Using Mechanical Event Simulation in the Design of Mechanisms

ABSTRACT

In this paper, we also focus on using MES to establish limits of operability for mechanisms that undergo large-scale motion. Generally, these limits are defined by the manifestation of permanent deformation. We use a centrifugal flyball governor to show how high rotation rates and the corresponding large-scale motion can induce permanent deformations. Flyball governors are designed to provide for physics-based feedback control in order to regulate the speed of the rotary steam engine. Thus, it is only appropriate that we also consider the effects of temperature on the motion of the governor. The resulting multiphysics problem includes a heat transfer analysis that considers both conduction and convection. Modern machinery is regulated using sophisticated, computer-based feedback control techniques. We end the paper examining how MES can be used to accurately analyze the motion and stresses of a mechanism whose movements are generated using computer-controlled actuators.

1.  INTRODUCTION

Before discussing why MES is capable of simultaneously solving for motion and stresses (using one model) under even large strain scenarios, we will first examine the standard means by which FEA is applied to mechanisms. The standard approaches consist of isolating each of the mechanism’s components and applying linear FEA methods to obtain stresses. Because the geometrical configuration of a mechanism generally changes dramatically throughout its operation, a single linear FEA analysis may not suffice. This is because linear FEA assumes that only infinitesimal changes in the body’s shape occur. Thus, it is necessary to consider only snapshots – frozen configurations – of the moving parts of the mechanism. Engineering experience is required to determine which configuration(s) result in the stresses critical to the design. Besides only using snapshots of the motion, the standard approaches suffer from other major drawbacks inherent to linear FEA such as:

1. Loads representing both the body’s motion and its interaction with other parts of the mechanism must be estimated.
2. Fictitious boundary conditions (constraints) may need to be introduced in order to avoid the presence of rigid body, non-infinitesimal motion.
3. Any large-strain (non-infinitesimal) material behavior is neglected.

Improvements in computing technology and numerical methods aided in the development of MES. The goal of MES is to simulate any physical mechanical event. In other words, MES is a tool for conducting virtual experiments. As mentioned above, MES solves for both the motion and the stresses simultaneously. Thus, it is particularly applicable to the analysis of mechanisms. One may think that standard, nonlinear FEA techniques would also be appropriate for the analysis of mechanisms. But, such techniques generally fall short when it comes to simulating large-scale motion. Many nonlinear FEA techniques utilize explicit time integration methods, which require the use of very small time-steps for the sake of stability. Thus, it is often impractical to use explicit methods when simulating (entire) events during which motion occurs over a relatively long period of time, as is the case with events involving many mechanisms or impact tests. MES utilizes an implicit time integration method that does not have nearly such strict time-step size restrictions. Implicit time integration allows for relatively large time-steps to be taken when non-critical events are occurring while still taking into account the inertial effects associated with motion. During critical times, such as when collisions are taking place, a smaller time-step size is warranted in order to accurately capture the underlying physics. In general, mechanical events contain a mixture of critical and non-critical periods. For instance, large-scale gross motion of a part can be classified as non-critical as long as no other competing phenomena are occurring. MES utilizes an automatic time-stepping scheme particularly designed to take advantage of implicit integration. This automatic time-stepping scheme is one of the reasons why MES is a viable tool in the analysis of physical events.

Before discussing other reasons underlying the success of MES, it is important to elaborate on the automatic time-stepping scheme. This scheme utilizes the local rate of convergence to ascertain whether a critical period is beginning. Imminent geometrical penetrations that occur as a result of surface interactions are also used to signal critical periods. If a critical period is detected, the scheme reduces the time-step size; whereas, if a critical period has ended, the time-step size is increased. Such a scheme is not only accurate, but is also computationally efficient. The implicit integration method invoked by the automatic time-stepping scheme is highly suitable for the analysis of events involving motion coupled with critical behavior, such as sudden impacts. Specifically, this method intelligently diminishes the importance of erroneous high frequencies common to FEA. This algorithmic dissipation does not result in a loss of accuracy because it does not affect the important low frequency modes. Furthermore, the order of the integration method is not reduced when algorithmic damping is introduced, as is the case with the classical Newmark method [1]. The fact that the integration order is maintained is one of the reasons why MES is capable of simulating long-lived events. The simulation of such events does require an efficient numerical implementation. For instance, care must be taken to ensure that models involving contacting surfaces are not computationally overwhelming. The dynamic sparse solver in MES only considers terms associated with locations currently making contact. This allows for efficient calculations of models where the location(s) and time(s) at which surface contact will occur are not known prior to the simulation.

