Structural Dynamics and Vibration

Conceptual overview

Dynamic analysis accounts for mass, damping and time-dependent forcing, rather than assuming that every load acts infinitely slowly. Two frames with similar static stiffness can have different natural periods if their mass distributions differ. A two-degree-of-freedom shear building is a compact way to see modal shapes, modal participation and frequency separation. The governing equation in linear structural dynamics is M ü + C u̇ + K u = p(t). Units of acceleration and force must be consistent. Modal analysis solves a generalized eigenproblem using both M and K. Time integration introduces numerical approximations, so the time step and integration method must be recorded alongside the physical assumptions.

Learning outcomes

After working through this unit you should be able to: (1) construct a physically consistent idealization, (2) express the relevant force, stiffness and displacement variables using compatible units, (3) solve a demonstrative problem by hand or a transparent numerical method, (4) interpret response patterns and their sensitivity to input parameters, and (5) identify what the simplified model cannot prove. In this unit the most important observation is not merely a numerical answer but the connection between structural behavior and the assumed support or constitutive conditions.

Unit engineering case — Office-floor vibration

A two-floor idealized shear structure has different floor masses and lateral story stiffnesses. Compute its two modal frequencies, explain mode shape normalization and compare a time-history forced vibration of a single ideal mode. Differentiate displacement amplitudes from acceleration amplitudes when discussing human comfort. A clean classroom oscillation does not establish performance under real wind, occupant walking or earthquakes without realistic forcing spectra and structural damping.

Investigation sequence

Begin by preparing a diagram that names every member, support, load, imposed movement and unknown. Choose an independent coordinate system and assign a sign to every generalized force. Derive a minimal reference calculation before using a computational model. Then run a sensitivity study in which only one physical assumption changes at a time. Record both the predicted movement and the reaction or internal-force counterpart. Check conservation of force and moment where the physical model supports those checks. Finally compare the calculation with a plausible measurement or with a more complete independent model. Label the evidence as a verification benchmark, an assumption or a genuinely observed fact.

Suggested written report

A useful report has a one-paragraph engineering question, a table of parameter values with units, an explicit statement of boundary conditions, the governing equations, a readable figure of the system, a table of numerical results, at least one independent verification, and a conclusion that identifies unverified assumptions. State what happens when the governing input doubles, when stiffness increases, and when a support is relaxed or stiffened. Explain whether the observed changes follow from algebraic scaling or from a change in load path. Keep the original units in the file even if a graph displays a converted unit.

7.1 Mass and stiffness idealization

Learning target and physical interpretation

A structural lumped-mass representation allocates inertia to chosen degrees of freedom. Story stiffness models summarize frame resistance but do not automatically capture rotational inertia, shear lag or flexible diaphragms.

Governing relation

For two lateral floor DOFs, M=diag(m1,m2), K=[[k1+k2,−k2],[−k2,k2]].

Worked engineering interpretation

A 50 tonne first floor and 35 tonne second floor generate very different inertia reactions under the same acceleration even if their story stiffnesses are equal.

What the idealization omits

Replacing tonnes with kilograms without converting stiffness units can shift predicted frequencies by a factor of √1000.

Verification and interpretation of results

Check units for ω²: stiffness (N/m) divided by mass (kg) must yield s⁻².

Applied case and decision boundary

Develop a simple comparison between heavy concrete floors and lightweight modular floors. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Replacing tonnes with kilograms without converting stiffness units can shift predicted frequencies by a factor of √1000. Explain which physical check exposes the error.

5. Professional judgment: Using the case Develop a simple comparison between heavy concrete floors and lightweight modular floors., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

7.2 Natural modes and periods

Learning target and physical interpretation

Set applied force and damping to zero and look for harmonic solutions. The characteristic condition det(K−ω²M)=0 gives squared frequencies; corresponding eigenvectors describe relative movement.

Governing relation

(K−ω_i² M)φ_i=0; natural period T_i=2π/ω_i.

Worked engineering interpretation

If a natural angular frequency is 10 rad/s, the corresponding cyclic frequency is approximately 1.592 Hz and the period 0.628 s.

What the idealization omits

Multiplying an eigenvector by −1 does not create a different physical mode; sign and normalization are arbitrary.

Verification and interpretation of results

Recover each modal residual (K−ω²M)φ and verify that the vector norm is nearly zero.

Applied case and decision boundary

Use Lab 11 to identify whether the fundamental mode is predominantly in-phase floor movement. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Multiplying an eigenvector by −1 does not create a different physical mode; sign and normalization are arbitrary. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 11 to identify whether the fundamental mode is predominantly in-phase floor movement., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

7.3 Modal superposition and participation

Learning target and physical interpretation

Under suitable linear assumptions, a response may be represented as a sum of modal coordinates. The excitation direction determines which modes are strongly excited.

