NL1040175C2 - A vibrating system with multiple and equal natural frequencies. - Google Patents

Description

A VIBRATING SYSTEM WITH MULTIPLE EQUAL NATURAL FREQUENCIES

Reducing component size offers many advantages in both industrial as well as consumer applications. As components become smaller, the need for low uncertainty 3D measurements increases. Moreover, on these components, more and more micro-sized features with high aspect ratio can be found that need to be dimensionally characterized. In the last decade various high precision 3D measurement probes have been developed and are currently commercially available. An extensive overview of probes used in dimensional (nano)metrology is given by Weckenmann et al [1,2]. Available tactile 3D microprobes are capable of measuring with nanometer repeatability, low contact forces and display low hysteresis [3,4], Despite these excellent specifications there are still factors that limit their operation, in particular when considering measurements on the micro scale [5]. On the micro scale, surface forces between probe and workpiece such as for instance capillary, Van der Waals and electrostatic forces can cause the probe to be pulled in and stick to the surface under test. Thus presenting limitations for single point measurements as well as for scanning measurements in which stick-slip behavior will become apparent. Another limitation of tactile 3D microprobes is due to the layout of most of these probes; a stylus is used to connect its probe tip to the probe measurement system, i.e. the stylus is part of the metrology loop. As a result the measurement sensitivity of these probes decreases if the stiffness of the stylus is in the same order of magnitude as that of the probe suspension, thus preventing the use of high aspect ratio (length of the stylus over its diameter) styli [5]. As a result existing 3D tactile microprobes are unable to measure high aspect ratio microfeatures, such as microholes or sides of microchannels. The device described here does not suffer from these difficulties and is still capable of measuring with high sensitivity in any direction.

The limitations mentioned above of existing tactile 3D microprobes can be overcome by a vibrating (micro)probe [6], The measurement principle of a vibrating probe is based on changes in its dynamic response as it interacts with the surface of a workpiece. The probe is excited close to or at its natural frequency. Due to probe tip-surface interactions the oscillation amplitude, phase and resonance frequency change. These changes are registered and used as measurement signals. This is displayed schematically in figure 1, on the left the probe oscillates freely and on the right the probe interacts with a surface as indicated schematically by the dashed circle. Also shown in this figure are the corresponding magnitude and phase response plots. As can be seen from the magnitude and phase responses, upon interaction these responses shift, which is used as the measurement signal. This change in the dynamic response of the system can be registered with very high sensitivity around the natural frequency. At this frequency the magnitude and phase responses respectively show a sharp peak and a steep slope which can be used for measurements with a high sensitivity.

Vibrating probes offer a solution for the difficulties a tactile probe experiences such as stick during point measurements and stick-slip during scanning measurements. A vibrating probe can interact with the surface of the workpiece in two ways, which are denoted by terms used in atomic force microscopy: the so-called intermittent or tapping contact mode and the non-contact mode. When operated in tapping contact mode, the vibrating probe touches (taps) the surface each oscillation. The oscillation amplitude is large enough to overcome sticking to the surface of the workpiece [7], In non-contact mode, the tip of the probe interacts via surface forces with the surface of the workpiece. In this mode no mechanical contact exists between probe-tip and workpiece and consequently the tip will not stick to the surface.

Similar to most tactile probes, the vibrating probe under consideration here uses a stylus to connect the probe tip to the main probe body where the actual measurements are performed. As a consequence the measurement sensitivity of a tactile probe is determined by the ratio of stylus stiffness to probe suspension stiffness, as discussed previously. However, for a vibrating probe this is somewhat different, not just the stiffness of the stylus and suspension but rather the dynamic behavior of these components determines the sensitivity with which measurements can be performed. To illustrate this, a model of a vibrating probe as shown in figure 2 is used to simulate its dynamic response. The model consists of two masses, the main probe body, mp, and the mass of the stylus (including probe-tip), mst, connected to some base structure via the probe suspension, kps, and to each other via the (lateral) stylus stiffness, kst. Damping is introduced by dps and dst. In this qualitative analysis the surface interaction forces are modeled with a linear spring, ksf. Note that the modeling of the interaction forces by a single linear spring is incomplete but will serve for this qualitative analysis. The probe is excited by translating the base, x0, and resulting translations of the main probe body and the stylus mass are denoted by xp and xsU respectively. This system can be characterized by two dynamic transferfunctions: H0p/ which describes the relation between the base excitation x0 and the translation of the main probe bodyxp and H0<st.p, which describes the relation between the base excitation x0and the difference in translation between the stylus mass and the main probe bodyxJt-xp, both as a function of the angular frequency ω:

H0,p is the response of the main probes body that is measured and changes in this response are used to determine interaction between probe-tip and workpiece. H0,st-P indicates how well the tip of the stylus follows the main probe mass and, consequently, how well information sensed at the probe-tip can be transferred via the stylus to the main probe body, i.e. the measurement system.