In this paper, we focus on applying MES to objects experiencing large-scale motion, including some mechanisms. In Section 2, we consider the accuracy of the MES method. Specifically, we examine its ability to model a rigid mechanism with a known analytical solution. Considering the rigid limit has the added advantage of showing how a nonlinear FEA method can accurately simulate large-scale motion. In Section 3, we show the results of a study that indicate how the analytical solution given in Section 2 is inappropriate when the parts of the mechanism deform enough to significantly affect its overall motion. Such large-strain deformation effects are easily captured by MES. Section 4 deals with the question of whether to consider multiphysics when modeling a mechanism. The close coupling of MES with standard heat transfer FEA facilitates such multiphysics analyses. Finally, in Section 5, we briefly describe how MES is used to simulate events involving two realistic mechanisms.

2.  VERIFYING THE ACCURACY OF MES

In this section, we utilize an analytical solution of a rigid mechanism to verify that MES is capable of accurately simulating such a situation. The problem considered is a simplified model of a flyball governor. The solution of this problem is discussed in [2]. Figure 1 is a sketch of the simplified model of a flyball governor assembly. The assembly consists of four identical arms (solid cylindrical rods), two spheres and a cylindrical base. Each of the rods weighs 10 N. The spheres each weigh 18 N and have a radius of gyration of 30 mm about their diameter. The base, which rotates with the assembly, weighs 20 N and has a radius of gyration along its axis of 50 mm. Initially, the system is rotating at speed ₁ of 500 rpm for = 45º. A force applied at the base in the Z direction maintains this configuration. We use MES not only to simulate these conditions, but to consider what happens when the force is changed such that decreases from 45º to 30º.

Let us focus on the event during which the angle decreases from 45º to 30º. During this event, no external moments act along the Z axis; thus, we have conservation of angular momentum about this axis. Note how each component of the governor assembly rotates about the Z axis. The solution of the problem is obtained by equating the total angular momentum of the system when it has an angle of 45° and an angular velocity of 500 rpm to when it has an angle of 30° and the unknown angular velocity, ₂. As a first step, we calculate the rotational inertia of each body about the Z axis.

First, consider the rod on the top right, which is shown in detail in Figure 2. The axes are principal axes of inertia for the rod, with their origin at the top joint. The axis is collinear with the Y axis, and both these axes are normal to the page. A rotation of angle about the axis relates the axes with the XYZ axes. The following expression shows how and each contribute to the rotational inertia of this rod about the Z axis,

It should be noted that is given by 1/2 Mr², where M is the mass of the rod and r is its radius. The rotational inertia about the other two principal axes is given by 1/3 Ml², where l is the length of the rod. These values for and assume that the rod is slender; which is the case in this example because it has an aspect ratio, l/r, of 40.

For each sphere, we are given the mass, the radius of gyration and the location of its center relative to the end of the rod, thus

We are also given the mass and the radius of gyration of the cylinder, thus

Exploiting the symmetry of the assembly, we can state that angular momentum along the Z axis is conserved if

Substituting the results of equations (1) through (3) into the above conservation statement, and solving for , gives = 882.9 rpm. It should be reiterated that this result was obtained assuming that the flyball governor assembly is composed entirely of rigid parts.

