Keywords

31.1 Introduction

As the life standards are going up and customer expectations are going towards reduced noise and vibration, manufacturing of washing machines has been an important issue for the industry. To satisfy customers, design criterion of any machine, as well as washing machines, has tendency toward minimized vibration. Therefore, there have been many studies about minimizing vibration and noise amplitudes of both horizontal and vertical type washing machines during operation.

Türkay et al. [1, 2] worked on a 3D dynamic model of horizontal washing machines by neglecting gyroscopic effects and assuming inertia as time invariant. They solved the nonlinear equations with time integration and studied on minimization of sum of vertical-horizontal vibration amplitudes. Li and Yam [3] studied another model by assuming small deflections where they try to decrease the maximum vibration amplitude below the requirements of quality control. Papadopoulos and Papadimitriou [4] created another simplified model by just considering the four legs of horizontal type washing machines. Using a linear model, they analyzed walking and jumping conditions of washing machines. To reduce vibration amplitudes, authors introduced balancing masses around the drum of the washing machine.

Argentini et al. [5, 6] considered both finite element model (FEM) and 6-degrees of freedom (DOF) mathematical model by linearizing gyroscopic terms and neglecting time dependency of terms in the equation of motion. After linearizing the system around its equilibrium position, FEM is solved and a validated numerical model is constructed. By introducing dry friction dampers, additional masses and a secondary suspension system, vibration amplitudes are reduced.

Lim et al. [7] created a dynamical model for the drum by considering the effects of bearings in the washing machine. Rotor dynamics equations are solved by time integration to observe the effects of parameters of the washing machine. They validated their numerical model by comparing the results with experiments. Authors obtain Campell diagram and forward whirling frequencies are also identified.

Nygårds and Berbyuk [8, 9] created another model utilizing dry friction and cubic stiffness as resilient elements. Pittner et al. [10,11,12] used another model by neglecting the translational motion of the mass center of the drum along the main rotation axis. By introducing two concrete blocks as extra masses and on-off dampers, vibration amplitudes are reduced.

Chen et al. [13, 14] mostly worked on stability analysis of the horizontal type washing machines using Poincaré mapping and by checking Floquet multipliers. Stability analysis is made in frequency domain by considering Torus bifurcation. In time domain, again, time integration is used to calculate vibration amplitude. Ball balancers are introduced to the system to reduce vibration amplitude. Generic algorithm is used to determine the amount of balls to be used in the gasket of the washing machine.

There are many studies aimed to reduce the vibration amplitude in horizontal type washing machines by using three dimensional dynamic models. Almost all of the studies consider the time response of the washing machine drum at a specific frequency. However, for washing machines working in a broadband frequency range, i.e. between 100 and 1000 rpm; it is to better analyze the vibration amplitude of the washing machine and obtain a frequency response function (FRF) to expedite the design procedure. In this work, a three dimensional dynamical model is introduced for horizontal-type washing machines without making any linearization. Resulting nonlinear differential equations of motion are converted into set of nonlinear algebraic equations utilizing Harmonic Balance Method (HBM). Newton’s method with arc-length is used to solve the resulting nonlinear equations. Effects of orientation positions and angles of the compliant elements, spring stiffness, damping constant and the type of the damper used on the vibration amplitude of the washing machine drum are studied under steady state conditions.

31.2 Analysis

3-D dynamic model for horizontal-type washing machines developed in this study including the drum, springs and dampers is shown in Fig. 31.1. Cabinet and cabinet side of the dampers and springs are assumed as fixed. A inertial reference frame F 0 and an non-inertial reference frame F d are placed at the geometrical center of the washing machine. Initial Frame Based (IFB) 1-2-3 sequence is used to describe the transformation matrix between non-inertial and inertial reference frames. Transformation matrix can be written for IFB 1-2-3 sequence by using exponential transformation matrices [15] as follows:

Fig. 31.1
figure 1

Model of washing machine

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$$ {\widehat{C}}^{\left(0,d\right)}={e}^{\overset{\sim }{u_1}{\theta}_3}{e}^{\overset{\sim }{u_2}{\theta}_2}{e}^{\overset{\sim }{u_3}{\theta}_1}, $$
(31.1)
$$ {e}^{\overset{\sim }{u_1}{\theta}_3}=\left[\begin{array}{ccc}1& 0& 0\\ {}0& \cos \left({\theta}_3\right)& \sin \left(-{\theta}_3\right)\\ {}0& \sin \left({\theta}_3\right)& \cos \left({\theta}_3\right)\end{array}\right]\kern1em {e}^{\overset{\sim }{u_2}{\theta}_2}=\left[\begin{array}{ccc}\cos \left({\theta}_2\right)& 0& \sin \left({\theta}_2\right)\\ {}0& 1& 0\\ {}\sin \left(-{\theta}_2\right)& 0& \cos \left({\theta}_2\right)\end{array}\right]\kern1em {e}^{\overset{\sim }{u_3}{\theta}_1}=\left[\begin{array}{ccc}\cos \left({\theta}_1\right)& \sin \left(-{\theta}_1\right)& 0\\ {}\sin \left({\theta}_1\right)& \cos \left({\theta}_1\right)& 0\\ {}0& 0& 1\end{array}\right]. $$
(31.2)

At first, viscous dampers are introduced. Forces on springs and viscous dampers are calculated in the base frame, F 0. The equation of motion is derived using Newton-Euler formulation as

$$ \sum \left(\overrightarrow{F_c}+\overrightarrow{F_s}\right)=m\overrightarrow{a}, $$
(31.3)
$$ \sum \left(\overrightarrow{M_c}+\overrightarrow{M_s}\right)={\check{J}}\cdot \overrightarrow{\alpha}+\overrightarrow{\omega}\times {\check{J}}\cdot \overrightarrow{\omega}. $$
(31.4)

However, in order to simplify the calculations, column matrix representations of vectors and inertia tensors are used as follows

$$ \sum \left({\overline{F_c}}^{(0)}+{\overline{F_s}}^{(0)}\right)=m{\overline{a}}^{(0)}, $$
(31.5)
$$ \sum \left({\overline{M_c}}^{(0)}+{\overline{M_s}}^{(0)}\right)={\widehat{J}}^{(0)}{\overline{\alpha}}^{(0)}+{\overset{\sim }{\omega}}^{(0)}{\widehat{J}}^{(0)}{\overline{\omega}}^{(0)}. $$
(31.6)

Overbars indicate the column matrix representations of vectors; where, supper scripts are used for the reference frames of which the vectors are resolved. Hats indicate matrix representation of the tensors and tilde represents the skew-symmetric form of the column matrices or simply cross product matrices. F c & M c are forces and moments coming from the viscous dampers, respectively; whereas, F s & M s represent the forces and moments of linear springs. Even though, it is preferred to work on the moving body frame, it is chosen to work in the inertial base frame, since, in eq. (31.6), to describe relative angular velocity and acceleration of the drum frame. Expressing relative angular velocity and acceleration in drum frame will lead lengthy equations; on the other hand, using inertial reference frame as the working frame, inertia matrices are needed to multiplied from both its left and right sides by transformation matrix and transpose of transformation matrix, respectively. Therefore, this decision is an engineering tradeoff, and base frame is selected as the working frame in this study.

Three dimensional, six degree of freedom, nonlinear differential equations of motions are converted into nonlinear algebraic equations utilizing HBM. In HBM, all nonlinear internal and external forces are represented by Fourier series. These representations are substituted in nonlinear differential equations and coefficients of similar harmonics are balanced to resolve all the displacements. Newton’s method with arc-length continuation is used to obtain the steady state response of the system. The number of harmonics used in the solution is determined based on the accuracy desired in the calculations.

An example case, where all compliant elements and the mass center of laundry (the reason of rotating unbalance) are placed at the same fictitious y–z plane that includes the point W, is considered. After converting the system to a two dimensional, three degrees of freedom dynamical model, the results are compared with another study [16] to validate the model developed. Results are obtained by using 8 harmonics. It is observed from Fig. 31.2 that in order to accurately model the system, using three or four harmonics is sufficient.

Fig. 31.2
figure 2

Responses of the system and force transmitted to the cabinet in logarithmic scale

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As it seen from Fig. 31.2, at high frequencies excessive forces occur since force on the viscous dampers are directly proportional to the excitation frequency. To overcome this problem, dry friction dampers may be introduced instead of viscous dampers. Dry friction dampers are acting like springs for small deflections, which occur at every frequency, but around resonance. Due to this behavior, force transmitted to the cabinet will be much more small compared to viscous damper case.

31.3 Conclusion

A 3D nonlinear dynamical model of a horizontal type washing machines is developed in this study. HBM is used to obtain frequency response function (FRF) of the 6-DOF system. Multiple harmonics are used in order to accurately capture the steady-state vibration response. Several case studies are performed to see the effects of system parameters or geometrical orientations of compliant elements on system FRF.