Dynamically variable negative stiffness structures

Science Advances  19 Feb 2016:
Vol. 2, no. 2, e1500778
DOI: 10.1126/sciadv.1500778


Variable stiffness structures that enable a wide range of efficient load-bearing and dexterous activity are ubiquitous in mammalian musculoskeletal systems but are rare in engineered systems because of their complexity, power, and cost. We present a new negative stiffness–based load-bearing structure with dynamically tunable stiffness. Negative stiffness, traditionally used to achieve novel response from passive structures, is a powerful tool to achieve dynamic stiffness changes when configured with an active component. Using relatively simple hardware and low-power, low-frequency actuation, we show an assembly capable of fast (<10 ms) and useful (>100×) dynamic stiffness control. This approach mitigates limitations of conventional tunable stiffness structures that exhibit either small (<30%) stiffness change, high friction, poor load/torque transmission at low stiffness, or high power active control at the frequencies of interest. We experimentally demonstrate actively tunable vibration isolation and stiffness tuning independent of supported loads, enhancing applications such as humanoid robotic limbs and lightweight adaptive vibration isolators.

  • Vibration isolation
  • impedance control
  • negative stiffness
  • nonlinear structures


We have developed an active variable stiffness vibration isolator capable of 100× stiffness changes and millisecond actuation times, independent of the static load. This performance surpasses existing mechanisms by at least 20× in either speed or useful stiffness change. Our design philosophy was to combine active vibration control and nonlinear negative stiffness (NS) into a benchtop demonstration system, although the idea is applicable at scales from microns to meters. We started with a classical quasi-zero–stiffness (QZS) system consisting of positive stiffness (PS) and NS springs in parallel, which together add up to zero stiffness (1). We then added a solid-state actuator to control the amount of compression in the nonlinear mechanism, which provides NS. This let us control stiffness rapidly and continuously between the stiffness of the positive spring and nearly zero while still retaining excellent isolation performance.

Our system builds on the capabilities of state-of-the-art variable stiffness systems in two ways: speed and amount of stiffness change. Mechanistic methods have inherently low performance at low stiffness, that is, zero force/torque at zero stiffness. This limits the practical stiffness change to about 5× when supporting a load (2). Pneumatic air springs are easily tunable but trade off friction for volumetric efficiency, whereas smart material–based methods are fast but are limited to about 30% stiffness change (3). Some NS systems have the ability to change stiffness but only for initial assembly and zero-stiffness matching (4, 5). To our knowledge, dynamic stiffness switching has not been attempted at this scale and speed.

Dynamic stiffness control could be useful to several engineering disciplines that must adapt to large variations in loading rate and frequency, as might be found in automotive (1, 3), aerospace, robotic, and assistive robotic systems. These platforms may use active isolators when vibrations are large, mixed with shocks, or vary in their frequency spectrum. Commercial active systems control dynamics through a counteracting force from an actuator, for example, hydraulics and voice coil (1, 2). Cost, parasitic weight, and narrow-band effect on vibration and shock have limited their widespread adoption.

Semiactive vibration mitigation manipulates the stiffness, mass, or damping of one or more of the system components. It is typically more efficient than active methods, which add energy to the system to directly react to a disturbance. The most common semiactive method is to tune damper properties through changes in fluid viscosity (that is, magnetorheological fluid) or damper valve systems (68). Tuning the mass or moment of inertia has been demonstrated in the case of dynamic vibration absorbers (9), but it is impractical for the isolation of larger structures and vehicles because a traditional isolator does not use a dynamic mass. Tuning of mechanical stiffness is typically performed either through the insertion of additional spring elements into the load path (for example, a jounce bumper on an automotive suspension) or through geometric changes to spring networks (2, 10). These methods work well when the supported mass is small, for example, with robust tuned mass damper networks (11), but may be unsuitable for load-bearing applications when the position of the isolated mass must be maintained independent of the stiffness characteristics.

The most ubiquitous tunable stiffness structures are our own joints, which use antagonistic muscle contractions to vary joint stiffness continuously. For example, your limbs will stiffen to lift a bowling ball but soften to paint with the tip of a brush. This functionality was initially identified in the field of humanoid robotics in software as impedance control (12), but because of stability problems (13), hardware solutions called variable stiffness or series elastic actuators arose. They consist of a pair of antagonistic nonlinear springs; the stiffness and load capacity of the joint are functions of the spring compression (2, 14, 15). The benefits of variable stiffness springs (or impedance control) include the ability to walk over uncertain terrain (16), safety while working with humans (17), energy efficiency (18, 19), and effective rehabilitation exoskeletons (20).

