Design of the symmetric tuned mass damper for torsional vibration control of the rotating shaft

Abstract

The shaft is one of the most important parts of the machine, and it is used to transmit torque. However, the shaft does not always rotate at constant angular velocity due to sudden acceleration or deceleration or due to unstable current. The rotation of the shaft varies with time, which causes torsional vibration on the rotating shaft. To the best of the author’s knowledge, there is no study on designing a symmetric tuned mass damper (STMD) for the rotating shaft with variable angular velocity. Therefore, the purpose of this study is to design an optimal STMD to reduce torsional vibration for the rotating shaft with variable angular velocity. First, the author designs an optimal STMD for the rotating shaft by the fixed-points theory. Second, the optimal parameters of the STMD are obtained by using the minimum quadratic torque method. The optimal parameters of the STMD are defined in analytic and explicit forms, helping researchers to easily design an optimal STMD when applying to reduce torsional vibration for the rotating shaft. Finally, to evaluate the reliability of the designed optimal STMD, Maple software is used to simulate the vibration of the rotating shaft attached with the optimal STMD, as well as to help the readers to have a visual view on the effect of reducing torsional vibration of the rotating shaft.

Keywords

Symmetric tuned mass damper torsional vibration minimum quadratic torque the rotating shaft fixed-points theory

Introduction

Rotating machines are often subjected to fluctuating torsional loads that can cause noise and torsional vibration. The torsional vibration of a rotating shaft is mainly due to sudden acceleration or deceleration or due to unstable current. Excessive torsional vibration can result in unwanted noise and powertrain component wear and, in severe cases, break the shaft. Therefore, they need to be eliminated or controlled immediately to ensure the safety of mechanical devices. Passive vibration control is widely used in reducing vibration in the main system due to its simplicity and acceptable effects.^1–20

A centrifugal pendulum vibration absorber (CPVA) has masses mounted on a rotor in such a way that they can freely move relative to the rotor along the prescribed paths related to rotating parts, and these movements are used to counteract the torsional moment, thus reducing torsional vibration for rotating systems. In 1929, Carter³ studied a form of the CPVA for diesel engines. Later, CPVAs with different designs were introduced for a wider range of operating conditions of the system. In 1936, a CPVA was proposed by Taylor¹⁵ for aircraft engines, in which Taylor designed the weight of the centrifugal mass so that the recovery force changes with speed. In 1937, Sarazin¹³ introduced a CPVA consisting of a pendulum with rollers applicable to aircraft engines. During the early years, most CPVAs were designed with circular path profiles. Later, dynamic vibration absorbers (DVAs) have been designed with various noncircular path types; for example, Chao et al.⁴ have studied the epicycloidal path, Denman⁶ has introduced the tautochronic curve, and Madden¹⁰ has considered the cycloidal path. In 2014, tautochronic pendulum vibration absorbers were studied by Mayet and Ulbrich,¹¹ in which they presented a general approach for the design and analysis of tautochronic bifilar and non-bifilar vibration absorbers. The configurations with cycloidal and epicycloidal paths were studied by Vitaliani et al.¹⁷ to highlight the vibration absorbing capabilities of CPVAs. A rigid rotor with moving components with capabilities of simultaneously tilting, translating, and rotating has been studied by Shi et al.¹⁴ to reduce the vibration in the system by using CPVAs.

A tuned mass damper (TMD) or DVA, which consists of a moving mass attached to the main structure through springs and dampers, is found to be an efficient, reliable, and low-cost vibration suppression device. To determine optimal parameters of the TMD and DVA to reduce the vibration of the main structure, there have been many optimization criteria to design the TMD and DVA. Den⁵ gave the fixed-points theory (FPT). Luft⁹ introduced the maximization of the equivalent viscous resistance method. Warburton¹⁹ studied and proposed the minimum quadratic torque method. Korenev and Reznikov⁷ gave the exact solution of a DVA’s optimal parameters based on stability maximization. The minimum kinetic energy method was studied by Truhar.¹⁶ These are the well-known optimization criteria for determining optimal parameters of the TMD and DVA to reduce the vibration of the main structure.

