Robust multi-modal vibration control of a rotating smart beam with end mass

Abstract

A coupled bending–torsional model of a rotating beam with an attached piezoelectric actuator and end mass is developed in this work. An integral sliding mode controller is designed to suppress coupled bending–torsional vibrations while maintaining the rotational speed, using a single collocated piezoelectric actuator–sensor pair. Since full-state feedback is required and direct measurement of torsional responses is challenging in rotating beams, a Kalman filter is incorporated to accurately estimate all system states. The proposed framework enables simultaneous control of bending and torsional vibrations in rotating smart beams subjected to external disturbances using minimal actuation and sensing. Numerical simulations demonstrate that the Kalman-filter-based integral sliding mode controller achieves superior vibration attenuation with reduced control effort, while ensuring robustness against uncertainties. The results highlight the potential of the proposed strategy as an efficient and robust approach for vibration control in rotating smart structures.

Keywords

rotating beam piezoelectric actuator sliding mode Kalman filter

1. Introduction

The rotating beam with an end mass is widely studied as a simplified model of structures such as helicopter and turbine blades. Recent advances in smart materials have enabled active vibration control by bonding them to the beam surface, forming a smart beam capable of suppressing vibrations.

Brockmann and Lammering (2006) developed a comprehensive model for a rotating thin-walled beam with integrated piezoelectric fibers and stated that helicopter rotor blades are a potential application for such a model. They also proved that through aeroelastic couplings torsional actuation may influence the blade dynamics to suppress vibrations.

Bhadbhade et al. (2008); Bhadbhade and Jalili (2006) investigated coupled bending–torsional vibrations of a rotating beam with a piezoelectric actuator and tip mass, neglecting the hub radius, and mainly focused on dynamic response rather than mode shapes or eigenvalue loci.

Khodaei et al. (2018) experimentally extended this work, showing good agreement in the stationary case but discrepancies under rotation.

Recently, El-Masry and El-Badawy (2025) studied the axial-bending–torsional vibrations of a rotating beam with end mass and piezoeelctric actuator attached. They demonstrated that the presence of the end mass is a significant source of coupling between the bending and torsional vibrations. Moreover, they proved that such coupling affects the natural frequencies and modeshapes of the vibrating model which consequently affects the design of controllers.

Earlier studies employed linear control techniques for rotating smart beam vibrations Sun et al. (2004) developed a discretized model using the assumed modes method and applied combined PD and velocity feedback via piezoelectric actuators to control flapwise bending.

Yang et al. (2004) derived the finite element model for the flexural vibration of a rotating flexible beam with a piezoelectric actuator attached then a PPF (positive position feedback) controller was applied to the beam and MEF (moment exchange feedback) controller to the rotating hub simultaneously. Later, Spier et al. (2009) determined the optimal locations for the piezo patches and controlled the smart beam using displacement feedback.

In a more recent study, Beache and Fenili (2016) controlled the tip vibrations of a rotating smart beam using PD and LQR controllers at different positions of the piezoelectric actuator and compared the results. They proved that LQR controller is more robust than the PD controller.

Recently, Hashemi et al. (2022) developed LQR controller to dampen the transverse vibrations of a wind turbine blade using piezoelectric materials. However, all the previous studies have employed linear controllers with multiple actuators for vibration suppression.

Song and Gu (2007) designed a sliding mode controller for an aluminum beam using piezoelectric actuators and sensors, enhancing robustness through natural frequency variation. Xue and Tang (2008) argued that torque control is unsuitable for rotorcraft blades and proposed a sliding mode controller using piezoelectric actuation, treating motor torque as an external input. Overall, sliding mode control effectively handles model uncertainties.

Qiu and Xu (2016) derived the dynamics of two connected rotating beams and developed fast terminal and fuzzy fast terminal sliding mode controllers, experimentally demonstrating effective vibration suppression with reduced chattering. Later, Vakilzadeh et al. (2017) investigated classical and fractional-order sliding mode control to simultaneously suppress beam vibrations and track hub angular position, using three piezoelectric layers as sensors and actuators.

In the previous studies, they relied on using more than one piezoelectric actuator and several sensors to effectively suppress the unwanted vibrations. Moreover, none of the previous studies used the Kalman filter to estimate the unmeasured states.

(Na et al., 2011) analyzed a rotating tapered thin-walled beam and developed a sliding mode controller and observer to suppress flapwise and chordwise vibrations, showing superior performance over a linear quadratic Gaussian controller with Kalman filter.

Atmeh and Hasan (2012) used a Kalman filter to estimate natural frequencies of a flexible appendage on a rotating hub, while Camino and Santos (2019) developed a Kalman-filter-based LQG controller for tip deflection control. However, both studies considered only flexural vibrations and did not address coupled bending–torsional dynamics.

Recently, Rodriguez et al. (2022) proposed a multimodal sliding mode controller using multiple sensors and actuators, while Cui et al. (2022) developed an LQR-based controller for vibration suppression under uncertainty. However, both studies considered non-rotating cantilever beams and relied on multiple actuators and sensors.

This study proposes a multimodal vibration control strategy using a single piezoelectric actuator–sensor pair to suppress coupled bending–torsional vibrations in a rotating smart beam with an end mass. While bending vibrations are measurable, torsional vibrations are not; therefore, a Kalman-filter-based sliding mode controller is developed to estimate all states. Numerical results demonstrate effective vibration attenuation with minimal control effort.

