Vibration reduction of a flexible robot link using a frictional damper

Abstract

One of the most important factors reducing flexible manipulator efficiency is the residual vibration occurrence. In this research, vibration reduction of flexible manipulators is investigated using an internal frictional damper. At first, the vibration equation of a manipulator is obtained using the finite element method with the Euler–Bernoulli beam element to study its vibrations in a reciprocal motion. In addition, an analytical model is developed to investigate the effect of the frictional damper on robot link vibrations. Using particle swam optimization, ICA, NSGA-II, and GWO methods, the optimal structure for the damper is obtained to maximize its effect. The optimally damped link is fabricated, and its dynamic characteristics are extracted from a modal test experiment. The modal test results show a considerable improvement in the damping ratio of the damped link in comparison with a simple link. The fabricated link samples are then tested in a realistic situation. The experimental results are in coincidence with the simulation results, certifying the performance of the proposed plan in vibration reduction of a robot link.

Keywords

Robot link flexible manipulator vibration reduction frictional damper optimization

1. Introduction

Nowadays, robot links are widely used in all industrial fields. The most common problems in efficiency of robot links are caused by their unwanted vibrations that become much more important at the tip of the links. For modeling and control of flexible manipulators’ vibrations, many attempts have been conducted so far (Dwivedy and Eberhard, 2006).

Dynamic link analysis for different initial and boundary conditions is performed by Ata et al. (2012). In addition, viscoelastic serial robotic manipulators with motors at the joints are theoretically and experimentally investigated using the Timoshenko beam theory with the assumed mode method and Gibbs–Appell formulation (Korayem et al., 2015).

Lochan et al. (2016) presented a review article on a variety of methods for modeling and controlling two-link arms. In addition, Shafei and Shafei (2015) conducted a study on flexible multiple robot links captured inside a closed space. They also presented an automatic approach for dynamic modeling of the oblique impact of a multi-flexible-link robotic manipulator (Shafei and Shafei, 2016) and a general approach for dynamic synthesis of closed-loop mechanisms with link flexibility (Shafei and Shafei, 2019).

To model a flexible robot link, another method is developed by Wei et al. (2017) that is known as the global mode method. Kumar and Pratiher (2019) performed a theoretical study on nonlinear modeling and vibration analysis, for a two-link flexible mechanism.

Because of the residual vibrations of flexible joints and links, trajectory tracking is an important problem in studies related to the flexible robotic systems, and it is investigated through a variety of control strategies. The control strategies can be applied to the joints as control torques (tracking) or to the links in the form of vibration control (disturbance rejection). As an example, application of control strategies in trajectory tracking, control of motor torques at joints, and piezoelectric actuators on links are investigated for a flexible robotic manipulator based on the Lyapunov stability theory by Bottega et al. (2008). A new displacement approach is also presented to determine the optimal trajectory of flexible link manipulators in point-to-point motion by Heidari et al. (2013). In addition, a path planning algorithm for nonholonomic mobile flexible manipulators is presented, computing the robot trajectory by maximizing pose stability measures (Heidari et al., 2017).

In the field of vibration control, Sabatini et al., 2012 studied the application of active damping strategies for vibration reduction of a space manipulator with flexible links during orbit operations. Modeling and nonlinear control of the flexible single link with the state-dependent Riccati equation method is studied by Shawky et al. (2013). Bakhti and Idrissi (2013) studied the flexible manipulator vibration reduction using Kalman optimal filtering. Dubay et al. (2014) conducted a study for modeling and vibration suppression of a one-link flexible manipulator applying a combination method with piezoelectric actuators. In another study directed for trajectory and vibration control of the flexible robot links, a distributed higher order differential feedback controller is applied (Agee et al., 2017). A method for vibration avoidance throughout trajectories of a flexible robot manipulator is also proposed comprising the hysteretic and leading mechanisms to maintain a steady pace (Zhang, 2018). Furthermore, a nonlinear-state feedback controller along with energy shaping is provided for vibration suppression and overshoot reduction of motor position (Yin et al., 2018). Liang et al. (2018) also used proportional–differential feedback controller with strain and strain rate feedback for vibration control and trajectory tracking of a 2 degree-of-freedom (DOF) flexible parallel manipulator using piezoelectric transducers.

Garcia-Perez et al. (2019) studied the modeling and vibration control of flexible robot links using the finite element method (FEM), modal analysis, and sliding mode control methods. Furthermore, an active vibration control algorithm with noncontact vibration measurement using a structural light sensor is proposed based on the recurrent neural network by Qiu and Zhang (2019). Development of an interval type-2 fuzzy logic controller for real-time trajectory and vibration control of a flexible joint manipulator is also presented by Kelekci and Kizir (2019).