MES is utilized to simulate an analogous change in configuration in an equivalent FEA model of the flyball governor described above. The flyball governor is modeled using flexible brick elements to represent the rods. Because we expect insignificant stresses to exist within the spheres and the (base) cylinder, we utilize 3-D kinematic elements to represent these components. Kinematic elements do not deform, yet possess inertia; thus their motion contributes to the stresses within the rods. In contrast to the analytical solution, which dealt with the abstract concept of the radius of gyration of the spheres and cylinder, the MES solution considers the motion of objects with detailed geometry. It is straightforward to find the physical dimensions of spheres and cylinders when given their radius of gyration. Specifically, a sphere with radius of gyration, k_sph, about its diameter of 30 mm has a physical radius, r_sph, of 0.04743 m (r_sph = 1.58114 k_sph), and a cylinder with radius of gyration, k_cyl, about its axis of 50 mm has a physical radius, r_cyl, of 0.07071 m (r_cyl = 1.41421 k_cyl). As a consequence of modeling a geometrically detailed mechanism, we also need to determine the density of all of the components of the governor. We are given the weight of each component, and we now know all pertinent physical dimensions. Thus, using the acceleration of gravity we can obtain the appropriate densities of the rods (19,219 kg/m³), the spheres (4,102.4 kg/m³) and the cylinder (1297.3 kg/m³). We also need mechanical properties for the flexible rods. We assume them to be composed of a material similar to steel with a Young’s modulus of 7.31×10¹⁰ N/m² and a Poisson’s ratio of 0.397.

The MES consists of three stages. At the onset of the first stage, all parts of the governor are given an initial axial rotation of 500 rpm about the Z axis. During this stage, the cylinder is maintained at a constant Z coordinate value – one that keeps at 45°. During the second stage, the bottom of the cylinder is moved until reaches 30°. The third stage is characterized by another constant Z coordinate value for the cylinder, and by the new faster rotation rate.

Figure 3 depicts the displacement time traces of two points on the governor. The dashed line corresponds to the displacement of the cylinder in the Z direction. Note how this displacement is constant during stages 1 and 3, but depends linearly on time during stage 2. The solid line corresponds to the Y displacement of a node at the end of a rod. This displacement is purely sinusoidal during stages 1 and 3. From the plot we obtain periods of motion of 0.1201 and 0.0681 seconds during stages 1 and 3, respectively. These periods correspond to 499.6 and 881.1 rpm. The first revolution rate simply confirms the ability of MES to simulate a constant motion of 500 rpm. The second revolution rate compares well with the analytical result of 882.9 rpm, and thus confirms the ability of MES to accurately simulate motion, including that experienced by an assembly with moving parts.

3. USING MES TO SIMULATE LARGE STRAINS

In Section 2, we showed how MES accurately describes the motion of a mechanism of rigid components. In this section, we consider the situation of a mechanism whose components deform significantly during its operation. MES was used in a study involving different models of the flyball governor. The models differed in two types of parameters: (1) the stiffness of the rods, and (2) the initial rate of rotation. Certainly, decreasing the stiffness of the rods will invoke greater deformations. Whereas, increasing the rotation rate will indirectly induce larger deformations through inertial effects. Equation (4) can easily be modified to give solutions for other initial rotation rates. In the study, we considered initial rotation rates of 500, 1000 and 2000 rpm, for which Equation (4) gives 882.9, 1,765.8 and 3,531.5 rpm for the final rotation rate, respectively. The results of the study are summarized in Table 1. As expected, note how as either the rotation rate is increased or the stiffness is decreased, the deviation from the analytical results increases. These deviations can be attributed to large-strain behavior. The length of a bottom rod was measured at the end of each simulation and compared with its original size. Table 1 also gives the elongation in the rod’s length. Note how the larger this change, the larger the discrepancy from the analytical solution. Permanent deformation occurs if this elongation is enough to cause the yield stress to be reached.

No special care was taken to ensure that large strains would be captured by the MES. Standard methods to analyze mechanisms would have been difficult to apply for the cases exhibiting large strains. This limitation of other methods to analyze mechanisms is due to the inability of most FEA techniques to simulate motion. Even the methods that utilize a separate dynamics package to solve for the motion and subsequently provide resultant forces to the FEA application are not intended for cases where large strains occur.

