Vibrations Fundamentals and Practice Appb

Vibrations Fundamentals and Practice Appb Maintaining the outstanding features and practical approach that led the bestselling first edition to become a standard textbook in engineering classrooms worldwide, Clarence de Silva''s Vibration: Fundamentals and Practice, Second Edition remains a solid instructional tool for modeling, analyzing, simulating, measuring, monitoring, testing, controlling, and designing for vibration in engineering systems. It condenses the author''s distinguished and extensive experience into an easy-to-use, highly practical text that prepares students for real problems in a variety of engineering fields.

de Silva, Clarence W “Appendix B”

Vibration: Fundamentals and Practice

Clarence W de Silva

Boca Raton: CRC Press LLC, 2000

Appendix B Newtonian and Lagrangian Mechanics

A vibrating system can be interpreted as a collection of mass particles In the case of distributedsystems, the number of particles is infinite The flexibility and damping effects can be introduced

as forces acting on these particles It follows that Newton’s second law for a mass particle formsthe basis of describing vibratory motions

System equations can be obtained directly by applying Newton’s second law to each particle

It is convenient, however, to use Lagrange’s equations for this purpose, particularly when the system

is relatively complex A variational principle known as Hamilton’s principle, which can be lished from Newton’s second law, is the starting point in the derivation of Lagrange’s equations.This appendix outlines some useful results of dynamics, in both Newtonian and Lagrangianapproaches The Newtonian approach uses forces, torques, and motions, which are vectors Hence,

estab-it is important to deal westab-ith vector mechanics in the Newtonian approach The Lagrangian approach

is based on energy, which is a scalar quantity Scalar energies can be expressed in terms of vectorialpositions and velocities

The subject of dynamics deals with forces (torques) and motions The study of motion alonebelongs to the subject of kinematics This appendix starts with vectorial kinematics, and thenaddresses Newtonian mechanics (dynamics) and finally Lagrangian dynamics

B.1 VECTOR KINEMATICS B.1.1 E ULER ’ S T HEOREM (S EE F IGURE B.1 )

Every displacement of a rigid body can be represented by a single rotation θ about some axis (ofunit vector υ)

The second term can be neglected for small rotations.

Note: Since this definition uses small rotations δθ, it follows that ω is always a vector

The velocity of P relative to O (see Figure B.1) is given by

δδ

Suppose that i1, i2, i3 are unit vectors along the three orthogonal axes of a frame rotating at ω The

rates are the velocities of these vectors about the origin of the frame Hence, from equation (B.3),

one obtains

(B.4)

Some special cases (natural frames of reference) are considered below

Cartesian Coordinates

Suppose that the frame is free to move independently in the x, y, and z directions only This

corresponds to a translatory motion (no rotation) The rates of the unit vectors in this moving frame

are 0; that is, ω = 0

d dt

d dt

Polar Coordinates (2-D)

Suppose that the frame is free to move independently in the r and θ directions, but an increment

in the r direction (i.e., δr) causes no rotation Hence,

From equation (B.4),

FIGURE B.3 Some natural coordinate frames: (a) Cartesian coordinates; (b) polar coordinates (2-D);

(c) spherical polar coordinates; and (d) tangential-normal coordinates (2-D).

ωω =θ˙i z

Spherical Polar Coordinates

The coordinates are (r, θ, φ) as in Figure B.3 To find natural ω for the frame:

Hence,

But

Hence,

(B.7)From equation (B.4),

(B.8)

Tangential-Normal (Intrinsive) Coordinates (2-D)

The coordinates are (s, ψ) as in Figure B.3 Note that s is the curvilinear distance along the path

of the particle (from a reference point P o), and ψ is the angle of slope of the path To find ω ofthe natural frame,

Hence,

d dt

d dt

r

r

i i

i i

Give to frame No rotation

Give to frame Rotation

Give to frame Rotation z

The derivatives of the tangential and normal unit vectors i t and i n are given by

(B.9)

Note: The vector cross-product of the two vectors

can be obtained by expanding the determinant

(B.10)

B.1.4 A CCELERATION E XPRESSED IN R OTATING F RAMES

Spherical Polar Coordinates

Substitute equation (B.8), and obtain

(B.11)Acceleration Hence, by differentiating equation (B.11) and using (B.8), one obtains

(B.12)

Tangential-Normal Coordinates (2-D) (See Figure B.4 )

The velocity is always tangential to the path Hence,

d dt d dt

t n

n t

i i

i i

d dt

r

r r

v v

v dv ds v

ds dt

vdv ds d

dt

d ds

ds dt v

ds d

ψ radius of curvature

B.2 NEWTONIAN (VECTOR) MECHANICS

B.2.1 F RAMES OF R EFERENCE R OTATING AT A NGULAR V ELOCITY ω

(S EE F IGURE B.5 )

Newton’s second law holds with respect to (w.r.t.) an inertial frame of reference (normally a frame

fixed on the earth’s surface or moving at constant velocity) The rate of change of vector B w.r.t.

an inertial frame is related to the rate of change w.r.t a frame rotating at ω by

(B.15)

From equation (B.15), the following results can be obtained for velocity v and acceleration a of

point B (see Figure B.5)

rel

frm

6 74 48

1444442444443 1 24 34

B.2.2 N EWTON ’ S S ECOND L AW FOR A P ARTICLE OF M ASS m (SEE FIGURE B.6 )

(B.18)

(Linear momentum principle)

Note: Linear momentum

Cross multiply equation (B.18) by r The torque about B is

Now,

Also,

This cross-product vanishes if either B is fixed (v B = 0) or B moves parallel to the velocity of m.

