Chapter 6 MACHINE DYNAMICS

The primary reasons for vibration profile variations are the dynamics of the machine, which are affected by mass, stiffness, damping, and degrees of freedom. However, care must be taken because the vibration profile and energy levels gener- ated by a machine also may vary depending on the location and orientation of the measurement.

MASS, STIFFNESS, AND DAMPING

The three primary factors that determine the normal vibration energy levels and the resulting vibration profiles are mass, stiffness, and damping. Every machine-train is designed with a dynamic support system that is based on the following: the mass of the dynamic component(s), a specific support system stiffness, and a specific amount of damping.

~ u

Mass is the property that describes how much material is present. Dynamically, it is the property that describes how an unrestricted body resists the application of an external force. Simply stated, the greater the mass the greater the force required to accelerate it. Mass is obtained by dividing the weight of a body (e.g., rotor assembly) by the local acceleration of gravity, g.

The English system of units is complicated compared to the metric system. In the English system, the units of mass are pounds-mass (Ibm) and the units of weight are pounds-force (lbf). By definition, a weight (i.e., force) of I lbf equals the force pro- duced by 1 Ibm under the acceleration of gravity. Therefore, the constant, go which has the same numerical value as g (32.17) and units of lbm-ft/lbf-sec 2, is used in the definition of weight:

7.6

Dymmles 27

Therefore,

Therefore,

b ' t # t n o u

W e i g h t = M a s s * g gc

Weigh t* g c M a s s = -

g

We igh t*gc l b f Ibm* f l M a s s = ....... = x . . . . Ibm

g ~ l b f , sec 2 - 2

s e c

Stiffness is a spring-like property that describes the level of resisting force that results when a body undergoes a change in length. Units of stiffness are often given as pounds per inch (lbffin.). Machine-trains have more than one stiffness property that must be considered in vibration analysis: shaft stiffness, vertical stiffness, and hori- zontal stiffness.

Shaft Stiffness

Most machine-trains used in industry have flexible shafts and relatively long spans between bearing-support points. As a result, these shafts tend to flex in normal opera- tion. Three factors determine the amount of flex and mode shape that these shafts have in normal operation: shaft diameter, shaft material properties, and span length. A small-diameter shaft with a long span will obviously flex more than one with a larger diameter or shorter span.

Vertical Stfl~ess

The rotor-bearing support structure of a machine typically has more stiffness in the vertical plane than in the horizontal plane. Generally, the structural rigidity of a bear- ing-support structure is much greater in the vertical plane. The full weight of and the dynamic forces generated by the rotating element are fully supported by a pedestal cross-section that provides maximum stiffness.

In typical rotating machinery, the vibration profile generated by a normal machine contains lower amplitudes in the vertical plane. In most cases, this lower profile can be directly attributed to the difference in stiffness of the vertical plane when compared to the horizontal plane.

Horizontal Stfl~ess

Most bearing pedestals have more freedom in the horizontal direction than in the ver- tical. In most applications, the vertical height of the pedestal is much greater than the horizontal cross-section. As a result, the entire pedestal can flex in the horizontal plane as the machine rotates.

28 Vibration lgundammtals

/ / / / / / / _ . / / _ /

- t Sprinq Static Des (X,j,) , , ,

Mass

Figure & l Undamped spring.mass system.

~ S

Mass

Weight (w)

This lower stiffness generally results in higher vibration levels in the horizontal plane. This is especially true when the machine is subjected to abnormal modes of operation or when the machine is unbalanced or misaligned.

Damping

Damping is a means of reducing velocity through resistance to motion, in particular by forcing an object through a liquid or gas, or along another body. Units of damping are often given as pounds per inch per second (lbf/in./sec, which is also expressed as lbf-sec/in.).

The boundary conditions established by the machine design determine the freedom of movement permitted within the machine-train. A basic understanding of this concept is essential for vibration analysis. Free vibration refers to the vibration of a damped (as well as undamped) system of masses with motion entirely influenced by their potential energy. Forced vibration occurs when motion is sustained or driven by an applied periodic force in either damped or undamped systems. The following sections discuss free and forced vibration for both damped and undamped systems.

