{"id":179583,"date":"2025-01-01T19:25:25","date_gmt":"2025-01-01T19:25:25","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=179583"},"modified":"2025-01-01T19:25:27","modified_gmt":"2025-01-01T19:25:27","slug":"zt-a-k-ke-rigid-vt-mass-m-c-7777777-l-2-l-2-figure-p2-3-4-12-for-the-system-shown-in-fig","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/01\/01\/zt-a-k-ke-rigid-vt-mass-m-c-7777777-l-2-l-2-figure-p2-3-4-12-for-the-system-shown-in-fig\/","title":{"rendered":"z(t) A k ke rigid v(t) mass = M c 7777777 L\/2 L\/2 Figure P2.3 4.12 For the system shown in Fig"},"content":{"rendered":"\n

z(t) A k ke rigid v(t) mass = M c 7777777 L\/2 L\/2 Figure P2.3 4.12 For the system shown in Fig. P2.3, use the complex-frequency-response method to obtain an ex- pression for the maximum steady-state force in spring AB due to harmonic excitation z(t) = Z cos St.<\/p>\n\n\n\n

\"\"<\/figure>\n\n\n\n

The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n

To solve for the maximum steady-state force in spring ABAB due to the harmonic excitation z(t)=Zcos\u2061(\u03c9t)z(t) = Z \\cos(\\omega t) using the complex-frequency-response method, we proceed as follows:<\/p>\n\n\n\n

\n\n\n\n

Solution:<\/strong><\/h3>\n\n\n\n
    \n
  1. System Dynamics and Parameters<\/strong>:\n
      \n
    • The mass MM is rigidly connected to a spring ABAB of stiffness kk.<\/li>\n\n\n\n
    • Excitation z(t)z(t) is applied as a harmonic input, z(t)=Zcos\u2061(\u03c9t)z(t) = Z \\cos(\\omega t).<\/li>\n\n\n\n
    • Force in spring ABAB is proportional to its extension or compression.<\/li>\n<\/ul>\n<\/li>\n\n\n\n
    • Displacement Relations<\/strong>: Let v(t)v(t) represent the displacement of the mass MM relative to the inertial frame. The relative extension of spring ABAB is \u0394x=v(t)\u2212z(t)\\Delta x = v(t) – z(t).<\/li>\n\n\n\n
    • Equation of Motion<\/strong>: From Newton’s second law, the force acting on the mass due to the spring is F=\u2212k(v(t)\u2212z(t))F = -k(v(t) – z(t)). Including damping (if present) and inertial effects, we write: Mv\u00a8(t)+kv(t)=kz(t).M\\ddot{v}(t) + kv(t) = kz(t).<\/li>\n\n\n\n
    • Steady-State Solution via Complex Frequency Response<\/strong>: Assume z(t)=Zej\u03c9tz(t) = Z e^{j\\omega t} (complex form). The solution for v(t)v(t) in steady state takes the form: v(t)=Vej\u03c9t.v(t) = V e^{j\\omega t}. Substituting into the equation, we obtain: \u2212M\u03c92V+kV=kZ\u2005\u200a\u27f9\u2005\u200aV=kZk\u2212M\u03c92.-M\\omega^2 V + kV = kZ \\implies V = \\frac{kZ}{k – M\\omega^2}.<\/li>\n\n\n\n
    • Force in Spring<\/strong>: The spring force is Fs=k(v(t)\u2212z(t))F_s = k(v(t) – z(t)). Substituting: Fs=k(Vej\u03c9t\u2212Zej\u03c9t)=k(V\u2212Z)ej\u03c9t.F_s = k\\left(V e^{j\\omega t} – Z e^{j\\omega t}\\right) = k(V – Z)e^{j\\omega t}. Using VV: Fs=k(kZk\u2212M\u03c92\u2212Z)ej\u03c9t=\u2212M\u03c92Zk\u2212M\u03c92ej\u03c9t.F_s = k\\left(\\frac{kZ}{k – M\\omega^2} – Z\\right)e^{j\\omega t} = \\frac{-M\\omega^2 Z}{k – M\\omega^2} e^{j\\omega t}.<\/li>\n\n\n\n
    • Maximum Force<\/strong>: The amplitude of the spring force is: Fs,max=\u2223\u2212M\u03c92Zk\u2212M\u03c92\u2223=M\u03c92Z\u2223k\u2212M\u03c92\u2223.F_{s,\\text{max}} = \\left|\\frac{-M\\omega^2 Z}{k – M\\omega^2}\\right| = \\frac{M\\omega^2 Z}{|k – M\\omega^2|}.<\/li>\n<\/ol>\n\n\n\n
      \n\n\n\n

      Explanation (300 words)<\/strong>:<\/h3>\n\n\n\n

      The complex-frequency-response method is a powerful tool for analyzing systems subjected to harmonic excitation. Here, the excitation z(t)=Zcos\u2061(\u03c9t)z(t) = Z \\cos(\\omega t) drives the mass MM, causing oscillations influenced by the spring ABAB. The governing equation combines inertial (Mv\u00a8(t)M\\ddot{v}(t)), elastic (kv(t)kv(t)), and input (kz(t)kz(t)) forces.<\/p>\n\n\n\n

      Assuming a steady-state response, we represent z(t)z(t) as a complex exponential Zej\u03c9tZ e^{j\\omega t}. This simplifies calculations since the steady-state solution v(t)v(t) also takes a harmonic form. Substituting these into the governing equation allows us to solve for the amplitude of v(t)v(t), representing the system’s frequency response.<\/p>\n\n\n\n

      The spring force FsF_s depends on the relative displacement between v(t)v(t) and z(t)z(t). Substituting v(t)v(t) and using algebraic manipulations, we derive an expression for FsF_s in terms of the system parameters. Importantly, the force amplitude depends on \u03c9\\omega, the excitation frequency.<\/p>\n\n\n\n

      The maximum force occurs when \u03c9\\omega approaches the system’s natural frequency (\u03c9n=k\/M\\omega_n = \\sqrt{k\/M}), leading to resonance. However, the denominator \u2223k\u2212M\u03c92\u2223|k – M\\omega^2| prevents singularity, ensuring the expression remains finite for practical damping.<\/p>\n\n\n\n

      This method demonstrates how harmonic excitation propagates through a mechanical system, emphasizing the critical relationship between driving frequency and system response.<\/p>\n","protected":false},"excerpt":{"rendered":"

      z(t) A k ke rigid v(t) mass = M c 7777777 L\/2 L\/2 Figure P2.3 4.12 For the system shown in Fig. P2.3, use the complex-frequency-response method to obtain an ex- pression for the maximum steady-state force in spring AB due to harmonic excitation z(t) = Z cos St. The correct answer and explanation is: […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-179583","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179583","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=179583"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/179583\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=179583"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=179583"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=179583"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}