Details on the use of the CPVA for reducing torsional vibration of the rotating systems are very rich in the literature. But, the use of the DVA and TMD to reduce torsional vibration for the rotating systems is very limited in the literature. In 2018, Vu et al. ¹⁸ studied the optimal parameter of the DVA to reduce torsional vibration for the shaft by the FPT. Also similar to the CPVA, when designing the TMD and DVA to reduce vibration for the main system, details on the shape of the absorbers are quite rich in the literature, depending on the type of structure to be installed. Since the machine shaft often has a large rotation speed, if the design of the absorber is not scientific, it will cause eccentric rotation, which in turn produces very large centrifugal and Coriolis forces. These two inertial forces destabilize the structure, leading to the destruction of shaft bearings. An optimal model for reducing torsional vibration of the machine shaft with the symmetric TMD in the form of mass–spring–pendulum has been studied in Ref. 12. This optimal model eliminates the centrifugal and Coriolis forces of the absorber acting on the rotating shaft and thus increases the stability of the rotating shaft. However, Chinh¹² only studied the rotating shaft with constant angular velocity. But, the shaft does not always rotate at constant angular velocity due to sudden acceleration or deceleration or due to unstable current. The rotation of the shaft varies with time, which causes torsional vibration on the rotating shaft. To the best of the author’s knowledge, there is no study on designing a symmetric tuned mass damper (STMD) for the rotating shaft with variable angular velocity. Therefore, to overcome the limitations and develop the research results in Ref. 12, in this article the author continues to find the optimal parameters of the STMD to reduce torsional vibration varying with time for the rotating shaft.

Equations of motion

Figure 1 shows the modeling of a rotating shaft attached with the STMD. The rotating shaft is connected to the STMD via a disc. The main system consists of a rotating shaft and a disc, with radius $ρ$ , mass $M$ , and torsional spring constants $k_{t}$ . The STMD has the mass–spring–pendulum type with length $2 L$ , mass $2 m$ , damping constant $c$ , and spring constant $k_{m}$ . The difference of this model compared with the models of the conventional CPVA is that the mass of the STMD is symmetrical at the center of the rotating shaft so that the centrifugal and Coriolis forces acting on the STMD are at the opposite direction and have the same magnitude and therefore, they cancel out each other. Another difference is that a damper is installed in the STMD, which dissipates energy for the system to eliminate vibration of the rotating shaft.

Figure 1.

Modeling a rotating shaft attached with a symmetric tuned mass damper.

The torsional vibration of the rotating shaft can be determined as

θ = φ_{1} - φ_{2}

(1)

where φ₁ and φ₂ are the angular displacement of the disc and shaft end, respectively.

The system has two degrees of freedom: θ and φ, in which independent generalized coordinates are relatively the angle of rotation of the disc to the shaft end θ and relatively the angle of rotation of the STMD to the disc φ. The kinetic energy, the potential energy, and the energy dissipation function are, respectively

T = \frac{1}{2} M ρ^{2} {(\dot{θ} + {\dot{φ}}_{2})}^{2} + \frac{1}{3} m_{t} L^{2} {(\dot{θ} + \dot{φ} + {\dot{φ}}_{2})}^{2} + m L^{2} {(\dot{θ} + \dot{φ} + {\dot{φ}}_{2})}^{2}

(2)

Π = \frac{1}{2} k_{m} φ^{2} + \frac{1}{2} k_{t} θ^{2} + const

(3)

Φ = 2 [\frac{1}{2} c {(L \dot{φ})}^{2}]

(4)

Using Lagrangian equations, the system equations of motion for the rotating shaft attached with the STMD can be written as

\begin{array}{l} \frac{d}{d t} (\frac{\partial T}{\partial \dot{θ}}) - \frac{\partial T}{\partial θ} = - \frac{\partial Π}{\partial θ} - \frac{\partial Φ}{\partial \dot{θ}} \\ \frac{d}{d t} (\frac{\partial T}{\partial \dot{φ}}) - \frac{\partial T}{\partial φ} = - \frac{\partial Π}{\partial φ} - \frac{\partial Φ}{\partial \dot{φ}} \end{array}

(5)

Substituting equations (2)–(4) into equation (5) yields

[3 M ρ^{2} + 2 (m_{t} L^{2} + 3 m L^{2})] \ddot{θ} + 2 (m_{t} L^{2} + 3 m L^{2}) \ddot{φ} + 3 k_{t} θ = - [3 M ρ^{2} + 2 (m_{t} L^{2} + 3 m L^{2})] {\ddot{φ}}_{2}

