r

~

~ a

i

=

=

= − +

=

=

i

_ i i

The values of the angular velocity and acceleration, respectively, are defined as

~ i

a ~ i

=

= =

o

o p

Arc length, transverse velocity, and transverse acceleration, respectively, are

s r

v r

a r

i

~ a

=

=

=

i i

The radial acceleration is given by

(towards the center of the circle) a_r=-r~²

Normal and Tangential Components

r

et

en

PATH x

y

Unit vectors e_t and e_n are, respectively, tangent and normal to the path with e_n pointing to the center of curvature. Thus

/ , where

instantaneous radius of curvature v t

a t v

v e

a e ² e

t

t t t n

t

=

= +

=

^

^ `

h

h j

Constant Acceleration

The equations for the velocity and displacement when acceleration is a constant are given as

a(t) = a₀

v(t) = a₀ (t – t₀) + v₀

s(t) = a₀ (t – t₀)²/2 + v₀ (t – t₀) + s₀, where

s = displacement at time t, along the line of travel s₀ = displacement at time t₀

v = velocity along the direction of travel v₀ = velocity at time t0

a₀ = constant acceleration t = time

t₀ = some initial time

For a free-falling body, a₀ = g (downward towards earth).

An additional equation for velocity as a function of position may be written as

v²= v₀²+ 2a s_{0_} - s_0i

For constant angular acceleration, the equations for angular velocity and displacement are

/ , where

t

t t t

t t t 2 t t

0

0 0 0

0 0

2

0 0 0

=

= - +

= - + - +

a a

~ a ~

i a ~ i

^

^ _

^ _ _

h

h i

h i i

θ = angular displacement

θ0 = angular displacement at time t₀ ω = angular velocity

ω0 = angular velocity at time t0

α0 = constant angular acceleration t = time

t₀ = some initial time

74 ^DYNAMICS An additional equation for angular velocity as a function of

angular position may be written as

2 2

0 2

0 0

= +

-~ ~ a i_{_} i_i

Projectile Motion

x y

v0

θ

The equations for common projectile motion may be obtained from the constant acceleration equations as

a_x = 0 v_x = v₀ cos(θ) x = v₀ cos(θ)t + x₀ a_y = –g

v_y = –gt + v₀ sin(θ)

y = –gt²/2 + v₀ sin(θ)t + y₀ Non-constant Acceleration

When non-constant acceleration, a(t), is considered, the equations for the velocity and displacement may be obtained from

v t a d v

s t v d s

t t

t

t t

t 0

0

0

0

x x

x x

= +

= +

^ ^

^ ^

h h

h h

#

#

For variable angular acceleration

t d

t d

t t

t

t t

t 0

0

0

0

= +

= +

~ a x x ~

i ~ x x i

^ ^

^ ^

h h

h h

#

#

where τ is the variable of integration CONCEPT OF WEIGHT

W = mg, where W = weight, N (lbf) m = mass, kg (lbf-sec²/ft)

g = local acceleration of gravity, m/s² (ft/sec²) PARTICLE KINETICS

Newton's second law for a particle is ΣF = d(mv)/dt, where

ΣF = the sum of the applied forces acting on the particle m = the mass of the particle

v = the velocity of the particle

For constant mass,

ΣF = m dv/dt = ma

One-Dimensional Motion of a Particle (Constant Mass) When motion exists only in a single dimension then, without loss of generality, it may be assumed to be in the x direction, and

a_x = Fx /m, where

F_x = the resultant of the applied forces, which in general can depend on t, x, and v_x.

If F_x only depends on t, then a t F t /m

v t a d v

x t v d x_t

x x

x x

t t

xt

x t

t 0

0

0

0

x x

x x

=

= +

= +

^

^

^ ^

^

^

h h

h h

h h

#

#

where τ is the variable of integration

If the force is constant (i.e., independent of time, displacement, and velocity) then

/

/2

a F m

v a t t v

x a t t v t t x

0

0 0

x x

x x xt

x xt t

2 0

0 0

=

= − +

= _ − + − +

_

_

j j

j

Normal and Tangential Kinetics for Planar Problems When working with normal and tangential directions, the scalar equations may be written as

/ /

F ma mdv dt

F ma m v

t t t and

n n t

2

= =

= = t

R

R ` j

PRINCIPLE OF WORK AND ENERGY

If T_i and V_i are, respectively, the kinetic and potential energy of a particle at state i, then for conservative systems (no energy dissipation or gain), the law of conservation of energy is

T₂ + V₂ = T₁ + V₁

If nonconservative forces are present, then the work done by these forces must be accounted for. Hence

T₂ + V₂ = T₁ + V₁ + U_1→2 , where

U_1→2 = the work done by the nonconservative forces in moving between state 1 and state 2. Care must be exercised during computations to correctly compute the algebraic sign of the work term. If the forces serve to increase the energy of the system, U_1→2 is positive. If the forces, such as friction, serve to dissipate energy, U_1→2 is negative.