Hence, the angular momentum principle:

p=m d R dt

dt

d dt

d dt

= × = ×r f r p= (r×p)− r×p

r× =p h B=angular momentum about B

d dt

= h +v ×p in general

B.2.3 S ECOND L AW FOR A S YSTEM OF P ARTICLES — R IGIDLY OR

F LEXIBLY C ONNECTED (S EE FIGURE B.7 )

(B.20)

(Linear momentum principle)

Now, using the procedure outlined before for a single particle, and summing the results, one obtains

FIGURE B.7 (a) Dynamics of a system of particles, and (b) dynamics with respect to the centroid.

: resultant external force

= total linear momentum

centroid of the particles

i i

d dt

d dt

ττ

1 24 3 123

Hence, we have the angular momentum principle

Note: Equation (B.20) for a system of particles is also the convenient form of the linear momentum

principle for rigid bodies But a more convenient form of equation (B.21) is possible using ω —the angular velocity of the rigid body

Angular momentum about O is a given by

Now, using the cross-product relation

one obtains

d dt

M

i i

,

)

d dt

= H +v ×P in general

d dt

where the inertia matrix is

(B.24)

Note: In Cartesian coordinates,

Angular momentum about the centroid:

FIGURE B.8 Motion of a rigid body with respect to a fixed frame.

2

2

11

m d dt

(B.23b)

Equations (B.23a) and (B.23b) can be written

(B.23)

B.2.5 M ANIPULATION OF I NERTIA M ATRIX (S EE FIGURE B.9 )

Parallel Axis Theorem — Translational Transformation of [I]

If the axes of the two frames are parallel,

(B.25)

Rotational Transformation of [I]

If a nonsingular square matrix [C] satisfies

Principal Directions (Eigenvalue Problem)

Principal directions ≡ Directions in which angular momentum is parallel to the angular velocity

i i i

H B I B

B B

Substitute equation (B.23):

If the corresponding direction vector is u (i.e., ω = ωu), then

(B.29)

Equation (B.29) represents an eigenvalue problem The nontrivial solutions for u are eigenvectors

and represent principal directions Since [I] is symmetric, three independent, real solutions

(u1, u2, u3) exist for u The corresponding values of λ are I1, I2, and I3 These are termed principal

moments of inertia The matrix of normalized eigenvectors

(B.30)

is an orthogonal matrix The orthogonal transformation of coordinates

(B.31)rotates the frame into the principal directions and hence diagonalizes the inertia matrix

T T T

r p=[ ]C r

C I C

I I I

00

Mohr’s Circle

In the two-dimensional (2-D) case of rigid bodies, the principal directions and principal moments

of inertia can be determined conveniently using Mohr’s circle In Figure B.9(b), the direct inertias

I xx and I yy are read on the horizontal axis, and the cross inertia I xy on the vertical axis If the inertia

matrix is given in the (x, y, z) frame, two diametrically opposite points of the circle are known These determine the Mohr’s circle The inertia matrix in any other frame (x′, y′, z′), rotated byangle θ about the common z-axis in the positive direction, is obtained by moving through 2θ counter

closewise on the circle This procedure also determines the principal moments I1 and I2, and theprincipal direction

B.2.6 E ULER ’ S E QUATIONS ( FOR A R IGID B ODY R OTATING ATω)

First consider a general body (rigid or not) for which angular momentum about a point B is expressed

in terms of directions of a frame that rotates at ω Then, from equation (B.15), one obtains

Consider two Cartesian frames F and F′ in different orientations One can rotate F to coincide with

F′ in three steps, as follows:

I I I

1 2 3

1 1

2 2

3 3

0 0

ωωω

ωωω

if

fixed, or moving parallel to (special case )

• For principal directions only

• fixed or moving parallel to centroid

Step 1: Rotate by angle ψ about the z-axis The orthogonal transformation matrix for this

Step 3: Rotate by angle φ about the new x-axis The orthogonal transformation matrix for

this rotation is:

about the z-axis of the original frame

about the y-axis of the intermediate frame

about the x-axis of the final frame.

It follows that the angular velocity expressed in F′ is

00

cos sinsin cos

00

000

0

00

Also, angular velocity expressed in F is

Note: The set of Euler angles described here is known as the (3, 2, 1) set or Type I Euler angles.