An alternative approach to manipulate stiffness is through NS structures. NS structures exploit instability, such as buckling, snap-through, or phase change, to create a valuable mechanical nonlinearity. Previously, they have been used in passive networks to create extreme properties in demonstration systems, such as extreme stiffness and damping in composites (21, 22), enhanced broadband energy harvesting (23, 24), QZS vibration isolation systems (1, 4, 25), enhancement of damping and energy management for earthquake engineering (5, 26), and enhanced MEMS (microelectromechanical systems) sensors (27). Despite considerable research interest, they have not found wide acceptance, in part because they are difficult to make and their nonlinear properties are highly sensitive to boundary conditions. We take advantage of that sensitivity, placing a solid-state actuator in the NS system to enable stiffness control of more than 100×, independent of the static load.


At small deflections, our variable stiffness mechanism can be described with a simple single-degree-of-freedom (DOF) model. Following the work of Alabuzhev and Lakes (1, 21), Fig. 1B illustrates a basic NS mechanism, also known as a “stiffness corrector.” A PS spring (k3) supports the payload, whereas a horizontal spring (k2) acts in the vertical direction through a snap-through mechanism. Held at its unstable point, this NS mechanism, which includes both the k2 spring and the parallelogram linkage mechanism, cancels out the PS to create a QZS or high-static–low-dynamic stiffness (HSLDS) system. Our novel addition to this work is to add a high-speed actuator to the NS mechanism, which continuously adjusts the k2 spring preload, x0. This creates a highly variable stiffness system, one that can exhibit stiffness ranging from the k3 spring alone to as close to zero as tolerance permits. At the small-deflection limit defined by eq. S5, the equation of motion for the vertical deflection (y) of the payload mass (m) isEmbedded Image(1)where c is an equivalent viscous damping coefficient, L is the length of the NS flexure beams, and F is an external force. Note that the equilibrium position y = 0 is constant for all stable x0.The transmissibility of this single-DOF system in frequency space can still be modeled using a classical spring-mass-damper model, with the definition of a variable stiffness tuning ratio ε = 1 − 2k2x0/(k3L)Embedded Image(2)

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.

We built the hardware in Fig. 1A to demonstrate this variable stiffness system; static force displacement characteristics for the individual NS and PS components, as well as the assembled system, are shown in Fig. 1D. Measured separately, the PS k3 spring has a stiffness of 100 N/mm. The NS component achieves any stiffness ranging from slightly positive 10 N/mm to a negative −100 N/mm, equaling that of the PS component, up to a displacement of about ±0.25 mm.

To measure the dynamic behavior of the system, we bolted it to an electromagnetic shaker fitted with two single-DOF accelerometers mounted at the base (input y0) and the sprung mass (output ym). We tuned the apparatus to its highest stiffness state (−30 V to the actuator) and gradually increased the voltage to increase x0. The resulting transmissibility data in Fig. 2A clearly show that the system’s resonance varies 10× from 160 to 16 Hz, which corresponds to a stiffness change of 100×. At our chosen excitation amplitude, the sensor noise floor lies at about −20 dB; at higher amplitudes, we expect the system to provide isolation up to 30 dB at frequencies below 400 Hz, the next structural resonance.

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.

Assigning zero deflection, x0 = 0, at this voltage and given the measured PS of the system with no preload, we estimated a dynamic mass of 0.106 kg. The hardware functions well with up to 5 kg of supported mass, but we chose this relatively small mass to bring the natural frequency into the range of our test equipment. Figure 2 shows the points surrounding each resonant peak (excluding points under −10 dB) fitted to the classical spring-mass-damper transmissibility (eq. 2) to estimate the system damping. These damping estimates, expressed as both the quality factor Embedded Image and the viscous damping equivalent c, are shown in Fig. 2B. The model above defines c as a constant, whereas Q is proportional to frequency with a slope m/c. As the fitted values show, this is a good assumption at most tuning values, with a constant value of c = 0.8 to 1 kg/s or dQ/dω = 0.9, valid for frequencies above 20 Hz. At the most extreme tuning, this assumption breaks down, and we measured increased damping from an unknown source, which could be interactions with the actuator power supply or stiction from bolted joints.

