Design of the mounting system of a grass trimmer engine for minimum handle vibration response based on frequency-dependent stiffness and loss factor properties

2.1. Mathematical model

The mathematical model for the rolling vibration of the engine, which includes the orientation angles and location of the engine mount, must first be defined. The engine is assumed to be a rigid mass, where all the engine particles are moving with the same velocity. The vibration of the rigid frame of the engine is ignored because it is negligible compared to the vibration of the engine itself [10]. There are two coordinate systems involved in the modelling: a local coordinate system and a global coordinate system. The dynamic properties of the individual engine mount are referred to as the individual mount properties (which also include the engine block), whereas the installation of the engine mounts for the whole engine mounting structure is referred to as the engine mounting system properties. The center of mass of the engine is considered to be the origin of the global coordinate system (Fig. 1); in this coordinate system the $X$- and $Y$-axes are parallel to the floor while the $Z$-axis is perpendicular to it. The $Y$-axis that aligns with the crankshaft rotation axis is considered to be the roll axis. The global coordinate system is labelled as $XYZ$ and the local coordinate system as $X\text{'}Y\text{'}Z\text{'}$, as in Fig. 1. There are three units of engine mounts in the original grass trimmer design, labelled as 1, 2 and 3, as shown in the schematic of Fig. 1.

The grass trimmer engine and the mounting system are modelled with six degrees of freedom (DOF) and represented by the following equation of motion:

where $\left[M\right]$ is the rigid mass matrix of the engine which consists $m$, mass of engine and $I_{xx}$, $I_{xy}$,..., $I_{zz}$, moment of inertia along the particular axis and is given by:

$\mathrm{}[x{]}^T=[\begin{array}{llllll}x& y& z& {\theta }_x& {\theta }_y& {\theta }_z\end{array}{]}^T$ is the translational and rotational displacement vector of the centre of mass, $\left[K\right(\omega \left)\right]$ is the frequency-dependent complex stiffness matrix which consisting of translational stiffness and rotational stiffness. $\left[K\right(\omega \left)\right]$ is defined as the stiffness modulus including the stiffness properties and the loss factor of the isolator as $k\left(\omega \right)\{1+j\eta (\omega \left)\right\}$. At here, $k\left(\omega \right)$ is the frequency-dependent stiffness and $\eta \left(\omega \right)$ is the frequency-dependent loss factor which representing the damping properties in the complex stiffness method [19]. $\left[f\right]$ is the vector of excitation forces and moments on the system and $\omega$ is the angular frequency.

Let $\left[x\right]=\left[X\right]\mathrm{e}\mathrm{x}\mathrm{p}\left(j\omega t\right)$ and $\left[f\right]=\left[F\right]\mathrm{e}\mathrm{x}\mathrm{p}\left(j\omega t\right)$. Therefore, the response of the system can be derived as:

The receptance function is the complex ratio of output response to the input force (Newland, 1989) and can be written as:

Since the rolling vibration is the angular acceleration of the rolling axis of the engine and the main parameter of this analysis, the accelerance (ratio of acceleration to force), $A\left(j\omega \right)$, is calculated in accordance to the ISO 10846-1:2008 standard, as follows:

Each individual engine mount is associated with a three-dimensional complex stiffness matrix, $\left[k\mathrm{\text{'}}\right(\omega ){]}_i$, comprising of frequency-dependent compression and shear stiffnesses in accordance to:

where $k_x\left(\omega \right)$, $k_y\left(\omega \right)$, $k_z\left(\omega \right)$ is the frequency-dependent stiffness and ${\eta }_x\left(\omega \right)$, ${\eta }_y\left(\omega \right)$, ${\eta }_z\left(\omega \right)$ is the frequency-dependent loss factor along the three orthogonal directions of the respective local coordinate system. $i=$ 1, 2, 3 which denoted numbering of the engine mount. These local complex stiffness properties of each individual engine mount are transformed to the global coordinate by using the Euler’s transformation matrix $\left[A\right]$ in accordance to:

After the transformation is applied, the dynamic properties for each individual mount part in the global coordinate system become:

where $k_{xx}$, $k_{xy}$, $k_{xz}$,..., $k_{zz}$ represent the individual mount stiffness along the particular axis in the global coordinate system. $[A{]}^T$ is the transpose matrix for the Euler transformation matrix. The dynamic properties for the whole engine mounting system are the summation of the product of each individual dynamic property in the global coordinate system with its respective position vector, expressed as:

where ${\left[r\right]}_i={\left[\begin{array}{ccc}0& -z& y\\ z& 0& -x\\ -y& x& 0\end{array}\right]}_i$ is the position matrix of each individual engine mount $i$ and $n$ is the number of the engine mount.

