The natural frequencies and normal mode shapes become important in devices that are subject to vibration and in those that operate at high speeds in the dynamic regime. Some devices, especially at micro scale in micro-electro-mechanical systems (MEMS), are intentionally operated at resonance frequencies to enhance their performance. Optimizing the structural elements in order to have desired frequencies is of interest in some applications at both the macro and micro scales. This &&inverse frequency'' problem is studied extensively [1], including the topology optimization problem [2]. The &&inverse mode shape'' problem, on the other hand, entails the determination of the geometry of the structure such that it will have prescribed mode shapes. There are applications where a particular mode shape is critical for the performance of the device. This is especially true in the design of resonance-based micro accelerometers and gyroscopes where sensing is accomplished through capacitance measurement, which is very sensitive to the geometry of elastically deforming structures. For instance, in the micro rate gyroscope [3], the mode shape of a ring structure can be optimized to improve the performance. Designing the shape of the cantilever probes on the atomic force microscope to attain a desired modal de#ection is another example [4]. In micro locomotion systems (e.g., swimming) where repeated changes in shape propel the whole entity forward [5], energy e$ciency can be achieved by designing the structure such that the normal mode shapes are the same as the required repetitive shape changes. At macro scale, the inverse mode shape problem can be used to design the tooling for manufacturing equipment such that vibratory displacement is minimized in certain directions. Currently, there do not appear to be any general systematic methods to design the geometry of the structures for desired mode shapes.
Compared to the inverse frequency problem, the inverse mode shape problem has received much less attention. The inverse mode shape problem arises in two di!erent settings. In the "rst category, the experimentally obtained eigendata (natural frequencies and mode shapes) of the structures is used for the characterization of their geometry and material density. E!orts in this direction are found in references [6}14]. In the second category, which is the focus of this paper, the geometry of the structure is designed for prescribed mode shapes using a given material. E!orts in this direction are found in references [4,15]. It should be noted that an arbitrarily speci"ed mode shape might not always be physically realizable with a given class of structures such as strings, cables, rods, and frames consisting of straight and curved beams, plates and membranes, shells and general 3-D structures. Consequently, in the second category of &&the design for desired mode shape problems'', the prescribed mode shape should be checked against a set of criteria that ensure physical realizability.