Figure 3 shows the reduction in response H0,p> the amplitude of the main probe mass, as a function of the stylus diameter for a typical vibrating system as shown in figure 2 using parameters (stiffness, damping, mass) of a Gannen XP [5,8], which interacts in a tapping-mode fashion with a workpiece made of an engineering material such as steel. The vertical axis in this figure shows the amount with which the amplitude of the main probe mass is reduced upon contact compared to the original amplitude (when there is no contact). For higher reductions, higher sensitive measurements can be performed. As can be seen, there are two regions where the reduction is large enough to be considered for high sensitive measurements. The first region is for large diameter styli, the larger the diameter of the stylus, the larger the stiffness connecting the probe tip to the main probe mass and both masses act as a single (rigid) mass. The second region is the region where the natural frequency is equal or close to the natural frequency of the probe suspension. For the system whose response is displayed in figure 3, this is the case for a stylus diameter of approximately 32 pm. In practice it can be quite challenging to match the natural frequency of the stylus to the natural frequency of the probe suspension, so this second region is in this instance not considered as a viable option. Considering the first region, the resolution with which the amplitude reduction can be registered determines the minimal diameter with which measurements can be performed. With typical amplitudes in tapping-contact mode of 100 nm [7,9] and stating that the minimal resolution that can be registered using an optical beam deflection method is 0.1 nm [10,11], a reduction in amplitude of 0.1% can be resolved. Which corresponds in figure 3 to a stylus diameter of 42 pm. Considering that the results shown in figure 3 are based on a stylus length of 6.8 mm this leads to a stylus aspect ratio of 162, which is a great improvement compared to existing tactile microprobes such as the Gannen XP and XM with a maximum aspect ratio of 22.5 and 40 respectively [5], Concluding it can be stated that compared to tactile microprobes, vibrating microprobes are better suited to perform high sensitivity measurements on high aspect ratio microfeatures [6],

Until now only measurement characteristics in one dimension are considered, the novel feature of the invention described here is to be found in its multi-dimensional measurement capabilities. Various vibrating probes exist that are capable of measuring in multiple dimensions as for instance the probes described in [12,13,14,15]. A drawback of these probes is that their natural frequencies are not equal and consequently these probes suffer from a decrease in measurement sensitivity for directions of vibration that do not correspond to one of the natural modes of the system. To illustrate this a model is presented of a two dimensional vibrating probe, shown in figure 4. It shows a model of a 2D vibrating probe consisting of a mass m, which is fixed in x and y direction by two springs kx and ky, respectively. Parallel to each spring, although not depicted, a damper is present, dx and dy, respectively. The contact between probe and surface under test is, as in the previous section, modeled by a single spring, ksf. Again the assumption is made that for the qualitative analysis performed here, this simplified representation of the probe tip-surface interactions will suffice. The direction in which the vibrating mass interacts with the surface under test is modeled by the angle Θ. The system is once again excited via excitations of the base, Xo and y0, and the position of the mass is expressed as x and y.

Figure 5 shows the normalized response of the mass vibrating in x-direction upon interaction with the surface forces for varying angle Θ. The response is taken as the difference in vibration of the free system (no surface interaction forces) and the vibration of the system with surface interaction forces. At 0 degrees ksf is in line with kxand a maximum response is registered, at 90 degrees the contribution in x-direction by ksf is zero and consequently, the response registered in x-direction is zero. This response compares well to experimental results obtained by [12], One remark needs to be made however, where the direction of vibration is parallel to the surface (at 90 and 270 degrees), a zero response is registered which in practice will not occur. As can be seen in [12], the response at these angles will be at a minimum, but a response will still be registered. This discrepancy is likely caused by the way in which the interaction between probe tip and surface under test are modeled. Effects such as squeeze film damping, which will affect vibrations near a surface, or friction of the probe over the surface workpiece need to be incorporated into the model to describe the response more fully at these angles.

As can be concluded from figure 5, the directional sensitivity of a one dimensional vibrating probe changes significantly as the direction of vibration with respect to the surface under test changes. As the probes considered here are multidimensional, it is straightforward that when one measures in y-direction one does not employ the x-response, as this response is zero in y-direction (at 90 and 270 degrees in figure 5). As a consequence the resulting response of the vibrating probe is improved considerably as the y-response is also included as shown in figure 6b. Figure 6b shows the response of a probe with two different natural frequencies in x and y direction and the corresponding dynamic magnitude response of this probe is shown in figure 6a. As can be seen from figure 6a, the natural frequencies of this probe differ by 10 percent and, consequently, the x-and y-response are different, next to the obvious 90 degrees shift in response, for the same disturbance ksf. Moreover, as the natural frequencies of the x and y mode are unequal and far apart, one cannot drive the system at a single frequency without losing sensitivity. One has to choose what response (x or y) is the most sensitive for a given direction of vibration (Θ) and drive at that corresponding natural frequency. As a result the total response of this system is highly non-isotropic as shown in figure 6b. Existing multidimensional vibrating probes such as described by [12,13,14,15] all have multiple, unequal natural frequencies and, consequently, will experience non-isotropic measurement sensitivity throughout their measurement range.