75 ^DYNAMICS

♦Kinetic Energy

Particle T = mv Rigid Body

T = mv + I

Plane Motion _{c c}

1 2 1 2

1 2

2

2 ω2

(( )

subscript c represents the center of mass

♦ Potential Energy

V= V_g + V_e , where V_g = Wy, V_e = 1/2 ks²

The work done by an external agent in the presence of a conservative field is termed the change in potential energy.

Potential Energy in Gravity Field V_g= mgh, where

h = the elevation above some specified datum.

Elastic Potential Energy

For a linear elastic spring with modulus, stiffness, or spring constant, k, the force in the spring is

F_s = k s, where

s = the change in length of the spring from the undeformed length of the spring.

In changing the deformation in the spring from position s1 to s₂, the change in the potential energy stored in the spring is

/2 V₂-V₁=k s` ₂²-s₁²j

♦Work

Work U is defined as U = ∫F • dr

Variable force U = F_F

∫

^cosθds Constant force U = F_{F c}cosθ ∆s Weeight U = _W − ∆W y

Couple moment UU = M _M

Spring  s₂²− s₁²

1   U = _s − 2k 1

2k

♦Power and Efficiency P = dU

dt = = P

P = U F v ^out U

in

out in

ε

•

IMPULSE AND MOMENTUM Linear

Assuming constant mass, the equation of motion of a particle may be written as

mdv/dt = F mdv = Fdt

♦Adapted from Hibbeler, R.C., Engineering Mechanics, 10th ed., Prentice Hall, 2003.

For a system of particles, by integrating and summing over the number of particles, this may be expanded to

F

v v

m_{i i t} m_{i i t} idt

t t

2 1

1 2

= +

R ^ h R ^ h R #

The term on the left side of the equation is the linear momentum of a system of particles at time t₂. The first term on the right side of the equation is the linear momentum of a system of particles at time t₁. The second term on the right side of the equation is the impulse of the force F from time t₁ to t₂. It should be noted that the above equation is a vector equation. Component scalar equations may be obtained by considering the momentum and force in a set of orthogonal directions.

Angular Momentum or Moment of Momentum

The angular momentum or the moment of momentum about point 0 for a particle is defined as

,

H r v

H

m I

or

0

0 0

#

=

= ~

Taking the time derivative of the above, the equation of motion may be written as

d I /dt

Ho₀= _ ₀~i =M , where₀

M₀is the moment applied to the particle. Now by integrating and summing over a system of any number of particles, this may be expanded to

H0i t H0i t M dt0i t

t

2 1

1

Σ_ i =Σ_ i +Σ#2

The term on the left side of the equation is the angular momentum of a system of particles at time t₂. The first term on the right side of the equation is the angular momentum of a system of particles at time t₁. The second term on the right side of the equation is the angular impulse of the moment M₀ from time t₁ to t₂.

Impact

During an impact, momentum is conserved while energy may or may not be conserved. For direct central impact with no external forces

,

v v v v

m1 1+m2 2=m1 1l+m2 l2 where m₁, m₂ = the masses of the two bodies

v₁, v₂ = the velocities of the bodies just before impact ,

v vl l1 2 = the velocities of the bodies just after impact

76 ^DYNAMICS For impacts, the relative velocity expression is

,

e v v

v v

where

n n

n n

1 2

2 1

=

-l - l

^ ^

^ ^

hh hh e = coefficient of restitution

(v_i)_n = the velocity normal to the plane of impact just before impact

vl_{i n}

_ i = the velocity normal to the plane of impact just after impact

The value of e is such that

0 ≤ e ≤ 1, with limiting values

e = 1, perfectly elastic (energy conserved) e = 0, perfectly plastic (no rebound)

Knowing the value of e, the velocities after the impact are given as

v m m

m v e m em v

v m m

m v e em m v

1 1

n

n n

n

n n

1 1 2

2 2 1 2 1

2 1 2

1 1 1 2 2

= +

+ +

-= +

+ -

-l l

^ ^ ^{^} ^ ^

^ ^ ^{^} ^ ^

h h ^h h h

h h ^h h h

Friction

The Laws of Friction are

1. The total friction force F that can be developed is independent of the magnitude of the area of contact.

2. The total friction force F that can be developed is proportional to the normal force N.

3. For low velocities of sliding, the total frictional force that can be developed is practically independent of the sliding velocity, although experiments show that the force F necessary to initiate slip is greater than that necessary to maintain the motion.

