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32.10 Control of Beam Vibrations

Back

Beam is a distributed-parameter system, which in theory has an infinite number of modes of vibration

with associated mode shapes and natural frequencies. In this sense, it is an “infinite order” system with

infinite DoF. Hence, the computation of modal quantities and associated control inputs can be quite

complex. Fortunately, however, just a few modes may be retained in a dynamic model without sacrificing

a great deal of accuracy, thereby facilitating simpler control. Some concepts of controlling vibrations in a

beam are considered in this section. The present treatment is intended as an illustration of the relevant

techniques and is not meant to be exhaustive. These techniques may be extended to other types of

continuous system such as beams with different boundary conditions and plates. Because the control

techniques that were outlined previously depend on a model, we will first illustrate the procedure of

obtaining a state-space model for a beam.

32.10.1 State-Space Model of Beam Dynamics

Consider a Bernoulli – Euler-type beam with Kelvin – Voigt-type internal (material) damping. The beam

equation may be expressed as

ELvðx; tÞ þ Ep L

›vðx; tÞ

›t þ rAðxÞ

›2vðx; tÞ

›t2 ¼ f ðx; tÞ ð32:146Þ

in which L is the partial differential operator given by

L ¼

›2IðxÞ

›x2

›2

›x2 ð32:147Þ

and

f ðx; tÞ ¼ distributed force excitation per unit length of the beam

vðx; tÞ ¼ displacement response at location x along the beam at time t

IðxÞ ¼ second moment of area of the beam cross section about the neutral axis

E ¼ Young’s modulus of the beam material

E p ¼ Kelvin – Voigt material damping parameter

Note that a general beam with nonuniform characteristics is assumed and, hence, the variations of IðxÞ

and rAðxÞ with x are retained in the formulation.

Vibration Design and Control 32-67

Using the approach of modal expansion, the response of the beam may be expressed by

vðx; tÞ ¼

i¼1

YiðxÞqiðtÞ ð32:148Þ

where YiðxÞ is the ith mode shape of the beam, which satisfies

LYiðxÞ ¼

rAðxÞ

v2i

YiðxÞ ð32:149Þ

and vi is the ith undamped natural frequency. The orthogonality condition for this general example of a

nonuniform beam is

ðl

x¼0

rAYiYjdx ¼

0 for i – j

aj for i – j

(

ð32:150Þ

Suppose that the forcing excitation on the beam is a set of r point forces ukðtÞ located at x ¼ lk;

k ¼ 1; 2; …; r: Then, we have

f ðx; tÞ ¼

k¼1

ukdðx 2 lkÞ ð32:151Þ

where dðx 2 liÞ is the Dirac delta function. Now, substitute Equation 32.148 and Equation 32.151 into

Equation 32.146, use Equation 32.149, multiply throughout by YjðxÞ; and integrate over x½0; l􀀉; using

Equation 32.150. This gives

q€j þgjq_jðtÞ þv2j

qj ¼

k¼1

ukYjðlkÞ for j ¼ 1; 2; … ð32:152Þ

where

gj ¼

v2j

ð32:153Þ

Now, define the state variables xj according to

x2j21 ¼ vjqj; x2j ¼ q_j for j ¼ 1; 2;… ð32:154Þ

Assuming that only the first m modes are retained in the expansion, we then have the state equations

x_2j21 ¼ vjx2j; x_2j ¼ 2vjx2j21 2gjx2j þ

k¼1

ukYjðlk Þ for j ¼ 1; 2; …; m ð32:155Þ

This can be put in the matrix – vector form of a state-space model

x_ ¼ Ax þ Bu ð32:131Þ

where

A ¼

0 v1

2v1 2g1

. .

0 vm

2vm 2gm

66666666664

77777777775

n£n

ð32:156Þ

32-68 Vibration and Shock Handbook

and

B ¼

0 0

Y1ðl1Þ=a1 · · · Y1ðlrÞ=a1

.. .

0 0

Ymðl1Þ=am · · · YmðlrÞ=am

66666666664

77777777775

n£r

ð32:157Þ

with n ¼ 2m; where m is the number of modes retained in the modal expansion. Note that, as the

number of modes used in this model increases, both the accuracy and the computational effort that is

needed for the control problem increase because of the proportional increase of the system order. At

some point, the potential improvement in accuracy by further increasing the model size would be

insignificant in comparison with added computational burden. Hence, a balance must be struck in this

tradeoff.

32.10.2 Control Problem

The state-space model (Equation 32.131) for the beam dynamics, with matrices (Equation 32.156 and

Equation 32.157), is known to be controllable. Hence, it is possible to determine a constant-gain feedback

controller u ¼ Kx that minimizes a quadratic-integral cost function of the form in Equation 32.141.