Other combinations are possible For example, if the first rotation is about the x-axis, the second rotation about the new y-axis, and the final rotation about the new x-axis, then one has the (1, 2, 1)

set

B.3 LAGRANGIAN MECHANICS

B.3.1 K INETIC E NERGY AND K INETIC C OENERGY

Consider a single particle

00

00

˙

˙

˙ψ

˙ ˙ sin

˙ cos ˙ sin cos

˙ cos cos ˙ sin

˙ cos cos ˙ sin

˙ cos sin ˙ cos

Note: For this reason, it is not necessary to distinguish between T and T* But in Lagrangian

mechanics, traditionally T* is retained

For a system of particles,

B.3.2 W ORK AND P OTENTIAL E NERGY

When the points of application r i of a set of forces f i move by increments δr i, the incremental workdone is given by

T=T*=1m ⋅ = mv

2

12

2

12

*= 1 + ⋅( × )= [ ] + ( × )⋅2

12

12

12

v v

From eq (B.23) Using vector identity

*=1 + [ ]2

12

2 ωω ωω(For a rigid body with centroid )

B.3.3 H OLONOMIC S YSTEMS , G ENERALIZED C OORDINATES , AND D EGREES OF F REEDOM

Holonomic constraints can be represented entirely by algebraic relations of the motion variables

Dynamic systems having holonomic constraints only are termed holonomic systems For any system (holonomic or non-holonomic), the number of degrees of freedom (n) equals the minimum number

of incremental (variational) generalized coordinates (δq1, δq2, …, δq n) required to completely describeany general small motion (without violating the constraints) The number of (nonincremental)

generalized coordinates required to describe large motions may be greater than n in general But, for

(B.53)

in which Q j are the nonconservative generalized forces corresponding to the generalized coordinates q j

For a motion trajectory, t0 and t f are the initial and the final times, respectively Hamilton’s principle

states that this trajectory corresponds to a natural motion of the system if and only if δH = 0 for

arbitrary δq j about the trajectory

B.3.5 L AGRANGE ’ S E QUATIONS

Note that L is a function of q j and in general, because V is a function of q j and T* is a function

of q j and Hence,

(B.54)Then, it follows from Hamilton’s principle [equation (B.53) with δH = 0], that:

(B.55)

These are termed Lagrange’s equations, and represent a complete set of equations of motion

Note: Newton’s equations of motion are equivalent to the Lagrange’s equations.

To determine Q j, give an incremental motion δq j to the system with the other coordinates fixed,and determine the work done δW j Then,

(B.56)

This gives Q j

E XAMPLE

Figure B.10 shows a simplified model that can be used to study the mechanical vibrations that are

excited by the control-loop disturbances in a single-link robot arm The length of the arm is l, the mass is M, and the moment of inertia about the joint is I The gripper hand (end effector) is modeled

as a mass m connected to the arm through a spring of stiffness k The joint has an effective viscous damping constant c for rotary motions The motor torque (applied at the joint) is τ(t).

Generalized Coordinates

This is a two-degree-of-freedom holonomic system The angle of rotation θ of the arm and spring

deflection x from the unstretched position are chosen as the generalized coordinates.

Generalized Nonconservative Forces

Keeping x fixed, increment θ by δθ The corresponding incremental work due to nonconservative

L q

Fθ=τ( )t −cθ˙

The total kinetic coenergy (= kinetic energy in these Newtonian systems) is

Note: is not exact (there is a nonlinear term that is neglected)

The potential energy (due to gravity and spring) is given by:

Note that the centroid of the arm is assumed to be halfway along the link It follows that theLagrangian is given by:

If the steady-state configuration of the link is assumed to be vertical, for small departures fromthis position (e.g., due to control loop disturbances), the equations of motion are obtained bylinearizing for small θ (i.e., sinθ = θ, cosθ = 1) The corresponding equations are

Note: For equilibrium at θ = 0, one needs τ = mgx0 In other words, this term represents the statictorque needed at the motor joint in order to maintain equilibrium at the (θ = 0, x = 0) position If

the system was symmetric, τstatic = 0

Note also that this set of equations of motion is of the form

and the matrices M, C, and K are symmetric The mass matrix M is not diagonal in the present

formulation It is possible, however, to make it diagonal simply by eliminating the term in thefirst equation of motion and the term in the second equation of motion through straightforwardalgebraic manipulation Natural frequencies and mode shapes of the undamped system can bedetermined in the usual manner (see Chapter 5) If the feedback gain of the control loop is suchthat at least a pair of eigenvalues is complex and has an imaginary part that is approximately equal

to a natural frequency of the structural system, then that mode can be excited in an undesirablemanner even by a slight disturbance in the control force Such situations can be avoided bymodifying the control system (e.g., by changing the gains) or the structural system (for example,

by adding damping, and by changing stiffness and mass) so that the structural natural frequencieswould not be near the resonances of the feedback control system Related issues of design andcontrol for vibration are addressed in Chapter 12

c x