Research ArticleMATERIALS ENGINEERING

Dynamically variable negative stiffness structures

See allHide authors and affiliations

Science Advances  19 Feb 2016:
Vol. 2, no. 2, e1500778
DOI: 10.1126/sciadv.1500778
  • Fig. 1 Design and performance of a dynamically variable stiffness structure.

    (A to D) Mechanical assembly (A and C), equivalent model (B), and measured force-displacement responses (D) for individual components. The vertical k3 (PS) spring is a simple steel plate. The horizontal k2 spring is principally the compliance of the piezoelectric stack. The snap-through mechanism transforms the compression of the k2 spring into a vertical NS spring with a controllable stiffness between +10 and −100 N/mm. The assembled system stiffness is the sum of the NS and PS springs, resulting in a system with highly variable stiffness. Unlike other variable stiffness technologies, load capacity and stiffness are completely decoupled; in this example, the supported load is 130 N.

  • Fig. 2 Dynamic performance.

    (A) Transmission data and estimated linear model with best-fit damping and natural frequency for each stiffness tuning voltage. The best-fit values are plotted in (B), in terms of both damping coefficient and quality factor. SDOF, single-DOF; f, frequency.

  • Fig. 3 Rapid stiffness switching.

    (A to D) Rapid softening (A and C) and stiffening (B and D) of the adaptive stiffness platform under white noise base excitation. The top row shows time histories of the mass acceleration (dots) and piezoelectric actuator motion (solid blue line). The bottom row shows the frequency response, for a short 0.5-s window before and after the stiffness switch. The window center for each spectrogram is indicated by the vertical lines on the top row. am, acceleration amplitude.

  • Fig. 4 Isolator transmissibility during a continuous stiffening then softening sweep through simultaneous tonal excitations of 50 and 100Hz.

    (A) 50-Hz line transfer function history during sweep times of 100, 500, and 3000 ms. (B) 100-Hz line history. (C) Piezo position (x0) history.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/2/2/e1500778/DC1

    Modeling dynamic stiffness control

    Eq. S1. Nonlinear system equation of motion.

    Eq. S2. Nonlinear nondimensional equation of motion.

    Eq. S3. Stiffness change ratio.

    Eq. S4. Maximum practical stiffness change.

    Eq. S5. Nonlinear frequency amplification factor.

    Eq. S6. Linear nondimensional equation of motion.

    Eq. S7. Linear system transmissibility.

    Table S1. Model parameters and nondimensional substitutions.

    Reference (28)

  • Supplementary Materials

    This PDF file includes:

    • Modeling dynamic stiffness control
    • Eq. S1. Nonlinear system equation of motion.
    • Eq. S2. Nonlinear nondimensional equation of motion.
    • Eq. S3. Stiffness change ratio.
    • Eq. S4. Maximum practical stiffness change.
    • Eq. S5. Nonlinear frequency amplification factor.
    • Eq. S6. Linear nondimensional equation of motion.
    • Eq. S7. Linear system transmissibility.
    • Table S1. Model parameters and nondimensional substitutions.
    • Reference (28)

    Download PDF

    Other Supplementary Material for this manuscript includes the following:

    Files in this Data Supplement:

Navigate This Article