The vector of excitation forces and their moments is given by:

where $F_x$, $F_y$ and $F_z$ are the excitation forces acting on the centre of mass of the engine in the $X$-, $Y$- and $Z$-directions, respectively, while $M_x$, $M_y$ and $M_z$ are the corresponding moments about the principal axes of the engine-mount system. These measured excitation forces and moments from the engine are inserted in Eq. (3) (as shown in a subsequent section), to calculate the response from the grass trimmer.

2.2. Experimental measurement of the engine force and moment

The excitation forces and moments of the engine are determined experimentally for use as input in Eq. (10). Since the angular acceleration along the $Y$-axis (rolling axis), which is parallel to the crankshaft axis of the engine, is the main interest in this study, and given the fact that the primary vibration is due to $M_y$ [20], the input parameters of $M_x$ and $M_z$ are neglected.

A four-component Kistler dynamometer (type: 9272) was used for measuring the three orthogonal excitation forces and the moment about the $Y$-axis, with the engine mounted vertically on the dynamometer at the fixed table, as shown in Fig. 2. The measurement is carried out at a speed of 6000 revolutions per minute (rpm).

Fig. 2Experiment setup for measuring the excitation forces and moment

2.3. Experimental measurement for rolling vibration acceleration of the engine to the handle

The rolling vibration angular acceleration of the handle and the engine are the main parameters that determine the effectiveness of the system in reducing the rolling vibration of the handle. The instruments used for experimental measurements include the data acquisition system and analyser (IMC, type: CS-3008-1) and two accelerometers (Kistler, type: 8776A50). The grass trimmer used in this study and the measurement setup are shown in Fig. 3. The accelerometers are placed on the engine and on the handle to measure the engine’s input acceleration and the handle’s transmitted output angular acceleration. The angular vibration is measured indirectly using two accelerometers that allow the measurement of the relative acceleration of the two locations (engine and handle), as the difference of the experimentally measured accelerations divided by the distance between the accelerometers, as illustrated in Fig. 3. In Fig. 3, $a_{e1}$ and $a_{e2}$ are the accelerometers at the engine and $a_{h1}$ and $a_{h2}$ are the accelerometers at the handle.

Fig. 3Measurement setup for the angular acceleration transmissibility from the engine to the handle

4.1. Input force and moment from grass trimmer

The excitation forces $F_x$, $F_y$, $F_z$ and moment $M_y$ at an engine operating speed of 6000 rpm (100 Hz) are measured and shown in Fig. 5. From this figure, the excitation forces measured at the $X$-, $Y$- and $Z$-axes at the operating speed are found to have peak values of 45 N, 43 N and 76 N, respectively. This shows that the excitation force in the vertical direction ($Z$-axis) is the highest, while the other forces acting in horizontal plane ($X$- and $Y$-axes) are approximately the same. In Fig. 5(b), there is the second peak for the excitation force in the $Y$-direction. This second peak is due to the second harmonic of the reciprocating motion, as discussed by Tao et al. (2000). The variation of the engine’s measured moment about the $Y$-axis is shown in Fig. 5(d). Its maximum value is measured to be 4.16 N.m. Force and moment values are used as the input excitation data for the calculation described in the section that follows.