Free Vibration---Undraped

To understand the interactions of mass and stiffness, consider the case of undamped free vibration of a single mass that only moves vertically, as illustrated in Figure 6.1. In this figure, the mass M is supported by a spring that has a stiffness K (also referred to as the spring constant), which is defined as the number of pounds of tension neces- sary to extend the spring 1 in.

M ~ Dynamics 29

The force created by the static deflection, X i, of the spring supports the weight, W, of the mass. Also included in Figure 6.1 is the free-body diagram that illustrates the two forces acting on the mass. These forces are the weight (also referred to as the inertia force) and an equal, yet opposite force that results from the spring (referred to as the spring force, Fs).

The relationship between the weight of mass M and the static deflection of the spring can be calculated using the following equation:

WffigX,

If the spring is displaced downward some distance, X 0, from X i and released, it will oscillate up and down. The force from the spring, F s, can be wTitten as follows, where a is the acceleration of the mass:

MQ F s = - K X = - - -

gc

It is common practice to replace acceleration a with d2X--- the second derivative of the dt 2 '

displacement, X, of the mass with respect to time, t. Making this substitution, the equation that defines the motion of the mass can be expressed as:

M d 2 X M d 2 X - - = - K X o r - - - - - - + K X = 0

gc d t 2 gc d t 2

Motion of the mass is known to be periodic in time. Therefore, the displacement can be descfibeA by the expression:

x ffi Xocos(,ot)

where

X = Displacement at time t X0 = Initial displacement of the mass r = Frequency of the oscillation (natural or resonant frequency) t = Tune.

If this equation is differentiated and the result inserted into the equation that defines motion, the natural frequency of the mass can be calculated. The first derivative of the equation for motion given previously yields the equation for velocity. The second derivative of the equation yields acceleration.

V e l o c i t y - d__X _ y~ _ _coXosin(tot ) dt

30 Vibration ~md~m~bDIs

A c c e l e r a t i o n = d2X--- = ~ =-r d t 2

Inserting the above expression for acceleration, or d2X--- into the equation for F s yields the following: dt 2 '

M d2X--- + K X = 0

gc d t 2

M 2Xoco s - - - o~ (cot) + K X = 0 gc

M M 2 - : eo2X + K X = - - -r + IC = 0 gc gc

Solving this expression for co yields the equation:

where

co = Natural frequency of mass K = Spring constant M - Mass.

Note that, theoretically, undamped free vibration persists forever. However, this never occurs in nature and all free vibrations die down after time due to damping, which is discussed in the next section.

Free Vibration--Damped

A slight increase in system complexity results when a damping element is added to the spring-mass system shown in Figure 6.2. This type of damping is referred to as viscous damping. Dynamically, this system is the same as the undamped system illus- trated in Figure 6.1, except for the damper, which usually is an oil or air dashpot mechanism. A damper is used to continuously decrease the velocity and the resulting energy of a mass undergoing oscillatory motion.

The system is still comprised of the inertia force due to the mass and the spring force, but a new force is introduced. This force is referw~ to as the damping force and is proportional to the damping constant, or the coefficient of viscous damping, c. The damping force is also proportional to the velocity of the body and, as it is applied, it opposes the motion at each instant.

equilibrium position

-

(a)

Lo+h

(b)

dX

Macl~e Dynamic, 31

x'-O

A (c) mg

Figure 6.2 Dampd spring.mass sysWm.

In Figure 6.2, the unelongated length of the spring is L o and the elongation due to the weight of the mass is expressed by h. Therefore, the weight of the mass is Kh. Figure 6.2(a) shows the mass in its position of stable equilibrium. Figure 6.2(b) shows the mass displaced downward a distance X from the equilibrium position. Note that X is considered positive in the downward direction.

