Numerical Integration Rigid Assemblages Giacomo Boffi Dipartimento - PowerPoint PPT Presentation

SbS methods PVD Giacomo Boffi Numerical Integration Rigid Assemblages Giacomo Boffi Dipartimento di Ingegneria Strutturale, Politecnico di Milano April 9, 2014

SbS methods PVD Giacomo Boffi Examples of SbS Methods Part I Numerical Integration

Outline SbS methods PVD Giacomo Boffi Examples of SbS Methods Examples of SbS Methods Piecewise Exact Method Central Differences Method Methods based on Integration Constant Acceleration Method Linear Acceleration Method Newmark Beta Methods Specialising for Non Linear Systems Modified Newton-Raphson Method

Piecewise exact method SbS methods PVD Giacomo Boffi Examples of SbS Methods Piecewise Exact Central Differences Integration ◮ We use the exact solution of the equation of motion Constant Acceleration Linear Acceleration for a system excited by a linearly varying force, so the Newmark Beta Non Linear Systems source of all errors lies in the piecewise linearisation of Newton-Raphson the force function and in the approximation due to a local linear model.

Piecewise exact method SbS methods PVD Giacomo Boffi Examples of SbS Methods Piecewise Exact Central Differences Integration ◮ We use the exact solution of the equation of motion Constant Acceleration Linear Acceleration for a system excited by a linearly varying force, so the Newmark Beta Non Linear Systems source of all errors lies in the piecewise linearisation of Newton-Raphson the force function and in the approximation due to a local linear model. ◮ We will see that an appropriate time step can be decided in terms of the number of points required to accurately describe either the force or the response function.

Piecewise exact method SbS methods PVD Giacomo Boffi Examples of SbS For a generic time step of duration h , consider Methods Piecewise Exact ◮ { x 0 , ˙ Central Differences x 0 } the initial state vector, Integration Constant Acceleration ◮ p 0 and p 1 , the values of p ( t ) at the start and the end Linear Acceleration Newmark Beta of the integration step, Non Linear Systems Newton-Raphson ◮ the linearised force p ( τ ) = p 0 + ατ, 0 ≤ τ ≤ h, α = ( p ( h ) − p (0)) /h, ◮ the forced response x = e − ζωτ ( A cos( ω D τ )+ B sin( ω D τ ))+( αkτ + kp 0 − αc ) /k 2 , where k and c are the stiffness and damping of the SDOF system.

Piecewise exact method SbS methods PVD Giacomo Boffi Examples of SbS Methods Piecewise Exact Central Differences Evaluating the response x and the velocity ˙ x for τ = 0 and Integration Constant Acceleration equating to { x 0 , ˙ x 0 } , writing ∆ st = p (0) /k and Linear Acceleration Newmark Beta δ (∆ st ) = ( p ( h ) − p (0)) /k , one can find A and B Non Linear Systems Newton-Raphson � 1 � x 0 + ζωB − δ (∆ st ) A = ˙ h ω D B = x 0 + 2 ζ δ (∆ st ) − ∆ st ω h substituting and evaluating for τ = h one finds the state vector at the end of the step.

Piecewise exact method SbS methods PVD Giacomo Boffi Examples of SbS With Methods Piecewise Exact Central Differences S ζ,h = sin( ω D h ) exp( − ζωh ) and C ζ,h = cos( ω D h ) exp( − ζωh ) Integration Constant Acceleration Linear Acceleration and the previous definitions of ∆ st and δ (∆ st ) , finally we can write Newmark Beta Non Linear Systems Newton-Raphson x ( h ) = A S ζ,h + B C ζ,h + (∆ st + δ (∆ st )) − 2 ζ δ (∆ st ) ω h x ( h ) = A ( ω D C ζ,h − ζω S ζ,h ) − B ( ζω C ζ,h + ω D S ζ,h ) + δ (∆ st ) ˙ h where � 1 B = x 0 + 2 ζ δ (∆ st ) � x 0 + ζωB − δ (∆ st ) − ∆ st , A = ˙ . ω h h ω D

Example SbS methods PVD Giacomo Boffi Examples of SbS Methods We have a damped system that is excited by a load in Piecewise Exact resonance with the system, we know the exact response Central Differences Integration and we want to compute a step-by-step approximation Constant Acceleration Linear Acceleration using different step lengths. Newmark Beta Non Linear Systems Newton-Raphson

Example SbS methods PVD Giacomo Boffi We have a damped system that is excited by a load in Examples of SbS Methods resonance with the system, we know the exact response Piecewise Exact Central Differences and we want to compute a step-by-step approximation Integration Constant Acceleration using different step lengths. Linear Acceleration Newmark Beta Non Linear Systems 0.02 Newton-Raphson Displacement [m] m=1000kg, 0.01 k=4 π 2 1000N/m, 0 Exact ω =2 π , -0.01 h=T/4 h=T/8 ζ =0.05, -0.02 h=T/16 p ( t ) = -0.03 4 π 2 5 N sin(2 π t ) 0 0.5 1 1.5 2 Time [s] It is apparent that you have a very good approximation when the linearised loading is a very good approximation of the input function, let’s say h ≤ T/ 10 .