Initial Rpm	1/10 of Original Stiffness	Base Original Stiffness	10 Times Original Stiffness
500	498.3 (0.34%) 867.1 (1.79%) 0.867 mm	499.6 (0.08%) 881.1 (0.20%) 0.089 mm	499.7 (0.06%) 882.4 (0.06%) 0.009 mm
1,000	985.5 (1.45%) 1,639.3 (7.16%) 2.748 mm	996.4 (0.36%) 1,734.1 (1.80%) 0.350 mm	997.5 (0.25%) 1,744.2 (1.22%) 0.036 mm
2,000	1,903.4 (4.83%) 2,906.0 (17.71%) 10.058 mm	1,980.6 (0.97%) 3,368.7 (4.61%) 1.297 mm	1,989.5 (0.52%) 3,457.4 (2.10%) 0.138 mm

Table 1: Summary of study conducted to determine effect of initial rotation rate and stiffness on mechanical behavior of flyball governor. The top line in each cell corresponds to the rotation rate observed in the MES before the base’s location was moved; the number in parenthesis corresponds to the difference from the input initial rotation rate. The second line corresponds to the final rotation rate observed in the MES after the base’s location was moved; the number in parenthesis corresponds to the difference from the rotation rate given by equation [4]. The last line is the extension of a lower rod during the course of the event.

4. WHEN TO CONSIDER MULTIPHYSICS EFFECTS

is used to calculate Nu for the rods. Values of 37.7 and 46.4 were obtained for the two rotation speeds. A correlation given in [4] is used to calculate Nu for the spheres,

Values of 128.2 and 177.2 are obtained for the two rotation speeds. Using these values of Nu, it is possible to calculate the appropriate heat transfer coefficients, h, using

where d is the pertinent diameter. Values of 66.0 and 81.2 W/(m²-°C) are obtained for the rods, and 42.1 and 58.1 W/(m²-°C) for the spheres. This information completes the data required by the heat transfer analyses. These analyses show that the greatest temperature gradients occur in the lower half of the bottom rods. The analysis corresponding to 500 rpm results in the temperature changing from 139.2 to 38.1°C along this portion of the rods. In the 1,000 rpm case, the temperature changes from 138.2 to 35.6°C. These results show that the temperature distribution is not highly sensitive to rotation speed. Yet, the temperature gradients may be significant enough to warrant the use of a temperature-dependent material model for the rods when conducting a corresponding MES.

5. SIMULATIONS INVOLVING REALISTIC MECHANISMS

Prior to this section, we have focused on a simplified model of a flyball governor. Now, we change our emphasis to models of realistic, more complex components and mechanisms. The MES results of these models not only include stresses and motion, but animations of the simulated events. These results will be shown in the presentation associated with this paper. The first realistic model is that of an actual flyball governor. Figure 4 illustrates the finite element mesh used by the MES for this model. It should be mentioned that this model originated in CAD. Its CAD geometry was seamlessly captured and converted to FEA data without the need of an intermediary file – a direct memory transfer was performed. The MES shows how the stresses in the spring are affected by the rotation speed and corresponding geometrical configuration of the governor.

The second realistic model analyzed is shown in Figure 5; the figure shows the finite element mesh used by the MES. This model is of an actuator clamp assembly. Once again, the geometry of this model was directly transferred from CAD. The MES includes surface contact effects, and is driven by actuator elements. These elements are used to specify the relative motion of two points of a structure or mechanism – just like actual computer-controlled actuators. It should also be noted that kinematic elements are used in both realistic models to describe components expected to experience insignificant strains. Kinematic elements can be constrained and loaded, and interact in surface contact just like their flexible counterparts. It is through changes in their inertia that kinematic elements affect the stresses in adjoining flexible elements. The advantage of using kinematic elements over flexible elements is that they result in many less unknowns in the underlying system of equations. Thus, much faster solution times are obtained when using kinematic elements. Finally, because the second model included surface contact effects, it took advantage of the dynamic sparse solver of MES. Thus, only points currently making contact contributed to the stiffness matrix – making the entire simulation significantly faster.

REFERENCES

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