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This paper focuses on theoretical investigation of active vibration control of a cantilever beam using heat actuation. The actuator is a thin metal bar rigidly bonded to the beam on one face and subject to heat input on the opposite face. The actuator then works like a piezoelectric actuator, and expands and contracts in response to applied heat. We assume that the actuator is insulated so that no heat is transferred to the beam, ensuring that the heat does not alter the beam’s thermal state. To avoid necessity of cooling, we consider two actuators working together at the same span-wise location, one on the upper and one on the lower face of the beam. Then, the beam can be bent up and down by applying heat to the lower and upper actuators, respectively. The governing equations are partial differential equations for one-dimensional heat conduction of the actuators and the bending vibration of the beam with attached actuators. For an approximate solution, Rayleigh-Ritz method replaces the partial differential equations with a system of ordinary differential equations. A control model is obtained from a low-dimensional representation of the system, and used to design feedback control and observer by means of LQR and Kalman-Bucy filtering techniques. The control signal obtained is introduced into the plant model, a high-dimensional representation of the system, to mimic the true system as closely as possible. In a numerical application, the response of the beam to an initial excitation is simulated, which demonstrates that the heat actuators are in fact effective in active vibration control of the beam.

Suppression of vibration in flexible structures remains to be a challenging problem in the related technical com- munity. The current trend is to use active means to suppress the vibration. The active control of vibration in structures is implemented by means of sensors, actuators, an accurate mathematical model of the whole system (flexible structure + sensors + actuators), and a computer to carry out all required computations in real time.

Sensors do all measurements, but usually the states of the system cannot be measured directly. Hence, the mathematical model of the system is used to estimate all of the states based on the available measurements, provided that the system is observable. The mathematical model is also used to compute the control signal to be inputed to the actuator based on these estimates of the states. The actuator produces what is needed to affect the true system to suppress the vibration. This can be force, moment, strain, etc. The actuators widely used in vibration control are hydraulic, pneumatic, and piezo-electric, with each having its own benefits and disadvantages depending on the specific applications in which they are used [

Another type of actuation that has started receiving some attention in vibration control is simple heat. Considering a structure initially at room temperature, an applied load causes temperature to vary over the body of the structure nonuniformly. Areas of the structure under tension are now cooler and areas under compression are hotter than the room temperature, creating a temperature gradient in the structure. This in turn causes a heat flow by conduction from hotter areas to cooler areas. Hence, a portion of the mechanical energy introduced to the structure is converted to heat through this process. Since the heat flow created is irreversible, the portion of the mechanical energy converted to heat is dissipated. This phenomenon is known as thermoelastic damping [

Pioneering research by Edberg [

References [

We will demonstrate our modeling and control approach on a cantilever beam, which is a fundamental element commonly encountered in structures. To make the presentation of our approach more clear and focus, we limit ourselves to a simple model by assuming the beam is uniform and there is only one pair of actuators, as shown in

where

where each subscript x indicates a partial derivative with respect to x. Then, the strain components of the beam can be expressed as

Since the ends of the actuators are free, the normal strains of the actuators will be constant at beam’s strains at its interface with the actuators [

Assuming a linear stress-strain relation, we can express the only nonzero stress component in beam as

where E is the Young’s modulus of the beam. On the other hand, the normal stresses in the actuators are

where

We approximate w,

where

are the matrices consisting of m and n number of shape functions and

are the vectors of generalized coordinates for elastic displacement, and temperatures of the upper and lower actuators. Note that we use the same matrix of shape functions for the upper and lower actuators because we assume the actuators are identical and placed symmetrically about the x axis. We choose the shape functions

where

Let us now express the total potential energy, which is the summation of potential energies of the beam and the lower and upper actuators:

where

the volumes of the actuators, and

Equation (7) into Equation (11), we rewrite the potential energy expression as

where

We note that K is the stiffness matrix.

Kinetic energy of the beam is

where the mass matrix is

where

Generalized force is obtained from the virtual work:

where

is the generalized force vector.

The equations of motion of the beam can be derived by means of Lagrange’s equations of motion, which can be written in the vector as

where

Equation (19) governs the elastic deflection of the beam. Since this equation is coupled with the temperature difference

The heat conduction in the upper actuator is described by the boundary-value problem defined by the following partial differential equation and boundary conditions:

where

Substituting the second of Equation (7), premultiplying both sides by

Similarly, substituting the second of Equations (7) into the boundary conditions, we get

Using the integration by parts, the second integral in Equation (22) can be written as

Substituting Equation (24), Equation (22) can be rewritten in the compact form

where

Equation (25) completely describes the temperature in the upper actuator for prescribed heat input

which then leads to the following scalar equations for the individual generalized coordinates:

Hence, the eigenvalues of the heat conduction are

The heat conduction in the lower actuator is similarly described by the boundary-value problem

Following the same steps, Equation (22) through Equation (24), we get

which governs the temperature in the lower actuator for prescribed heat input

Equations (19), (25) and (30) completely describe the motion of the uniform cantilever beam with the upper and lower actuators for prescribed inputs

which can be rewritten in the state-space form

where

are the state and input vectors, and

are the coefficient matrices. Note that the dimensions of

We will call the state-space model of Equation (32) the plant model, which, in absence of an experimental model, will be used to simulate the behavior of the actual system and to test the performance of our control design. In order for this model to approximate the behavior of the actual system better, we will choose m and n (the number of shape functions used in the expansions of Equations (7)) as high as possible. Control design, on the other hand, will be based on a control model that uses a less number of shape functions, as explained next.