2. Dynamic model

2.1. Kinematics

In this study, the coupled flapwise bending and torsional vibrations of a rotating smart beam are considered. The analytical methodology proposed by El-Masry and El-Badawy (2025) is adopted, excluding axial vibrations, with some key steps revisited here to derive the equations of motion for the present model.

Two coordinate frames are used to model this beam, a fixed frame with an origin at the center axis of the hub (X0 Y0 Z0) and a moving frame that rotates with the beam (X1 Y1 Z1). In Figure 1, { $u_{x}$ , $u_{y}$ , $u_{z}$ } are the components of the displacement vector $U$ that represents the displacement of any point from the undeformed position $P_{0}$ to the deformed position $P$ . As shown in Figure 2, the Euler–Bernoulli beam considered in this study is of length $L$ , width $w_{b}$ and thickness t, with collocated piezoelectric actuator-sensor pair bonded near the root of the beam. It is attached to a rigid hub of radius $r$ and rotates with a constant angular speed ω about the Z-axis. A third coordinate system (X2 Y2 Z2) that has the same origin as the rotating frame of reference is used to represent the angle of torsion $β$ of the cross-section of the beam about the X1 axis as shown in Figure 3. A tip mass of a cube shape is attached to the end of the beam. Tables 1 –3 show the physical properties of the beam, piezoelectric actuator and the end mass.

Figure 1.

Model of flexible smart rotating beam with tip mass.

Figure 2.

Top view showing the dimensions and the rotation of the smart beam with angular velocity ω.

Figure 3.

Side view showing the twist of the cross section by the angle of torsion $β$ .

Table 1.

Physical and geometrical properties of the beam.

Parameter	Value	Unit
$ρ_{b}$	1840	$k g / m^{3}$
$L$	1	$m$
$w_{b}$	0.03	$m$
$t$	0.003	$m$
$E_{b}$	34.64	$G p a$
$G_{b}$	13	$G p a$
$r$	0.02	$m$

Table 2.

Physical and geometrical properties of the piezoelectric actuator.

Parameter	Value	Unit
$ρ_{p}$	7650	$k g / m^{3}$
$w_{p}$	0.03	$m$
$l_{1}$	0.001	$m$
$l_{2}$	0.015	$m$
$d_{31}$	−190* $10^{- 12}$	$m / V$
$E_{p}$	63	$G p a$
$G_{p}$	25.565	$G p a$

Table 3.

Physical and geometrical properties of the tip mass.

Parameter	Value	Unit
$l_{m}$	0.008	$m$
$w_{m}$	0.008	$m$
$h_{m}$	0.008	$m$

2.2. Equations of motion

The governing equations of motion are derived using the extended Hamilton’s principle stated as (Meirovitch, 2001)

\int_{t 1}^{t 2} (δ T - δ V + δ W n c) d t = 0

(1)

The total kinetic energy of this model consists of that of the beam, the piezoelectric actuator and the tip mass. Products of inertia are neglected as the beam’s cross section is symmetric

T = \int_{0}^{L} \frac{1}{2} ρ (x) {V_{p}}^{T} . V_{p} d x + \frac{1}{2} m {V_{m}}^{T} . V_{m} + \frac{1}{2} ω^{2} J_{m a s s} + \frac{1}{2} {(\frac{\partial β [x, t]}{\partial t})}^{2} I_{m a s s}

(2)

Where

ρ (x) = ρ_{b} A_{b} + S (x) ρ_{p} A_{p}

and

S (x)

is the Heaviside function known as

S (x) = H (l_{1} - x) - H ((l_{2} - x)

(3)

The total potential energy also consists of that of the beam and that of the piezoelectric actuator

V = \int_{0}^{L} \frac{1}{2} E I (x) {(\frac{\partial^{2} w [x, t]}{\partial x^{2}})}^{2} + \frac{1}{2} G J (x) {(\frac{\partial β [x, t]}{\partial x})}^{2} + \frac{1}{2} ρ (x) ω^{2} (r (L - x) + \frac{1}{2} (L^{2} - x^{2})) {(\frac{\partial w (x, t)}{\partial x})}^{2} d x

(4)

Where

G J (x) = G_{b} J_{b} + S (x) G_{p} J_{p}

E I (x) = E_{b} I_{b} + S (x) E_{p} I_{p}

In the Hamiltonian approach, the actuator moment is represented as non-conservative work (Jalili, 2010)

δ W n c = \int_{0}^{L} \frac{\partial^{2} M p}{\partial x^{2}} δ w d x

(5)

Where Mp is the piezoelectric actuator moment and represented as

M_{p} = - \frac{1}{2} b E_{p} d_{31} (t_{b} + t_{p)} V_{a} (t) S (x)

(6)

By substituting equations (2), (4), and (5) in equation (1), the equations of motion are derived as

ρ (x) \frac{\partial^{2} w (x, t)}{\partial t^{2}} - ρ I (x) \frac{\partial^{2} w (x, t)}{\partial x \partial t} + 2 ρ J (x) ω \frac{\partial^{2} β (x, t)}{\partial x \partial t} + ρ I (x) ω^{2} \frac{\partial^{2} w (x, t)}{\partial x^{2}} + E I (x) \frac{\partial^{4} w (x, t)}{\partial x^{4}} - ω^{2} [\frac{1}{2} (L - x) (L + 2 r + x)] \frac{\partial^{4} w (x, t)}{\partial x^{4}} = - \frac{\partial^{2} M_{p} (x, t)}{\partial x^{2}}