Stiffness control largely divorces actuator bandwidth from excitation frequency, but most applications require stiffness to be changed at some minimum rate. For example, the system may need to react to a sudden change in the excitation frequency/amplitude or protect itself from an anticipated shock load. Our piezoelectric-driven platform can change stiffness on demand at extremely high rates, yet still lets us simulate slower, cheaper actuators. To stiffen, energy is released from the system, so the only limit on transition speed is the inertia of the components. To soften, energy is added to the system, so actuator power becomes the limiting factor.

First, we show the system response at the most rapid switching rate under the same broadband noise excitation as in Fig. 2. During softening in Fig. 3A, the system reaches 90% of its target position in only 5 ms. Improved isolation is immediate, and acceleration amplitude drops from 5 to <0.5 g. This is also reflected in the power spectrum in Fig. 3C, where the resonance peak shifts below 20 Hz and drops by 10 dB owing to increased damping. The stiffening transformation in Fig. 3B is faster (<1 ms) because energy is being released from the system into the new 160-Hz natural frequency, which rings immediately after the stiffness change.

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.

When changing stiffness in a tonal excitation environment, it is valuable to “hop” over troublesome excitation frequency bands without resonating. This sets a lower limit on actuator power. Whereas creating a single rule to cover all possible environments depends on the tone spectrum and system damping, we have found that for the case of low damping, excitation and resonance regions should overlap no more than 10 oscillation periods.

To illustrate this effect, the base excitation was changed from constant-power white noise to a dual-sine excitation with tones at 50 and 100 Hz. Next, the actuator was swept between 20 and 150 Hz at various rates of voltage change. Figure 4 shows time histories of the 50- and 100-Hz spectrum lines, as measured with a moving flat top window, 0.5 s wide (512 samples). The system starts at a natural frequency of 20 Hz, so the 50- and 100-Hz tones are isolated with 8 and 25 dB, respectively. In the time period of 4 to 6 s, the system is in its stiff state, and the tones sit below isolator resonance so transmission is close to 0 dB at 50 Hz and slightly higher at the closer 100-Hz tone. At 6 s, the isolator is softened, and the system returns to its isolating state. When softened quickly, in 100 ms (black line), the system does not resonate measurably, but at slower rates, the tones are increasingly transmitted, first at 100 Hz and then at 50 Hz. At the lowest rate, the isolator risks nonlinear behavior (stiffness amplification), as described in eq. S5, due to the excessive displacement amplitude at resonance.

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.


Advanced lightweight materials are increasingly finding their way into transportation platforms to achieve low mass and high stiffness. They are expected to isolate multiple different vibration environments, but their increased structural efficiency tends to only increase transmission of shock and vibration to the payload. To compensate, we seek new mechanisms of vibration control that are inherently energy efficient and scalable to a variety of loads and environmental frequencies. In the area of robotics, we also need high-quality variable stiffness joints for safety, power efficiency, and natural movement.

We have introduced a new concept of dynamic stiffness control using NS coupled with an actuator to build a system that controls stiffness through two orders of magnitude. This method manages high-frequency dynamic loads with simple, low-power, low-frequency inputs, which can inform new solutions to long-standing problems. We imagine robot manipulators that can dynamically vary stiffness to successfully manipulate a scalpel and lift a patient from their bed with the same joint or semipassive mechanical isolators that avoid multiple input frequencies as encountered in high-efficiency transportation platforms.

The scope of this paper demonstrates the basic concept of active stiffness control and experimentally validates a simple model for the system’s dynamic behavior. Ongoing work focuses on multiaxis loading, actuator efficiency and optimization, and robust handling of low-frequency load variations. Scaling this approach to meet the needs of these varied applications will require individually tailored NS components as well as actuators and control systems.


