Advanced damper with high stiffness and high hysteresis ... lakes/ damper with high stiffness and high hysteresis damping based on negative structural stiffness ... which it is intendedthat

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  • International Journal of Solids and Structures 50 (2013) 24162423

    Contents lists available at SciVerse ScienceDirect

    International Journal of Solids and Structures

    journal homepage: www.elsevier .com/locate / i jsolst r

    Advanced damper with high stiffness and high hysteresis dampingbased on negative structural stiffness

    Liang Dong a,, Roderic Lakes b,c,a Department of Materials Science and Engineering, University of Virginia, Charlottesville, VA 22904, USAb Materials Science Program, University of Wisconsin, Madison, WI 53706, USAc Department of Engineering Physics, University of Wisconsin, Madison, WI 53706, USA

    a r t i c l e i n f o a b s t r a c t

    Article history:Received 19 July 2012Received in revised form 8 March 2013Available online 8 April 2013

    Keywords:DampingStiffnessColumnsBucklingUniaxial CompressionExperimental techniquesStructures

    0020-7683/$ - see front matter 2013 Elsevier Ltd. Ahttp://dx.doi.org/10.1016/j.ijsolstr.2013.03.018

    Corresponding authors. Tel.: +1 608 346 7275 (L. DonE-mail addresses: dongliang@uwalumni.com (L.

    (R. Lakes).

    High structural damping combined with high stiffness is achieved by negative stiffness elements. Nega-tive incremental structural stiffness occurs when a column with flat ends is subjected to snap-throughbuckling. Large hysteresis (i.e., high damping) can be achieved provided the ends of the column undergotilting from flat to edge contact. The column configuration provides high structural stiffness. Stable axialdampers with initial modulus similar to that of the parent material and with enhanced damping weredesigned built and tested. Effective damping of approximately two and stiffness-damping product ofapproximately 200 GPa were achieved in such dampers consisting of stainless steel columns. This is a sig-nificant improvement for this figure of merit (i.e., the stiffness-damping product), which generally cannotexceed 0.6 GPa for currently used damping systems.

    2013 Elsevier Ltd. All rights reserved.

    1. Introduction classes of material at ambient temperature. The diagonal line in

    Mechanical damping is of vital importance as one can dampvibrations in mechanical systems so as to prolong the service lifeof components, also reduce acoustic noise. The measure of damp-ing in linear systems is tand, with d as the phase angle betweenstress and strain sinusoids. Damping of structures can be achievedvia layers of high-damping materials, typically polymers, by exter-nal lumped dampers that may contain a viscous device, or usingmaterials with intrinsically high damping. Structural metals suchas steel, brass and aluminum alloys exhibit very low damping of103 or less (steel: 0.0001; brass: 9 105; aluminum alloy:

  • Fig. 1. Stiffness-damping map for several classes of materials in the linear regime atambient temperature (adapted from Lakes, 2009). The diagonal line (i.e., the thickblue line) represents a constant figure of merit |E|tand = 0.6 GPa. Commercialdamping layers typically have a lower figure of merit. The solid diamond symbolsrepresent the properties of the stainless steel damper modules in the nonlinearregime. (For interpretation of the references to color in this figure legend, the readeris referred to the web version of this article.)

    Fig. 2. Sketch of stainless steel axial damper module. The damper module iscomposed of a frame and one or more press contact flat ends columns (the quantityis represented by n). The frame is supported by two dog-bone support rodsclamped in the disk-shaped base plates.

