Dynamics

Dynamic problems – the transient response of a structure to time-varying loads, wave propagation, and similar – extend the balance of linear momentum with an inertial term.

Equation of Motion

The strong form of the dynamic balance of linear momentum is $var element = document.getElementById("moose-equation-844e2064-63e5-4027-869f-45ec79b7a76d");katex.render(" \\rho\\, \\ddot{u}_i + \\eta\\, \\rho\\, \\dot{u}_i = \\sigma_{ij,j} + b_i", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-1c3d6903-6bfc-4559-92bf-90bd235710ba");katex.render("\\rho", element, {displayMode:false,throwOnError:false});$ is the mass density, $var element = document.getElementById("moose-equation-d815c4a6-3c52-4e3c-be14-fb7fb72477fa");katex.render("\\ddot{u}_i", element, {displayMode:false,throwOnError:false});$ the acceleration, $var element = document.getElementById("moose-equation-149df996-d9ac-4af6-b5a4-7a0dbe8149c6");katex.render("\\dot{u}_i", element, {displayMode:false,throwOnError:false});$ the velocity, $var element = document.getElementById("moose-equation-78bd3754-3fe8-4092-bbef-02d595143f22");katex.render("\\eta", element, {displayMode:false,throwOnError:false});$ the mass-proportional damping coefficient, $var element = document.getElementById("moose-equation-fc587cf0-7da6-4f91-a1d4-cd16ea776f8c");katex.render("\\sigma_{ij}", element, {displayMode:false,throwOnError:false});$ the stress, and $var element = document.getElementById("moose-equation-aa8a4c48-9bf7-451a-ba25-ef39c2499ce7");katex.render("b_i", element, {displayMode:false,throwOnError:false});$ the body force per unit volume. This is the static balance with the inertial force $var element = document.getElementById("moose-equation-5581da45-d9c1-40b4-bdbe-ed6e85d643b5");katex.render("\\rho\\ddot u_i", element, {displayMode:false,throwOnError:false});$ and a mass-proportional damping force $var element = document.getElementById("moose-equation-b92fd4e4-858f-48a4-ba61-edca9e78245d");katex.render("\\eta\\rho\\dot u_i", element, {displayMode:false,throwOnError:false});$ added; it reduces to the static problem when $var element = document.getElementById("moose-equation-f2df1cab-e21b-4750-984b-0930fb55dcf6");katex.render("\\dot u_i = \\ddot u_i = 0", element, {displayMode:false,throwOnError:false});$ , and it holds on the reference or current configuration following the same total/updated convention as the static case.

Each term is supplied by a separate object, one per displacement component:

Term	Meaning	Object
$var element = document.getElementById("moose-equation-cd8b1310-6947-4f36-8f0e-9a1b71252af4");katex.render("\\rho\\,\\ddot{u}_i", element, {displayMode:false,throwOnError:false});$	inertia	`InertialForce`
$var element = document.getElementById("moose-equation-c5c63812-f870-4abf-8fdc-7f088e8ec70c");katex.render("\\eta\\,\\rho\\,\\dot{u}_i", element, {displayMode:false,throwOnError:false});$	mass-proportional damping	`InertialForce` (`eta`)
$var element = document.getElementById("moose-equation-63c5e320-47c2-448f-94e0-ea88a06963c4");katex.render("\\sigma_{ij,j}", element, {displayMode:false,throwOnError:false});$	internal force	Lagrangian stress-divergence kernel + constitutive model
$var element = document.getElementById("moose-equation-ca8b7e7e-9377-4b7f-921c-85cf962e680b");katex.render("b_i", element, {displayMode:false,throwOnError:false});$	body force	e.g. `Gravity`

Stiffness-proportional damping does not appear as a separate term: it is carried inside $var element = document.getElementById("moose-equation-9fb3ea77-9aed-4551-9b6e-8fa81fc325d1");katex.render("\\sigma_{ij}", element, {displayMode:false,throwOnError:false});$ as a viscous stress.

Time Integration

The velocity and acceleration are related to the displacement by a time-integration scheme. The implicit Newmark (Newmark, 1959) and Hilber-Hughes-Taylor (HHT) (Hughes, 2000) schemes are described below, along with experimental support for explicit dynamics.