(7)

ρ I (x) ω^{2} β (x, t) - ρ J (x) \frac{\partial^{2} β (x, t)}{\partial t^{2}} - 2 ρ I (x) ω \frac{\partial^{2} w (x, t)}{\partial x \partial t} + G J (x) \frac{\partial^{2} β (x, t)}{\partial x^{2}} = 0

(8)

Where

ρ J (x) = ρ_{b} J_{b} + S (x) ρ_{p} J_{p}

ρ I (x) = ρ_{b} I_{b} + S (x) ρ_{p} I_{p}

The boundary conditions at the clamped end ( $x = 0$ )

w (0, t) = 0, \frac{\partial w (0, t)}{\partial x} = 0, β (0, t) = 0

(9)

While the boundary conditions at the free end (

x = L

)

{(ρ J (x) ω \frac{\partial β [x, t]}{\partial x} + ρ I (x) ω^{2} \frac{\partial w [x, t]}{\partial x} + E_{b} I_{b} \frac{\partial^{3} w [x, t]}{\partial x^{3}} + E_{p} I_{p} G [x] \frac{\partial^{3} w [x, t]}{\partial x^{3}}) |}_{x = L} + M (ω^{2} (L + r)) = 0

(10)

{(G_{b} J_{b} \frac{\partial β [x, t]}{\partial x} + G_{p} J_{p} \frac{\partial β [x, t]}{\partial x}) |}_{x = L} + J_{m a s s} \frac{\partial^{2} β (L, t)}{\partial t^{2}} = 0

(11)

{(- E_{b} I_{b} \frac{\partial^{2} w [x, t]}{\partial x^{2}} - E_{p} I_{p} G [x] \frac{\partial^{2} w [x, t]}{\partial x^{2}}) |}_{x = L} + M \frac{\partial^{2} w (L, t)}{\partial t^{2}} = 0

(12)

2.3. Non dimensional analysis

The governing equations presented in equations (7) and (8) and their boundary conditions in equations (9)–(12) are non-dimensionalized to simplify the mathematical analysis and be able to compare results with other references.

By introducing other non-dimensional parameters

w_{d} = \frac{w_{b}}{L} {, ω}_{d} = L^{2} \sqrt{\frac{ρ_{b} A_{b}}{E_{b} I_{b}}} ω, α = \frac{{ρ_{b} J}_{b}}{ρ_{b} A_{b} L^{2}}, α_{p} = \frac{{ρ_{p} J}_{p}}{ρ_{b} A_{b} L^{2}}, v = \frac{G_{b} J_{b}}{E_{b} I_{b}} {, v}_{p} = \frac{G_{p} J_{p}}{E_{b} I_{b}}, κ = \frac{ρ_{b} I_{b}}{ρ_{b} A_{b} L^{2}} {, κ}_{p} = \frac{ρ_{p} I_{p}}{ρ_{b} A_{b} L^{2}}, b = \frac{E_{p} I_{p}}{E_{b} I_{b}}, c = \frac{ρ_{p} A_{p}}{ρ_{b} A_{b}} {, M}_{d} = \frac{m_{L}}{ρ_{b} A_{b} L} {, α}_{m} = \frac{J_{m a s s}}{ρ_{b} A_{b} L^{3}} {, η}_{1} = \frac{l_{1}}{L}, η_{2} = l_{2} / L {, κ}_{m} = \frac{I_{m a s s}}{ρ_{b} A_{b} L^{3}}

(13)

The non-dimensional equations of motion are represented as

\frac{\partial^{2} w_{d}}{\partial {T_{d}}^{2}} + c S (x_{d}) \frac{\partial^{2} w_{d}}{\partial {T_{d}}^{2}} + κ ω_{d} \frac{\partial^{2} β}{\partial x_{d} \partial T_{d}} + κ_{p} S (x_{d}) ω_{d} \frac{\partial^{2} β}{\partial x_{d} \partial T_{d}} + κ {ω_{d}}^{2} \frac{\partial^{2} w_{d}}{\partial {x_{d}}^{2}} + κ_{p} S (x_{d}) {ω_{d}}^{2} \frac{\partial^{2} w_{d}}{\partial {x_{d}}^{2}} + \frac{\partial^{4} w_{d}}{\partial {x_{d}}^{4}} + b S (x_{d}) \frac{\partial^{4} w_{d}}{\partial {x_{d}}^{4}} = Q_{d}

(14)

α \frac{\partial^{2} β}{\partial {T_{d}}^{2}} + α_{p} S (x_{d}) \frac{\partial^{2} β}{\partial {T_{d}}^{2}} + κ ω_{d} \frac{\partial^{2} w_{d}}{\partial x_{d} \partial T_{d}} + κ_{p} S (x_{d}) ω_{d} \frac{\partial^{2} w_{d}}{\partial x_{d} \partial T_{d}} - κ {ω_{d}}^{2} \frac{\partial^{2} w_{d}}{\partial {x_{d}}^{2}} - κ_{p} S (x_{d}) {ω_{d}}^{2} \frac{\partial^{2} w_{d}}{\partial {x_{d}}^{2}} - v \frac{\partial^{2} β}{\partial {x_{d}}^{2}} - v_{p} \frac{\partial^{2} β}{\partial {x_{d}}^{2}} = 0

