In this paper, an experimental model of a horizontal cantilever beam with a rotating/oscillating attached to the shaker for harmonic excitation at the one end and a gyrostabilizer at the other end is built to verify the equations of the Lagrangian model. The primary focus of the study was to investigate the parameters of excitation amplitude, natural frequency, rotating mass (disk mass), and disk speed of gyro that would minimize the amplitude of the beam to identify these effects. Numerical and experimental results indicate that the angular momentum of the gyrostabilizer is the most effective parameter in the reduction of beam displacement.

The application of rotating and oscillating types of vibration control systems can be found in several systems such as space structures, robotics, aircraft, ships, etc. [

Currently, a large number of researchers remain focused on flexible structures with passive dynamic vibration absorbers such as rotating mass (flywheel), beam, liquid, pendulum, etc., used for several engineering applications [

Meanwhile, several researches have been investigating different types of vibration control apparatuses in range from simple to complex; for instance the use of gyrostabilizers, being a complex yet effective vibration control apparatus is used for several engineering applications such as stabilization of wheeled vehicles [

For the modeling of antennas, robotic manipulators, appendages of aircrafts, spacecrafts or vehicles, etc., beams carrying a tip mass with (or without) rotary inertia can be used. In this study, the vibration control results of a horizontal cantilever beam with a rotating/oscillating tip mass (gyrostabilizer) at one end and harmonic excitation at the other end has been investigated through theoretical and experimental methods. Parameters of excitation amplitude, natural frequency of the beam tip mass system, rotating mass (disk mass) and disk speed that would minimize the amplitude of the beam were identified alongside the effects to the system. The apparatus included an oscillated mass connected to the free end of the beam with a small ball bearing and disk with DC electrical motor mounted to the oscillating tip, while the other end was mounted to a shaker.

Experimental studies were performed on an apparatus set-up in the Mechanical Engineering Department laboratory at Texas Tech University as seen in

Symbol | Numerical values | Description |
---|---|---|

E | 210e9 N/m^{2} |
Young’s modulus |

h | 0.0016 m | Cantilever beam thickness |

b | 0.0254 m | Width of the cantilever beam |

g | 9.81 m/s^{2} |
Gravitational acceleration |

L | 0.312 m, 0.505 m | Length of the cantilever beam |

I | 8.670e−12 m^{4} |
Geometrical moment of inertia of the cantilever beam |

c_{b} |
0.05 N.s/m | Equivalent coefficient of dissipation of cables on beam |

I_{t} |
5.4242e−04 kg.m^{2} |
Mass moment of inertia of free end of the cantilever beam |

c | 0.007 N.m.s/rad | Equivalent coefficient of torsion damping of bearing on gyro |

k | 5e−05 N.m/rad | Stiffness of torsion spring of gyro |

m | 0.0263 kg, 0.0344 kg | Mass of rotating disk |

r | 0.0196 m | Radius of disk |

M_{t} |
0.128 kg | The weight of the all gyro mechanism excluding flywheel |

I_{p} |
1.334e−5 kg.m^{2} |
Rotary inertia of disk |

I_{o} |
6.670e−6 kg.m^{2} |
Mass moment of inertia of disk |

I_{fx} |
12e−4 kg.m^{2} |
Mass moment of inertia of gyroscope at x-direction |

I_{fy} |
6e−4 kg.m^{2} |
Mass moment of inertia of gyroscope at y-direction |

I_{fz} |
7e−4 kg.m^{2} |
Mass moment of inertia of gyroscope at z-direction |

Ω | 0–25000 rpm | Rotating speed of disk |

The experimental apparatus set-up consists of a waveform generator (Siglent SDG1025), a vertical/horizontal shaker (Calidyne Inc., Model A88), two accelerometers (TE Connectivity, 4623M3-010-040, 4600-010-060) and MATLAB software (National Instruments, USB 6343) as seen in a schematic diagram form shown in

Experimental studies were performed for different parameters such as length of the cantilever beam system, rotating mass, rotating speed of the mass and forcing amplitude. In order to determine the relationship between varying parameters, one parameter was changed and the rest of were kept stable for each experimental run.

The Euler-Bernoulli beam theory based on thin cantilever beams was used to obtain governing equations of motion of the cantilever beam with a tip mass. Shear deformation and rotational inertia of the beam were neglected from the governing equations due to thin cantilever beam assumption (L/h > 20). In the governing equation, the parameters of the cantilever beam mass density ρ, cross-sectional area A, Young’s modulus E and moment of inertia.

The mathematical model of the cantilever beam with tip mass (gyrostabilizer) was derived through the displacement model of the cantilever as shown in _{t}) at the free end is subject to a harmonic base excitation (z(t)) from the opposing end, which is attached to the shaker. In the equation models used for these experiments, the denotations are as follows: arc-length (s), harmonic excitation (z = z_{0}cos(ωt)) with varying forcing frequencies (ω) and forcing amplitudes (z_{0}). Due to the beam displacement, horizontal and vertical displacements at the free end and orientation angle were denoted as (u), (v) and (φ), respectively.