Newmark

Newmark integration expresses the acceleration and velocity at $var element = document.getElementById("moose-equation-05ccadcd-9311-4b72-a4de-b894d7627432");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false});$ in terms of the state at $var element = document.getElementById("moose-equation-661771af-5de5-45d6-8238-cd4ebd5403e0");katex.render("t", element, {displayMode:false,throwOnError:false});$ and the displacement at $var element = document.getElementById("moose-equation-65939088-66e4-45a5-b998-b88c126496ef");katex.render("t+\\Delta t", element, {displayMode:false,throwOnError:false});$ : $var element = document.getElementById("moose-equation-e3f4913e-edb4-42dd-bbee-fa435d5ba4e8");katex.render("\\begin{aligned} \\ddot{u}_i(t+\\Delta t) &= \\frac{u_i(t+\\Delta t)-u_i(t)}{\\beta \\Delta t^2} - \\frac{\\dot{u}_i(t)}{\\beta \\Delta t} + \\frac{\\beta - \\tfrac{1}{2}}{\\beta}\\,\\ddot{u}_i(t), \\\\ \\dot{u}_i(t+\\Delta t) &= \\dot{u}_i(t) + (1-\\gamma)\\Delta t\\, \\ddot{u}_i(t) + \\gamma \\Delta t\\, \\ddot{u}_i(t+\\Delta t), \\end{aligned}", element, {displayMode:true,throwOnError:false});$ with parameters $var element = document.getElementById("moose-equation-b2202803-9a93-4e91-8939-2654bc2be421");katex.render("\\beta", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-5253e511-0472-4719-bafe-085d31f55fc2");katex.render("\\gamma", element, {displayMode:false,throwOnError:false});$ . Common choices:

$var element = document.getElementById("moose-equation-51150b77-f426-4121-865d-819c86ea8306");katex.render("\\beta = \\tfrac{1}{4}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-839933a3-87e4-4f1e-807e-fe2475f7d3c7");katex.render("\\gamma = \\tfrac{1}{2}", element, {displayMode:false,throwOnError:false});$ : constant average acceleration – unconditionally stable with no numerical damping (recommended with a constant time step).
$var element = document.getElementById("moose-equation-858dd2be-dd56-49ea-80a3-7aef2e1829a6");katex.render("\\beta = \\tfrac{1}{6}", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-f74479ef-1eac-4b12-976e-dbe81e3873a6");katex.render("\\gamma = \\tfrac{1}{2}", element, {displayMode:false,throwOnError:false});$ : linear acceleration.
$var element = document.getElementById("moose-equation-bbd924f9-b521-480a-a9a7-e30aeb001da8");katex.render("\\beta = 0", element, {displayMode:false,throwOnError:false});$ , $var element = document.getElementById("moose-equation-049525a8-20e1-4fc6-99e0-4d45677285d3");katex.render("\\gamma = \\tfrac{1}{2}", element, {displayMode:false,throwOnError:false});$ : the central-difference method in theory (but see Explicit Dynamics).

note

For $var element = document.getElementById("moose-equation-6c3ca401-6834-489e-80ca-f3896c1004ea");katex.render("\\gamma = \\tfrac{1}{2}", element, {displayMode:false,throwOnError:false});$ the Newmark method is second-order accurate and, for $var element = document.getElementById("moose-equation-4cde4f99-5fab-41e6-95b0-5c06768773b6");katex.render("\\tfrac{1}{2} \\le \\gamma \\le 2\\beta", element, {displayMode:false,throwOnError:false});$ , unconditionally stable; it is first-order accurate for other values of $var element = document.getElementById("moose-equation-cc1adee5-ad90-4fa1-a53b-e193fd986ef3");katex.render("\\gamma", element, {displayMode:false,throwOnError:false});$ .