The formula expressing the Laws of Friction is F ≤ µN, where

µ = the coefficient of friction.

In general

F < µs N, no slip occurring

F = µ_s N, at the point of impending slip F = µ_k N, when slip is occurring Here,

µs = the coefficient of static friction µ_k = the coefficient of kinetic friction

PLANE MOTION OF A RIGID BODY Kinematics of a Rigid Body

Rigid Body Rotation For rigid body rotation θ

ω = dθ/dt α = dω/dt αdθ = ωdω

Instantaneous Center of Rotation (Instant Centers)

An instantaneous center of rotation (instant center) is a point, common to two bodies, at which each has the same velocity (magnitude and direction) at a given instant. It is also a point in space about which a body rotates, instantaneously.

O2

2 A

3

4 B

I12

I23

I34

I14 ∞ 1GROUND

θ2

The figure shows a fourbar slider-crank. Link 2 (the crank) rotates about the fixed center, O2. Link 3 couples the crank to the slider (link 4), which slides against ground (link 1). Using the definition of an instant center (IC), we see that the pins at O_2, A, and B are ICs that are designated I12, I₂₃, and I₃₄. The easily observable IC is I₁₄, which is located at infinity with its direction perpendicular to the interface between links 1 and 4 (the direction of sliding). To locate the remaining two ICs (for a fourbar) we must make use of Kennedy's rule.

Kennedy's Rule: When three bodies move relative to one another they have three instantaneous centers, all of which lie on the same straight line.

To apply this rule to the slider-crank mechanism, consider links 1, 2, and 3 whose ICs are I12, I₂₃, and I₁₃, all of which lie on a straight line. Consider also links 1, 3, and 4 whose ICs are I₁₃, I₃₄, and I₁₄, all of which lie on a straight line.

Extending the line through I₁₂ and I₂₃ and the line through I₃₄ and I₁₄ to their intersection locates I₁₃, which is common to the two groups of links that were considered.

I14 ∞

I14 ∞

I24

I12

I34

I13

I23

1GROUND

2 3

4

Similarly, if body groups 1, 2, 4 and 2, 3, 4 are considered, a line drawn through known ICs I12 and I₁₄ to the intersection of a line drawn through known ICs I₂₃ and I₃₄ locates I₂₄.

The number of ICs, c, for a given mechanism is related to the number of links, n, by

c n n 2

= ^ - 1h

77 ^DYNAMICS Kinetics of a Rigid Body

In general, Newton's second law for a rigid body, with constant mass and mass moment of inertia, in plane motion may be written in vector form as

, where

F a

M

M a

m I

I m

c c

p c pc c

c

# a

a t Σ

Σ Σ

=

=

= +

F are forces and a_c is the acceleration of the body's mass center both in the plane of motion, Mc are moments and α is the angular acceleration both about an axis normal to the plane of motion, I_cis the mass moment of inertia about the normal axis through the mass center, and ρpcis a vector from point p to point c.

♦Mass Moment of Inertia I=

∫

r dm² Parallel-Axis Theorem I = I_c + md² Radius of Gyration mI

rm=

♦Equations of Motion

Rigid Body = m

Plane Motion

x x

ΣF a_c

ΣM_c = I_cα or ΣM_P = Σ(M_k)_P

y = m y

ΣF a_c

Subscript c indicates center of mass Mass Moment of Inertia

The definitions for the mass moments of inertia are

I y z dm

I x z dm

I x y dm

2 2

2 2

2 2

x y z

= +

= +

= ` +

` _

j i j

#

#

#

A table listing moment of inertia formulas for some standard shapes is at the end of this section.

Parallel-Axis Theorem

The mass moments of inertia may be calculated about any axis through the application of the above definitions.

