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Machine Frame, Dynamic Analysis

MDA performed dynamic analysis of a drop test for this machine frame. Nodal masses were used to represent the weights of the diesel engine and hydraulic pump. These nodal masses were attached to their respective brackets using rigid elements. Accelerometer data, retrieved from field tests in the severe environment in which the machine is operated, was used in the boundary/load conditions. A simulated drop was performed which replicated the real-life physics of the event. This simulation demonstrated high stresses in the machine where field failures were being witnessed. Simulation results were used to strengthen the machine in areas showing weakness.

Generator Skid, Dynamic Analysis

MDA performed dynamic analysis on this diesel generator skid. The generator is chained to a flatbed railcar for transportation by rail. The analysis must demonstrate the generator skid can withstand a 5 [ mph ] impact of the rail car with another object without exceeding the ultimate material strength. The combined center of mass of the diesel engine and generator are represented by a single nodal mass rigidly connected to the mounts. Gap and rod elements are used to model the chains holding the generator skid to the flatbed railcar which is represented by the plane shown in grey.

Robotic Transport Unit, Dynamic Analysis

MDA participated in the new product development of this robotic transport unit. A dynamic analysis was used for the emergency-stop (e-stop) event; the most harsh operational event the transport unit will encounter. The simulation demonstrated the unit will decelerate from 1 [ m / s ] to zero in a little over 1.2 [ in ] within about 0.06 [ s ]. Additionally, loads were pulled from elements representing the transport rail and guide blocks and used to specify these components.

Conveyor Arm, Dynamic Analysis

MDA was retained to perform dynamic analysis on this lowering-in arm used in a pipe concrete coating facility. The pipe rolls into the arm from a table with an initial velocity of approximately 18 [ in / sec ]. The pipe impacts the arm and the arm then lowers the pipe into a conveyor. The black line shown about 2/3 to the right of the arm is a beam element representative of the hydraulic cylinder used to operate the arm. The arm is pinned to a frame at the left which was not included in the analysis. The chart below the video shows the axial force on the beam element as the pipe impacts the arm at approximately 0.2 [ sec ]. The purpose of the analysis was threefold:

  • Check stresses in arm caused by pipe impact
  • Use beam element dynamic loads to specify a hydraulic cylinder
  • Pull dynamic loads from pin and provide to client so linear static analysis may be performed on the frame which supports the arm
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About Dynamic Analysis

Most of our clients could care less whether the analysis results are obtained using an "explicit" or "implicit" numerical technique, just so long as it is quick and accurate. On our side, we care whether it is implicit or explicit because choosing the wrong numerical path can mean the difference between success or an angry client. Aside from speed and accuracy concerns, our clients are likely interested in what dynamic analysis is used for and what it can do for them. Dynamic analysis (explicit or implicit depending on the physics of the problem) is frequently used for:

  • High speed events
  • Transient events
  • Drop and direct impact events
  • Highly nonlinear problems
  • Complex contact problems
  • Post buckling and snap-through analysis
  • Nonlinear materials
  • Material yielding such as metal forming
  • Mechanical event simulation
  • Dynamic stress and deflection

A brief list of things which may be learned from a dynamic analysis are:

  • Accelerations/decelerations caused by impacts or other time varying events
  • Dynamic loads transferred to interfacing equipment may be captured and used to specify vendor supplied components
  • Point during event where structure exceeds design limits
  • Time required for an event to occur
  • Dynamic internal loads created by the event
Time-Dependent vs. Time-Independent Analysis

A geometric, material, or boundary nonlinearity requires an analysis that implements incremental load (or displacement) steps. At the end of each increment, the structure geometry changes, possibly boundary conditions have changed, and perhaps the material is nonlinear or has yielded. Considering geometry, material, and boundary changes, the stiffness matrix is updated for the next time increment in the analysis. Explicit and implicit analyses are two different ways of implementing this time step procedure. In more simplistic terms, the physics/time relationship must be broken down to solve a mathematical problem. This is accomplished by forming two groups of problems: "time-dependent" and "time-independent". These problems are commonly solved using explicit and/or implicit methods.

When the effects of acceleration are pronounced and cannot be neglected, the problems are referred to as time-dependent. For example, in a drop test, the highest force occurs within the first few milliseconds after impact as the item decelerates to a halt. In this case, the effect of such a deceleration must be taken into account.

In contrast, when loads are slowly applied to a structure or surface (i.e., placing a laptop on a table), the loading may be considered "quasi-static" or "time-independent". Because of this slow loading of the table, acceleration effects are negligible.