Also, a similar controller can be determined that places the eigenvalues of the system at specified locations

thereby achieving not only specified levels of modal damping but also a specified set of natural

frequencies. However, there is a practical obstacle to achieving such an active controller. Note that, in the

model given in Equation 32.156 and Equation 32.157, the state variables are proportional to the modal

variables qi and their time derivatives q_i: They are not directly measurable. However, the displacements

and velocities at a set of discrete locations along the beam can usually be measured. Let these locations (s)

be denoted by p1; p2; …; ps: Thus, in view of the modal expansion (Equation 32.148), the measurements

can be expressed as

vðpj; tÞ ¼

i¼1

YiðpjÞqiðtÞ; v_ðpj; tÞ ¼

i¼1

YiðpjÞq_iðtÞ for j ¼ 1; 2;…; s ð32:158Þ

Now, define the output (measurement) vector y according to

y ¼ ½vðp1; tÞ; v_ðp1; tÞ;…; vðps; tÞ; v_ðps; tÞ􀀉T ð32:159Þ

In view of Equation 32.158 and the definitions of the state variable in Equation 32.154, we can write

y ¼ Cx ð32:160Þ

with

C ¼

Y1ðp1Þ=v1 0 · · · Ymðp1Þ=vm 0

0 Y1ðp1Þ · · · 0 Ymðp1Þ

.. .

· · · .. .

.. .

Y1ðpsÞ=v1 0 · · · YmðpsÞ=vm 0

0 Y1ðpsÞ · · · 0 YmðpsÞ

66666666664

77777777775

2s£n

ð32:161Þ

Hence, an active controller is possible of the form:

u ¼ Hy ð32:162Þ

Vibration Design and Control 32-69

which is an output feedback controller. Therefore, in view of Equation 32.160, we have

u ¼ HCx ð32:163Þ

This is not the same as complete state feedback u ¼ Kx where K can take any real value (and, hence, the

LQR solution in Equation 32.142 and the complete pole placement solution cannot be applied directly).

In Equation 32.163, only H can be arbitrarily chosen, and C is completely determined according to

Equation 32.161. The resulting product HC will not usually correspond to either the LQR solution or the

complete pole assignment solution. Still, the output feedback controller in Equation 32.162 can provide a

satisfactory performance. However, a sufficient number of displacement and velocity sensors (s) have to

be used in conjunction with a sufficient number of actuators (r) for active control. This will increase the

system complexity and cost. Furthermore, due to added components and their active nature, the

reliability of fault-free operation may degrade somewhat. A satisfactory alternative would be to use

passive control devices such as dampers and dynamic absorbers, which is illustrated below. Note that in

the matrices B and C given by Equation 32.157 and Equation 32.161, both the actuator locations li and

the sensor locations pj are variable. Hence, there exists an additional design freedom (or optimization

parameters) in selecting the sensor and actuator locations in achieving satisfactory control.

32.10.3 Use of Linear Dampers

Now, consider the use of a discrete set of linear

dampers for controlling beam vibration. Suppose

that r linear dampers with damping constants bj

are placed at locations lj; j ¼ 1; 2; …; r along the

beam, as schematically shown in Figure 32.34. The

damping forces are given by

uj ¼ 2bjv_ðlj; tÞ for j ¼ 1; 2;…; r ð32:164Þ

Substituting the truncated modal expansion

(m modes)

v_ðlj; tÞ ¼

i¼1

YiðljÞq_iðtÞ ð32:165Þ

we get, in view of Equation 32.154, the passive feedback control action

u ¼ 2Kx ð32:166Þ

with

K ¼

0 b1Y1ðl1Þ · · · 0 b1Ymðl1Þ

.. .

· · · .. .

.. .

0 brY1ðlrÞ · · · 0 brYmðlrÞ

6664

7775

r£n

ð32:167Þ

By substituting Equation 32.166 into Equation 32.131, we have the closed-loop system equation

x_ ¼ ðA 2 FÞx ¼ Acx ð32:168Þ

Linear

Damper

y, v(x,t)

Beam

FIGURE 32.34 Use of linear dampers in beam

vibration control.

32-70 Vibration and Shock Handbook

where F ¼ BK and is given by

F ¼

0 0 · · · 0 0

biY11ðliÞ=a1 · · · 0

biY1mðliÞ=a1

.. .

· · · .. .

.. .

0 0 · · · 0 0

biYm1ðliÞ=am · · · 0

biYmmðliÞ=am

666666666664

777777777775

n£n

ð32:169Þ

with

YijðxÞ ¼ YiðxÞYjðxÞ ð32:170Þ

In this case, the controller design involves the selection of the damping constants bi and the damper

locations lj to achieve the required performance. This may be achieved, for example, by seeking to make

the eigenvalues of the closed-loop system matrix Ac reach a set of desired values. This achieves the desired

modal damping and natural frequency characteristics. However, given that the structure of the F matrix is

fixed, as seen in Equation 32.169, this is not equivalent to complete state feedback (or complete output

feedback). Hence, generally, it is not possible to place the poles of the system at the exact desired

locations.