Central differences SbS methods PVD Giacomo Boffi Examples of SbS To derive the Central Differences Method, we write the eq. Methods of motion at time τ = 0 and find the initial acceleration, Piecewise Exact Central Differences Integration x 0 = 1 Constant Acceleration m ¨ x 0 + c ˙ x 0 + kx 0 = p 0 ⇒ ¨ m ( p 0 − c ˙ x 0 − kx 0 ) Linear Acceleration Newmark Beta Non Linear Systems Newton-Raphson On the other hand, the initial acceleration can be expressed in terms of finite differences, x 0 = x 1 − 2 x 0 + x − 1 = 1 ¨ m ( p 0 − c ˙ x 0 − kx 0 ) h 2 solving for x 1 x 1 = 2 x 0 − x − 1 + h 2 m ( p 0 − c ˙ x 0 − kx 0 )

Central differences SbS methods PVD Giacomo Boffi We have an expression for x 1 , the displacement at the end of the Examples of SbS step, Methods x 1 = 2 x 0 − x − 1 + h 2 Piecewise Exact m ( p 0 − c ˙ x 0 − kx 0 ) , Central Differences Integration Constant Acceleration but we have an additional unknown, x − 1 ... if we write the finite Linear Acceleration differences approximation to ˙ x 0 we can find an approximation to Newmark Beta Non Linear Systems x − 1 in terms of the initial velocity ˙ x 0 and the unknown x 1 Newton-Raphson x 0 = x 1 − x − 1 ˙ ⇒ x − 1 = x 1 − 2 h ˙ x 0 2 h Substituting in the previous equation x 0 + h 2 x 1 = 2 x 0 − x 1 + 2 h ˙ m ( p 0 − c ˙ x 0 − kx 0 ) , and solving for x 1 x 0 + h 2 x 1 = x 0 + h ˙ 2 m ( p 0 − c ˙ x 0 − kx 0 )

Central differences SbS methods PVD Giacomo Boffi x 0 + h 2 Examples of SbS x 1 = x 0 + h ˙ 2 m ( p 0 − c ˙ x 0 − kx 0 ) Methods Piecewise Exact Central Differences To start a new step, we need the value of ˙ x 1 , but we may Integration Constant Acceleration approximate the mean velocity, again, by finite differences Linear Acceleration Newmark Beta Non Linear Systems x 0 + ˙ ˙ x 1 = x 1 − x 0 x 1 = 2( x 1 − x 0 ) Newton-Raphson ⇒ ˙ − ˙ x 0 2 h h The method is very simple, but it is conditionally stable . The stability condition is defined with respect to the natural frequency, or the natural period, of the SDOF oscillator, ω n h ≤ 2 ⇒ h ≤ T n π ≈ 0 . 32 T n For a SDOF this is not relevant because, as we have seen in our previous example, we need more points for response cycle to correctly represent the response.

Methods based on Integration SbS methods PVD Giacomo Boffi Examples of SbS We will make use of an hypothesis on the variation of the Methods acceleration during the time step and of analytical Piecewise Exact Central Differences integration of acceleration and velocity to step forward Integration Constant Acceleration from the initial to the final condition for each time step. Linear Acceleration Newmark Beta In general, these methods are based on the two equations Non Linear Systems Newton-Raphson � h x 1 = ˙ ˙ x 0 + ¨ x ( τ ) dτ, 0 � h x 1 = x 0 + x ( τ ) dτ, ˙ 0 which express the final velocity and the final displacement in terms of the initial values x 0 and ˙ x 0 and some definite integrals that depend on the assumed variation of the acceleration during the time step.

Integration Methods SbS methods PVD Giacomo Boffi Examples of SbS Methods Piecewise Exact Central Differences Depending on the different assumption we can make on the Integration Constant Acceleration variation of velocity, different integration methods can be Linear Acceleration Newmark Beta derived. Non Linear Systems Newton-Raphson We will see

Integration Methods SbS methods PVD Giacomo Boffi Examples of SbS Methods Piecewise Exact Central Differences Depending on the different assumption we can make on the Integration Constant Acceleration variation of velocity, different integration methods can be Linear Acceleration Newmark Beta derived. Non Linear Systems Newton-Raphson We will see ◮ the constant acceleration method,

Integration Methods SbS methods PVD Giacomo Boffi Examples of SbS Methods Piecewise Exact Central Differences Depending on the different assumption we can make on the Integration Constant Acceleration variation of velocity, different integration methods can be Linear Acceleration Newmark Beta derived. Non Linear Systems Newton-Raphson We will see ◮ the constant acceleration method, ◮ the linear acceleration method,