The invention described here is a vibrating system which has multiple and equal or near-equal natural frequencies, in which near-equal is defined as having a difference in frequency of less than 5%. This allows the system to be driven at a single frequency resulting in a well defined vibration, with or without employing a suitable control algorithm, and to perform highly sensitive measurements in any direction. Figures 7a and 7b show a system with near-equal natural frequencies and its measurement response. As can be seen from figure 7a, the system has two natural frequencies that differ by 1 percent allowing the system to be driven at a single frequency whilst operating in the sensitive part of both the x and y dynamic response of the probe. As both the vibrations in x and y direction of the system are driven at the same frequency, their responses can be combined resulting in the total response shown in figure 7b. This system is capable of measuring with nearly equal measurement sensitivity in any given direction. Reducing the difference between the natural frequencies to zero results in the system shown in figure 8a and its corresponding response in figure 8b. This system is capable of performing measurements with isotropic sensitivity throughout its measurement range as indicated by the total response in figure 8b.

Examples of the layout of a system with multiple natural frequencies that differ less than 5% in frequency are shown in figure 9. It consists of a platform (1) suspended via one or more flexures (2) to a base (3). On the platform a protruding element (4) can be mounted, such as a probe stylus which in turn can be outfitted with a probe tip (5) of spherical or other shape. The system is further outfitted with one or more actuators to excite the platform. One or more sensors are included in the system to determine the position, orientation or a combination thereof of the platform. A control algorithm is included to move a point on the protruding element in a predefined motion, where the predefined motion is a linear translation, a curved translation, a rotation or a combination thereof. The direction of vibration is set such that it is substantially orthogonal to the surface of the artefact with which it interacts. A control algorithm can be included to influence the mass, damping, stiffness or a combination thereof to obtain a vibrating system in which the natural frequencies differ less than 5% in frequency. Furthermore one can alter the properties of the system such as mass, damping, stiffness or a combination thereof by a machining procedure to obtain a vibrating system in which the natural frequencies differ less than 5% in frequency. The system can be mounted on a coordinate measuring system or a coordinate measuring machine in order to serve as a measuring probe.

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[8] Xpress Precision Engineering website, www.xpressoe.com. visited on 10-07-2013.

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Claims (14)

A measuring instrument for detecting contact with a workpiece, consisting of: a platform that has at least three natural frequencies that differ by less than five percent in frequency; one or more elastic elements from which the platform is suspended; one or more protruding elements which are on the platform; one or more actuators which excite the platform one or more sensors which measure the position, orientation, rotation or a combination thereof of the platform. A vibrating system according to claim 1, characterized in that the dynamic response such as the amplitude response, phase response, natural frequencies or a combination thereof, changes as a result of interaction of the platform or a projecting element with an external artifact. A vibrating system according to claim 2, characterized in that changes in the dynamic response are used to detect interaction, contact or a combination thereof between platform or protruding element and an external artifact. A vibrating system according to claim 2, characterized in that changes in the dynamic response are used to measure the position of the platform or protruding element relative to an external artifact. A vibrating system according to any one of claims 1-4, characterized in that the system is excited on one frequency. A vibrating system according to claim 5, characterized in that the frequency at which the system is excited is within five percent of a natural frequency of the vibrating system. A vibrating system according to any of claims 1-6, characterized in that the direction of the vibration of a point on the projecting element is controlled to realize a predetermined movement. A vibrating system according to claim 7, characterized in that the predetermined movement relates to a linear translation, a curved translation, a rotation or a combination thereof. A vibrating system according to claims 7 or 8, characterized in that the direction of the vibration is substantially orthogonal to the surface of the artifact with which the system interacts. A vibrating system according to any of claims 1-9, characterized in that the natural frequencies of the system, its eigenvectors or a combination thereof are manipulated with the aim of obtaining a vibrating system whose natural frequencies are less than five percent in frequency of differ. A vibrating system as claimed in claim 10, characterized in that, based on the position, speed or acceleration of the platform or a combination thereof, the forces supplied by the actuators are manipulated with the aim of obtaining a vibrating system whose natural frequencies less than five percent in frequency. A vibrating system according to claim 11, characterized in that the mechanical properties of the system such as the mass, damping, stiffness or a combination thereof are changed by means of an operation procedure to obtain a vibrating system whose natural frequencies do not exceed five percent in frequency. A vibrating system as claimed in any one of claims 1-12, characterized in that the vibrating system is used as a measuring instrument or forms part of a measuring instrument and is used to determine the position, shape, displacement, distortion, or a combination thereof of an artifact to decide. A vibrating system according to any one of claims 1-13, characterized in that the vibrating system is used in a coordinate measuring machine.