Sun et al. (2019) introduced a set-point regulation scheme of proxy-based sliding mode control for vibration suppression of flexible joint robot using link side energy feedback. A vibration control method is also presented for a two-link flexible manipulator that is a hybrid of active and passive control methods (Mishra and Singh, 2019). A trajectory planning method is also presented to suppress vibration of spatial flexible manipulators by Cui et al., 2020, solving the trajectory planning as an optimization problem with the particle swam optimization (PSO) method. In addition, vibration control of an inflatable link robot is investigated by Oliviera et al. using a built-in vision sensor and an active mechanism (Oliveira et al., 2020).

Because implementing vibration absorbers on robot links involves some difficulties, virtual vibration absorbers are developed that are inspired by passive vibration absorbers (Nissing, 2000; Shang-Teh and Yi-Chih, 2003). Virtual vibration absorbers’ frequency and damping coefficients are actively adjusted by simple control algorithms applied to the actuators (Bian and Gao, 2017). However, a magnetorheological elastomer vibration absorber is also proposed for semi-active vibration control of a flexible arm (Bian et al., 2018).

Most of the proposed methods are based on active control. Active control is more efficient than passive control for these types of systems, although complexity of active control is a challenge in application. In addition, active systems need professional maintenance. Therefore, the passive vibration suppression methods may be more accepted for industrial intentions because of simplicity. As an example, the passive compliant joint that is composed of a magnetorheological damper and a rotary spring is developed for vibration control of a safe arm (Kim et al., 2004).

In this study, application of a frictional damper is investigated for passive vibration reduction of a flexible manipulator. The plan is similar to the one proposed in studies of Ziegert et al. (2006) and Kim et al. (2006) that is further developed by Madoliat et al. (2011a, 2011b). The structure of these dampers is composed of a solid core press fitted inside a hollow multifingered cylinder. This set is again press-fitted inside an axial hole inside the main beam-shaped structure to increase its internal damping. The mentioned compound leads to energy dissipation in bending vibration of the robot link. Such dampers are implemented inside manipulators needing no maintenance and change in the appearance of the structure.

In Section 2, the dynamical equations governing the problem are derived based on the FEM. Subsequently, an initial model validation is performed. In Section 3, the design and optimization of the friction damper have been investigated based on mathematical modeling and various evolutionary optimization algorithms. The optimally damped link is then fabricated. In Section 4, modal test and motion tests are performed. The experimental results along with simulation results certify the advantages of the proposed strategy. In Section 5, conclusions of the research are stated.

2. Dynamic model of the flexible manipulator

In this section, the dynamic governing equation of a flexible manipulator is derived.

2.1. Deriving dynamic equation

The dynamic governing equation of a flexible manipulator is obtained using the FEM. A schematic of the manipulator with corresponding coordinate systems is shown in Figure 1. The link is modeled as a clamped-free Euler–Bernoulli beam which is imposed by inertial forces and couples. In Figure 1(a), a flexible link with payload is shown. Figure 1(b) shows a schematic of inertial load distribution on the link. In Figure 1(c), a cutoff section of the beam, with equivalent forces and moments applied to its center, is shown, which causes equivalent forces and couples on the cross-section area. In Figure 1(d), the link divides into some beam elements, and the equivalent forces and couples on each node of the link are shown. The complete description of Figure 1 is presented at the following.

To analyze the vibrations of the link, it is divided to n − 1 Euler–Bernoulli beam elements via n nodes with two DOF.

It should be noted that the payload mass effect must also be included in the mass matrix. For this purpose, the following 2 × 2 matrix $(M_{p b})$ that contains the mass properties of the payload is added to the block corresponding to the 1st node

M_{p b} = [\begin{matrix} M_{p} & 0 \\ 0 & I_{p} \end{matrix}]

(1)

where

M_{p}

and

I_{p}

denote the payload mass and moment of inertia respectively. Because the link has a homogeneous structure, mass and stiffness matrices for all elements are the same. After assembling the matrices, the total properties of the system can be obtained by applying the boundary conditions. Because the link is clamped at the last (nth) node, the columns and rows that are related to the nth node must be omitted. Applying the boundary conditions, the equation of motion for the link with payload can be obtained as follows