Governing relation

u(t)=Σ φ_i q_i(t); Γ_i=(φ_iᵀ M r)/(φ_iᵀ M φ_i) for an influence vector r.

Worked engineering interpretation

An in-phase first mode often dominates roof displacement under uniform base excitation, but a higher mode can matter for story shear.

What the idealization omits

Summing unsigned modal maxima directly generally overestimates correlated response; spectral combination rules need defined assumptions.

Verification and interpretation of results

Check mode-shape mass orthogonality and explain the effect of retaining only one mode.

Applied case and decision boundary

Discuss why a tall flexible frame and a short stiff frame respond differently to the same ground motion. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Summing unsigned modal maxima directly generally overestimates correlated response; spectral combination rules need defined assumptions. Explain which physical check exposes the error.

5. Professional judgment: Using the case Discuss why a tall flexible frame and a short stiff frame respond differently to the same ground motion., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

7.4 Damping and harmonic response

Learning target and physical interpretation

Viscous damping is a linear idealization that dissipates energy in proportion to velocity. A sinusoidal load can create resonance-like amplification as excitation approaches natural frequency.

Governing relation

For SDOF: m ü + 2ζmωn u̇ + k u = F sin(ωt), with ωn=√(k/m).

Worked engineering interpretation

With ζ=0.05 and ω≈ωn, the steady-state displacement amplification is approximately 1/(2ζ)=10 relative to F/k.

What the idealization omits

A transient numerical integration starting from rest does not instantly show the asymptotic steady-state amplitude; it contains a free-vibration component.

Verification and interpretation of results

Compare measured peak after successive cycles with the closed-form steady-state transfer-function amplitude, allowing for transient decay.

Applied case and decision boundary

Use Lab 12 to explore vibration sensitivity of a slender footbridge span. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: A transient numerical integration starting from rest does not instantly show the asymptotic steady-state amplitude; it contains a free-vibration component. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 12 to explore vibration sensitivity of a slender footbridge span., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

7.5 Time integration and numerical stability

Learning target and physical interpretation

Newmark average acceleration is widely used for linear structural dynamics because it is unconditionally stable for ordinary positive-definite linear systems; accuracy still depends on time step.

Governing relation

With β=1/4 and γ=1/2, predictor and corrector equations solve effective stiffness m+γΔt c+βΔt²k for each scalar acceleration update.

Worked engineering interpretation

Reducing the time step from T/30 to T/90 should reduce phase error when resolving a smooth harmonic response.

What the idealization omits

Unconditional linear stability is not a promise of physical accuracy, nor does it establish stability for arbitrary nonlinear or irregular implementations.

Verification and interpretation of results

Halve Δt and compare peak responses; report the relative difference before accepting a plotted dynamic result.

Applied case and decision boundary

Prepare a validation note for a vibration-monitoring sensor that records displacement at a fixed sampling interval. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Unconditional linear stability is not a promise of physical accuracy, nor does it establish stability for arbitrary nonlinear or irregular implementations. Explain which physical check exposes the error.

5. Professional judgment: Using the case Prepare a validation note for a vibration-monitoring sensor that records displacement at a fixed sampling interval., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

End-of-unit self-check

1. Which unknown in the five lesson models is governed principally by equilibrium, and which requires a deformation condition? Explain for two different examples.
2. Identify a boundary-condition assumption in this unit that could change if a real bearing loses contact or a joint begins to slip.
3. Form a dimensionless or unit-consistent ratio from a model response and a characteristic load or span, then explain how that ratio changes with structural stiffness.
4. Explain how a model could satisfy overall force balance and nevertheless predict the wrong deformation field.
5. Design a two-point numerical verification plan that does not reuse the same solver for both checks.
6. Identify the limitation of treating code design or material yielding as if it were included in these ideal elastic analyses.

Short answer guidance: A complete answer should use a named force path or deformation coordinate rather than generic phrases. A verification answer should specify the numerical quantity being compared and the expected reference; a validation answer should identify a relevant observation or tested physical condition.

Unit 7 — Structural dynamics and vibration

Conceptual overview

Dynamic analysis accounts for mass, damping and time-dependent forcing, rather than assuming that every load acts infinitely slowly. Two frames with similar static stiffness can have different natural periods if their mass distributions differ. A two-degree-of-freedom shear building is a compact way to see modal shapes, modal participation and frequency separation. The governing equation in linear structural dynamics is M ü + C u̇ + K u = p(t). Units of acceleration and force must be consistent. Modal analysis solves a generalized eigenproblem using both M and K. Time integration introduces numerical approximations, so the time step and integration method must be recorded alongside the physical assumptions.