Figure 6.2(c) is a free-body diagram of the mass, which has three forces acting on it. The weight (Mg/gc), which is directed downward, is always positive. The damping

(cdX force ~ ~-~-/, which is the damping constant times velocity, acts opposite to the direc-

tion of the velocity. The spring force, K(X + h), acts in the direction opposite to the

displacement. Using Newton's equation of motion, where ' ~ F = Ma, the sum of ,L,d

32 Vibration PmKlummtals

the forces acting on the mass can be represented by the following equation, remem- bering that X is positive in the downward direction:

Dividing by M. ge

M d2X._._ = ~ d X _ K(X + h) g'-c dt 2 gc - c - ~

- . _ dX M d2X = Kh - c - ~ - KX - gh gc dt 2

M d2X dX

E

d2X Cgc dX KgcX

dt 2 M dt M

To look up the solution to the preceding equation in a differential equations table (such as in the CRC Handbook of Chemistry and Physics) it is necessary to change the form of this equation. This can be accomplished by defining the relationships, cg c IM = 21~ and Kg c IM = to 2, which converts the equation to the following form:

d2X___ = dX 2 X dt 2 -2 p~-~- -to

Note that for undamped free vibration, the damping constant, c, is zero and, therefore, is also zero.

d2X = _co 2X dt 2

d2 X---. + o~2 X = 0 dt 2

The solution of this equation describes simple harmonic motion, which is given below:

X = Acos(cot) + Bsin(mt)

dX Substituting at t = 0, then X = X 0 ands- = 0, then

X = X0cos (mt)

This shows that free vibration is periodic and is the solution for X. For damped free vibration, however, the damping constant, c, is not zero.

d2X - 2 dX 2 X d t 2 ~ - ~ -co

~ ~ 33

o r

o r

dX d2 X + 2 t t -~ + eo2 X = 0 dt 2

D 2 + 2 t t D + t o 2 = 0

which has a solution of:

X = Aedlt+Be eft

where

•/tt 2 dl ffi _ tt + 2 _ r

d 2 - - i t - ~/tt 2 - 03 2

There are different conditions of damping: critical, overdamping, and underdamping. Critical damping occurs when rt = co. Overdamping occurs when I~ > o). Under- damping occurs when rt < co.

The only condition that results in oscillatory motion and, therefore, represents a mechanical vibration is underdamping. The other two conditions result in aperiodic motions. When damping is less than critical (~t < (o), then the following equation applies:

X = X----Oe-~tt(CLlCOSCLlt + ~ s i n e t l t ) ot 1

where

~t I = ~/r 2 - It 2

Forced Vibration--Undamped

The simple systems described in the preceding two sections on free vibration are alike in that they are not forced to vibrate by any exciting force or motion. Their major con- tribution to the discussion of vibration fundamentals is that they illustrate how a sys- tem's natural or resonant frequency depends on the mass, stiffness, and damping characteristics.

The mass-stiffness-damping system also can be disturbed by a periodic variation of external forces applied to the mass at any frequency. The system shown in Figure 6.1 is increased in complexity by the addition of an external force, F 0, acting downward on the mass.

34 Vibration Fundamentals

In undamped forced vibration, the only difference in the equation for undamped free vibration is that instead of the equation being equal to zero, it is equal to F 0 sin(tot):

M d2X -- - - - + KX ffi Fosin(cot ) gc dt 2

Since the spring is not initially displaced and is "driven" by the function F 0 sin(o~t), a particular solution, X = X o sin(co0, is logical. Substituting this solution into the above equation and performing mathematical manipulations yields the following equation for X:

Xst X = C! sin(cont)+ C2cos(cont)+ ~ ,,,(~/co ; 2 s i n ( c o t ) i c o n )

where

X = Spring displacement at time, t X~t = Static spring deflection under constant load, F0 co = Forced frequency co n = Natural frequency of the oscillation t = Tune

CI ,C 2 = Integration constants determined from specific boundary conditions.

In the above equation, the first two terms are the undamped free vibration, and the third term is the undamped forced vibration. The solution, containing the sum of two sine waves of different frequencies, is itself not a harmonic motion.

Forced Vibration--Damped

In a damped forced vibration system such as the one shown in Figure 6.3, the motion of the mass M has two parts: (1) the damped free vibration at the damped natural fre- quency and (2) the steady-state harmonic motions at the forcing frequency. The damped natural frequency component decays quickly, but the steady-state harmonic associated with the external force remains as long as the energy force is present.