[M] {\begin{matrix} {\ddot{y}}_{1} \\ {\ddot{θ}}_{1} \\ ⋮ \\ {\ddot{y}}_{n - 1} \\ {\ddot{θ}}_{n - 1} \end{matrix}} + [C] {\begin{matrix} {\dot{y}}_{1} \\ {\dot{θ}}_{1} \\ ⋮ \\ {\dot{y}}_{n - 1} \\ {\dot{θ}}_{n - 1} \end{matrix}} + [K] {\begin{matrix} y_{1} \\ θ_{1} \\ ⋮ \\ y_{n - 1} \\ θ_{n - 1} \end{matrix}} = {\begin{matrix} F_{y (1)} \\ T_{(1)} \\ ⋮ \\ F_{y (n - 1)} \\ T_{(n - 1)} \end{matrix}} + {\begin{matrix} M_{p} L α \\ I_{p} α \\ 0 \\ ⋮ \\ 0 \end{matrix}}

(2)

[M] and [K] are the final (2n − 2) × (2n − 2) mass and stiffness matrices, and the damping matrix C is obtained from equation (3). Load vectors and response vectors also have 2n − 2 components. Parameter $F_{y (j)}$ denotes the lateral inertial force, and $T_{(j)}$ is the inertial torque imposed to the link at the jth node. Nodal loads are imposed to the link because of its inherent mass and moment of inertia. Likewise, $M_{p} L α$ is the inertial payload induced force in the axial and lateral direction, and $I_{p} α$ denotes the payload-induced inertial torque imposed to the link at its 1st node in payload presence. Parameter α is the angular acceleration of the angular motion

[C] = [φ] [M] [diag (2 ξ_{i} ω_{i})] [M] {[φ]}^{T}

(3)

In this equation, $[φ] = [\begin{matrix} {φ_{1}} & {φ_{2}} & \begin{matrix} \dots & {φ_{n - 1}} \end{matrix} \end{matrix}]$ is the orthonormalized modal matrix of the system. Here, $ω_{i}$ and $φ_{i}$ are obtained from the solution of the system’s characteristic equation, and $ξ_{i}$ is the structural damping of the ith mode extracted from the modal test.

A robot link angular motion generally causes eccentric and angular accelerations that impose inertial loads to the link. Intensities of the distributed axial and lateral loads are achieved as derivations of the related forces with respect to location variable z. Adapting the load distribution with FEM, it can be rewritten in a discrete form for the jth element as follows

W_{y (j)} = ρ A (L - [(j - 1) L_{e} + \bar{z}]) α

(4)

where L is the distance from the link end to the center of the rotating hub

(L = (L_{l} + R))

\bar{z}

is the location variable calculated from the beginning of each element to its end

(0 \leq \bar{z} \leq L_{e})

, and

L_{e} = L_{l} / (n - 1)

gives the length of elements which has a constant value for all elements. Parameter ρ denotes the density of the link, and A is the cross-section area of the link. The nodal loads are then obtained integrating the product of load intensities and the shape functions.

2.2. FEM validation

In this section, the first natural frequency of the link obtained via FEM with different numbers of elements is compared with its value from the exact solution of the Euler–Bernoulli beam. The geometrics and material properties of a typical robot link investigated in the present study are given in Table 1.

Table 1.

Geometrics and material properties of the investigated link.

Material	Polyvinyl chloride
Section shape	Circular
Density $(ρ)$	$1200 (kg / {cm}^{3})$
Link length $(L_{l})$	$0.4 m$
Link diameter (D = 2r₃)	$0.04 m$
Elastic modulus (E)	$1.3 GPa$

In Figure 2, convergence of the FEM results to the exact analytical result is shown with regard to the increase in the number of elements.

As it can be seen, for more than five elements, the FEM results converge to the exact solution with a rational error percentage of 4.63%. According to the performed validation, the simulations are all performed with five elements.

3. Design of the damper

In this section, the damper mechanism is discussed, and a mathematical model is presented to investigate its damping effect. To maximize the damper effect, an optimization process based on PSO, imperialist competitive algorithm (ICA), nondominated sorting genetic algorithm II (NSGA II), and grey wolf optimizer (GWO) is applied to the model.

3.1. Modeling of the damper

In Figure 3, the mechanism and structure of the damper is shown. As mentioned above, the damper consists of a core cylinder press-fitted inside a multifingered hollow cylinder press fitted-inside the link body. During bending deflection, the link body and damper parts are deflected all together (Figure 3(a)). As shown in Figure 3(b), the neutral axes of the fingers are not located on neutral axes of the core and link body. Therefore, bending-induced displacements of the system components on the contact lines will be different, and this difference causes sliding of the parts over the rough contact surfaces which are imposed by press-fitting pressure. When the system components slide over each other, an opposing frictional force occurs over the contact surfaces. The imposed frictional force over the sliding contact surfaces causes a dissipation of energy during the bending vibrations (Figure 3(c)).