(6)

(2 m_{t} L^{2} + 6 m L^{2}) \ddot{θ} + 2 (2 m_{t} L^{2} + 6 m L^{2}) \ddot{φ} + 6 c L^{2} \dot{φ} + 3 k_{m} φ = - (2 m_{t} L^{2} + 6 m L^{2}) {\ddot{φ}}_{2}

(7)

To write the equations of motion in a nondimensional form, the following parameters are presented

\begin{array}{l} μ = \frac{3 m + m_{t}}{3 M}, ω_{D}^{2} = \frac{k_{t}}{M ρ^{2}}, ω_{d}^{2} = \frac{3 k_{m}}{2 (3 m + m_{t}) L^{2}}, \\ ξ = \frac{3 c}{2 (3 m + m_{t}) ω_{d}}, ω_{d} = α ω_{D}, L = γ ρ, ω = β ω_{D} \end{array}

(8)

The parameters in equation (8) are described in Table 1.

Table 1.

Parameters used to write the nondimensional equations of motion.

Symbol	Description
µ	Ratio between the mass of the STMD and the mass of the main system
α	Tuning ratio of the STMD
ξ	Damping ratio of the STMD
ω _D	Natural frequency of vibration of the main system
ω _d	Natural frequency of vibration of the STMD
ω	Frequency of angular acceleration of the rotating shaft
$β$	Frequency ratio
γ	Ratio between the length of the pendulum and the radius of gyration of the main system

STMD: symmetric tuned mass damper.

Substituting the expressions of equation (8) into equations (6) and (7), the matrix equation of the system can be rewritten as

[\begin{matrix} 1 + 2 μ γ^{2} & 2 μ γ^{2} \\ 1 & 1 \end{matrix}] [\begin{array}{l} \ddot{θ} \\ \ddot{φ} \end{array}] + [\begin{matrix} 0 & 0 \\ 0 & 2 ξ α ω_{D} \end{matrix}] [\begin{array}{l} \dot{θ} \\ \dot{φ} \end{array}] + [\begin{matrix} ω_{D}^{2} & 0 \\ 0 & ω_{D}^{2} α^{2} \end{matrix}] [\begin{matrix} θ \\ φ \end{matrix}] = \{\begin{matrix} - 1 - 2 μ γ^{2} \\ - 1 \end{matrix}\} {\ddot{φ}}_{2}

(9)

The right-hand side of equation (9) represents the inertial force caused by the angular acceleration of the rotating shaft; this inertial force causes torsional vibration for the rotating shaft. Therefore, equation (9) is used to design the STMD.

Determination of optimal parameters by the FPT

Suppose that the rotating shaft according to the law of harmonic

{\ddot{φ}}_{2} = A_{0} e^{i ω t}

(10)

Combining equation (9) with (10), we obtain the equation of vibration as follows

θ (t) = \frac{[2 α β ξ (2 γ^{2} μ + 1) i + (2 α^{2} γ^{2} μ + α^{2} - β^{2})] A_{o}}{[2 α β ξ (2 μ β^{2} γ^{2} + β^{2} - 1) i + (2 α^{2} β^{2} γ^{2} μ + α^{2} β^{2} - β^{4} - α^{2} + β^{2})] ω_{D}^{2}} e^{i ω t}

(11)

We consider the largest amplitude of the rotating shaft to calculate the optimal parameters of the STMD when e^iωt = 1. Substituting the value e^iωt = 1 into expression (11) leads to

θ_{Max} = \frac{[2 α β ξ (2 γ^{2} μ + 1) i + (2 α^{2} γ^{2} μ + α^{2} - β^{2})] A_{o}}{[2 α β ξ (2 μ β^{2} γ^{2} + β^{2} - 1) i + (2 α^{2} β^{2} γ^{2} μ + α^{2} β^{2} - β^{4} - α^{2} + β^{2})] ω_{D}^{2}}

(12)

The magnitude of $θ_{Max}$ is

| θ_{Max} | = \sqrt{\frac{{(4 α β γ^{2} μ ξ + 2 α β ξ)}^{2} + {(2 α^{2} γ^{2} μ + α^{2} - β^{2})}^{2}}{[{(4 α β^{3} γ^{2} μ ξ + 2 α β^{3} ξ - 2 α β ξ)}^{2} + {(2 α^{2} β^{2} γ^{2} μ + α^{2} β^{2} - β^{4} - α^{2} + β^{2})}^{2}]}} | \frac{A_{o}}{ω_{D}^{2}} |