With damped forced vibration, the only difference in its equation and the equation for damped free vibration is that it is equal to F 0 sin(cot) as shown below instead of being equal to zero.

M d2X + dX g'-c dt 2 c - ~ + KX ffi F0sin(cot)

With damped vibration, damping constant c is not equal to zero and the solution of the equation gets quite complex assuming the function, X = X 0 sin(cot- ~). In this equa- tion, ~ is the phase angle, or the number of degrees that the external force, F0 sin(o~t), is ahead of the displacement, X o sin(cot- ~). Using vector concepts, the following

, / / / / / / / / , /

l, l , , , f

t+ Mass

Figure 6.3 Damped forced vibration system.

Mass

equations apply, which can be solved because there are two equations and two unknowns:

Vertical vector component: KXo-Mm2x Focos ~ = 0 gc o -

Horizontal vector component: cruX o - F o sin~ = 0

Solving these two equations for the unknowns X o and ~:

F o +o=j, c

F o

K

l( ~~ (2~c ~-~2 1 - + •

where

2 c m X - - CO) Cc (On

taln l~ --'-- ...... ----" -

M 2 '2' 2 .... K - - - m 1 - (m /COn)

gc

c = Damping constant

c c = Critical damping = 2Mton gc

36 Vibration gundamentab

c/c~ = Damping ratio F o = External force

Folk = Deflection of the spring under load, F o (also called static deflection, X~) o = Forced frequency to n = Natural frequency of the oscillation

o~F0 . = Frequency ratio.

For damped forced vibrations, three different frequencies have to be distinguished:

the undamped natural frequency, co. = K~~c/M; the damped natural frequency,

q = ~,2MI ; and the frequency of maximum forced amplitude, sometimes

referred to as the resonant frequency.

DEGREES OF FREEDOM

In a mechanical system, the degrees of freedom indicate how many numbers are required to express its geometrical position at any instant. In machine-trains, the rela- tionship of mass, stiffness, and damping is not the same in all directions. As a result, the rotating or dynamic elements within the machine move more in one direction than in another. A clear understanding of the degrees of freedom is important in that it has a direct impact on the vibration amplitudes generated by a machine or process system.

One Degree of Freedom

If the geometrical position of a mechanical system can be defined or expressed as a single value, the machine is said to have one degree of freedom. For example, the position of a piston moving in a cylinder can be specified at any point in time by mea- suring the distance from the cylinder end.

A single degree of freedom is not limited to simple mechanical systems such as the cylinder. For example, a 12-cylinder gasoline engine with a rigid crankshaft and a rig- idly mounted cylinder block has only one degree of freedom. The position of all of its moving parts (i.e., pistons, rods, valves, cam shafts, etc.) can be expressed by a single value. In this instance, the value would be the angle of the crankshaft.

However, when mounted on flexible springs, this engine has multiple degrees of free- dom. In addition to the movement of its internal parts in relationship to the crank, the entire engine can now move in any direction. As a result, the position of the engine and any of its internal parts require more than one value to plot its actual position in space.

The definitions and relationships of mass, stiffness, and damping in the preceding sec- tion assumed a single degree of freeAom. In other words, movement was limited to a

M ~ h ~ D y m m ~ 37

Figure 6.4 TorsioMI one-degree-of-frsedom system.

- j

single plane. Therefore, the formulas are applicable for all single-degree-of-freedom mechanical systems.

The calculation for torque is a primary example of a single degree of freedom in a mechanical system. Figure 6.4 represents a disk with a moment of inertia,/, that is attached to a shaft of torsional stiffness, k.

Torsional stiffness is defined as the externally applied torque, T, in inch-pounds needed to turn the disk one radian (57.3 degrees). Torque can be represented by the following equations:

N~. Torque = Moment of inertia • angular acceleration = I d-~- = I~ dt 2 g l d

38 Vibration Fundamentals

In this example, three torques are acting on the disk: the spring torque, damping torque (due to the viscosity of the air), and external torque. The spring torque is minus (-)k~, where ~ is measured in radians. The damping torque is minus (-)c~, where c is the damping constant. In this example, c is the damping torque on the disk caused by an angular speed of rotation of one radian per second. The external torque is To sin(tot).