    L. Dong, R. Lakes / International Journal of Solids and Structures 50 (2013) 24162423 2417

    For example a slender bar in a post-buckled S shaped configurationis in unstable equilibrium (Bazant and Cedolin, 1991). If it is con-strained laterally, it may be stabilized and the negative stiffnessmeasured. Structures containing buckled tubes (Lakes, 2001a) exhi-bit negative stiffness that is observed under the constraint of dis-placement control. These structures exhibit extremely highstructural damping under small oscillations when the pre-strain isappropriately tuned. The high damping is understood in the contextof composite theory of Reuss (series) systems (Lakes, 2001a). Com-posite materials (Lakes, 2001b) (not structures) containing negativestiffness constituents are also predicted to exhibit extreme lineardamping and anomalies in the stiffness. Such effects have been ob-served experimentally (Lakes et al., 2001; Jaglinski et al., 2007;Dong et al., 2011). Instability of a negative stiffness structure alsogives rise to a nonlinear snap through effect that generates higheffective damping. Structures based on lateral force on S shapedbeams or axial force on buckled flexible tubes provide a negativestiffness region (Lakes, 2001a) by which high structural dampingcan be attained via positive and negative stiffness in series (Lakes,2001a). Follow-on efforts have further explored this concept (Kash-dan et al., 2009, 2011; Haberman et al., 2012) but such structuresare not very stiff. However, high stiffness is usually desired in struc-tural applications. Therefore dampers were developed based on thenegative stiffness of axially loaded columns with flat ends (Dongand Lakes, 2012). The design of the columns enables a snap-throughbuckling that gives rise to negative stiffness. The abrupt snap effectconverts the low frequency input to a much higher frequency. En-ergy loss mechanisms are more effective at higher frequency. In-deed the damping ultimately converts the energy to thermalvibrations of atoms at frequencies greater than 10 GHz.

    The ends are in a press fit condition and are free to tilt duringbuckling. A polymeric damper based on this concept was demon-strated (Dong and Lakes, 2012); stiffness comparable to that ofthe parent material was achieved, with greatly enhanced damping.In the present study a steel damper is developed with the aim ofobtaining higher stiffness. In the present work, the post-bucklingproperties of flat ends press contact stainless steel (SS) columns

    were experimentally studied; stable modules consisted of clampedand press contact SS columns were then constructed. The stiffnessof the module is similar to that of the parent material but withgreatly enhanced damping. Furthermore, it was found that withappropriate pre-strain and peak to peak displacement (denotedu(P-P) in the figures), the effective damping of the damper modulecan attain values as high as approximately two, and a maximumstiffness-damping product of approximately 200 GPa was achieved,a significant improvement for this figure of merit.

    2. Methods

    Commercial SS (17-4PH, McMaster-Carr, tight-tolerance hard-ened) rods were used. SS columns of nominal 3.175 mm (1/800)diameter with different lengths of 184, 150, and 125 mm werecut with a diamond saw and machined with a lathe to obtain a flatsurface for both ends. Force displacement relationship measure-ments were performed at room temperature using a servohydrau-lic (maximum force capacity 100 kN, MTS system Corp. Mpls. MN)test system under compression in displacement control with0.5 Hz sinusoidal waveforms as the input. A sufficient displace-ment enables post-buckling to occur during loading. The forceand displacement waveforms were captured by a digital oscillo-scope. Stable axial dampers were designed and built as proof ofconcept, taking advantage of the effects in post-buckling of presscontact flat-ends columns. The frame of the damper module iscomposed of two 40 mm diameter 8 mm thick SS disk-shaped baseplates supported by two identical dog-bone shaped SS rodsclamped in between the two SS disk plates. The middle part ofthe dog-bone support rod is 92 mm in length and 3.175 mm indiameter; the two ends have effective length of 46 mm and diam-eter of 6.35 mm. The design of this module is shown in fig. 2. Engi-neering strain and stress were used in the present study.

    Damping is inferred from load-deformation curves as follows.Damping (tand) is proportional to the area that is enclosed bythe Lissajous figure (i.e., the closed curve of force vs. displacementsinusoidal). When the Lissajous figure is elliptic as it is in linearmaterials and structures, damping equals to the ratio of the widthof the elliptic Lissajous figure to the length of the projection of thiselliptic Lissajous figure onto the displacement axis. When the

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