(13)

Set the amplitude magnification factor E as follows

E = \sqrt{\frac{{(4 α β γ^{2} μ ξ + 2 α β ξ)}^{2} + {(2 α^{2} γ^{2} μ + α^{2} - β^{2})}^{2}}{[{(4 α β^{3} γ^{2} μ ξ + 2 α β^{3} ξ - 2 α β ξ)}^{2} + {(2 α^{2} β^{2} γ^{2} μ + α^{2} β^{2} - β^{4} - α^{2} + β^{2})}^{2}]}}

(14)

Figure 2 shows the graphs of the amplitude magnification factor E versus the frequency ratio β corresponding to some different values of the damping ratio ξ. We can observe from these graphs that there exist two fixed points A and B. Therefore, it is possible to apply the FPT to determine the optimal parameters. According to Den⁵, we have

\frac{\partial E}{\partial ξ} = 0

(15)

E_{A} = E_{B}

(16)

Figure 2.

Graphs of the amplitude magnification factor versus the frequency ratio β

Combining equations (15) and (16) with equation (14), the optimal parameters of $α$ and $β$ are defined as

α_{opt}^{FPT} = α = \frac{\sqrt{1 - μ γ^{2}}}{2 μ γ^{2} + 1}

(17)

β_{1}^{2} = \frac{\sqrt{μ} γ + 1}{(2 μ γ^{2} + 1)}

(18)

β_{2}^{2} = \frac{1 - \sqrt{μ} γ}{2 μ γ^{2} + 1}

(19)

In the next step, the STMD’s damping ratio ξ is determined as

\frac{\partial E}{\partial β} = 0

(20)

Using equations (17)–(20) and (14), we get the following results

ξ_{1}^{2} = \frac{(1 + 2 μ γ^{2}) (μ γ + \sqrt{μ}) (μ γ^{2} - 4 \sqrt{μ} γ + 3) γ^{2}}{4 (μ^{2} γ^{4} - 2 μ γ^{2} + 1) (4 μ^{2} γ^{4} + 4 μ γ^{2} + 1)}

(21)

ξ_{2}^{2} = - \frac{\sqrt{μ} γ^{2} (μ γ + 3 \sqrt{μ})}{4 (2 μ γ^{2} - 1) (2 μ γ^{2} + 1)}

(22)

The optimal damping ratio of the STMD is found as²

ξ_{opt}^{FPT} = ξ_{opt} = \sqrt{\frac{ξ_{1}^{2} + ξ_{2}^{2}}{2}}

(23)

Substituting equations (21) and (22) into (23), we obtain the optimal value of $ξ$ as follows

ξ_{opt}^{FPT} = \frac{γ}{2} \sqrt{\frac{3 μ}{(1 + 2 μ γ^{2}) (1 - μ γ^{2})}}

(24)

From equations (17) and (24), we obtain the optimal parameters of the STMD to reduce the torsional vibration of the rotating shaft by using the FPT.

Determination of optimal parameters by the minimum quadratic torque method (MQT)

The minimum quadratic torque method is used for the determination of quadratic torque; this method has been given by Warburton.¹⁹ From equation (9), the equation of state is determined as

[\begin{array}{l} \dot{θ} \\ \dot{φ} \\ \ddot{θ} \\ \ddot{φ} \end{array}] = [\begin{matrix} B_{11} & B_{12} & B_{13} & B_{14} \\ B_{21} & B_{22} & B_{23} & B_{24} \\ B_{31} & B_{32} & B_{33} & B_{34} \\ B_{41} & B_{42} & B_{43} & B_{44} \end{matrix}] [\begin{matrix} θ \\ φ \\ \dot{θ} \\ \dot{φ} \end{matrix}] + [\begin{matrix} 0 \\ 0 \\ - 1 \\ 0 \end{matrix}] \ddot{φ_{2}}