Learning outcomes

After working through this unit you should be able to: (1) construct a physically consistent idealization, (2) express the relevant force, stiffness and displacement variables using compatible units, (3) solve a demonstrative problem by hand or a transparent numerical method, (4) interpret response patterns and their sensitivity to input parameters, and (5) identify what the simplified model cannot prove. In this unit the most important observation is not merely a numerical answer but the connection between structural behavior and the assumed support or constitutive conditions.

Unit engineering case — Office-floor vibration

A two-floor idealized shear structure has different floor masses and lateral story stiffnesses. Compute its two modal frequencies, explain mode shape normalization and compare a time-history forced vibration of a single ideal mode. Differentiate displacement amplitudes from acceleration amplitudes when discussing human comfort. A clean classroom oscillation does not establish performance under real wind, occupant walking or earthquakes without realistic forcing spectra and structural damping.

Investigation sequence

Begin by preparing a diagram that names every member, support, load, imposed movement and unknown. Choose an independent coordinate system and assign a sign to every generalized force. Derive a minimal reference calculation before using a computational model. Then run a sensitivity study in which only one physical assumption changes at a time. Record both the predicted movement and the reaction or internal-force counterpart. Check conservation of force and moment where the physical model supports those checks. Finally compare the calculation with a plausible measurement or with a more complete independent model. Label the evidence as a verification benchmark, an assumption or a genuinely observed fact.

Suggested written report

A useful report has a one-paragraph engineering question, a table of parameter values with units, an explicit statement of boundary conditions, the governing equations, a readable figure of the system, a table of numerical results, at least one independent verification, and a conclusion that identifies unverified assumptions. State what happens when the governing input doubles, when stiffness increases, and when a support is relaxed or stiffened. Explain whether the observed changes follow from algebraic scaling or from a change in load path. Keep the original units in the file even if a graph displays a converted unit.

7.1 Mass and stiffness idealization

Learning target and physical interpretation

A structural lumped-mass representation allocates inertia to chosen degrees of freedom. Story stiffness models summarize frame resistance but do not automatically capture rotational inertia, shear lag or flexible diaphragms.

Governing relation

For two lateral floor DOFs, M=diag(m1,m2), K=[[k1+k2,−k2],[−k2,k2]].

Worked engineering interpretation

A 50 tonne first floor and 35 tonne second floor generate very different inertia reactions under the same acceleration even if their story stiffnesses are equal.

What the idealization omits

Replacing tonnes with kilograms without converting stiffness units can shift predicted frequencies by a factor of √1000.

Verification and interpretation of results

Check units for ω²: stiffness (N/m) divided by mass (kg) must yield s⁻².

Applied case and decision boundary

Develop a simple comparison between heavy concrete floors and lightweight modular floors. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Replacing tonnes with kilograms without converting stiffness units can shift predicted frequencies by a factor of √1000. Explain which physical check exposes the error.

5. Professional judgment: Using the case Develop a simple comparison between heavy concrete floors and lightweight modular floors., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

7.2 Natural modes and periods

Learning target and physical interpretation

Set applied force and damping to zero and look for harmonic solutions. The characteristic condition det(K−ω²M)=0 gives squared frequencies; corresponding eigenvectors describe relative movement.

Governing relation

(K−ω_i² M)φ_i=0; natural period T_i=2π/ω_i.

Worked engineering interpretation

If a natural angular frequency is 10 rad/s, the corresponding cyclic frequency is approximately 1.592 Hz and the period 0.628 s.

What the idealization omits

Multiplying an eigenvector by −1 does not create a different physical mode; sign and normalization are arbitrary.

Verification and interpretation of results

Recover each modal residual (K−ω²M)φ and verify that the vector norm is nearly zero.

Applied case and decision boundary

Use Lab 11 to identify whether the fundamental mode is predominantly in-phase floor movement. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Multiplying an eigenvector by −1 does not create a different physical mode; sign and normalization are arbitrary. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 11 to identify whether the fundamental mode is predominantly in-phase floor movement., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

7.3 Modal superposition and participation

Learning target and physical interpretation

Under suitable linear assumptions, a response may be represented as a sum of modal coordinates. The excitation direction determines which modes are strongly excited.

Governing relation

u(t)=Σ φ_i q_i(t); Γ_i=(φ_iᵀ M r)/(φ_iᵀ M φ_i) for an influence vector r.

Worked engineering interpretation

An in-phase first mode often dominates roof displacement under uniform base excitation, but a higher mode can matter for story shear.

What the idealization omits

Summing unsigned modal maxima directly generally overestimates correlated response; spectral combination rules need defined assumptions.

Verification and interpretation of results

Check mode-shape mass orthogonality and explain the effect of retaining only one mode.