However, once the moments of inertia have been determined about an axis passing through a body's mass center, it may be transformed to another parallel axis. The transformation equation is

I_new = Ic + md², where

I_new =the mass moment of inertia about any specified axis I_c =the mass moment of inertia about an axis that is parallel

to the above specified axis but passes through the body's mass center

m = the mass of the body

d = the normal distance from the body's mass center to the above-specified axis

Mass Radius of Gyration

The mass radius of gyration is defined as r_m= I m

Without loss of generality, the body may be assumed to be in the x-y plane. The scalar equations of motion may then be written as

,

F a

F a

M m m

I where

x xc

y yc

zc zc

=

=

= a

R R R

zc indicates the z axis passing through the body's mass center, a_xcand a_ycare the acceleration of the body's mass center in the x and y directions, respectively, and α is the angular acceleration of the body about the z axis.

Rotation about an Arbitrary Fixed Axis

♦Rigid Body Motion About a Fixed Axis

= Variable

d d

d d

d d

t t

α

α ω

ω θ

ω ω α θ

=

=

t = tan

2 1 2

Cons 0

0 0

0 0 0 2

2 0

2 0 0

α α

ω ω α

θ θ ω α

ω ω α θ θ

-t t

= + +

t

= +

= + _ i

For rotation about some arbitrary fixed axis q M_q= I_qa

R

If the applied moment acting about the fixed axis is constant then integrating with respect to time, from t = 0 yields

α = M_q/I_q ω = ω₀ + α t

θ = θ₀ + ω₀ t + α t²/2

where ω₀ and θ₀are the values of angular velocity and angular displacement at time t = 0 , respectively.

The change in kinetic energy is the work done in accelerating the rigid body from ω₀to ω

/ /

Iq 2 Iq 2 M dq 2

0 2

0

= +

~ ~ i

i

#i

Kinetic Energy

In general the kinetic energy for a rigid body may be written as

/ /

T= mv²2+ I_c~²2

For motion in the xy plane this reduces to

/ /

T= m v` _cx² + v_cy²j 2+ I_c~²_z2 For motion about an instant center,

T = I_ICω²/2

♦ Adapted from Hibbeler, R.C., Engineering Mechanics, 10th ed., Prentice Hall, 2003.

78 ^DYNAMICS

♦ Principle of Angular Impulse and Momentum Rigid Body

(Plane Motion)

(H_c)₁ + Σ

∫

^{Mc dt = (Hc)2}

where H_c = I_cω (H₀)₁ + Σ

∫

^{M0 dt = (H0)2}

where H₀ = I₀ω Subscript c indicates center of mass

♦ Conservation of Angular Momentum Σ(syst. H)₁ = Σ(syst. H)₂

Free Vibration

The figure illustrates a single degree-of-freedom system.

δst

m

k

x

Position of Undeformed Length of Spring

Position of Static Equilibrium

The equation of motion may be expressed as mxp = mg- k x_{_} + dst_i

where m is mass of the system, k is the spring constant of the system, δ_stis the static deflection of the system, and x is the displacement of the system from static equilibrium.

From statics it may be shown that mg = kδst

thus the equation of motion may be written as ,

/ mx kx x k m x

0 0 + = or

+ =

p

p ^ h

The solution of this differential equation is x(t) = C₁ cos(ω_nt) + C₂ sin(ω_nt)

where ω_n = k m is the undamped natural circular frequency and C₁ and C₂ are constants of integration whose values are determined from the initial conditions.

If the initial conditions are denoted as x(0) = x₀ and ,

xo^ h0 = v₀ then

/

cos sin

x t^h= x₀ _~_nti+_v₀ ~_ni _~_nti

♦ Adapted from Hibbeler, R.C., Engineering Mechanics, 10th ed., Prentice Hall, 2003.

It may also be shown that the undamped natural frequency may be expressed in terms of the static deflection of the system as

/

n= g st

~ d

The undamped natural period of vibration may now be written as

/

mk g

2 2 2

n n

st

x r ~ r

d

= = = r

Torsional Vibration

k_t

I

θ

For torsional free vibrations it may be shown that the differential equation of motion is

, k I_t 0 where

+ =

ip _ ii

θ = the angular displacement of the system k_t = the torsional stiffness of the massless rod I = the mass moment of inertia of the end mass The solution may now be written in terms of the initial conditions i_^0_h= i₀andio_^0_h= io₀as

/

cos sin

t = _{0 n}t + _{0 n n}t

i_^h i _{_}~ i _i ~o i _~ i

where the undamped natural circular frequency is given by /

n= k It

~

The torsional stiffness of a solid round rod with associated polar moment-of-inertia J, length L, and shear modulus of elasticity G is given by

k_t = GJ/L

Thus the undamped circular natural frequency for a system with a solid round supporting rod may be written as

/ GJ IL

n=

~

Similar to the linear vibration problem, the undamped natural period may be written as

/

I k

IL

2 2 2GJ

n n

t

x = r ~ = r = r

79 DYNAMICS

L x

y

z c

x y

z

c R

x y

z

h c R

y

h R₁ c

R₂

x

x y

R c

Figure Mass & Centroid Mass Moment of Inertia (Radius of Gyration)² Product of Inertia M = ρLA

xc = L/2 yc = 0 zc = 0

A = cross-sectional area of rod

ρ = mass/vol.