(25)

in which

\begin{array}{l} B_{11} = 0; B_{21} = 0; B_{31} = - ω_{D}^{2}; B_{41} = ω_{D}^{2}; \\ B_{12} = 0; B_{22} = 0; B_{32} = 2 μ γ^{2} α^{2} ω_{D}^{2}; B_{42} = - (1 + 2 μ γ^{2}) α^{2} ω_{D}^{2}; \\ B_{13} = 1; B_{23} = 0; B_{33} = 0; B_{43} = 0; \\ B_{14} = 0; B_{24} = 1; B_{34} = 4 μ γ^{2} ξ α ω_{D}; B_{44} = - 2 (1 + 2 μ γ^{2}) ξ α ω_{D} \end{array}

(26)

Suppose that the angular acceleration of the rotating shaft, ${\ddot{φ}}_{2}$ , has a form of white noise of spectral density λ. According to Warburton,¹⁹ the quadratic torque matrix P is a solution of the Lyapunov equation (27)

B P + P B^{T} + λ H H^{T} = 0

(27)

where

B = [\begin{matrix} B_{11} & B_{12} & B_{13} & B_{14} \\ B_{21} & B_{22} & B_{23} & B_{24} \\ B_{31} & B_{32} & B_{33} & B_{34} \\ B_{41} & B_{42} & B_{43} & B_{44} \end{matrix}], H = [\begin{matrix} 0 \\ 0 \\ - 1 \\ 0 \end{matrix}]

(28)

Substituting equations (26) and (28) into (27), the matrix $P$ can be determined as

P = [\begin{matrix} P_{11} & P_{12} & P_{13} & P_{14} \\ P_{21} & P_{22} & P_{23} & P_{24} \\ P_{31} & P_{32} & P_{33} & P_{34} \\ P_{41} & P_{42} & P_{43} & P_{44} \end{matrix}]

(29)

in which

P_{11} = \frac{2 λ [\frac{1}{16} + α^{4} {(\frac{1}{2} + μ γ^{2})}^{4} + 2 {(μ γ^{2} + \frac{1}{2})}^{2} [μ (ξ^{2} + \frac{1}{4}) γ^{2} + \frac{1}{2} ξ^{2} - \frac{1}{4}] α^{2}]}{ω_{D}^{3} α γ^{2} μ ξ}

(30)

P_{12} = \frac{(μ γ^{2} + \frac{1}{2}) [{(μ γ^{2} + \frac{1}{2})}^{2} α^{2} + \frac{1}{2} μ γ^{2} - \frac{1}{4}] λ}{ω_{D}^{3} α γ^{2} μ ξ}; P_{13} = 0; P_{14} = \frac{{(μ γ^{2} + \frac{1}{2})}^{2} λ}{μ γ^{2} ω_{D}^{2}}

(31)

P_{21} = \frac{(μ γ^{2} + \frac{1}{2}) [{(μ γ^{2} + \frac{1}{2})}^{2} α^{2} + \frac{1}{2} μ γ^{2} - \frac{1}{4}] λ}{ω_{D}^{3} α γ^{2} μ ξ}; P_{22} = \frac{1}{2} \frac{[{(μ γ^{2} + \frac{1}{2})}^{2} α^{2} + \frac{1}{2} μ γ^{2}] λ}{α^{3} γ^{2} μ ξ ω_{D}^{3}}

(32)

P_{23} = - \frac{(\frac{1}{2} + μ γ^{2}) λ}{μ γ^{2} ω_{D}^{2}}; P_{24} = 0; P_{31} = 0; P_{32} - \frac{(μ γ^{2} + \frac{1}{2}) λ}{μ γ^{2} ω_{D}^{2}}

(33)

P_{33} = \frac{λ [\frac{1}{8} + {(μ γ^{2} + \frac{1}{2})}^{3} α^{4} + 2 (μ γ^{2} + \frac{1}{2}) (μ ξ^{2} γ^{2} + \frac{1}{2} ξ^{2} - \frac{1}{4}) α^{2}]}{μ γ^{2} ξ α ω_{D}}

(34)

P_{34} = \frac{1}{8} \frac{λ [4 (μ γ^{2} + \frac{1}{2}) α^{2} - 1]}{μ γ^{2} ξ α ω_{D}}; P_{41} = \frac{λ {(μ γ^{2} + \frac{1}{2})}^{2}}{μ γ^{2} ω_{D}^{2}}; P_{42} = 0

(35)

