A new trifilar pendulum approach to identify all inertia parameters of a rigid body or assembly
Introduction
For the sake of dynamic analysis and subsequent structural design of many practical engineering applications, it is often necessary to know the inertia properties of a structure. The structure, which is regarded as rigid, will be referred in the following sections as “the target body” or “the body” for simplicity. The widely used inertia properties include 10 parameters, namely the mass, the coordinates of the centroid, and the six independent entries of the inertial tensor. Quite a few approaches have been developed to identify these parameters. They can be roughly classified into four categories as follows: (1) direct approaches based on moment equilibrium, a complex pendulum or trifilar pendulum [1], [2], [3], (2) numerical integration method based on a 3D model of the target body [4], (3) parameter identification procedures based on vibration tests [5], [6], [7] and (4) other methods such as one based on motion equation of an unconstrained target body [8].
A critical analysis revealed the advantages and disadvantages of various experimental methods of the first category [9]. Error analysis pointed out that the inertia tensor identified by a trifilar pendulum method is most sensitive to the measured period of the pendulum vibration [10].
According to the viewpoint of the present authors, a direct approach using a trifilar pendulum might be the most competitive one in terms of cost, simplicity and reliability at current time for odd-shaped bodies, complex structures or assemblies. In addition, the vibration period can now be measured with high enough accuracy.
For traditional methods using a trifilar pendulum, it is usually required that the centroid of the target body lie on the rotational axis of the pendulum. This is time consuming due to repetitive configuration adjustments of a heavy target body. The requirement also implies that the mass and the position of the centroid have to be determined by other approaches in advance. In other words, a traditional method can only identify six independent entries of the inertia tensor.
It is widely acknowledged that the target body has to be put on a trifilar pendulum at least six different orientations to figure out the inertia tensor. But there seems to be no clearer discussion in literature about the exact number of the orientations to ensure a compromise between accuracy and test labor.
For each orientation during the test based on a traditional approach, the azimuths of the rotational axis of the trifilar pendulum have to be measured with respect to a predefined reference frame attached to the body. Unfortunately, however, it is not easy to figure out with a satisfying accuracy the angle between a line and a surface or between two spatial lines when only simple and crude tools are available due to limited budget, especially when the target body is odd-shaped and heavy.
To circumvent the above difficulties or disadvantages, a new approach is put forward in this paper, which can be used to identify all the 10 inertia parameters of the target body by means of a trifilar pendulum. In the new method, besides measuring vibration periods of the pendulum system as before, measurements are only conducted on the distances between geometrical points. From the measured distances, coordinate values of the involved points are determined with respect to two coordinate frames and then used to calculate the transform matrices. Balancing weights as well as load cells are used to reduce the manpower for the configuration adjustment.
Section snippets
Principal apparatus
A trifilar pendulum is implemented as shown in Fig. 1, where a sketch map is also presented. The trifilar pendulum consists of a lower disk suspended to an upper frame by three parallel cables. The cables are attached equidistantly around a circle near the periphery of the disk or the upper frame. The centroid of the circular disk coincides with its geometrical center. Certain calculagraph and rulers are needed for subsequent testing.
Different from a traditional trifilar pendulum established
Definition of reference frames
Four reference frames are defined to locate the rotational axis of the trifilar pendulum in relation to the body at any configuration.
The first frame, notated as Xi (i = 1, 2, 3), is attached to the lower disk of the trifilar pendulum as shown in Fig. 3. The origin coincides with the center of the disk. The X1-axis is parallel to the vector starting from point 1 to point 2, while the X2-axis is perpendicular to the X1-axis and towards point 3. The three points are the joints through which the
Configuration adjustment
The balancing weights and load cells, as depicted by Fig. 2 in Section 1, are employed for easier adjustments on the configuration of the target body.
After putting the body at a desired orientation to a proper location of the lower disk, one or several balancing weights are located on the disk so that the three load cells have identical readings within a permitted error tolerance. In other words, the balancing weight is used as an inching switch to make sure the rotational axis of the trifilar
Parameter identification
The mass of the target body can be obtained by subtracting the masses of the lower disk, the balancing weight and other accessories from the summation of the readings of the three load cells at any one of the designed configurations. The average of the masses corresponding to all configurations can be used as the final estimation.
For any one configuration, substituting the coordinates of any referential point in both frames xi and Xi into Eq. (5), ultimately leads to Xxo, the coordinate vector
Calibration and error analysis
Before conducting tests on the target body, some pre-tests or extra experiments are performed for calibration.
The error with identifying the mass is quite easy to evaluate. But a complete error analysis on the centroid coordinates and inertia tensor is quite complicated, as there are three error sources: (1) error occurring for one configuration, (2) error of figuring out the coordinate transform matrices and (3) error transfer during a series of matrix manipulations. For the concision of the
Examples
For the aforementioned new method, a special program has been implemented with some supplementary techniques to assure the accuracy of the method. More than 35 applications conducted by the authors during the past 5 years, including vehicle axles, engine, transmission or a whole powertrain, have validated the rightness and efficiency of the method. In order to demonstrate the method and its accuracy, two examples are presented in this section. In the first example, the inertia parameters of a
Summary
The idea and procedures of an improved method are presented, by which a trifilar pendulum with newly incorporated load cells and balancing weights can be used to identify all 10 inertia parameters of an odd-shaped body, structure or assembly. The main difference of the method from previous ones is to describe the configuration of the target body by means of coordinate transformation matrices, and the matrices are constructed from coordinates of referential points determined by distance
Acknowledgements
The above contents are extracted from several years’ work of the authors and their colleagues aiming at engine mounting design. Special thanks belong to Mr. Liu Yang, Mr. Xu Yang and Prof. Xu Shi’an for their pioneer work or suggestions. The authors also appreciate the help supplied by engineers from Chang’ an Auto in China with performing the tests on the rectangular block.
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