Fig. 5Variational plots of measured input force versus frequency of the grass trimmer engine at 6000 rpm

a)$X$-direction

b)$Y$-direction

c)$Z$-direction

d) Measured moment about the $Y$-axis

Fig. 6Design of the new engine mounting system based on the optimum orientation angle and location values

a) Top view

b) Isometric view

c) Front view

d) Right view

4.2. Optimisation results

The results of the original and the optimised parameters are listed in Table 3. From Table 3, the peak value of angular acceleration of the rolling axis of the engine is reduced from 307 rad/s² to 82 rad/s² in the optimised system, representing a 73 % reduction. Each optimised engine mount has its dominant orientation angle aligning with the $Y$-axis, as shown in Table 3. Specifically, indicative values of 55°, 55°, 55° and –55° are recorded for engine mounts EM1, EM2, EM3 and EM4, respectively. The location of each optimised engine mount is observed to be symmetrical about the $X$ and $Y$ axes as shown in Fig. 6, depicting the front, top, right and isometric views of the optimised design. The optimised engine mounting system parts are then fabricated and installed, as shown in Fig. 6, for validation purposes.

The system responses of the optimised and original systems are shown in Fig. 7. The peak acceleration response in the $X$-direction reduces from 46 m/s² to 12 m/s² after the optimisation, an effective percentage reduction of 73 %, as shown in Fig. 7(a). For the acceleration response along the $Y$-direction, there is a reduction of 62 % from a value of 28 m/s² to 10 m/s². The reduction of the vibration is the smallest along the $Z$-axis, as shown in Fig. 7(c), noting a change in value compared to the original response from 48 m/s² to 36 m/s². The objective of this study was to minimize the vibration about the rolling axis of the engine by searching for the optimum engine mount parameters. The optimum results show that the highest percentage of vibration reduction is 73 %, with a noted reduction in rotational acceleration from 307 rad/s² to 82 rad/s², as shown in Fig. 7(d). These reduction levels are significant and are expected to reduce the vibration level of the handle. Such findings indicate the effectiveness of vibration reduction with the choice of the proper orientation angle and location of the engine mounts.

4.3. Comparison of the rolling vibration for the existing and the newly proposed design of the engine mounting system

The rolling vibration at both the engine and the handle of the grass trimmer are measured at both the original and the optimised systems to demonstrate the practical application of an optimised design. Fig. 8 shows the measured rolling vibration of the original and optimised engine mounting system when the engine is operating at 6000 rpm. For the rolling vibration of the engine the peak rolling vibration is reduced from 44 rad/s² to 31 rad/s², as shown in Fig. 8(a). For the rolling vibration of the handle, the peak rolling vibration at 100 Hz is reduced from 18.5 rad/s² to 6.6 rad/s², as shown in Fig. 8 (b). The reduction in the rolling vibration for both the engine and handle justifies the effectiveness of the optimised system. Noted also is a second peak of 40 rad/s² at 24 Hz in the optimised system. However, this is outside the operating speed of the engine. The vibration on the grass trimmer handle is caused by the rolling vibration of the engine. The rolling vibration of engine is subsequently transmitted to the grass trimmer handle and causes HAV. From Fig. 8, the results showed that the rolling vibration transmitted from the engine to the handle is reduced significantly. At the operating speed of 100 Hz, the rolling vibration transmissibility from the engine to the handle is 0.42 for the original engine mounting system and 0.21 for the new design. This shows that the rolling vibration transmission is reduced by 50 % in the new design of the engine mounting system.

Fig. 7Comparison plots of the Log translational acceleration response versus frequency for the original and optimised system (––– original system, - - - - optimised system)

a)$X$-axis

b)$Y$-axis

c)$Z$-axis

d) Rotational acceleration response along the $Y$-axis

The mean angular acceleration transmissibility, for both the original and newly designed systems, is calculated over the measured frequency range. The results are summarised in Table 4. The mean angular acceleration transmissibility about the rolling axis is successfully reduced by 75 % from 1.661 to 0.420. This implies that the optimum engine mount parameters not only reduce the vibration about the engine rolling axis but also reduce the transmissibility of angular acceleration from the engine to the handle. Furthermore, the mean transmissibility of the optimised system has a value of less than unity, indicating that the vibration is not amplified when it is transmitted from the engine to the handle.

Fig. 8Comparison plots of the angular acceleration versus frequency of the a) engine, and b) handle, for the original and optimised systems for the grass trimmer

a)

b)