P_{43} = \frac{1}{8} \frac{λ [4 (μ γ^{2} + \frac{1}{2}) α^{2} - 1]}{μ γ^{2} ξ α ω_{D}}; P_{44} = \frac{1}{8} \frac{λ (1 + 2 μ γ^{2})}{α ω_{D} γ^{2} μ ξ}

(36)

In the quadratic torque matrix P, the vibration response of the rotating shaft is P₁₁. The smaller the vibration response of the rotating shaft is, the faster the torsional vibration of the system turns off. The conditions for the vibration response of the rotating shaft, P₁₁, yield minimum values as follows

{\frac{\partial P_{11}}{\partial α} |}_{α_{opt}^{MQT} = α} = 0

(37)

{\frac{\partial P_{11}}{\partial ξ} |}_{ξ_{opt}^{MQT} = ξ} = 0

(38)

Solving equations (30), (37), and (38), we obtain the optimal parameters of the STMD as

α_{opt}^{MQT} = \frac{\sqrt{1 - μ γ^{2}}}{1 + 2 μ γ^{2}}

(39)

ξ_{opt}^{MQT} = \frac{γ \sqrt{μ (2 + 3 μ γ^{2} - 2 μ^{2} γ^{4})}}{2 (1 + 2 μ γ^{2}) \sqrt{1 - μ γ^{2}}}

(40)

From equations (39) and (40), we obtain the optimal parameters of the STMD to reduce the torsional vibration of the rotating shaft by using the minimum quadratic torque method. The optimal expressions determined in the equations (17), (24), (39), and (40) for the STMD are different from the optimal expressions found in Ref. 12. This asserts that when the shafts rotate with the constant and variable angular velocities, various optimal parameters can be determined to eliminate the torsional vibration of the rotating shafts.

Numerical simulation

To evaluate the effect of reducing torsional vibration for the rotating shaft with optimal design parameters according to equations (17), (24), (39), and (40), numerical simulation is considered under three cases: Under the influence of the inertial force caused by the angular acceleration, the rotating shaft has initial deflections and initial velocities.

Case 1: The rotating shaft has initial deflections

The author simulates the vibration of the system with the input parameters of the rotating shaft and the STMD which are listed in Table 2.

Table 2.

Input parameters of the rotating shaft and the symmetric tuned mass damper for case 1.

Parameter	M	ρ	k _t	m _t	m	L	$μ$	γ
Value	500 kg	1.0 m	10⁵Nm/rad	15 kg	10 kg	0.9 m	0.03	0.9

Combining equation (8) and Table 2, the natural frequency of the rotating shaft is found as follows

ω_{D} = \sqrt{\frac{k_{t}}{{M ρ}^{2}}} = 14.142 (rad / s)

(41)

Using equations (8), (17), (24), and (39)–(41) and Table 2, we infer the optimal parameters of the STMD, shown in Table 3.

Table 3.

Optimal parameters of the symmetric tuned mass damper for case 1.

Fixed-points theory
Parameter	$α_{opt}^{FPT}$	$ξ_{opt}^{FPT}$	$c$	k _m
Value	0.942	0.134	53.340 Ns/m	4312.537 Nm/rad

Minimum quadratic torque method
Parameter	$α_{opt}^{MQT}$	$ξ_{opt}^{MQT}$	$c$	k _m
Value	0.942	0.108	43.287 Ns/m	4312.537 Nm/rad

The parameters of the rotating shaft and the STMD are listed in Tables 2 and 3. The initial deflection of the rotating shaft is set up with $θ_{0} = 0.02 (rad)$ . The Maple software is used to simulate the torsional vibration of the rotating shaft and the STMD, as shown in Figure 3.

Figure 3.

Vibration of the rotating shaft in the case with initial torsional vibration $θ_{0} = 0.02 (rad)$ .

From Figure 3, we find that the optimal parameters of the STMD defined in this article have a good effect on reducing torsional vibration of the rotating shaft in the time domain with respect to case 1.

Case 2: The rotating shaft has initial velocity

The vibration of the system is simulated with the input parameters of the rotating shaft and the STMD which are given in Table 4.

Table 4.

Input parameters of the rotating shaft and symmetric tuned mass damper for case 2.

