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2.2  Frequency domain test cases

• Example lists
• Spring mass examples
• Beam problem with local non-linearity
• Lap joint problem with contact
• Hyperelastic bushing

2.2.1  Example lists

• d_hbm('TestDuffing2dof') spring mass duffing example detailed in section 2.3.1
• model=d_hbm('TestDofSet1') provides a test case with enforced displacement.
• d_hbm('TestBeamNL') is a simple beam with a rotation spring. This was analyzed in detail in [22].
• d_hbm('TestBeamVNL') is similar but implements the rotation spring as a single volume element. This is used to validate the implementation of volume non-linearities described section 1.2.3.
• d_hbm('TestSPlate') illustrates a plate on non-linear supports.
• t_hbm TestLapJoint Bj2 (provided by AGI as part of project CLIMA)

2.2.2  Spring mass examples

The 2DOF duffing model provided by AGI is implemented in d_hbm('TestDuffing2dof').

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Figure 2.1: Two DOF duffing oscillator with cubic non-linearity

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sdtweb d_hbm('TestDuffing2Dof'); % Open source code of example
[mo1,opt,Z,XF]=d_hbm('TestDuffing2dof'); % Run and display

2.2.3  Beam problem with local non-linearity

sdtweb t_hbm beamnl

As a first example clima16('BeamNLKrange'), one seeks to demonstrate the sensitivity to a variable stiffness. The non-linear stress strain relation is defined by

Fu=@(NL,fc,model,u,v,a,opt,Case,RO)kcur.*(NL.unl-loss*NL.vnl/opt.w);

which combines a constant stiffness kcur and a loss factor defined in the time domain using a frequency dependent velocity contribution. Taking q=cos(ω t), one assumes the equality of the complex stiffness and viscous damping forms in the stress/strain relationship s=Re(k(1+iη)e^{iω t})=Re((k+iω c e^{iω t})) which leads to c=−η/ω. In the resulting frequency responses below, one clearly sees a frequency response having a transition from a lower frequency at 4 Hz for kcur=10e6 (in model units) and an upper frequency at 15 Hz for kcur=1e8. The figure also clearly shows that damping decreases close to the limits as expected (see [22] for details).

As a second example clima16('BeamNLARange'), one seeks to illustrate the amplitude dependence obtained for a stiffening spring.

2.2.4  Lap joint problem with contact

Current simple test is found in t_contact('LapJzt'). Variants include one or 3 bolts. Different strategies to generate the response.

The current test clima16('LJEB').

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Figure 2.2: Lap joint with contact surface

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2.2.5  Hyperelastic bushing

This example is a functional demonstration of capabilities associated with hyper-viscoelastic behavior. It is based on the RotDamper example.

[mo1,hopt]=d_hbm('TestBeamVNL');