3 12 0

2 2

ML I I

ML I

I I I

z y

z y

x x

c c

c

=

=

=

=

=

=

3 12 0

2 2 2

2 2 2

2 2

L r r

L r r

r r

z y

z y

x x

c c

c

=

=

=

=

=

=

0 etc.

0 etc.

=

= , I

, I

xy y x_{c c}

M = 2πRρA

xc = R = mean radius yc = R = mean radius zc = 0

A = cross-sectional area of ring

ρ = mass/vol.

2 2 2

2

3

2 3

2

MR I

MR I

I MR I

MR I

I

z y x z

y x

c c c

=

=

=

=

=

=

2 2

2 2 2

2 22

2 2 2

3

2 3

2

R r

R r r

R r

R r r

z y x

y x

c c c

=

=

=

=

=

=

0 0 etc.

2

=

=

=

=

yz xz

z z

y x

I I

MR I

, I

c c

c c

M = πR²ρh xc = 0 yc = h/2 zc = 0

ρ = mass/vol.

( )

(

^{3 2 4}

)

¹²

12 3

2 2 2

2 2

h R M I I

MR I I

h R M I I

z x

y y

z x

c c c

+

=

=

=

=

+

=

=

( )

(

^{3 2 4}

)

¹²

12 3

2 2 2 2

2 2 2

2 2 2 2

h R r r

R r r

h R r r

z x

y y

z x

c c c

+

=

=

=

=

+

=

=

0 etc.

0 etc.

=

= , I

, I

xy y x_{c c}

( )

vol.

mass00 2

22 12

=

ρ ====π − ρ

c c c

z h y x

h R R

M

( )

( )

(

^{3 3 4}

)

¹²

2 12 3

3

2 22 12

22 12

2 22 12

h R R M

I I

R R M I I

h R R M I I

z x

y y

z x

c c c

+ +

=

=

+

=

=

+ +

=

=

( )

( )

(

^{3 3 4}

)

¹²

2

12 3

3

2 2 2 2 1 2 2

2 2 2 1 2 2

2 2 2 2 1 2 2

h R R r r

R R r r

h R R r r

z x

y y

z x

c c c

+ +

=

=

+

=

=

+ +

=

=

0 etc.

0 etc.

=

= , I

, I

xy y x_{c c}

M = π ³ρ 3

4 R

xc = 0 yc = 0 zc = 0

ρ = mass/vol.

5 2

5 2

5 2

2 2 2

MR I

I

MR I

I

MR I

I

z z

y y

x x

c c c

=

=

=

=

=

=

5 2

5 2

5 2

2 2 2

2 2 2

2 2 2

R r r

R r r

R r r

z z

y y

x x

c c c

=

=

=

=

=

= I , etc. =0

c cy x

Housner, George W., and Donald E. Hudson, Applied Mechanics Dynamics, D. Van Nostrand Company, Inc., Princeton, NJ, 1959. Table reprinted by permission of G.W. Housner & D.E. Hudson.

z

z

80 MECHANICS OF MATERIALS

UNIAXIAL STRESS-STRAIN Stress-Strain Curve for Mild Steel

♦

The slope of the linear portion of the curve equals the modulus of elasticity.

DEFINITIONS Engineering Strain

ε = ∆L/Lo, where

ε = engineering strain (units per unit)

∆L = change in length (units) of member L_o = original length (units) of member Percent Elongation

% Elongation = L

L 100

o #

cD m

Percent Reduction in Area (RA)

The % reduction in area from initial area, A_i, to final area, A_f , is:

%RA = A

A A

100

i

i f

- #

e o

Shear Stress-Strain γ = τ/G, where γ = shear strain τ = shear stress

G = shear modulus (constant in linear torsion-rotation relationship)

,

G v

E

2 1 where

= ^ + h

E = modulus of elasticity (Young's modulus) v = Poisson's ratio

= – (lateral strain)/(longitudinal strain)

STRESS, PSI STRESS, MPa

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