Hilber-Hughes-Taylor

The constant-average-acceleration Newmark scheme introduces no numerical damping, which can leave spurious high-frequency content. HHT damps the high frequencies by evaluating the non-inertial terms at a weighted time using the parameter $var element = document.getElementById("moose-equation-02ad8928-a069-4c94-870f-9480a5482c5c");katex.render("\\alpha", element, {displayMode:false,throwOnError:false});$ : $var element = document.getElementById("moose-equation-2c91d940-874e-45cc-ab66-b983f174b6c2");katex.render(" \\rho\\,\\ddot{u}_i(t+\\Delta t) + \\eta\\rho\\left[(1+\\alpha)\\dot{u}_i(t+\\Delta t) - \\alpha\\,\\dot{u}_i(t)\\right] = (1+\\alpha)\\left[\\sigma_{ij,j} + b_i\\right]_{t+\\Delta t} - \\alpha\\left[\\sigma_{ij,j} + b_i\\right]_{t}.", element, {displayMode:true,throwOnError:false});$ Choosing $var element = document.getElementById("moose-equation-808f72ad-75a4-4bdb-813f-acf8e08c6a54");katex.render("-\\tfrac{1}{3} \\le \\alpha \\le 0", element, {displayMode:false,throwOnError:false});$ with $var element = document.getElementById("moose-equation-4b4e3aae-16b9-4e13-96cc-517aaca8de0d");katex.render("\\beta = \\tfrac{(1-\\alpha)^2}{4}", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-0f7c0c33-46ba-4d49-8fb0-232aa8dae692");katex.render("\\gamma = \\tfrac{1}{2} - \\alpha", element, {displayMode:false,throwOnError:false});$ keeps the scheme second-order accurate and unconditionally stable while damping frequencies above $var element = document.getElementById("moose-equation-e5c9d2bd-f1ba-4d49-80db-311da72e68f4");katex.render("\\tfrac{1}{2\\Delta t}", element, {displayMode:false,throwOnError:false});$ . The inertial term is not weighted; the internal force, damping, and external loads are, so loads such as gravity and pressure are evaluated at $var element = document.getElementById("moose-equation-bf3e6d88-2173-47da-aad4-bfb9a8454b12");katex.render("t + (1+\\alpha)\\Delta t", element, {displayMode:false,throwOnError:false});$ . Setting $var element = document.getElementById("moose-equation-dc62ccd3-734d-49db-a964-063738afbfbe");katex.render("\\alpha = 0", element, {displayMode:false,throwOnError:false});$ recovers Newmark.

Explicit Dynamics

warning:Experimental

Explicit dynamics support is under active development and should be considered experimental.

The Newmark and HHT schemes above are implicit. For wave-propagation and high strain-rate problems, an explicit central-difference scheme – which advances the solution without a nonlinear solve, at the cost of a conditionally stable (critical) time step – is available through the ExplicitMixedOrder time integrator.

note:Newmark beta = 0 is not an explicit solve

Setting $var element = document.getElementById("moose-equation-68f30c9c-4dbc-43bf-b938-1afcde8368e2");katex.render("\\beta = 0", element, {displayMode:false,throwOnError:false});$ in the Newmark integrator is the central-difference method in theory, but the implementation still assembles and solves the full system every step – it is not a genuine explicit solve. For explicit dynamics, use ExplicitMixedOrder.

Damping

Rayleigh damping combines a mass-proportional and a stiffness-proportional contribution, giving a damping ratio that varies with the modal frequency $var element = document.getElementById("moose-equation-9e9fbf4f-92b8-4f99-9570-0910ceda7feb");katex.render("\\omega", element, {displayMode:false,throwOnError:false});$ as $var element = document.getElementById("moose-equation-c3fcf90e-4f8e-4723-9432-4d8f254ce1de");katex.render(" \\xi(\\omega) = \\frac{\\eta}{2\\omega} + \\frac{\\zeta}{2}\\,\\omega,", element, {displayMode:true,throwOnError:false});$ where $var element = document.getElementById("moose-equation-a25e9bb2-092f-43ee-b182-f093f22a0bef");katex.render("\\eta", element, {displayMode:false,throwOnError:false});$ is the mass-proportional and $var element = document.getElementById("moose-equation-4f90fa7f-bd23-4789-92f5-f9212ef88593");katex.render("\\zeta", element, {displayMode:false,throwOnError:false});$ the stiffness-proportional coefficient. To obtain a target damping ratio $var element = document.getElementById("moose-equation-5b93fd2e-0d6e-4c8e-af8b-6dfe0837f2ce");katex.render("\\xi_t", element, {displayMode:false,throwOnError:false});$ at two frequencies $var element = document.getElementById("moose-equation-51da7522-a4ab-4974-b6cb-1ae9233f9dbf");katex.render("\\omega_1", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-8dcd2d23-cd46-4eb2-b057-31d0f2d759bd");katex.render("\\omega_2", element, {displayMode:false,throwOnError:false});$ , solving $var element = document.getElementById("moose-equation-436ce96a-77e0-45c6-a9d4-804ad9cf8b6e");katex.render("\\xi(\\omega_1) = \\xi(\\omega_2) = \\xi_t", element, {displayMode:false,throwOnError:false});$ gives $var element = document.getElementById("moose-equation-e075dc6d-7cc4-4fb3-b23b-b88f77882fd2");katex.render(" \\eta = 2\\,\\xi_t\\,\\frac{\\omega_1 \\omega_2}{\\omega_1 + \\omega_2}, \\qquad \\zeta = \\frac{2\\,\\xi_t}{\\omega_1 + \\omega_2}.", element, {displayMode:true,throwOnError:false});$ Between $var element = document.getElementById("moose-equation-2233fb4e-bc76-41c6-8225-0a5e60d42821");katex.render("\\omega_1", element, {displayMode:false,throwOnError:false});$ and $var element = document.getElementById("moose-equation-0e50be92-b4f1-451a-b28c-fb15d562ad67");katex.render("\\omega_2", element, {displayMode:false,throwOnError:false});$ the actual ratio dips slightly below $var element = document.getElementById("moose-equation-7f352013-8704-49ef-ab6e-fdd33fee5b8e");katex.render("\\xi_t", element, {displayMode:false,throwOnError:false});$ ; outside the band it rises.