In this section, the aim is to find a mathematical model for computing the amount of dissipated energy. This model is used for optimization of the damper structure to maximize its damping effect.

Considering the bending of the damped link structure under the effect of a single force at the tip of the link, the damped link components including the link body (denoted with index b), damper fingers (index fi for ith finger), and the core (with index c) will have the same deflections with respect to z as follows

\frac{F_{b}}{E_{b} I_{b}} (- z^{3} + 3 z L_{l}^{2} - 2 L_{l}^{3}) = \frac{F_{f i}}{E_{f i} I_{f i}} (- z^{3} + 3 z L_{l}^{2} - 2 L_{l}^{3}) = \frac{F_{c}}{E_{c} I_{c}} (- z^{3} + 3 z L_{l}^{2} - 2 L_{l}^{3})

(5)

where

F_{b}

F_{f i}

, and

F_{c}

are respectively the portions of the total force

(F_{t})

imposed to the link body at its tip, ith finger, and the core. It is obvious that the summation of these force portions must be equal to the amount of the total force as follows

F_{t} = F_{b} + F_{c} + \sum_{i = 1}^{N_{f}} F_{f i}

(6)

where

N_{f}

denotes the number of damper fingers. Combining equations (5) and (6) gives the following equation

\frac{F_{b}}{E_{b} I_{b}} = \frac{F_{f i}}{E_{f i} I_{f i}} = \frac{F_{c}}{E_{c} I_{c}} = \frac{F_{t}}{E_{b} I_{b} + E_{c} I_{c} + \sum_{j = 1}^{N_{f}} E_{f j} I_{f j}}

(7)

The displacement along the z-direction for the core surface, the inner surface of the link, and the internal and external surfaces of the fingers ith (respectively, $δ_{c}$ , $δ_{b}$ , $δ_{I f i}$ , $δ_{E f i}$ ) are obtained integrating the strain functions with respect to z as follows (Madoliat et al., 2011b)

δ_{c} = \frac{F_{c} r_{1} \sin γ_{1}}{2 E_{c} I_{c}} (L_{l}^{2} - z^{2})

(8)

δ_{b} = \frac{F_{b} r_{2} \sin γ_{2}}{2 E_{b} I_{b}} (L_{l}^{2} - z^{2})

(9)

δ_{I f i} = \frac{F_{f i} (r_{1} sin γ_{1} - {\bar{Y}}_{f i})}{2 E_{f i} I_{f i}} (L_{l}^{2} - z^{2})

(10)

δ_{E f i} = \frac{F_{f i} (r_{2} sin γ_{2} - {\bar{Y}}_{f i})}{2 E_{f i} I_{f i}} (L_{l}^{2} - z^{2})

(11)

where

r_{1}

is the core radius and

r_{2}

is the outer radius of the fingers. Index

I f_{i}

deals with the internal surface of the fingers, and

E f_{i}

index deals with the external surface of the finger. Subtracting the obtained displacement values, the amount of total slip between the contact surfaces is obtained that will be used for computing the frictional work of the damper. In addition,

(r_{2} sin γ_{2} - {\bar{Y}}_{f i})

and

(r_{1} sin γ_{1} - {\bar{Y}}_{f i})

are respectively the vertical distance between the center of the ith finger and any contact point on external and internal contact surfaces. Noting Figure 3(b),

{\bar{Y}}_{f i}

is the vertical distance between the overall neutral axis and center of the ith finger.

The frictional work is obtained integrating the product of the frictional force and slip between the components over the contact surfaces. The frictional force is a function of the pressure between the surfaces, the coefficient of friction, and the contact surface size. The internal contact surface is located at the radius of $r_{1}$ in angles of $ψ_{i} - (π / N_{f}) \leq γ_{1} \leq ψ_{i} + (π / N_{f}) - (s / r_{1})$ where $ψ_{i}$ deals with the angle between the link neutral axis (X) and a radius passing the link center and the ith finger center. Parameter s also denotes the width of the slits that are necessarily carved to shape the fingers. Similarly, the external contact surface is located at the radius of $r_{2}$ in angles of $ψ_{i} - (π / N_{f}) \leq γ_{2} \leq ψ_{i} + (π / N_{f}) - (s / r_{2})$ . The amount of frictional work for the damper on its external and internal surfaces is then calculated by integrating a product of the contact pressure, frictional coefficient, and absolute slip value over the contact surface area element as follows