Applied case and decision boundary

Discuss why a tall flexible frame and a short stiff frame respond differently to the same ground motion. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Summing unsigned modal maxima directly generally overestimates correlated response; spectral combination rules need defined assumptions. Explain which physical check exposes the error.

5. Professional judgment: Using the case Discuss why a tall flexible frame and a short stiff frame respond differently to the same ground motion., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

7.4 Damping and harmonic response

Learning target and physical interpretation

Viscous damping is a linear idealization that dissipates energy in proportion to velocity. A sinusoidal load can create resonance-like amplification as excitation approaches natural frequency.

Governing relation

For SDOF: m ü + 2ζmωn u̇ + k u = F sin(ωt), with ωn=√(k/m).

Worked engineering interpretation

With ζ=0.05 and ω≈ωn, the steady-state displacement amplification is approximately 1/(2ζ)=10 relative to F/k.

What the idealization omits

A transient numerical integration starting from rest does not instantly show the asymptotic steady-state amplitude; it contains a free-vibration component.

Verification and interpretation of results

Compare measured peak after successive cycles with the closed-form steady-state transfer-function amplitude, allowing for transient decay.

Applied case and decision boundary

Use Lab 12 to explore vibration sensitivity of a slender footbridge span. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: A transient numerical integration starting from rest does not instantly show the asymptotic steady-state amplitude; it contains a free-vibration component. Explain which physical check exposes the error.

5. Professional judgment: Using the case Use Lab 12 to explore vibration sensitivity of a slender footbridge span., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

7.5 Time integration and numerical stability

Learning target and physical interpretation

Newmark average acceleration is widely used for linear structural dynamics because it is unconditionally stable for ordinary positive-definite linear systems; accuracy still depends on time step.

Governing relation

With β=1/4 and γ=1/2, predictor and corrector equations solve effective stiffness m+γΔt c+βΔt²k for each scalar acceleration update.

Worked engineering interpretation

Reducing the time step from T/30 to T/90 should reduce phase error when resolving a smooth harmonic response.

What the idealization omits

Unconditional linear stability is not a promise of physical accuracy, nor does it establish stability for arbitrary nonlinear or irregular implementations.

Verification and interpretation of results

Halve Δt and compare peak responses; report the relative difference before accepting a plotted dynamic result.

Applied case and decision boundary

Prepare a validation note for a vibration-monitoring sensor that records displacement at a fixed sampling interval. Before drawing a design conclusion, identify which operating condition governs and which variation in span, load, support restraint or stiffness most affects the answer. If the calculation is to be used beyond teaching, a competent engineer must apply the relevant local code, material specification, site conditions, partial factors and independent review. This book's interactive demonstrations do not implement those obligations. Use the result to formulate better questions about the structure, not to replace a complete engineering assessment.

Practice and reflection

1. Derivation: Starting from the stated model, identify the equilibrium equation and the additional compatibility or constitutive equation required. Rewrite the derivation with explicit coordinate signs.

2. Numerical perturbation: Repeat the worked interpretation after increasing the principal load parameter by 10%, keeping the other inputs fixed. Predict the response before calculating it and state whether proportionality is theoretically expected.

3. Boundary-condition test: Replace one ideal support or restraint with a more flexible or more restrictive condition. Explain qualitatively which actions and movements are likely to change and what additional data are required.

4. Error diagnosis: Construct a plausible incorrect solution based on the warning in this lesson: Unconditional linear stability is not a promise of physical accuracy, nor does it establish stability for arbitrary nonlinear or irregular implementations. Explain which physical check exposes the error.

5. Professional judgment: Using the case Prepare a validation note for a vibration-monitoring sensor that records displacement at a fixed sampling interval., identify one observable site measurement that would be needed to validate the idealization and one design-code question the classroom model cannot answer.

Model-check prompt: State in one sentence what would falsify the predicted response. State separately what numerical residual could be tested without any field data.

End-of-unit self-check

1. Which unknown in the five lesson models is governed principally by equilibrium, and which requires a deformation condition? Explain for two different examples.
2. Identify a boundary-condition assumption in this unit that could change if a real bearing loses contact or a joint begins to slip.
3. Form a dimensionless or unit-consistent ratio from a model response and a characteristic load or span, then explain how that ratio changes with structural stiffness.
4. Explain how a model could satisfy overall force balance and nevertheless predict the wrong deformation field.
5. Design a two-point numerical verification plan that does not reuse the same solver for both checks.
6. Identify the limitation of treating code design or material yielding as if it were included in these ideal elastic analyses.

Short answer guidance: A complete answer should use a named force path or deformation coordinate rather than generic phrases. A verification answer should specify the numerical quantity being compared and the expected reference; a validation answer should identify a relevant observation or tested physical condition.