Structural Dynamics Problem Characterization

Types of problems: stiffness * displacement = force, and mass * acceleration + damping * velocity + stiffness * displacement = force are fundamentally expressed using partial differential equations (PDE’s). These PDE's may be formulated as linear or nonlinear matrix equations which are readily solved using a computer. In most structural problems, nonlinearities fall into three categories:

  • Material nonlinearity: Large deformations and strains (i.e., rubber and plastic materials)
  • Geometric nonlinearity: Small strains but large rotations (i.e., thin structures such as aircraft wings)
  • Boundary nonlinearity: Flexible boundary conditions (i.e., kinematic constraints and contact problems)

For linear static problems, the matrix equation is:

[ K ] { x } = { f }

Nonlinear static problems are characterized by the stiffness being a function of displacemnt:

[ K ( x ) ] { x } = { f }

For dynamic problems, mass, acceleration, velocity, and damping must also be considered:

[ M ] { x´´ } + [ C ] { x´ } + [ K ] { x } = { f }

where,

{ f } = force vector
{ x } = displacement vector
{ x‘ } = velocity vector (first derivative of displacement vector)
{ x" } = acceleration vector (second derivative of displacement vector)
[ M ] = mass matrix
[ K ] = stiffness matrix
[ K ( x ) ] = stiffness matrix as a function of displacement
[ C ] = damping matrix
Explicit and Implicit Dynamic Analyses

The numerical methods named below used in the overriding explicit and implicit methods are "for example" but are applicable to the task they are assigned. Different software use different numerical methods and there are many. For further reading on numerical methods used in explicit and implicit analysis, perform a search on the most common methods: Central Difference, Forward Euler, Backward Euler, Newmark, Newmark-Beta, Newmark-Euler, Newton-Rhapson, Gaussian Elimination, Bathe, Noh-Bathe, Linear-Acceleration, Constant-Acceleration, Trapezoidal Rule, Houbolt, Wilson-Theta, Runge-Kutta, and many more.

Explicit

The explicit method is based on dynamic equilibrium and makes use of time steps to discretize the calculation. Explicit analyses solve for nodal accelerations { x´´ } directly (not iteravely) at step ( n ). The accelerations are equal to the inverse of the mass matrix [ M ]−1 times the net nodal force vector { f } which includes contributions from all external sources: body forces, applied pressure, contact, etc., as well as the internal effects of element stress, damping, bulk viscosity, and hourglass control. Stated mathematically: { x´´ } = [ M ]−1 * { f } . In most cases, the mass is considered “lumped” resulting in a diagonal mass matrix. Inversion of a diagonal matrix is a straightforward process of simply making the diagonal elements reciprocal. After calculating nodal accelerations at step ( n ), nodal velocities { x' } are determined at step ( n + 1 / 2 ) followed by calculating nodal displacements { x } at step ( n + 1 ). Finally, the stiffness matrix [ K ] is updated based on geometry changes (if applicable), material changes (if applicable), and boundary changes (if applicable). A new stiffness matrix is then constructed, but not inverted because displacements have already been determined. From displacements comes strain. From strain comes stress, and the next increment of load (or displacement) is applied to the system. The direct integration scheme (such as the Forward Euler method) used for these calculations is not unconditionally stable and thus very small time steps are required. To be more precise, the time step in an explicit analysis must satisfy the "Courant" number. The Courant number in one dimension is defined as C ≡ u * ∆t / ∆x where u is the characteristic wave speed of the system, ∆t is the time step of the numerical model, and ∆x is the spacing of the grid in the numerical model.

The Courant number is a measure of how much information traverses u a computational grid cell ∆x in a given time-step ∆t. For example, in an explicit Eulerian method, the Courant number cannot be greater than unity because a Courant number greater than one means information is propagating through more than one grid cell at each time step. This causes the time integrator to not have time to properly interpret what is physically happening resulting in the solution becoming unstable and diverging. In most cases, a very small time step of around 10−6 seconds is required to achieve a Courant number sufficient to satisfy the stability condition.

In explicit analyses, the results will be accurate if the time step increments are small enough. One problem with this method is that many small increments are needed for good accuracy which is time consuming. If the incremental step is too large, the solution tends to diverge. Additionally, explicit analysis cannot solve some problems such as cyclic loading and problems of snap through or snap back. Perhaps most importantly, this method does not enforce equilibrium of the internal structural forces with the externally applied loads.