\begin{array}{l} W_{E} = \sum_{i = 1}^{N_{f}} \int_{ψ_{i} - (π / N_{f})}^{ψ_{i} + (π / N_{f}) - (s / r_{2})} \int_{0}^{L_{l}} μ P_{E} r_{2} | δ_{E f i} - δ_{b} | d z d γ \\ = μ P_{E} r_{2} \sum_{i = 1}^{N_{f}} \int_{ψ_{i} - (π / N_{f})}^{ψ_{i} + (π / N_{f}) - (s / r_{2})} \int_{0}^{L_{l}} | \frac{r_{2}}{2} (\frac{F_{f i}}{E_{f i} I_{f i}} - \frac{F_{b}}{E_{b} I_{b}}) sin γ - \frac{F_{f i} {\bar{Y}}_{f i}}{2 E_{f i} I_{f i}} | (L_{l}^{2} - z^{2}) d z d γ \\ = \frac{L_{l}^{3}}{3} μ P_{out} r_{2} (\frac{2 π}{N_{f}} - \frac{s}{r_{2}}) \sum_{i = 1}^{N_{f}} \frac{F_{f i} | {\bar{Y}}_{f i} |}{E_{f i} I_{f i}} \end{array}

(12)

\begin{array}{l} W_{I} = \sum_{i = 1}^{N_{f}} \int_{ψ_{i} - (π / N_{f})}^{ψ_{i} + (π / N_{f}) - (s / r_{1})} \int_{0}^{L_{l}} μ P_{I} r_{1} | δ_{I f i} - δ_{c} | d z d γ \\ = μ P_{I} r_{1} \sum_{i = 1}^{N_{f}} \int_{ψ_{i} - (π / N_{f})}^{ψ_{i} + (π / N_{f}) - (s / r_{1})} \int_{0}^{L_{l}} | \frac{r_{1}}{2} (\frac{F_{f i}}{E_{f i} I_{f i}} - \frac{F_{c}}{E_{c} I_{c}}) sin γ - \frac{F_{f i} {\bar{Y}}_{f i}}{2 E_{f i} I_{f i}} | (L_{l}^{2} - z^{2}) d z d γ \\ = \frac{L_{l}^{3}}{3} μ P_{in} r_{1} (\frac{2 π}{N_{f}} - \frac{s}{r_{1}}) \sum_{i = 1}^{N_{f}} \frac{F_{f i} | {\bar{Y}}_{f i} |}{E_{f i} I_{f i}} \end{array}

(13)

The parameter μ denotes the friction coefficient of the surfaces, $P_{I}$ is the contact pressure on the internal surface, and $P_{E}$ is the contact pressure on the external surface. The total work is equal to the summation of these works. The parameter ${\bar{Y}}_{f i}$ for each ith finger can be easily obtained regarding the related parameters and their summation is expressed as follows

\sum_{i = 1}^{N_{f}} | {\bar{Y}}_{f i} | = (\frac{2 N_{f} (r_{2}^{3} - r_{1}^{3})}{3 π (r_{2}^{2} - r_{1}^{2})}) \sin (\frac{π}{N_{f}}) \sum_{i = 1}^{N_{f}} | \sin ψ_{i} |

(14)

Calculation of the

\sum_{i = 1}^{N_{f}} | \sin ψ_{i} |

value is a bit controversial as it depends on the angular position of the fingers with respect to neutral axis (X), and it has a range of values. Table 2 denotes the maximum, minimum, and average values for this parameter regarding numbers of fingers.

Table 2.

Maximum, minimum, and average values of $\sum_{i = 1}^{N_{f}} | sin ψ_{i} |$ regarding the number of fingers.

Number of fingers	${(\sum_{i = 1}^{N_{f}} \| \sin ψ_{i} \|)}_{\min}$	${(\sum_{i = 1}^{N_{f}} \| \sin ψ_{i} \|)}_{\max}$	${(\sum_{i = 1}^{N_{f}} \| \sin ψ_{i} \|)}_{ave}$
2			1
2	0	2	1
3			$1 + (\sqrt{3} / 2)$
3	$\sqrt{3}$	22	$1 + (\sqrt{3} / 2)$
4			$1 + \sqrt{2}$
4	2	$2 \sqrt{2}$	$1 + \sqrt{2}$
$N_{f}$			$(1 / 2) [\sum_{i = 1}^{N_{f}} \| \sin ((π (2 i - 2) / N_{f})) \| + \sum_{i = 1}^{N_{f}} \| \sin ((π (2 i - 2) / N_{f})) \|]$
$N_{f}$	$\sum_{i = 1}^{N_{f}} \| \sin ((π (2 i - 2) / N_{f})) \|$	$\sum_{i = 1}^{N_{f}} \| \sin ((π (2 i - 2) / N_{f})) \|$