(15)

Where

Q_{d} = d V_{a} (t) (\emptyset_{i}^{'} (l_{2}) - \emptyset_{i}^{'} (l_{1}))

and

d = \frac{\frac{1}{2} * b * (t_{p} + t_{b} - 2 z_{n}) E_{p} d_{31}}{E_{b} I_{b}}

The non-dimensional boundary conditions at ( $x = 0$ ) are

w_{d} = 0, \frac{\partial w_{d}}{\partial x_{d}} = 0, β = 0

(16)

and the non-dimensional boundary conditions at (

x = L

) are

{((α + α_{p}) ω \frac{\partial β}{\partial x_{d}} + (κ + κ_{p}) ω^{2} \frac{\partial w_{d}}{\partial x_{d}} + \frac{\partial^{3} w_{d}}{\partial {x_{d}}^{3}} + b S (x_{d}) \frac{\partial^{3} w_{d}}{\partial {x_{d}}^{3}}) |}_{x = L} + M_{d} (ω^{2} (L + r)) = 0

(17)

{(v \frac{\partial β}{\partial x_{d}} + v_{p} \frac{\partial β}{\partial x_{d}}) |}_{x = L} + α_{m} {ω_{d}}^{2} \frac{\partial^{2} β}{\partial {T_{d}}^{2}} = 0

(18)

{(- \frac{\partial^{2} w_{d}}{\partial {x_{d}}^{2}} - b S (x_{d}) \frac{\partial^{2} w_{d}}{\partial {x_{d}}^{2}}) |}_{x = L} + M_{d} \frac{\partial^{2} w_{d}}{\partial {T_{d}}^{2}} = 0

(19)

Where

S (x_{d}) = H (η_{1} - x_{d}) - H ((η_{2} - x_{d})

2.4. Discretization

A non-dimensional reduced-order model is derived using the assumed modes method. Two modes (i.e., j = 2) are retained for bending and torsion, as they capture the dominant dynamics. Higher modes mainly introduce high-frequency effects with limited impact on control performance while increasing computational cost. This choice is validated in Section 4 using a three-mode model.

The bending and torsional assumed modes are chosen as

w [x_{d}, t_{d}] = \sum_{j = 1}^{n} \emptyset_{j} (x_{d}) p_{j} (t_{d})

(20)

β [x_{d}, t_{d}] = \sum_{j = 1}^{n} Ψ_{j} (x_{d}) q_{j} (t_{d})

(21)

Where

\emptyset_{j} (x_{d})

and

Ψ_{j} (x_{d})

are the bending and torsion non-dimensional trial functions, respectively, while

p_{j} (t_{d})

and

q_{j} (t_{d})

are the generalized coordinates.

The admissible trial functions used here are the bending and torsion eigen functions of the conventional cantilever beam without rotation which are denoted as

\emptyset_{j} (x_{d}) = \sin β_{j} x_{d} - \sinh β_{j} x_{d} + \frac{\cos β_{j} x_{d} (- Sin β_{j} - Sinh β_{j})}{\cos β_{j} - \cosh β_{j}} - \frac{\cosh β_{j} x_{d} (- \sin β_{j} - \sinh β_{j})}{\cos β_{j} - \cosh β_{j}}

(22)

Ψ_{j} (x_{d}) = \sin (\frac{(2 j - 1) π x_{d}}{2})

(23)

By substituting equations (22) and (23) in equations (20) and (21) and substituting the resulting discretized modes in the non-dimensional governing equations (14) and (15), the discretized equations of motion become

M_{1} \ddot{p} (t_{d}) + 2 ω_{d} G_{1} \dot{p} (t_{d}) + (K_{1} + D_{1} {ω_{d}}^{2}) p (t_{d}) = Q

(24)

M_{2} \ddot{q} (t_{d}) + 2 ω_{d} G_{2} \dot{q} (t_{d}) + (K_{2} + D_{2} {ω_{d}}^{2}) q (t_{d}) = 0

(25)

Where

p (t_{d})

and

q (t_{d})

are

n \times 1

vectors of modal coordinates. The other matrices are of dimension

n \times n

and are defined in Appendix A. Calculations are done using Mathematica® version 11 (Wolfram Research, Inc., 2017).

3. Controller design

Previous studies often use multiple actuators to control different vibration modes, partly because torsional vibrations in rotating structures are difficult to measure for feedback (Khodaei et al., 2018; Qiu et al., 2021). To address this, a Kalman-filter-based controller is developed to estimate unmeasured states, enabling suppression of coupled bending–torsional vibrations using a single actuator.

For comparison, an integral sliding mode controller is first designed assuming full-state measurement, then combined with a Kalman filter where only one bending mode is measured and the remaining states are estimated. Since low-frequency dynamics dominate, the first two bending and torsional modes are considered. The framework is implemented in MATLAB/Simulink.

The general non-linear dynamics of the model considered in this study are represented in state space form as

\dot{x} (t) = f ((x (t)) + g ((x (t)) u (t) + d (t) + \bar{w} (t)

(26)

y (t) = h (x (t)) + \bar{v} (t)

(27)

Where

f ((x (t))

represents the non-linear dynamics

g ((x (t))

is the input matrix

h ((x (t))

is the output matrix

u (t)

is the input

d (t)

is the external disturbance applied to the system which is assumed to be bounded.

