An improved pendulum method for the determination of the center of gravity and inertia tensor for irregular-shaped bodies
Highlights
► Present an improved trifilar torsional pendulum measuring inertia tensor of a body. ► The body is suspended with a universal joint. ► Easy adjustment of the C.G. of the body in line with the pendulum axis. ► Use data from a tri-coordinate measuring machine for calculating inertia tensor.
Introduction
The Center of Gravity (C.G.) and inertia tensor (the three moments of inertia and the three products of inertia) are the prerequisite design parameters for designing the dynamic performances of a mechanical system, such as dynamic characteristics of automotive Powertrain Mounting System (PMS) by which the vehicle comfort is determined [1]. The methodologies for determining the C.G. and inertia tensor of an irregular-shaped body are divided into three categories: calculation method based on three-dimensional solid model [2], model parameter identification method [3], [4], Trifilar Torsional Pendulum (TTP) method [5], [6], [7], [8], [9], and multi-cable pendulum [10].
It is a huge task to accomplish the three-dimensional solid modeling of an irregular-shaped body with all details, so the calculation method based on the CAD model is seldom used.
The methodology based on experimental modal analysis is simple in positioning postures of the body when identifying the C.G. and the inertia tensor, but with this method too many parameters need to be identified, the principle is complicate, the error analysis is difficult and the hardware requirements are expensive [8], [12]. Moreover, the modal analysis method is sensitive to measurement noise, selection of response measurement points and excitation conditions [4]. Hence, this method is still seldom used in practical engineering.
However, TTP method with simple structure and theory, has been widely used in engineering field [5], [6], [7]. The shortcoming is that it requires the C.G. of the measured irregular-shaped body must coincide with the pendulum axis [5], [6], [7], and it is always a repeated and skilled task to make the C.G. of a complex body in line with the pendulum axis. In addition, for the published PPT apparatus [5], [6], [7], in order to compute the C.G. and inertia tensor of the powertrain, it is required to measure the distance between two predefined points, which is often measured by using a ruler or caliper, thus with low precision.
To reduce the limitations of the current available methods [5], [6], [7], this paper proposes an improved TTP to determine the C.G. and the inertia tensor. Enhancements of the proposed TTP to the conventional TTP are: (1) a body with irregular-shaped is suspended under the TTP through an universal joint, thus the C.G. of the body can naturally lie on the pendulum axis of the TTP; (2) the methods for calculating C.G. and inertia tensor of the body are based on the coordinate and vector transformation.The coordinate and vector are measured with a tri-coordinate measuring machine and so with high precision. (3) The coordinates and vectors obtained in each measurement can be used simultaneously both for calculating the C.G. and the inertia tensor of an irregular-shaped body.
The errors of the proposed TTP come from two categories. One is from the precision of the experimental setup [13]. The length of the TTP wire and the empty mass of the TTP are two important parameters for the precision of the experimental setup. The method for identifying the length and the empty mass of the TTP are described and analyzed in detail. Another one is from data processing procedure for obtaining C.G. and inertia tensor of the body. To validate the effectiveness of the proposed procedure, the C.G. and inertia tensor for a regular body that merged with two different regular rectangular bodies are calculated with measured coordinates and vectors. The results are compared with the theoretical solution obtained from the CAD model. The comparisons show that the deviation between the measured C.G. and the theoretical C.G. is less than 1.5 mm, and the relative error for the measured moment of inertia around an axis is less than 1%.
Finally, the C.G. and the inertia tensor for an automotive powertrain (an engine plus a transmission) are measured with the developed TTP and the proposed data processing procedure by using two different approaches. One approach is to directly measure and calculate the parameters from the powertrain. Another approach is to calculate the parameters based on synthesizing the C.G. and the inertia tensor from the engine and the transmission, which are identified from the measurement on the engine and the transmission, respectively. The parameters from the two approaches agree reasonably well. It is demonstrated that the developed TTP and the data processing procedure has good characteristics for repeat measurement and with high precision. The experimental method, developed TTP and data processing technologies proposed in this paper can be used for getting the C.G. and the inertia tensor for an irregular-shaped body with high precision and less skill requirements.
Section snippets
Definition of coordinate systems
The inertial properties of a body consist of the mass, the Center of Gravity (C.G.), the moments of inertia and the products of inertia. Where, the moment of inertia and the products of inertia are the components of inertia tensor. The mass is unrelated to any coordinate system and can be measured easily. But it is a rather arduous task to determine of the C.G. and the inertia tensor. In addition, the C.G. and the inertia tensor should be defined in one specific coordinate system. To calculate
Geometrical configuration
Fig. 2 shows a conventional TTP, which is composed of a lower disk hanging from a fixed frame using three parallel wires. The 3 steel wires are used to connect the lower disk to the upper frame through 3 holes. The 3 holes are at the same radial distance and separated 120 degrees. The length of the three wires is the same. The steel wires must be firmly attached to avoid any distance variation during the loading of the bottom disk. The C.G. of the lower disk coincides with its geometrical
Errors estimate
The errors in the TTP are divided into two categories. The fist category of the error is determined by the precision of the experimental setup, i.e., tri-coordinate measurement system, photo-electronic sensors, and pendulum parameters (empty pendulum mass, steel wire length and lower disk radius). Another category of the error is caused by the improper centering of the body, i.e., the mass center of the body is not coincident with the pendulum axis. This situation will produce another
Example
To verify the measurement and data processing procedures, an example is performed by means of the identification of the C.G. and inertia tensor of a vehicle powertrain. Error analyses are also carried out to demonstrate the reliability and precision of the proposed TTP.
This section presents two methods to obtain the C.G. and the inertia tensor of the powertrain. The first method is to measure and calculate the C.G. and the inertia tensor with a powertain directly. The second method is to
Conclusion
An improved TTP to determine C.G. and inertia tensor of an irregular-shaped body is presented. Compared to the conventional TTP measurement method, this new apparatus has several advantages: (1) both C.G. and the inertia tensor can be obtained from each measurement, thereby improving the measurement efficiency. (2) A universal joint is employed to facilitate the adjustment of making the mass center of the body coincide with the pendulum axis, which shortens the test cycle. (3) By using
Acknowledgments
The authors gratefully acknowledge the financial support of the Natural Science Foundation of China (Project Nos. 50575073, 50975091), Science Creation Project of Beijing Forestry University (Funding No. YX201110) and New Faculty Support Funding of Beijing Forestry University (Funding No. 2010BLX11).
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