The average of

\sum_{i = 1}^{N_{f}} | \sin ψ_{i} |

obtained from Table 3 with respect to

N_{f}

can be used to find the average value of the total frictional work as follows

{(W_{T})}_{ave} = \frac{2 N_{f} L_{l}^{3}}{9 π} μ (P_{I} r_{1} (\frac{2 π}{N_{f}} - \frac{s}{r_{1}}) + P_{E} r_{2} (\frac{2 π}{N_{f}} - \frac{s}{r_{2}})) \times (\frac{F_{t}}{E_{b} I_{b} + E_{c} I_{c} + \sum_{j = 1}^{N_{f}} E_{f j} I_{f j}}) \times \frac{(r_{2}^{3} - r_{1}^{3})}{(r_{2}^{2} - r_{1}^{2})} \sin (\frac{π}{N_{f}}) \times {(\sum_{i = 1}^{N_{f}} | \sin ψ_{i} |)}_{ave}

(15)

Table 3.

Best optimization results for different algorithms.

Optimization parameter	PSO	ICA	NSGA-II	GWO
$N_{f}$	6	6	6	6
$r_{1} (mm)$	5.5	5.1	5	5
$r_{2} (mm)$	15.2	14.9	15	15
$W_{0}$	6.93	6.54	7.85	7.85

PSO: particle swam optimization; ICA: imperialist competitive algorithm; NSGA II: nondominated sorting genetic algorithm II; GWO: grey wolf optimizer.

Solving the stress equilibrium and applying boundary conditions for the press fitting induced strain, the contact pressures on internal and external contact surfaces ( $P_{I}$ and $P_{E}$ respectively) are obtained. The final results are summarized as follows, giving these terms with respect to radii and elasticity moduli of the involved parts which are considered to be equal $(E_{b} = E_{c} = E_{f j} = E)$

P_{I} = \frac{β_{2} e_{1} - β_{1} e_{2}}{β_{2} α_{1} - β_{1} α_{2}}

(16)

P_{E} = \frac{α_{1} e_{2} - α_{2} e_{1}}{β_{2} α_{1} - β_{1} α_{2}}

(17)

where

α_{1} = - β_{1} = \frac{2 r_{1} r_{2}^{2}}{E (r_{2}^{2} - r_{1}^{2})}

(18)

α_{2} = \frac{- 2 r_{2} r_{1}^{2}}{E (r_{2}^{2} - r_{1}^{2})}

(19)

β_{2} = \frac{2 r_{2}}{E} (\frac{r_{3}^{2}}{(r_{3}^{2} - r_{2}^{2})} + \frac{r_{1}^{2}}{(r_{2}^{2} - r_{1}^{2})})

(20)

where the interference between the internal and external contact surfaces is

e_{1} = e_{2} = 1 \times 10^{- 5} m

, that is a constant value assigned regarding the limitations in accuracy of manufacturing instruments. These small values are assigned to avoid plastic deformation during press fitting that might cause lack of slip. The parameter

r_{3}

also deals with external radius of the investigated robot link that is equal to 0.02 m.

3.2. Optimization

In the present section, maximizing the damper effect is performed using variety of optimization algorithms. At first, equation (15) is simplified as follows to obtain a proper objective function omitting the constants

W_{O} = [r_{1} P_{I} (\frac{2 π}{N_{f}} - \frac{s}{r_{1}}) + r_{2} P_{E} (\frac{2 π}{N_{f}} - \frac{s}{r_{2}})] (\frac{r_{2}^{3} - r_{1}^{3}}{r_{2}^{2} - r_{1}^{2}}) N_{f} \sin (\frac{π}{N_{f}}) {(\sum_{i = 1}^{N_{f}} | \sin ψ_{i} |)}_{ave}

(21)

Equation (21) gives the damper performance index with respect to three damper design parameters ( $N_{f}$ , $r_{1}$ , and $r_{2}$ ) as the optimization variables. Ranges and constraints of the optimization variables are assigned with respect to manufacturing processes and physical limitations. Consulting a manufacturing expert, avoiding a too thin wall for the link, the axial hole inside the link must not have a diameter larger than 0.03 m. The core diameter also must not be smaller than 0.01 m as it may cause problems during the press-fitting process. Furthermore, the feasibility limitations can be stated in brief as $0.005 m \leq r_{1} < r_{2} \leq 0.015 m$ .