32.10.3.1 Design Example

In realizing a desirable modal response of a beam using a set of linear dampers, one may seek to minimize

a cost function of the form

J ¼ Reðl 2 ldÞTQ Reðl 2 ldÞ þ Imðl 2 ldÞTRðl 2 ldÞ ð32:171Þ

where l are the actual eigenvalues of the closed-loop system matrix (Ac), and ld are the desired

eigenvalues that will give the required modal performance (damping ratios and natural frequencies).

“Re” denotes the real part and “Im” denotes the imaginary part. Weighting matrices Q and R, which are

real and diagonal with positive diagonal elements, should be chosen to relatively weight various

eigenvalues. This allows the emphasis of some eigenvalues over others, with real parts and the imaginary

parts weighting separately.

Various computational algorithms are available for minimizing the cost function (Equation 32.171).

Although the precise details are beyond the scope of this book, we will present an example result.

Consider a uniform simply supported 12 £ 5 American Standard beam, with the following pertinent

specifications: E ¼ 2 £ 108 kPa (29 £ 106 psi), rA ¼ 47 kg/m (2.6 lb/in.), length l ¼ 15.2 m (600 in.),

I ¼ 9 £ 1025 m4 (215.8 in.4). The internal damping parameter for the jth mode of vibration is given by

EpðvjÞ ¼ ðg2=vjÞ þ g2 ð32:172Þ

in which vj is the jth undamped natural frequency given by

vj ¼ ðjp=lÞ2 ffiffiffiffiffiffiffiffi

EI=rA

ð32:173Þ

The numerical values used for the damping parameters are g1 ¼ 88 £ 104 kPa (12.5 £ 104 psi) and

g2 ¼ 3.4 £ 104 kPa s (5 £ 103 psi s). For the present problem, YiðxÞ ¼

ffiffi

2 p sinðjpx=lÞ and aj ¼ rAl for all j:

First, vj and gj are computed using Equation 32.173 and Equation 32.153, respectively, along with

Equation 32.172. Next, the open-loop system matrix A is formed according to Equation 32.156 and its

eigenvalues are computed. These are listed in Table 32.2, scaled to the first undamped natural frequency

(v1). Note that in view of the very low levels of internal material damping of the beam, the actual natural

frequencies, as given by the imaginary parts of the eigenvalues, are almost identical to the undamped

natural frequencies.

Next, we attempt to place the real parts of the (scaled) eigenvalues all at 2 0.20 while exercising no

constraint on the imaginary parts (i.e., damped natural frequencies) by using: (a) single damper, and

Vibration Design and Control 32-71

(b) two dampers. In the cost function (Equation 32.171), the first three modes are more heavily weighted

than the remaining three. Initial values of the damper parameters are b1 ¼ b2 ¼ 0.1 lbf sec/in.

(17.6 N sec/m) and the initial locations l1=l ¼ 0:0 and l2=l ¼ 0:5: At the end of the numerical

optimization, using a modified gradient algorithm, the following optimized values were obtained:

1. Single-damper control

b1 ¼ 36:4 lbf sec=in: ð6:4 £ 103 N sec=mÞ

l1=l ¼ 0:3

The corresponding normalized eigenvalues (of the closed-loop system) are given in Table 32.3.

2. Two-damper control

b1 ¼ 22:8 lbf sec=in: ð4:0 £ 103 N sec=mÞ

b2 ¼ 12:1 lbf sec=in: ð2:1 £ 103 N sec=mÞ

l1=l ¼ 0:25; l2=l ¼ 0:43

The corresponding normalized eigenvalues are given in Table 32.4.

It would be overly optimistic to expect perfect assignment all real parts at 2 0.2. However, note that

good levels of damping have been achieved for all modes except for Mode 3 in the single-damper control

and Mode 4 in the two-damper control. In any event, because the contribution of the higher modes

towards the overall response, is relatively smaller, it is found that the total response (e.g., at point

x ¼ l=12) is well damped in both cases of control.

TABLE 32.2 Eigenvalues of the Open-Loop (Uncontrolled) Beam

Mode Eigenvalue (rad/sec) (Multiply by 26.27)

1 2 0.000126 ^ j1.0

2 2 0.000776 ^ j4.0

3 2 0.002765 ^ j9.0

4 2 0.007453 ^ j16.0

5 2 0.016741 ^ j25.0

6 2 0.033.75 ^ j36.0

TABLE 32.4 Eigenvalues of the Beam with Optimized Two Dampers

Mode Eigenvalue (rad/sec) (Multiply by 26.27)

1 2 0.216 ^ j0.982

2 2 0.233 ^ j3.974

3 2 0.174 ^ j8.997

4 2 0.079 ^ j15.998

5 2 0.145 ^ j24.999

6 2 0.354 ^ j35.989

TABLE 32.3 Eigenvalues of the Beam with an Optimal Single Damper

Mode Eigenvalue (rad/sec) (Multiply by 26.27)

1 2 0.225 ^ j0.985

2 2 0.307 ^ j3.955

3 2 0.037 ^ j8.996

4 2 0.119 ^ j15.995

5 2 0.355 ^ j24.980

6 2 0.158 ^ j35.990

32-72 Vibration and Shock Handbook

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