Parameter	M	ρ	k _t	m _t	m	L	μ	γ
Value	450 kg	1.0 m	2 × 10⁵Nm/rad	10 kg	8 kg	0.9 m	0.025	0.9

From equation (8) and Table 4, the natural frequency of the rotating shaft can be determined as follows

ω_{D} = \sqrt{\frac{k_{t}}{{M ρ}^{2}}} = 21.082 (rad / s)

(42)

Combining equations (8), (17), (24), (39), (40), and (42) and Table 4, the optimal parameters of the STMD are derived, shown in Table 5.

Table 5.

Optimal parameters of the symmetric tuned mass damper for case 2.

Fixed-points theory
Parameter	$α_{opt}^{FPT}$	$ξ_{opt}^{FPT}$	$c$	k _m
Value	0.95	0.123	55.666 Ns/m	7379.117 Nm/rad

Minimum quadratic torque method
Parameter	$α_{opt}^{MQT}$	$ξ_{opt}^{MQT}$	$c$	k _m
Value	0.95	0.10	45.219 Ns/m	7379.117 Nm/rad

From the parameters in Tables 4 and 5, the initial velocity of the rotating shaft is set up with ${\dot{θ}}_{0} = 0.9 (r a d / s)$ ; using the Maple software, the torsional vibration of the rotating shaft is simulated, which is shown in Figure 4.

Figure 4.

Vibration of the rotating shaft in the case with initial torsional vibration $θ_{0} = 0.0 (rad)$ and initial angular velocity ${\dot{θ}}_{0} = 0.9 (rad / s)$ .

From Figure 4, it is clear that the torsional vibration amplitudes of the rotating shaft with the optimal STMD are significantly lower than those of the rotating shaft without the STMD in the time domain with respect to case 2.

Case 3: The rotating shaft has initial deflection and initial velocity

The input parameters of the rotating shaft and the STMD are used for simulation, which are given in Table 6.

Table 6.

Input parameters of the rotating shaft and symmetric tuned mass damper for case 3.

Parameter	M	ρ	k _t	m _t	$m$	L	μ	γ
Value	400 kg	0.9 m	3 × 10⁵Nm/rad	12 kg	9 kg	0.8 m	0.033	0.89

By using equation (8) and Table 6, the natural frequency of the rotating shaft is found as follows

ω_{D} = \sqrt{\frac{k_{t}}{{M ρ}^{2}}} = 30.429 (rad / s)

(43)

From equations (8), (17), (24), (39), (40), and (43) and Table 6, we infer the optimal parameters of the STMD, shown in Table 7.

Table 7.

Optimal parameters of the symmetric tuned mass damper for case 3.

Fixed-points theory
Parameter	$α_{opt}^{FPT}$	$ξ_{opt}^{FPT}$	c	k _m
Value	0.939	0.137	101.849 Ns/m	13580.956 Nm/rad

Minimum quadratic torque method
Parameter	$α_{opt}^{MQT}$	$ξ_{opt}^{MQT}$	c	k _m
Value	0.939	0.111	82.623 Ns/m	13580.956 Nm/rad

The parameters of the rotating shaft and the STMD are listed in Tables 6 and 7.^1–20 The initial deflection and velocity of the rotating shaft are set up $θ_{0} = 0.02 (r a d)$ and ${\dot{θ}}_{0} = 0.9 (r a d / s)$ , respectively. The Maple software is used to simulate the torsional vibration of the rotating shaft and the STMD, which is shown in Figure 5.

Figure 5.

Vibration of the rotating shaft in the case with initial torsional vibration $θ_{0} = 0.02 (rad)$ and initial angular velocity ${\dot{θ}}_{0} = 0.9 (rad / s)$ .

From Figure 5, again it is clear that the optimal STMD significantly reduces the vibration amplitudes of the rotating shaft in comparison with those of the rotating shaft without the STMD in the time domain with respect to case 3.

Figures 3–5 show for the first 1.2 s, the vibration amplitudes of the rotating shaft attached with the optimal STMD obtained by using the minimum quadratic torque method are smaller than those of the FPT. But, from 1.2 s onward, the vibration amplitudes of the FPT are smaller than those of the minimum quadratic torque method. Overall, however, the STMD designed by using the FPT has a greater effect on reducing torsional vibration for the rotating shaft. It is clear that the torsional vibration amplitudes of the rotating shaft are eliminated with the optimal STMD under different initial conditions.