The mass-proportional force $var element = document.getElementById("moose-equation-c74bc34e-af7c-4620-bfaa-a465cbf206ac");katex.render("\\eta\\rho\\dot{u}_i", element, {displayMode:false,throwOnError:false});$ is applied by the InertialForce kernel through its eta parameter. The stiffness-proportional contribution is treated as a viscous stress and can be written as a rate-dependent constitutive model.

Implementation and Usage

A dynamic model adds, for each displacement component:

a Lagrangian stress-divergence kernel for the internal force,
an InertialForce kernel for $var element = document.getElementById("moose-equation-160c48cd-ce75-4bd5-94fe-2571967158c2");katex.render("\\rho\\ddot{u}_i", element, {displayMode:false,throwOnError:false});$ (and, via eta, the mass-proportional damping), and
a time integrator, with the Newmark beta/gamma and, for HHT, the alpha parameter.

The SolidMechanics/QuasiStatic action sets up the stress-divergence kernels and the strain calculator; the inertial kernel and time integrator are added alongside it. A material density is required, and stiffness-proportional damping, when needed, is supplied as the viscous constitutive stress above.

note

For dynamic problems, prescribe boundary motion with PresetDisplacement and PresetAcceleration rather than a plain Dirichlet condition on the displacement.

Static Initialization

To start from a statically loaded state (for example under gravity), run an initial static step with the inertial contribution disabled, then enable it for the transient. Activating the InertialForce kernel only after the first step (for instance with a Controls block) makes the first solve return the static equilibrium under the sustained load.

References

T. J. R Hughes. The finite element method:Linear static and dynamic finite element analysis. Dover Publications, Mineola, NY, 2000.

BibTeX

@book{hughes2000fem,
    author = "Hughes, T. J. R",
    title = "The finite element method:{L}inear static and dynamic finite element analysis",
    year = "2000",
    publisher = "Dover Publications, Mineola, NY"
}

RIS

TY  - BOOK
AU  - Hughes, T. J. R
TI  - The finite element method:Linear static and dynamic finite element analysis
PY  - 2000
PB  - Dover Publications, Mineola, NY
ER  -

Plain Text

T. J. R Hughes. The finite element method:Linear static and dynamic finite element analysis. Dover Publications, Mineola, NY, 2000.

N. M. Newmark. A method of computation for structural dynamics. Journal of Engineering Mechanics, 85(EM3):67–94, 1959.

BibTeX

@article{newmark1959amethod,
    author = "Newmark, N. M.",
    title = "A method of computation for structural dynamics",
    journal = "Journal of Engineering Mechanics",
    publisher = "ASCE",
    volume = "85",
    number = "EM3",
    pages = "67--94",
    year = "1959"
}

RIS

TY  - JOUR
AU  - Newmark, N. M.
TI  - A method of computation for structural dynamics
JO  - Journal of Engineering Mechanics
PY  - 1959
VL  - 85
IS  - EM3
PB  - ASCE
SP  - 67
EP  - 94
ER  -

Plain Text

N. M. Newmark. A method of computation for structural dynamics. Journal of Engineering Mechanics, 85(EM3):67–94, 1959.

Equation of Motion
Time Integration
Damping
Implementation and Usage
Static Initialization
References