On cross effects of seismic responses of structures

Lee, G.C.; Liang, Zach

doi:10.1016/s0141-0296(97)00089-8

Cited by 8 publications

(8 citation statements)

References 1 publication

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“…The sign of the discriminant determines the type of solution for Eq. (12); the roots of a cubic polynomial can be [15,16] • three distinct real roots (D < 0)…”

Section: Resultsmentioning

confidence: 99%

“…From Eq. (13), the discriminant can be expressed as a function of the muscle damping as follows [15]: The function is shown for both wrist (top) and elbow (bottom). For j < 8 the discriminant is positive, independent of the value of C P h , which translate in a free oscillatory response of the system.…”

Section: Resultsmentioning

confidence: 99%

“…(14) is always positive and the expression within the parenthesis is a bi-quadratic form. Hence, to find the solutions to D ¼ 0, we can substitute C P ð Þ 2 ¼ v and solve the following quadratic equation [15]:…”

Section: Resultsmentioning

confidence: 99%

“…(5) into Eq. (4), and multiplying both members by s þ ðK During a free response, the external force is null (i.e., g D n D'; t ð Þ¼0) so the governing equation of a PT model in the time domain is [15] …”

Section: Methodsmentioning

confidence: 99%

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Critical Damping Conditions for Third Order Muscle Models: Implications for Force Control

Piovesan

Pierobon

Ivaldi

2013

Journal of Biomechanical Engineering

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Experimental results presented in the literature suggest that humans use a position control strategy to indirectly control force rather than direct force control. Modeling the muscle-tendon system as a third-order linear model, we provide an explanation of why an indirect force control strategy is preferred. We analyzed a third-order muscle system and verified that it is required for a faithful representation of muscle-tendon mechanics, especially when investigating critical damping conditions. We provided numerical examples using biomechanical properties of muscles and tendons reported in the literature. We demonstrated that at maximum isotonic contraction, for muscle and tendon stiffness within physiologically compatible ranges, a third-order muscle-tendon system can be under-damped. Over-damping occurs for values of the damping coefficient included within a finite interval defined by two separate critical limits (such interval is a semiinfinite region in second-order models). An increase in damping beyond the larger critical value would lead the system to mechanical instability. We proved the existence of a theoretical threshold for the ratio between tendon and muscle stiffness above which critical damping can never be achieved; thus resulting in an oscillatory free response of the system, independently of the value of the damping. Under such condition, combined with high muscle activation, oscillation of the system can be compensated only by active control.

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“…The sign of the discriminant determines the type of solution for Eq. (12); the roots of a cubic polynomial can be [15,16] • three distinct real roots (D < 0)…”

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Resultsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

See 2 more Smart Citations

Critical Damping Conditions for Third Order Muscle Models: Implications for Force Control

Piovesan

Pierobon

Ivaldi

2013

Journal of Biomechanical Engineering

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“…The sign of the discriminant determines the type of solution for equation (13). The roots of a cubic polynomial can be [4, 5]: Three distinct real roots (Δ < 0)Three real roots, at least two of which are equal (Δ = 0)One real root and two complex conjugate roots (Δ > 0)…”

Section: System's Oscillatory Behaviormentioning

confidence: 99%

Third-Order Muscle Models: The Role of Oscillatory Behavior in Force Control

Piovesan

Pierobon

Mussa-Ivaldi³

2012

Volume 2: Biomedical and Biotechnology

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This paper presents the analysis of a third-order linear differential equation representing a muscle-tendon system, including the identification of critical damping conditions. We analytically verified that this model is required for a faithful representation of muscle-skeletal muscles and provided numerical examples using the biomechanical properties of muscles and tendon reported in the literature. We proved the existence of a theoretical threshold for the ratio between tendon and muscle stiffness above which critical damping can never be achieved, thus resulting in an oscillatory free response of the system, independently of the value of the damping. Oscillation of the limb can be compensated only by active control, which requires creating an internal model of the limb mechanics. We demonstrated that, when admissible, over-damping of the muscle-tendon system occurs for damping values included within a finite interval between two separate critical limits. The same interval is a semi-infinite region in second-order models. Moreover, an increase in damping beyond the second critical point rapidly brings the system to mechanical instability.

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