Dynamic, Nonlinear and Stability Analysis
Comprehensive university textbook unit covering fundamental principles, governing relationships, and engineering applications of Dynamic, Nonlinear and Stability Analysis.
§7.1 Mass matrix and free vibration
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `Kϕ=ω² Mϕ`. **Field study:** Estimate frequency of a suspended platform then compare an FE eigenmode.
§7.2 Damping and forced response
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `Mü+Cẋu+Ku=F(t)`. **Field study:** Explore a single-degree-of-freedom frequency response near resonance.
§7.3 Geometric stiffness and buckling
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `(K+λ K_G)ϕ=0`. **Field study:** Compare Euler column prediction with a geometric eigen-buckling mode.
§7.4 Newton iteration and load increments
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `K_t Δu=-R(u)`. **Field study:** Track residual norm over increments for a softening structural component.
§7.5 Time integration, stability and fatigue context
**Target competency.** Derive the appropriate approximate relation, apply it with coherent units, and identify one situation in which it would produce a misleading engineering result. **Primary relation:** `Δt < Δt_critical (explicit)`. **Field study:** Study a vibration transient under different timestep choices.
Step-by-Step Finite Element Analysis and Computational Mechanics Solutions
Three original questions connect calculations, definitions and interpretation. Open each solution to follow the reasoning.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **natural frequency of one mode**, based on `f=sqrt(k/m)/(2pi)`. Illustrative inputs are **Equivalent stiffness=80000 N/m, Modal mass=35 kg**. Converting the displayed units into SI inside the calculation gives **7.609062 Hz**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **natural frequency of one mode**, based on `f=sqrt(k/m)/(2pi)`. Illustrative inputs are **Equivalent stiffness=80000 N/m, Modal mass=35 kg**. Converting the displayed units into SI inside the calculation gives **7.609062 Hz**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **natural frequency of one mode**, based on `f=sqrt(k/m)/(2pi)`. Illustrative inputs are **Equivalent stiffness=80000 N/m, Modal mass=35 kg**. Converting the displayed units into SI inside the calculation gives **7.609062 Hz**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **viscous damping ratio**, based on `zeta=c/(2sqrt(km))`. Illustrative inputs are **Damping coefficient=130 N·s/m, Stiffness=90000 N/m, Mass=24 kg**. Converting the displayed units into SI inside the calculation gives **0.0442269 —**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.
Problem Statement:
Worked Engineering Derivation:
A small hand-checkable model associated with this unit uses **viscous damping ratio**, based on `zeta=c/(2sqrt(km))`. Illustrative inputs are **Damping coefficient=130 N·s/m, Stiffness=90000 N/m, Mass=24 kg**. Converting the displayed units into SI inside the calculation gives **0.0442269 —**. This number comes from the stated elementary relation and is *not* a full FE simulation or a measured response. Recreate the substitution independently, confirm the dimensional units, and explain the physical meaning of the output sign and magnitude. The associated browser model provides bounded parameter changes and a sensitivity plot. For negative outcomes such as compressive stress or reverse heat flow, retain the sign in the balance even if a subsequent design comparison uses absolute magnitude. Compute a second scenario by changing the dominant parameter by ten percent while holding all others fixed. If the governing relation is nonlinear in that parameter, do not assume the output changes by ten percent. For example, beam deflection scales with length cubed under Euler–Bernoulli assumptions; this makes a modest length error important. An area or thickness variation can also change local stress and global stiffness differently. When the relevant approximation instead assumes constant geometry, explain why replacing the original problem by a new geometry requires re-analysis rather than simply scaling the final result.