Following the variable stiffness hardware pictured in Fig. 1A, the PS (k3) spring lies at the base and consists of a laser-cut steel planar spring. A snap-through mechanism lies directly above the k3 spring. It consists of a parallel bar linkage system with notch flexure hinges at the ends to provide a frictionless pinned connection. The bar length is 24.1 mm; the flexure notches are 0.15 mm thick. The left-hand side of the mechanism is grounded, whereas the other side is connected to a piezoelectric actuator, model APC PSt 150/10/20 VS15, held in a horizontal parallel flexure translation stage. The right-hand side of the actuator is grounded through a fine-thread adjustment screw, which adjusts the initial system preload x0. This preload ensures that the actuator is in constant contact with the translation stage, as well as operating in the best displacement range for maximum stiffness change. As the actuator expands, it increases the horizontal compression force acting on the snap-through mechanism, thus increasing the NS measured in the vertical direction. Lateral stiffeners are designed to increase the first global bending mode frequency of the entire device, expanding the bandwidth of single-DOF operation. Most of the contribution to the horizontal spring k2 is from the stiffness of the piezoelectric stack itself, which was measured to have an open-circuit stiffness of 15 N/μm. Together, the piezo and snap-through mechanism form the NS component. The center of the snap-through mechanism and the k3 spring are connected in parallel by a vertical-threaded rod. Nuts along the rod adjust the k3 preload, y0.

Before full dynamic experiments, the individual quasistatic force-displacement measurements in Fig. 1D were measured in a uniaxial hydraulic load frame (MTS model 858). First, the k3 spring was measured alone with other components removed. It had a PS of 100 N/mm. Next, the k3 spring was removed, and the NS component was tested at various levels of preload x0. With the actuator removed entirely, the flexures alone were measured to add 43 N/mm stiffness. Next, the piezo stack was set to its minimum −30 V and manually loaded with a set screw to about x0 = 50 μm. Force-displacement curves in the bottom of Fig. 1D were taken adjusting the screw, only changing x0 through the actuator from about 50 to 240 μm. The NS component showed stiffness ranging from slightly positive to −100 N/mm, enough to perfectly cancel the stiffness of the k3 spring. Finally, the k3 spring was coupled with a small vertical preload y0 to demonstrate load-holding ability. The top of Fig. 1D shows the static response of the fully assembled system.

For dynamic experiments, the validation system was bolted to an electromagnetic shaker (Unholtz-Dikie; rated at 900 N, 85 g, and 50 mm displacement) and fitted with two single-axis accelerometers mounted at the base (input) and the sprung mass (output).

For the data in Fig. 2, the system was excited with an open-loop random input spanning 0 to 400Hz. Deflections were kept sufficiently low (<0.5 mm) so as not to create any amplitude-dependent properties (Ω ≅ 1 from eq. S5). At each stiffness (or actuator voltage), the transmissibility of the system was reported as the ratio of accelerations between the base and supported mass, from 0 to 200Hz.

For Fig. 3, two separate power supplies were set to +150 and −30 V. A single-pull double-throw switch between the supplies and the actuator allowed rapid switching between the two supplies, whereas diodes protected each supply from reverse-current surge. Time resolution of system stiffness was limited by the digital sample window length because a good natural frequency measurement cannot be resolved in the relatively small 0.5-s window range. To compensate, we used a strain gauge directly attached to the actuator to directly measure end motion, providing a separate real-time readout of system state.

For Fig. 4, the actuator was attached to a voltage-controlled power supply (ThorLabs MDT693; maximum current, 60 mA). Instead of random noise, the shaker was excited with dual tones at 50 and 100 Hz, still taking care to keep the absolute amplitude for yb below ±0.5 mm. The stiffness was changed continuously by commanding the power supply with a linear voltage ramp between 0 and 150 V at rates corresponding to 100−, 500−, and 3000−ms sweeps.


Supplementary material for this article is available at

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)

This is an open-access article distributed under the terms of the Creative Commons Attribution-NonCommercial license, which permits use, distribution, and reproduction in any medium, so long as the resultant use is not for commercial advantage and provided the original work is properly cited.


Acknowledgments: We appreciate the continued support of our LLC partners Boeing and General Motors in helping to motivate this research. We acknowledge J. Ensberg for performing structural analyses, which improved the performance of the demonstration hardware. We also appreciate the helpful discussions with and technical assistance of our colleagues C. Henry, G. Herrera, J. Mikulsky, G. Chang, W. Carter, and L. Momoda. Funding: This work was performed at HRL Laboratories LLC, using internal funding. Author contributions: G.P.M., C.B.C., and D.W.S. provided the research motivation and system models. C.B.C., S.P.S., and A.C.K. designed, assembled, and tested the variable stiffness hardware. Competing interests: The authors declare that they have no competing interests. Data and materials availability: All data needed to evaluate the conclusions in this paper are present in the paper and/or the Supplementary Materials. Additional data and raw data are available upon request from C.B.C. (cbchurchill{at}
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