Our objective in this paper is to control the deflection of the beam using

where

Control design must be as simple as possible for easy implementation and fast real-time computations. This in turn requires a low-order control model. To that end, we first note that the second integral of the expression for

which clearly means the first column is the only non-zero column in

where

are the control state vector and the control input, and

The dimensions of

The system to be controlled is inherently underactuated because cantilever beam is a distributed-parameter system with infinite degrees of freedom while the number of control inputs that can be used must always be finite. This inherent property of the system remains a fact even though we use a finite-order approximation of the actual system, namely the control model, because the order of the control model is

where

where

The control design described above assumes that the state vector

where

in which V is the state excitation noise intensity matrix.

We assume deflection measurements at the points

where

is the output matrix relating the actual state to the output. The estimate of the output, on the other hand, is written as

where

is the output matrix relating the estimated state to the estimated output.

Once

The modeling approach and the control design presented above will be numerically demonstrated for a uniform cantilever beam. We will use the same cantilever beam and the related parameters used in [

The matrices

To demonstrate the simulation of the closed-loop system, we choose

where “diag” means diagonal matrix and

The corresponding eigenvalues of open-loop plant and control models, closed-loop control model, and ob- server dynamics are listed in

Parameter | Value | Unit |
---|---|---|

H | 1 | mm |

B | 4 | cm |

L | 1 | m |

h | 0.5 | mm |

b | 4 | cm |

l | 40 | cm |

7860 | kg/m^{3} | |

N/m^{2} | ||

1/K | ||

m^{2}/s | ||

k | 45 | W/(m×K) |

0.6288 | kg/m | |

m^{4} | ||

m^{2} | ||

0.016 | m^{2} |

A | |||
---|---|---|---|

0 | 0 | −3.5823 | −0.0203 |

−509.27 | |||

−2037.1 | |||

−4583.4 | |||

will be answered by the simulation of the closed-loop plant.

Block diagram shown in

so that the beam is initially at rest in the shape of

In this paper, we assume each actuator is insulated in all faces except the face that is subject to the heat, including the faces normal to x and y axes. This assumption clearly means the temperatures everywhere in the actuators increase monotonically until the heat inputs are ceased after vibration amplitude reduces to a desired value. As an example, temperatures

We can also compare the total heat injected into the volumes in the two papers. we compute the total heat input to the actuators within t seconds as the integral of total heat rate:

This paper investigates active vibration control of a structure using a new type of actuator. The structure of interest is a cantilever beam. The actuator, on the other hand, is a thin metal bar rigidly bonded to the beam on one face and subject to heat input on the opposite face. The actuator then can work like a piezoelectric actuator, and expand and contract in response to applied heat. We assume that the actuator is insulated so that no heat is transferred to the beam. This ensures that the heat applied to the actuator does not alter the thermal state of the beam. One problem that needs to be addressed is that expansion of the actuator requires heating while contraction requires cooling. However, cooling is not practical, and to avoid it, we consider two actuators working together at the same spanwise location, one on the upper and one on the lower face of the beam. Then, the beam can be bent down by applying heat to the upper actuator, and bent up by applying heat to the lower actuator.

In the first part of the paper, we obtain a comprehensive model of the system that can be used to design control and simulate the response of the closed-loop system. Euler-Bernoulli beam theory is assumed to be valid to describe the motion of the beam with the attached actuators. Then, the Lagrange’s equations of motion are used to derive the governing equations for the vibration of the beam with the actuators. Since we assume the elastic motions of the actuators only depend on the elastic motion of the beam, the actuators do not bring any additional elastic degrees of freedom into the governing equations. One-dimensional heat conduction equation is used as

the equation governing the temperature in each actuator. We also assume the temperatures of the actuators are not affected by the elastic motion. The governing equations are a set of three partial differential equations, each of which has an infinite dimension. For an approximate solution, we use Rayleigh-Ritz method to replace the partial differential equations with a desired number of ordinary differential equations, obtaining a finite, but otherwise high-dimensional system. The higher the dimension, the more accurate the model is.

We obtain the plant model by using a high-dimensional representation of the system so that it can be repre- sented as accurately as possible. On the other hand, we obtain the control model by using a low-dimensional representation of the system so that estimates of the states as well as control inputs can easily be computed in real time. Since we do not have an experimental model, we use the plant model to mimic the true system as closely as possible, which is the reason to choose a high dimensional model. The control model is used to design feedback control and observer by means of LQR and Kalman-Bucy filtering techniques. The observer design is based on only two displacement measurements, one in the middle and one at the free end of the beam.

In a numerical application, control signal is computed using the control and observer gains based on the control model, and inputted into the plant. Then, the response of the beam to an initial excitation is simulated. The numerical simulation demonstrates that the actuators are in fact effective in active vibration control of the beam. The results obtained are comparable to some of the results of [

For simplicity of the models, we assume that each actuator is insulated in all faces except the face it is subject to heat. With this assumption, the heat conduction is one-dimensional (only in z-direction), and also we do not have to make assumptions about the convection and/or radiation heat transfer from the faces normal to x and y axes to outside. However, in real-life applications, this assumption is hard to enforce and it must be relaxed. Our future work will aim to make design alterations to relax this assumption to make the work more applicable to real-life situations.

The author thanks the editor and the referee for their comments.

This paper is dedicated to the memory of Prof. Leonard Meirovitch.

Ilhan Tuzcu, (2016) Vibration Control Using Heat Actuators. World Journal of Mechanics,06,223-237. doi: 10.4236/wjm.2016.68018