This invention is related to the constraint of substrates, optics, or other mechanical components with low distortion and low levels of vibration.
In many mechanical assemblies, it is desirable to limit the deformation and position uncertainty of a body within the assembly. For example, it is often necessary to mount or manipulate a mirror, lens, or substrate with low distortion and low uncertainty in its location.
It is therefore common to apply exact-constraint (often referred to as kinematic) design principles in the design of structures, mounts, or grippers. In its simplest form, each body is considered rigid, and therefore each body has six degrees of freedom: displacement in three directions and rotation about three axes. To constrain a body according to exact-constraint design principles requires six constraints with the constraints applied at a discrete set of contact locations, where the contact locations are small compared to the overall dimensions of the body. See Blanding, D. L., Exact constraint: machine design using kinematic principles. ASME Press, 1999.
Real bodies are flexible and therefore susceptible to vibration, which often limits the performance of a machine or assembly. Among the most common methods for limiting the vibration of a body is to design it so that its resonant frequencies are as high as possible. But because exact-constraint design principles limit the number of locations where a flexible body is supported, there is often a difficult trade off between achieving a design that approximates exact constraint and attainment of high resonance frequencies.
A common requirement in precision machines is to constrain a plate-like structure or substrate, such as a silicon wafer, a mask, or a mirror. For plate-like structures, it is common to violate exact-constraint principles in the plane of the structure where it is relatively stiff, but to obey the exact-constraint principles in the direction normal to that plane. In such instances, we can restrict discussion to provision of three constraints in the direction normal to the plane of the structure.
For example, a silicon wafer 1 with radius R as shown in
The present invention provides a method and embodiments for constraining flexible bodies according to exact-constraint design principles while attaining the theoretical maximum resonant frequency. For plate-like structures, a set of three pivot rockers placed to preserve the free modes of the structure attain the maximum resonant frequency.
The Courant Minimax Principle is a mathematical principles summarized as follows: “If a linear constraint is applied to a system, each natural frequency increases, but does not exceed the next natural frequency of the original system.” See Gladwell, G. M. L., Inverse problems in vibration. Springer, Netherlands, 2005. p. 40-46. Therefore, if the natural frequencies of a flexible body before addition of a constraint are written in sequence as λ1, λ2, λ3 . . . and the natural frequencies of the flexible body after addition of a constraint are written as λ′1, λ′2, λ′3 . . . , the new natural frequencies must obey
λ1≤λ′1≤λ2, λ2≤λ′2≤λ3, λ3≤λ′3≤λ4 (1)
and so on. From the Courant Minimax Principle, we further find that a constraint which renders λ′1=λ2 must exist. This is referred to as the limit natural frequency.
In the first embodiment, a circular substrate (e.g., a silicon wafer) 1 is exactly constrained in the plane normal to the wafer surface with maximum natural frequency. Before these constraints are added, the circular substrate is free to vibrate in the direction normal to its surface, and the first three free mode shapes have zero natural frequency. The first three free mode shapes with non-zero natural frequency are shown in
According to the Courant Minimax Principle, three constraints that render the first two natural frequencies of the constrained system equal to the first two non-zero free natural frequencies of
In the first embodiment we choose the four points 8, 9, 10, and 11 as shown in
w8=−w9, w9=−w10, and w10=−w11 (2)
A drawing of such a system is shown in
The contacts 12, 13, 14, and 15 are attached to pivot rockers 16 and 17 by means of hourglass flexures 18. These allow the pivot rockers to rotate about flexural pivots 19 and 20 attached to base 21 without causing the contacts 12, 13, 14, and 15 to slip against the substrate 1. The pivot rockers 16 and 17 are connected by means of a third pivot rocker 22 by means of hourglass flexure pivots 23 and 24, and pivot rocker 22 is attached to base 21 by means of flexural pivot 25.
A three-dimensional finite-element analysis of the embodiment as shown in
The hourglass flexure 18 may be constructed as shown in
The flexure pivots 19, 20, and 25 may be constructed as shown in
The condition given by Eq. 2 could be realized with points 8, 9, 10, and 11 at any radius from the center, and therefore the constraint mechanism can avoid constraining the modes of
A second embodiment is now described with application to square plate-like structures. The first three free mode shapes with non-zero natural frequencies are shown in
In a third embodiment, the supports of the rocker arm are placed to fall on the nodal line 36 of the third free mode shape with non-zero natural frequency of
A rectangle with aspect ratio of less than 1.4:1 has mode shapes qualitatively like those of the square, and therefore the same constraint method as shown in the previous embodiments yields the same result by modification of the dimensions of the rocker arms.
It should also be noted that whereas in many instances it is advantageous to limit the number of contact points to the flexible body, an equivalent set of constraints could be obtained with shorter rocker arms with each rocker arm contacting the flexible body at two locations. Such a configuration could be advantageous in limiting the sag of the flexible body under gravity loading or in minimizing the lengths of the rocker arms.
The scope of this invention extends beyond the examples shown herein. The plate-like structure could be integral or affixed to the rocker arms or be a removable substrate supported under gravity or by other means such as vacuum clamps. The pivot rocker arrangement could form a static structure or a dynamically actuated mechanism where actuators are included at any of the supports. Embodiments involving only one or two pivot rockers and some number of fixed supports may also be used in some cases. The geometry of the plate-like structure could be considerably more complex than the simple circle or square and could have almost any exterior shape, holes, ribs, or reinforcement.
This application claims the benefit of U.S. Provisional Application No. 62/249,282 filed Nov. 1, 2015.
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https://books.google.com/books?id=fbyCcHPI0nsC&pg=PA133&lpg=PA133&dq=%22free+mode+shape%22&source=bl&ots=ak7_tpjc48&sig=ACfU3U2ZHUNQkFfHR048xHx9BCOv6459EQ&hl=en&sa=X&ved=2ahUKEwijqvO5wKLnAhVUj3IEHSVMBLAQ6AEwB3oECAgQAQ#v=onepage&q&f=false, Tony L. Schmitz & K. Scott Smith published on Sep. 17, 2011 (Year: 2011). |
Number | Date | Country | |
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20170122400 A1 | May 2017 | US |
Number | Date | Country | |
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62249282 | Nov 2015 | US |