The gyrostabilizer mass (M_{t}) has a rotating mass (m) within the structure consisting of a DC electrical motor which controls rotational speed of the mass (m) about the geometric axis as shown in _{t}) are denoted by Ω and θ, respectively. The ball bearing shows resistance from the torsional spring (

The kinetic energy equation of the cantilever beam with gyrostabilizer tip mass can be written in the form shown below in

The kinetic energy of the gyrostabilizer can be expressed in the form of

The potential energy of the cantilever beam with gyrostabilizer can be defined in the form of

The dissipation function (D) is defined in the form of

Orientation angle of the beam is defined in the form of

The displacement of the beam is expressed in

From

Derivative of

Beam deformation with respect to arc-length and time can be expressed by accepting the first mode of the shape function ψ(s) [

If

Therefore, the kinetic energy of the system can be defined by using the above equations, while the transverse displacement (v) at the free end can be expressed as in

The potential energy of the system in

Constants G_{1} through G_{9} used in the above equations can be expressed as in

Energy method has been used to obtain equations of motion for the cantilever beam with gyrostabilizer tip mass. The kinetic energy (

The energy method was also used to obtain equations of motion for the gyrostabilizer. The kinetic energy (

Theoretical and experimental studies were performed for varying parameters such as the natural frequency of the cantilever beam system, rotating mass, rotating speed of the mass and forcing amplitude. In order to accurately determine the relationship between the parameters, one parameter was changed as the rest were kept stable for each individual test.

Cantilever beam with gyrostabilizer system frequency response curves were obtained theoretically and experimentally for the following parameters of natural frequency of the system (ω_{n}), rotating mass (m), rotating mass speed (Ω), and forcing amplitude z(t). All theoretical and experimental results were plotted graphically with respect to forcing frequency

Theoretical and experimental frequency response curves of the beam with gyrostabilizer tip-mass system were plotted graphically as seen in _{n }=_{ }2.12 Hz and ω_{n }=_{ }4.13 Hz), forcing amplitude (z_{0 }=_{ }1.3 mm) and rotating mass (disk mass) (m = 0.0344 kg) and variable rotating mass speeds of Ω = 0, 11250 and 25000 rpm. The maximum amplitude of the system was observed at the resonance frequency when the rotating mass speed was equal to zero (non-active gyrostabilizer); as the rotational mass was initiated with speed (active gyrostabilizer), the maximum amplitude of the system was dramatically reduced, while the natural frequency of the system was increased (shifted to the right) due to the inertia moment of the gyrostabilizer. Hence, theoretical and experimental results concluded in agreement with all parametric factors displayed in

The relationship between rotational mass speed and maximum amplitude of the system (amplitude at resonance frequency) with two different natural frequencies of ω_{n }=_{ }2.12 Hz and 4.13 Hz are shown in

Theoretical and experimental frequency response curves of the beam with gyrostabilizer system were obtained with constant parameters of the system natural frequency (ω_{n }=_{ }4.13 Hz), forcing amplitude (z_{0 }=_{ }1.3 mm) and rotating mass (m = 0.0263 kg) and varying disk speeds of Ω = 0, 11250 and 25000 rpm as seen in

The gyrostabilizer with lower disk mass also reduced beam amplitude sıgnificantly as seen from

Theoretical and experimental frequency response curves of the beam system with constant parameters of ω_{n} = 4.13 Hz, m = 0.0344 kg and Ω = 11250 rpm were obtained with variable parameters of forcing amplitude z_{0 }=_{ }0.6 mm, 1.3 mm and 2.0 mm as seen in the

Equations of motion of a cantilever beam with gyrostabilizer tip-mass at one end and harmonic base excitation at the other end have been obtained in this paper. The experimental apparatus was built strictly to specifications listed in

The natural frequency of the beam gyrostabilizer system is significantly increased by disk speed and is fairly effected by the mass of the disk. Higher disk speed of the gyrostabilizer in turn increased the stiffness of the beam caused by the moment of inertia of rotating disk.

The investigation clearly shows the amplitude of beam at the free end is highly influenced by the disk speed of the gyrostabilizer and is fairly effected by the mass of the disk. Disk speed exponentially reduced the amplitude of the system. However, it is observed that disk speed of the gyrostabilizer becomes minimally effective on amplitude after the disk speed of Ω = 20000 rev/min. The gyro reduces the displacement amplitude of beam nearly by 100% by using the disk speed of Ω = 30000 rev/min.

the gravitational acceleration

Young’s modulus

cantilever beam thickness

width of the cantilever beam

length of the cantilever beam

geometrical moment of inertia of the cantilever beam

_{t}

mass moment of inertia of free end of the cantilever beam

equivalent coefficient of torsion damping of bearing on gyro

_{b}

equivalent coefficient of dissipation of cables on beam

stiffness of torsion spring of gyro

mass of rotating disk

_{t}

the weight of the all gyro mechanism excluding flywheel

total kinetic energy of system

_{gyro}

the kinetic energy of gyro

total dissipation energy of system

total potential energy of system

the coordination in the world coordination system

the horizontal displacement of beam at the free end

the vertical displacement of beam at the free end

the precession angle of gyro

the orientation angle of beam

arc-length of beam

the first mode of the shape function of beam

the harmonic base excitation

_{0}

the amplitude of the base excitation

the angular frequency of base excitation

_{n}

the natural frequency of beam

_{p}

the rotary inertia of disk

_{o}

the mass moment of inertia of disk

_{fx,}I

_{fy,}I

_{fz}

the principal moments of inertia of gyroscope (Taken at the center of mass of the gimbal)

the rotational velocity of disk

^{∞}control of gyroscopic ship stabilization systems.