Conclusions

This article designs the optimal parameters of the STMD to reduce the torsional vibration varying with time for the machine shaft. First, the optimal parameters of the STMD are determined by the FPT and are expressed according to equations (17) and (24). Then, the minimum quadratic torque method is applied to solve the optimal design problem of the STMD; the parameters of the optimal STMD are given in equations (39) and (40). To evaluate the effect on reducing vibration, the Maple software is used to simulate the torsional vibration of the rotating shaft. Through vibration simulation, we find that within the first 1.2 s, the vibration amplitudes of the rotating shaft attached with the optimal STMD obtained by using the minimum quadratic torque method are smaller than those of the FPT. But, from 1.2 s onward, the vibration amplitudes of the FPT are smaller than those of the minimum quadratic torque method. Overall, however, the STMD designed by using the FPT has a greater effect on reducing torsional vibration for the rotating shaft. It is clear that the torsional vibration amplitudes of the rotating shaft are eliminated with the optimal STMD under different initial conditions.

Footnotes

Acknowledgments

The author would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the manuscript.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research is funded by Hung Yen University of Technology and Education under grand number UTEHY.L.2021.04.

ORCID iD

Duy-Chinh Nguyen

References

Alsuwaiyan

Shaw

. Performance and dynamic stability of general-path centrifugal pendulum vibration absorbers. Journal of Sound and Vibration 2002; 252(5): 791–815.

Brock

. A note on the damped vibration absorber. Trans ASME, Journal of Applied Mechanics 1946; 13: A–284.

Carter

. Rotating pendulum absorbers with partly solid and liquid inertia members with mechanical or fluid damping. British: Patent 337, 1929.

Chao

Shaw

Lee

. Stability of the unison response for a rotating system with multiple tautochronic pendulum vibration absorbers. J Appl Mech 1997; 64: 149–156.

Den Hartog

. Mechanical vibrations. 4th Edition. New York: McGraw-Hill, 1956.

Denman

. Tautochronic bifilar pendulum torsion absorbers for reciprocating engines. Journal of Sound Vibration 1992; 159: 251–277.

Korenev

Reznikov

. Dynamic vibration absorbers: Theory and technical applications. New York: Wiley, 1993.

Lee

Shaw

. A subharmonic vibration absorber for rotating machinery. J Vib Acoust 1997; 119: 590–595.

Luft

. Optimal tuned mass damper for building. Journal of the Structural Division 1979; 105(12): 2766–2772.

10.

Madden

. Constant frequency Bifilar vibration absorber. USA: Patent 4218187, 1980.

11.

Mayet

Ulbrich

. Tautochronic centrifugal pendulum vibration absorbers: general design and analysis. Journal of Sound Vibration 2014; 333(3): 711–729.

12.

Chinh

. Determination of optimal parameters of the tuned mass damper to reduce the torsional vibration of the shaft by using the principle of minimum kinetic energy. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multibody Dynamics 2019; 233(2):327–335.

13.

Sarazin

RRR

. Means adapted to reduce the torsional oscillations of Crankshafts. USA: Patent 2079226, 1937.

14.

Shi

Shaw

Parker

. Vibration reduction in a tilting rotor using centrifugal pendulum vibration absorbers. Journal of Sound Vibration 2016; 385(22): 55–68.

15.

Taylor

. Eliminating crankshaft torsional vibration in radial aircraft engines. SAE paper 360105, 1936.

16.

Truhar

. An efficient algorithm for damper optimization for linear vibrating systems using Lyapunov equation. J Comput Appl Math 2004; 172: 169–182.

17.

Vitaliani

Di Rocco

Sopouch

. Modelling and simulation of general path centrifugal pendulum vibration absorbers. SAE paper 2015-24-2387. doi: 10.4271/2015-24-2387.

18.

Chinh

Khong

, et al. Closed-form solutions to the optimization of dynamic vibration absorber attached to multi-degree-of-freedom damped linear systems under torsional excitation using the fixed-point theory. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multibody Dynamics 2018; 232(2): 237–252.

19.

Warburton

. Optimum absorber parameters for various combinations of response and excitation parameters. Earthquake Engineering and Structural Dynamics 1982; 10: 381–401.

20.

Chinh

. Optimal parameters of tuned mass dampers for an inverted pendulum with two degrees of freedom. Proceedings of the Institution of Mechanical Engineers, Part K: Journal of Multibody Dynamics 2020; 0(0): 1–13.