\bar{w} (t)

and

\bar{v} (t)

are the process and measurement noises, respectively, assumed to be zero mean stochastic uncertainty. These terms account for model uncertainties and sensor noise.

For the controller design, the non-linear model is linearized and discretized as

M \ddot{η} (t) + G \dot{η} (t) + (K + D ω^{2}) η (t) = Q

(28)

Equation (28) is the assembled form of equations (24 and 25), where, $η (t)$ is a $2 n \times 1$ vector of modal coordinates while $M$ , $G$ , $D$ , and $K$ are the generalized mass, gyroscopic, centrifugal stiffness and stiffness matrices, respectively. $Q$ is the external force vector.

The disturbance term includes the model parameter uncertainties, which are multiplicative variations of the mass, gyroscopic and stiffness matrices in equation (28) represented as

M (I + Δ M (t)) \ddot{η} (t) + G (I + Δ G (t)) \dot{η} (t) + K (I + Δ K (t)) + D ω^{2}) η (t) = Q

(29)

Where

Δ M (t)

Δ G (t)

, and

Δ K (t)

represent time varying structured bounded parametric uncertainties in the mass, gyroscopic, and stiffness matrices, respectively. Each uncertain matrix is assumed to be within

\pm

10% of the nominal matrices

M

G

, and

K

Uncertainties are modeled as multiplicative to represent parametric variations in material properties, piezoelectric coefficients, end mass, and rotation-dependent terms. In rotating beams, centrifugal stiffening and gyroscopic effects scale system matrices, making multiplicative modeling more physically accurate than additive approaches.

Moreover, the disturbance includes a finite time pulse input, where in real life this simulates the effect of air turbulence or a shock and is represented as

d (t) = {\begin{cases} 1, 5 < t < 6 \\ 0, o t h e r w i s e \end{cases}

(30)

From equation (29), defining $M (I + Δ M (t))$ as $M_{a c t u a l}$ , $G (I + Δ G (t))$ as $G_{a c t u a l}$ and $K (I + Δ K (t))$ as $K_{a c t u a l}$ , the state vector is defined as

x (t) = {[η (t) \dot{η} (t)]}^{T}

(31)

Where

η

(t) is the generalized displacement vector and its size depends on the number of modes included in the discretized model.

Accordingly, the state equations are represented as

\dot{x} (t) = [\begin{array}{c} \dot{η} (t) \\ {M_{a c t u a l}}^{- 1} Q - {M_{a c t u a l}}^{- 1} G_{a c t u a l} \dot{η} (t) - {M_{a c t u a l}}^{- 1} (K_{a c t u a l} + D ω^{2}) η (t) \end{array}]

(32)

Then, the linearized state space form is represented as

\dot{x} (t) = A x (t) + B u (t) + d (t) + \bar{w} (t)

(33)

Where the system dynamics matrix is

A = [\begin{array}{l} 0 & I \\ - {M_{a c t u a l}}^{- 1} (K_{a c t u a l} + D ω^{2}) & - {M_{a c t u a l}}^{- 1} G_{a c t u a l} \end{array}]

(34)

and the input matrix is

B = [\begin{array}{l} 0 \\ {M_{a c t u a l}}^{- 1} Q \end{array}]

(35)

While the output equation is represented as

y (t) = C x (t) + \bar{v} (t)

(36)

3.1. Integral sliding mode controller

In this section, an integral sliding mode controller is designed based on the model in equations (35) and (36) to ensure robustness against the bounded disturbance in equation (30). It assumes full-state measurement of bending and torsional deflections and serves as a benchmark for comparison with the Kalman-based approach in Section 3.2.

The sliding surface of each mode is formulated following the approach presented by (Utkin and Shi, 1996). However, to be able to simultaneously control all modes using a single actuator, the individual mode sliding surfaces are combined into a single composite sliding surface that governs the overall system dynamics and is defined as

s = \sum_{i = 1}^{n} {\dot{e}}_{i} + λ_{i} e_{i} + {λ_{i}}^{2} \int_{0}^{t} e_{i} (τ) d τ

(37)

Where

λ_{i}

is a positive constant,

e_{i}

is the tracking error for each mode of vibration represented as

e_{i} = x_{i} - x_{i, d}

(38)

Where

x_{i}

is a vector of the generalized coordinates that needs to be controlled.

x_{i, d}

is the desired reference input that is assumed to be zero.

Since the controller is based on Lyapunov stability theory, the Lyapunov function is selected as

V = \frac{1}{2} s^{2}

(39)

To ensure stability, the time derivative of the Lyapunov function should be negative definite

\dot{V} = s \dot{s},

(40)

Where

\dot{s}

is the derivative of the sliding surface

s

and is obtained as

\dot{s} = \sum_{i}^{n} {\dot{s}}_{i}

(41)

{\dot{s}}_{i}

represents the derivative of each sliding surface and is represented as

{\dot{s}}_{i} = a_{i} + b_{i} u - {\ddot{x}}_{d} + λ_{i} {\dot{e}}_{i} + {λ_{i}}^{2} e_{i}

(42)

Where each mode’s second order dynamics is represented in affine form as

{\ddot{x}}_{i} = a_{i} (x) + b_{i} (x) u

(43)

Summing over all modes yields

\dot{s} = \bar{a} + \bar{b} u

(44)

With

\bar{a} = \sum_{i}^{n} a_{i} - {\ddot{x}}_{i, d} + λ_{i} {\dot{e}}_{i} + {λ_{i}}^{2} e_{i} \bar{, b} = \sum_{i}^{n} b_{i}

(45)

It is assumed that $\bar{b} \neq 0$ for all operating conditions, ensuring the existence of ${\bar{b}}^{- 1}$ .