Implicit

The implicit method solves the system of equations for the nodal displacements { x } at the current time step directly by inverting the stiffness matrix [ K ] (using a numerical technique such as Gaussian elimination) and solving the coupled system (using a direct integration method such as the Newmark method). Once displacements are known, the accelerations { x´´ } and velocities { x´ } are calculated for the current time step ( n ) using results from the previous time step ( n - 1 ). The Newmark method is implemented implicitly based on Newton-Rhapson iterations and requires equilibrium enforcement at the current time step which guarantees unconditional stability of the method. Equilibirium is usually enforced to some user specified tolerance. The unconditional stability of the Newmark method facilitates one to two orders of magnitude larger time steps than explicit analysis. Additionally, equilibrium enforcement enables implicit analysis to better handle problems such as cyclic loading and snap through or snap back so long as control methods such as arc length control or generalized displacement control are used. The disadvantage of calculating the inverse of the stiffness matrix [ K ]−1 is that large amounts of storage space and in-core memory are required to accomplish the task. Additionally, material, boundary, and geometric nonlinearities cause the stiffness matrix to become a function of displacement [ K ( x ) ] making updating and/or factoring the stiffness matrix prior to inverting it another computationally intensive process. Another drawback of the implicit method is that the stiffness matrix must be reconstructed and updated for every Newton-Raphson iteration which is a computationally expensive process. As a result, modified Newton-Raphson methods are used in an attempt to mitigate this cost. If done correctly, the modified Newton-Raphson iterations will have a quadratic rate of convergence which is very desireable.

Implicit analysis assumes the problem is quasi-static. The displacement and load conditions are broken up into several increments. For each increment, a set of nonlinear static equilibrium equations are solved with standard numerical methods such as those mentioned above. The implicit method enables a full static solution of the deformation problem with convergence control. However, the calculation time requirement for each increment is large. Additionally, memory requirements are high due to the stiffness matrix inversion and accurate integration schemes employed. The implicit method also suffers the problem of divergence of the solution because of changes in contact, friction, and boundary states creating a stiffness matrix that is a function of displacement. Even though the method is unconditionally stable, it may still yield incorrect results.

Differences Between Explicit and Implicit Analysis

Explicit analysis is used to calculate the state of a given system at a future time based on conditions of the current time step. Calculations per time step are relatively fast, but many time steps are required. In contrast, an implicit analysis finds a solution for the current time step using only the results of the previous time step. Fewer time steps are required than the explicit method, but the calculations per time step are time consuming. These solutions are unconditionally stable and facilitate one to two orders of magnitude larger time steps than explicit analysis, but often these gains in time step size are negated by the high computational time required per tiime step.

In implicit analysis, solution of each step requires a series of trial solutions (iterations) to establish equilibrium within a certain tolerance. In explicit analysis, no iteration is required as the nodal accelerations are solved directly.

Explicit analysis time steps must be determined such that their resulting Courant number is less than unity which imposes the requirement of very small time steps on the order of 10−6 seconds. Implicit analysis has no inherent limit on time step size. As such, implicit time steps can be several orders of magnitude larger than explicit time steps but must be carefully calculated by the analyst.

Implicit analysis requires a numerical solver to invert the stiffness matrix once or even several times over the course of a load/time step. This matrix inversion is an expensive operation, especially for large models. Explicit doesn't require this operation.

When to Use Explicit Analysis

The explicit method should be used for short timed events with nonlinearities such as large deformations, nonlinear material properties, and/or contact intensive events. Additionally, the explicit method should be used when the strain rates and/or velocities are over 10 [ units / second ] or 10 [ m / s ], respectively. Some examples of these types of physics are: automotive crashes, ballistic events, drop tests, energy deposition, and etc. In these cases, strain rate must also be included as part of the material model because the rates conribute significantly to structural behavior.

When to Use Implicit Analysis

The implicit method should be used for steady-state/quasi-static, and low velocity or long time events where the strain rates and/or velocities are under 10 [ units / second ] or 10 [ m / s ], respectively. The strain rates must be slow such that material response is static. This class of problems covers most structural dynamics problems, certain metal forming problems, crush analysis, earthquake response, and biomedical problems.

MDA and Dynamic Analyses

Nonlinear analysis takes a great deal of experience and thorough understandings of both what needs to be accomplished and the anlaysis capabilities of the software being used. Over the years, MDA has tackled a number of nonlinear static and dynamic problems using both implicit and explicit methods. A brief list is given below:

  • Machine frames
  • Skids
  • Robotic components
  • Heavy machinery
  • Drop tests of consumer products
  • Conveying equipment
  • Component parts
  • Complex assemblies

Our nonlinear analysis expertise, experience, and capabilities provide our clients confidence in their products and help them specify vendor supplied components that interface with their equipment or products. Click here to view an example of an analysis report we provide.

Our FEA Capabilities
  • Linear
  • Static stress & deflection
  • Dynamic stress & deflection
  • Critical buckling load
  • Nonlinear
  • Thermal
  • Topology optimization
  • Size optimization
  • Geometric
  • Material
  • Contact
  • Postbuckling (Riks)
  • Mechanical event simulation
  • Dynamic stress & deflection
  • Dynamic
  • Modal analysis
  • Frequency response
  • Time response
  • Response spectrum
  • Random vibration
  • Transient stress
  • Explicit & implicit
  • Drop and direct impact events
  • Rotor dynamics
  • Shock and seismic
  • Power train vibration analysis
  • Fatigue life & durability