I~ = ~ Torque = - c+ - kdp + T0sin(o)t)

or

I~ + c~ + k~ --" Tosin((ot)

Two Degrees of Freedom

The theory for a one-degree-of-freedom system is useful for determining resonant or natural frequencies that occur in all machine-trains and process systems. However, few machines have only one degree of freedom. Practically, most machines will have two or more degrees of freedom. This section provides a brief overview of the theo- ries associated with two degrees of freedom. An undamped two-degree-of-freedom system is illustrated in Figure 6.5.

The diagram of Figure 6.5 consists of two masses, M l and M 2, which are suspended from springs, K 1 and K 2. The two masses are tied together, or coupled, by spring K 3, so that they are forced to act together. In this example, the movement of the two masses is limited to the vertical plane and, therefore, horizontal movement can be ignored. As in the single-degree-of-freedom examples, the absolute position of each mass is defined by its vertical position above or below the neutral, or reference, point. Since there are two coupled masses, two locations (i.e., one for M 1 and one for M 2) are required to locate the absolute position of the system.

To calculate the free or natural modes of vibration, note that two distinct forces are acting on mass, M~: the force of the main spring, K t, and that of the coupling spring, K 3. The main force acts upward and is defined as -KtXt. The shortening of the cou- pling spring is equal to the difference in the vertical position of the two masses, Xt - X2. Therefore, the compressive force of the coupling spring is K3(X~ - X2). The compressed coupling spring pushes the top mass, M~, upward so that the force is negative.

Because these are the only tangible forces acting on Mr, the equation of motion for the top mass can be written as:

gc

o r

Maehlae Dyamaks 39

///// / .... / , / , , , , / ]

!

////

):I

_/I/1

\\\\\ \

K 2

\ \ \

M 2

Figure 6.5 Undamped two.degree-of.freedom system with a spring couple.

M, ----2"~ I + (K I + K3)X I - K3X 2 = 0 gc

The equation of motion for the second mass, M 2, is derived in the same manner. To make it easier to understand, turn the figure upside down and reverse the direction of

Xl and X 2. The equation then becomes:

M_M__2 f~ 2 -. - K2X2 - K3 ( X I - X2 ) gc

o r

M~ " ~ 2 + (K2 + K3)X2- K3XI - 0 gc

40 Vibration Fundamentals

If we assume that the masses Mt and M 2 undergo harmonic motions with the same frequency, co, and with different amplitudes, A t and A 2, their behavior can be repre- sented as follows:

X 1 ffi A 1 s i n ( c o t )

X 2 ffi A2sin(cot )

By substituting these into the differential equations, two equations for the amplitude

A1 ratio, ~2 ' can be found:

A 1 -K 3

A2 M...lco2 _ K 1 - K 3 gc

M2co2 _ K2 _ K3 AI gc

A 2 -K 3

For a solution of the form we assumed to exist, these two equations must be equal:

-K 3 M

lco2_ K1 _ K3 gc

M,~._..,., co2 _ K2_ K3 gc

= ..... -g-3 -

or

4_co2fK1 ..... + K3 K2 + K31 KIK2 +MIM2'K2K3 + KIK3 co ~ M1/gc + M2/g ~_ + ............ __._-

2 ge

= 0

This equation, known as the frequency equation, has two solutions for r 2. When sub- stituted in either of the preceding equations, each one of these gives a definite value

A1 for ~2" This means that there are two solutions for this example, which are of the

form A 1 sin(cot) and A2sin(cot). As with many such problems, the final answer is

Machine I)~mmsk~ 41

the superposition of the two solutions with the final amplitudes and frequencies deter- mined by the boundary conditions.

Many Degmu of Freedom

When the number of degrees of freedom becomes greater than two, no critical new parameters enter into the problem. The dynamics of all machines can be understood by following the rules and guidelines established in the one- and two-degree-of-free- dom equations. There are as many natural frequencies and modes of motion as there are degrees of freedom.