Since we have only one actuator, we have a single control input $u$ that is designed as

u = u_{e q} + u_{s w}

(46)

Where

u_{e q}

and

u_{s w}

are chosen to be

u_{e q} = - {\bar{b}}^{- 1} \bar{a}, u_{s w} = - {\bar{b}}^{- 1} K \tanh (s / φ)

(47)

By substituting equation (47) in equation (46), the control law is represented as

u = {\bar{b}}^{- 1} (- \bar{a} - K \tanh (s / φ))

(48)

Where

K > 0

is the sliding gain.

ϕ > 0

is the boundary layer thickness, reducing chattering by using

\tanh (s / φ)

In this study, the state vector is ordered as in equation (31) such that the first four states ( $x_{1}$ to $x_{4}$ ) correspond to generalized displacements, and the last four states ( $x_{5}$ to $x_{8}$ ) correspond to generalized velocities. It is assumed that all system states are measurable to provide a benchmark for comparison with the observer-based controller presented in section 3.2.

3.2. Kalman-filter-based integral sliding mode control

A Kalman filter is employed to estimate all system states for the integral sliding mode controller, as shown in Figure 4.

Figure 4.

Kalman filter-based SMC diagram.

Let the estimation error be defined as

\tilde{x} = x - \hat{x}

(49)

The estimation error dynamics are governed by

\dot{\tilde{x}} = (A - K_{f} (t) C) \tilde{x}

(50)

Where

K_{f} (t)

is the Kalman gain obtained from the solution of the Riccati equation.

For asymptotic convergence, the estimation error dynamics must be stable. Under the detectability condition, the Kalman gain $K_{f} (t)$ , obtained from equation (52), converges to a steady-state value, leading the estimation error to zero. Although the system is not fully observable, eigenvalue analysis shows that all unobservable modes are stable with negative real parts, satisfying detectability. This is confirmed in Figure 5, where the estimated states $\hat{x}$ closely match the actual states $x$ , consistent with standard Kalman filtering results (Yonezawa, 1980).

Figure 5.

Real and estimated states (a) non-dimensional first mode bending deflection; (b) non-dimensional first mode torsional deflection.

3.2.1. Kalman filter design

A continuous-time Kalman–Bucy filter is designed to estimate the unmeasured states of the system in equations (26) and (27), accounting for process disturbances and measurement noise, both modeled as zero-mean Gaussian white noise with covariance matrices $Q$ and $R$ .

The continuous Kalman filter estimates the system states as

\dot{\hat{x}} (t) = A \hat{x} (t) + B u (t) + K_{f} (t) (y (t) - C \hat{x} (t))

(51)

K_{f} (t) = P (t) C^{T} R^{- 1}

(52)

The covariance matrix $P (t)$ evolves according to the Riccati differential equation

\dot{P} (t) = A P (t) + P (t) A^{T} - P (t) C^{T} R^{- 1} C P (t) + Q

(53)

The process noise covariance matrix $Q$ is defined as a diagonal matrix

Q = d i a g (0, 0, 0, 0, 1 e^{- 5}, 0, 1 e^{- 5}, 0)

(54)

The process noise is applied only to the torsional rate states to reflect uncertainties in the torsional dynamics, which are not directly measured. The measurement noise covariance $R$ is defined as

R = d i a g (5 e^{- 4}, 5 e^{- 4})

(55)

Measurement noise is included for the bending displacement sensors to account for sensor inaccuracies and unmodeled disturbances. The values of the covariance matrices $Q$ and $R$ are relatively small, consistent with the non-dimensional formulation of the system, as discussed in the results section.

As shown in Figure 5, the estimated states closely follow the actual states obtained from the simulations, confirming the effectiveness of the designed Kalman filter.

3.2.2. Integral sliding mode controller design

In this section, an integral sliding mode controller is designed based on the estimated states obtained from the Kalman filter.

For a system with $n$ vibration modes, the sliding surface, $s$ , is chosen to be the summation of the integral sliding surfaces, $s_{i}$ , of each mode

s = \sum_{i = 1}^{n} s_{i}

(56)

Where each sliding surface,

s_{i}

, is chosen as

s_{i} = {\dot{\hat{e}}}_{i} + λ_{i} {\hat{e}}_{i} + {λ_{i}}^{2} \int_{0}^{t} {\hat{e}}_{i} (τ) d τ

(57)

Where

λ_{i}

is a positive constant,

{\hat{e}}_{i}

is the tracking error for each mode of vibration based on the estimated states

{\hat{x}}_{i}

{\hat{e}}_{i} = {\hat{x}}_{i} - x_{i, d}

(58)

Where

x_{i, d}

is the desired reference and is assumed to be zero.