The number of fingers that is equal to the number of carved slits must be logically an integer with a minimum value of two. As carving the slits causes a decrease in contact surface areas, their width (s) must be decreased as much as possible. Consulting a manufacturing expert, the minimum possible value for the slit width is chosen as s = 0.0015 m. Therefore, the number of slits must be limited as increase in their number leads to lack of contact especially on the internal contact surface. This limitation can be expressed as $s N_{f} < 2 π r_{1}$ that can be summarized as $2 \leq N_{f} < 4188.79 r_{1}$ where $N_{f} \in ℕ$ .

Having the objective function and the constraints, PSO, ICA, NSGA-II, and GWO algorithms are used to maximize the damper effect. Each optimization algorithm is applied for 50 times, and the best results are tabulated in Table 3. The results reveal that the best damper performance is obtained for the minimum possible value of $r_{1}$ (0.005 m), maximum possible value of $r_{2}$ (0.015 m), and six fingers $(N_{f} = 6)$ .

To assure the optimization result, the objective function value is shown in Figure 4 as polar plots with respect to $r_{1}$ and $r_{2}$ for different values of $N_{f}$ from 2 to 9. The polar plots make the objective function variation with respect to the effective parameters more comprehensible. It is observed that decrease in $r_{1}$ and increase in $r_{2}$ both lead to growth in objective function value, and the maximum value of the objective function occurs at $N_{f} = 6$ .

4. Experiments and simulations

In this section, the influence of the proposed frictional damper on vibration reduction of a robot link is investigated through experiments and simulations.

4.1. Modal test

To perform an experimental investigation, a specimen of the damped link is fabricated noting the optimization results. The axial holes and fingers are carved using a computer numerical control wire cut machine with the ultimate achievable accuracy. The damped link parts are then assembled as shown in Figure 5, beside a simple link.

The modal test is used to obtain the modal properties of both links with a setup that is shown in Figure 5(b). Preparing the clamped-free boundary condition, the links are fastened inside a cylindrical hub with a radius of $R = 0.06 m$ and fixed inside a fixture.

Two frequency response functions (FRFs) are achieved for different directions, including G_xx in X and G_yy in Y directions. For each FRF, the modal properties including the first natural frequency and the associated damping ratio are obtained applying the line fitting approach. The results are presented in Table 4 for both links.

Table 4.

Modal test results for both links.

FRF direction	Simple link properties		Damped link properties
FRF direction	$ω_{1}$ (rad/s)	$ξ_{1}$	$ω_{1}$ (rad/s)	$ξ_{1}$
G_xx	241.8132	0.0150	241.8432	0.0339
G_yy	241.8114	0.0152	241.8410	0.0327
Average	241.8123	0.0151	241.8421	0.0333

FRF: frequency response function.

The results show a notable increase in the damping ratio. The experimentally obtained G_xx FRFs for both links are shown in Figure 6. The presented FRFs reveal that using the damped link can reduce the vibration amplitude of the link for more than 50% in the first natural frequency.

4.2. Angular motion test and simulations

Having an initial realization of the damper effect on vibration of an elastic robot link, a time-domain simulation is performed using Simulink of MATLAB. The numerical results are obtained solving the simulated models with the ode45 variable step algorithm. A harmonic reciprocal motion is applied to the links with amplitude of $π / 4$ with the following resulting angular acceleration

α = - \frac{π}{4} Ω^{2} sin Ω t

(22)

Considering $ξ_{i} = ξ_{1}$ for all modes, the damping matrix is obtained through equation (3) for both simple and damped links. The equation of vibration (equation (2)) is then simulated in the Simulink software environment for both links that are different only in their damping matrices.

To investigate the link vibration in its most critical situation, the link vibration is simulated to have a reciprocal movement with its first natural frequency, $Ω = ω_{1}$ . For a link without payload, the first natural frequency is obtained equal to 36.4 Hz via FEM. The lateral tip deflections of the links at first resonance are shown in Figure 7 with respect to the time in the absence of payload.

The simulation results of Figure 7 show that a 47.3% decrease in the magnitude of the steady-state vibration response is achieved using the proposed damped link. For a link with a 1 kg payload $(M_{p} = 1 kg and I_{p} = 9.1 \times 10^{- 4} kg m^{2})$ , the first natural frequency is obtained equal to 13 Hz via FEM. The simulation is performed again for the same links at the resonance frequency in the presence of payload. The simulation results showing lateral deflections of the link tips in the presence of payload are exposed in Figure 8.