The sliding mode controller follows the same stability conditions and control law as in Section 3.1, with the actual tracking error $e_{i}$ replaced by the estimated error ${\hat{e}}_{i}$ . For a fair comparison, Gaussian zero-mean measurement noise is included. The control parameters are selected to satisfy the reaching condition under bending–torsional coupling and uncertainties. Specifically, $K = 2$ and $φ = 0.1$ are chosen based on the non-dimensional states to ensure robust convergence to the sliding surface while mitigating chattering. The sliding surface slopes are set to $λ = 5$ for the first modes and $λ = 25$ for the second modes, providing a balance between fast dynamic response and smooth displacement tracking.

Substituting the control law into the sliding surface dynamics equations (56)–(58) yields

\dot{s} = - K \tanh (\frac{s}{φ})

(59)

which implies

\dot{V} = s \dot{s} = s (- K \tanh (\frac{s}{φ})) \leq 0

(60)

Therefore, the Lyapunov stability condition in equation (48) is satisfied ensuring that the states of the system are driven toward the sliding surface (Figure 6).

Figure 6.

Sliding surface time response.

Furthermore, Figure 5 shows the time evolution of the sliding variable. The pulse observed at $t = 5$ s is caused by the applied external disturbance, but then it converges back toward $s = 0$ demonstrating the robustness and effectiveness of the proposed controller.

4. Results

The proposed controller suppresses coupled bending–torsional vibrations using a single piezoelectric actuator while estimating unmeasured states, reducing reliance on multiple sensors without altering system dynamics. It also demonstrates effective multimodal vibration suppression.

To evaluate performance, a rotating beam with an attached piezoelectric actuator–sensor pair is tested at a non-dimensional speed $ω_{d} = 10$ under zero initial conditions, impulse disturbance, and measurement noise. As shown in Figures 7 –10, the Kalman-filter-based sliding mode controller outperforms the sensor-based integral sliding mode controller in suppressing both bending and torsional vibrations.

Figure 7.

Uncontrolled and controlled (a) non-dimensional first mode bending deflection; (b) non-dimensional second mode bending deflection.

Figure 8.

Uncontrolled and controlled (a) non-dimensional first mode torsional deflection; (b) non-dimensional second mode torsional deflection.

Figure 9.

Uncontrolled and controlled (a) non-dimensional first mode bending velocity; (b) non-dimensional second mode bending velocity.

Figure 10.

Uncontrolled and controlled (a) non-dimensional first mode torsional velocity; (b) non-dimensional second mode torsional velocity.

Figure 11 shows that this improved performance is achieved with lower and smoother control effort. High-frequency activity in the control signal arises from bending–torsional coupling and model uncertainties but remains continuous and bounded.

Figure 11.

Control effort for the integral sliding mode controller and the proposed controller.

The Kalman-based SMC achieves up to 80% reduction in vibration amplitude, compared to 30–50% for ISMC, with significantly reduced chattering.

Figure 12 shows the frequency-domain response of the bending displacement before and after control. A dominant peak at approximately 3.6 Hz, corresponding to the first bending mode, is clearly observed in the uncontrolled case and significantly attenuated under the proposed controller. The amplitude is expressed in non-dimensional form.

Figure 12.

Frequency-domain response of the transverse displacement for uncontrolled and controlled cases.

To validate multimodal performance, the controller is applied to a three-mode model under identical conditions. Figures 13 and 14 show effective suppression of both bending and torsional vibrations despite additional high-frequency dynamics.

Figure 13.

Uncontrolled and controlled (a) non-dimensional first mode bending displacement; (b) non-dimensional first mode torsional displacement for the 3 modes model.

Figure 14.

Closer view for the controlled response (a) non-dimensional first mode bending displacement; (b) non-dimensional first mode torsional displacement for the 3 modes model.

A comparison with a classical PID controller is also conducted. Due to its SISO structure, PID mainly regulates bending with limited torsional suppression, whereas the proposed method achieves effective attenuation of both responses using a single actuator, as shown in Figure 15.

Figure 15.

PID vs Kalman based SMC controlled response (a) non-dimensional first mode bending displacement; (b) non-dimensional first mode torsional displacement for the 2 modes model.

5. Conclusion

This study presents an effective vibration suppression strategy for a rotating smart beam with an end mass using a single piezoelectric actuator and an integral sliding mode controller. A key contribution is the use of a single actuator to control coupled bending–torsional vibrations, typically requiring multiple actuators. To address unmeasured torsional states, a continuous-time Kalman filter is used for full-state estimation.

Numerical results demonstrate significant attenuation of both bending and torsional vibrations, with the Kalman-filter-based controller achieving superior performance and lower control effort. By exploiting gyroscopic and end-mass-induced coupling, the approach enables efficient multimodal control with minimal actuation. This provides a practical solution for smart structures with sensor and actuator limitations.

Footnotes

ORCID iDs

Menna El-Masry

Ayman El-Badawy

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Appendix

References

Atmeh

Hasan

(2012) Parameter estimation of a rotating beam under a kalman filtering framework In: Mechanics of Solids, Structures and Fluids. Presented at the ASME 2012 International Mechanical Engineering Congress and Exposition. Houston, Texas, USA: American Society of Mechanical Engineers, Volume 8: pp. 461–471. https://doi.org/10.1115/IMECE2012-88720

Beache

KVA

Fenili

(2016) Active Vibration Control of a Smart Beam Under Rotation. Proceedings of the XXXVII Iberian Latin-American Congress on Computational Methods in Engineering (CILAMCE 2016). Brasília, DF, Brazil: ABMEC.

Bhadbhade

Jalili

(2006) Coupled flexural/torsional vibrations of a piezoelectrically-actuated vibrating beam gyroscope In: Dynamic Systems and Control, Parts A and B. Presented at the ASME 2006 International Mechanical Engineering Congress and Exposition. Chicago, Illinois, USA: ASMEDC, pp. 1675–1683. https://doi.org/10.1115/IMECE2006-16049

Bhadbhade

Jalili

Nima Mahmoodi

(2008) A novel piezoelectrically actuated flexural/torsional vibrating beam gyroscope. Journal of Sound and Vibration 311: 1305–1324. https://doi.org/10.1016/j.jsv.2007.10.017

Brockmann

Lammering

(2006) Beam finite elements for rotating piezoelectric fiber composite structures. Journal of Intelligent Material Systems and Structures 17: 431–448. https://doi.org/10.1177/1045389X06058632

Camino

Santos

(2019) A periodic linear–quadratic controller for suppressing rotor-blade vibration. Journal of Vibration and Control 25: 2351–2364. https://doi.org/10.1177/1077546319853358

Cui

Liu

Jiang

, et al. (2022) Active vibration optimal control of piezoelectric cantilever beam with uncertainties. Measurement and Control 55: 359–369. https://doi.org/10.1177/00202940221091244

El-Masry

El-Badawy

(2025) Modeling and analysis of axial, bending, and torsional vibrations of a rotating smart beam with end mass. Shock and Vibration 2025: 5533783. https://doi.org/10.1155/vib/5533783

Hashemi

Jang

Hosseini-Hashemi

(2022) Smart active vibration control system of a rotary structure using piezoelectric materials. Sensors 22: 5691. https://doi.org/10.3390/s22155691

10.

Jalili

(2010) Piezoelectric-Based vibration-control: From Macro to Micro/nano Scale Systems. New York: Springer.

11.

Khodaei

Mehrvarz

Candelino

, et al. (2018) Theoretical and experimental analysis of coupled flexural–torsional vibrations of rotating beams. In: Proceedings of the ASME 2018 Dynamic Systems and Control Conference. Atlanta, USA: Stanford University, V003T42A004. https://doi.org/10.1115/DSCC2018-9050

12.

Meirovitch

(2001) Fundamentals of Vibrations. Boston: McGraw-Hill.

13.

Lee

Choo

, et al. (2011) Robust dynamic sliding mode control of a rotating thin-walled composite blade. Journal of Aerospace Engineering 24: 298–308. https://doi.org/10.1061/(ASCE)AS.1943-5525.0000076

14.

Qiu

(2016) Vibration control of a rotating two-connected flexible beam using chattering-free sliding mode controllers. Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 230: 444–468. https://doi.org/10.1177/0954410015592436

15.

Qiu

Wang

Zhang

(2021) Sliding mode predictive vibration control of a piezoelectric flexible plate. Journal of Intelligent Material Systems and Structures 32(1): 3–17. https://doi.org/10.1177/1045389X20948597

16.

Rodriguez

Collet

Chesné

(2022) Active vibration control on a smart composite structure using modal-shaped sliding mode control. Journal of Vibration and Acoustics 144: 021013. https://doi.org/10.1115/1.4053358

17.

Song

(2007) Active vibration suppression of a smart flexible beam using a sliding mode based controller. Journal of Vibration and Control 13: 1095–1107. https://doi.org/10.1177/1077546307078752

18.

Spier

Bruch

Sloss

, et al. (2009) Placement of multiple piezo patch sensors and actuators for a cantilever beam to maximize frequencies and frequency gaps. Journal of Vibration and Control 15: 643–670. https://doi.org/10.1177/1077546308094247

19.

Sun

Mills

Shan

, et al. (2004) A PZT actuator control of a single-link flexible manipulator based on linear velocity feedback and actuator placement. Mechatronics 14: 381–401. https://doi.org/10.1016/S0957-4158(03)00066-7

20.

Utkin

Shi

(1996) Integral sliding mode in systems operating under uncertainty conditions. In: Proceedings of the 35th IEEE Conference on Decision and Control (CDC). Japan: Kobe, 4591–4596.

21.

Vakilzadeh

Eghtesad

Nami

, et al. (2017) Vibration suppression of a rotating HUB-beam system with a flexible support using fractional order sliding mode control. Transactions of the Canadian Society for Mechanical Engineering 41: 627–643. https://doi.org/10.1139/tcsme-2017-0044

22.

Wolfram Research, Inc. (2017) Mathematica, Version 11.0. Champaign, IL, USA: Wolfram.

23.

Xue

Tang

(2008) Vibration control of nonlinear rotating beam using piezoelectric actuator and sliding mode approach. Journal of Vibration and Control 14: 885–908. https://doi.org/10.1177/1077546307085354

24.

Yang

Jiang

Chen

(2004) Dynamic modelling and control of a rotating euler–bernoulli beam. Journal of Sound and Vibration 274: 863–875. https://doi.org/10.1016/S0022-460X(03)00611-4

25.

Yonezawa

(1980) Reduced-order kalman filtering with incomplete observability. Journal of Guidance and Control 3: 280–282. https://doi.org/10.2514/3.55985