The results of Figure 8 show that using the proposed damper in a robot link reduces the steady-state link vibration amplitude up to 53.2% in the presence of the payload. Therefore, simulation of the links at first resonance shows that using the proposed damped link can reduce the link vibration amplitude up to half in comparison with a regular robot link.

Further investigating the damped link performance, an experimental test is applied to both links in the presence and absence of the payload. The test equipment and setup are shown in Figure 9. The setup system consists of two main subsystems for actuation and condition monitoring.

The tests are performed for the links at $Ω = 3 Hz$ in the presence and absence of payload. Performing the actuation, the link must be connected to a servomotor shaft via its hub. The servomotor is controlled through a feedback control system consisting of a digital signal processing control circuit and an inverter. The motor program is given to the control circuit from a PC system using code composer software. The control circuit takes feedback signals including current, voltage, and position from the motor. Taking feedback signals and the motor program, the control circuit sends gate signals to the inverter insulated-gate bipolar transistor switches whose function is to turn the DC power source on or off on the motor.

The sensing system includes a strain gauge sensor, an analyzer, and a PC system. The strain gauge is placed close to the clamped end of the robot link at z = 0.37 m, giving the $ε$ value at that point. The strain gauge data are then amplified and transferred to the PC where the data are recorded using condition monitoring software. The link tip deflection can be extracted with respect to the measured strain (Korayem et al., 2012).

The first experiment is performed in the absence of payload, and the results are shown in Figure 10. Here, the simulation results reveal an 18.67% decrease, and the experimental results show a 19.45% decrease in vibration amplitude.

To investigate the damper effect in the presence of payload, the experiments and simulations are repeated in the presence of the aforementioned payload, and the results are shown in Figure 11. In this case, simulations and experiments respectively show close results of a 33.91% and 35.04% decrease in vibration amplitude.

As can be seen, the experimental and simulation results are in very good agreement with each other, affirming the proposed frictional damper performance in vibration reduction of a robot link. Furthermore, the results show that growth in vibration amplitude raises the damper effect. The reason is that when the vibration amplitude increases, a larger slip between the contact surfaces of the damper and the link happens. Increase in slip leads to rise in the amount of dissipated energy that causes a larger vibration reduction effect.

5. Conclusions

In this study, a passive frictional damper is proposed for reducing lateral vibrations of a flexible robot link, and its effectiveness is investigated. The plan contains an economic passive strategy that is comparable with active strategies with no complications. The robot link is modeled as a clamped-free beam in angular motion. A finite element model is used for analytical study of the link vibrations using five Euler–Bernoulli beam elements.

An analytical model is also presented to evaluate the proposed damper effect in vibration reduction. This model is constructed to calculate the amount of dissipated energy during the link deflection by computing the friction force and sliding between contact surfaces. Noting the constructed model, an objective function is assigned, and it is used for optimization of the damper. Maximizing the damper effect, the objective function is maximized noting all constraints and limitations via PSO, ICA, NSGA-II, and GWO algorithms. The optimization results reveal that the maximum damper effect is achieved by setting $r_{1}$ equal to its minimum possible value, $r_{2}$ equal to its maximum possible value, and $N_{f}$ equal to 6.

Applying the modal test, it is observed that the first mode damping ratio of the link increased from 0.0151 to 0.0333, meaning a 120% improvement. Having the damping ratio of the links, their reciprocal angular motion is simulated using the constructed finite element model. The simulations are preformed for both links in the presence and absence of payload in the first resonance frequency. The simulation results show a 47.3% and 53.2% reduction in steady-state vibration amplitude in the absence and presence of the payload respectively.

The links are finally tested in a reciprocal motion with a frequency of 3 Hz. The experiment in the absence of payload shows a 19.45% reduction in link tip deflection, and in the presence of payload, the experiment shows a 35.04% reduction. The test results also validate the simulation results properly. In addition, the results show that as the dynamic excitation conditions become harsher, the damper effect becomes much more sensible, which is because of growth in the sliding of contact surfaces.

The proposed damper is used for a circular cross-section link because its shape is circular. However, it can be applied for any beam-shaped structures with other solid cross sections. For flat, hollow, or other forms, the plan may be adapted using multiple dampers implemented in solid areas of the cross section. Furthermore, the effect of such dampers can be raised via increasing the damper layers and contact pressure.

Footnotes

Declaration of conflicting interests

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

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Masoud Golmohammadi

Sajad Hayati

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